This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
x'. By means of (5.5.7) and (5.5.20), the equation of the isotropic cone Cx of the CO(2, 2)-structure associated with the given three-web is 2(x2 + y2)dx'dy' + (y1 - x')(dx'dy2 - dx2dy') = 0.
To find the tensors C. and CO of the CO(2, 2)-structure determined by the quadratic form on the left-hand side of this equation, we will differentiate the forms 11 ° and w a in (5.5.20) and find consequently the forms wb and the components of the torsion tensor abc = d(baC; next we take exterior derivatives of the forms wb and find the covariant differentials Dab and the components bbce of the curvature tensor of the given three-web. For the three-web (5.5.19) this work has been done in Goldberg [Co 85, 86[ (see also Goldberg [Go 881, pp.
422-425). We will write the values of those quantities indicated above which we will need for the computation the tensors Ca and CO, 2
p2i=921=0, p1I=-91i(xlyt)2; b2
= -b121 = (x1
2?y1)2,
2 bill = -(a;' y1)z,
and the remaining components of the curvature tensor are equal to 0. This implies that all components of the tensor Ca vanish and that the tensor CO has only one nonvanishing component: 8(x2 + y2)
bo = - (x1 -
y1)2'
Hence the CO(2, 2) -structure associated with the given three-web is anti-self-
dual, the three-web itself is isoclinic, and the equation C0(p) = 0 takes the form
bop' = 0.
200
5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES
Thus this equation has a quadruple root at p = 0, and as a result the isotropic fiber bundle E0 has only one integrable distribution Op(0). By (5.5.13), the equations of this distribution can be written as
W' =dx' =0, Z' = dy' =0, Put differently, this distribution is determined by the equations x' = const, y' = const, y1 > x'.
The system (5.5.12) for the given three-web takes the form
dx' + ady' = 0, (x2
+ y2)(-dx' + Ady') + (y' - x')(dx2 + Ady2) = 0,
dA + yJ2Ax,(dx' + dy') = 0.
This system is completely integrable and determines a three-parameter family of integral surfaces of the isotropic fiber bundle Ea. Example 5.5.2 Consider the three-web W(3, 2, 2) defined on the manifold R4 of variables x1, z2, y', and y2 by the closed form equations
z' = x' + y', z2 = -x'y2 + x2y'
(5.5.21)
(see Goldberg [Go 85, 86] or Goldberg [Go 88], pp. 425-428).
From (5.5.21) we find that dz' = dx' + dy',
dz2 = -yzdx' + y'dx2 -XI dy2 + x2dy',
and hence W1
= dx',
j2 = -yzdx' + y'dx2,
2' = dy',
Z2 = -x'dy2 + x2dy'.
5522
1
The conditions of solvability of these equations with respect to dx', dx2 and dy', dy2 are
Di =
_1Y2 Y
A2=1X2 _x'
1=--100-
These conditions are satisfied, for example, if x' > 0 and y' > 0. All further constructions we will make in this open domain of the space R4. By means of (5.5.7) and (5.5.22), the equation of the isotropic cone Cx of the CO(2, 2)-structure associated with the given three-web is (x2 +Y 2)dxIdYI - x'dx'dy2 - y'dx2dy' = 0.
5.5
Four-Dimensional Webs and CO(2,2)-Structures
201
For the three-web (5.5.21) computations show (see Goldberg [Go 85, 86] or
Goldberg [Go 88], pp. 425-428) that in this case p = q = 0. As a result the tensor C,, = 0, the CO(2, 2)-structure is anti-self-dual, and the three-web itself is isoclinic. The nonvanishing components of the tensor CO are (X2
y2
x1
bo
=
y1
-x
461
1
(x 1)2
y1
(yl)2
Hence we have
Cp(l)=
\y1
x1/(x1
y2)p+y1
+x1J14 3
Thus the equation C0(p) = 0 has a triple root at p = 0, and this root determines the integrable distribution Op(0) on the isotropic fiber bundle Ep. In addition the fourth root p -Z Y of the equation C0(µ) = 0 determines 1-1- 1+ 1
another distribution of Ep whici, in general, is not necessarily integrable.
Example 5.5.3 Consider the three-web W(3, 2, 2) defined on the manifold R4 of variables x1, x2, y1, and y2 by the closed form equations
z1 = x1 +y' + (x1)2y2, z2 = x2 +y2 - 2x1(y2)2
(5.5.23)
(see Goldberg [Go 87] or Goldberg [Go 88], pp. 431-432).
From (5.5.23) we find that
dz' = Odx1 + dy' + I (x1)2dy2,
dz2 = - (y2)2dx1 + dx2 + (2 - O)dy2, where 0 = 1 +x1y2, and consequently
i1 = Mdx',
w2 = dx2 - 2(y2)2dx1,
(5.5.24)
Z1 = dy1 + 1(x1)2dy2,
w2 2
= (2 - O)dy2.
The condition of solvability of these equations with respect to dx1, dx2 and dy', dy2 is 0 i4 0, 2. As a domain in which these conditions are satisfied we take an open domain of the space R4 defined by the inequalities:
-1<xIy2 r9. This metric is obtained from (5.6.1) if we set
e2' = I - rgr ,
e2"
= r2 sin2 B, k = 0, e2 *2 =
1-I r9, r
e2" = r2.
5.6
Conformal Structures of Some Metrics in General Relativity
207
Then equations (5.6.3), (5.6.8), and (5.6.9) take, respectively, the form w1 = r - r9
4=
1
T2
r
dt - dr,
w2 = sin 9 dp - idO', (5.6.13)
dt +
1
T2 - r9r
dr,
w3 = sin B dip + idB,
B=0e=83=B1 =BI =0,
01= lw1-r9 r
r
(5.6.14)
4
'
2(U)2 + W3) Cot
02 = and
dB1_r-3rgw1Aw4
I 1
2r
(5.6.15)
dB2 = 2 w2 A w3.
The components of the tensor of conformal curvature C. of the isotropic fiber bundle E,, of the Schwarzschild metric take the values ao = 0, a1 = 0, a2 =
T9
4r' a3 = 0, a4 = 0,
and the components b of the tensor of conformal curvature CO have the same values. Hence the Schwarzschild metric belongs to type D of the Petrov classification (see Subsection 5.4.6). Geometry of space-time outside of a spherically symmetric body with an electrical charge Q is described by the Reissner-Nordstrom metric b2
ds2 = 2(dt)2 -
(dr)2 -
(5.6.16)
(dB2))
(see Chandrasekhar [Cha 831, Ch. 5, §38, Eq. (48)), where 6 = r2 - r9r + Q2 and Q = const. This metric is obtained from the axially symmetric metric (5.6.1) if we set z
e2v =
2,
e2'1'
= r2 sin2 9, k = 0, e2P2 = a ,
e211a
= r2.
Equations (5.6.10) for the components of the tensor CQ of the isotropic fiber bundle Ea of the Reissner-Nordstrom metric take the form
ryr - Q2 ao = 0, a1 = 0, a2 =
4r2
,
a3 = 0, a4 = 0,
and the components b of the tensor of conformal curvature Cp have the same values. Hence the Reissner-Nordstrom metric also belongs to type D of the Petrov classification.
208
5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES
For the Schwarzschild metric and the Reissner-Nordstrbm metric, the relative conformal curvatures of the a- and p-planes are calculated according to the following formulas:
C. (,\) = 6a2A2, CO(p) = 6a2µ2. Thus, for the flat isotropic distributions, we have a2
A2
a2µ2 = 0,
= 0.
(5.6.17)
Each of equations (5.6.17) has two double roots:
Al=A2=0, A3=1\4=0o and µl={12=0, µ3=µ4= 00. Hence, on a manifold endowed with a conformal structure defined by the metrics (5.6.12) and (5.6.16), through every point there pass two pairs of totally isotropic submanifolds corresponding to the following values of the parameters A and p:
A=p=0 and A=p=oo.
For the Schwarzschild metric, integrability of isotropic distributions on the isotropic fiber bundle Ea, which corresponds to the values A = 0 and A = 00 of the parameter A, can be immediately verified by applying formulas (5.6.13). In the same way one can verify integrability of isotropic distributions on the isotropic fiber bundle Ep.
5.7
Conformal Structures on a Four-Dimensional Hypersurface
1. As we have shown in Section 4.3, on a tangentially nondegenerate hypersurface V^ of a projective space P1+1, a conformal structure which is called the asymptotic pseudoconformal structure is induced by its relatively invariant quadratic form 'P(2) = bijw'w1,
(5.7.1)
which is defined in a second-order neighborhood of V'. If n = 4, the quadratic form f(2) can have the signatures (2, 2), (1, 3), and (4, 0). Suppose that the signature is constant at all points x E V4. Then, in the first case, a conformal structure induced on V4 is the CO(2, 2)-structure; in the second case, it is the CO(1, 3)-structure; and in the third case, it is the CO(4)structure. Geometrically hypersurfaces of these three types are distinguished from one another by the fact that in the first case, their asymptotic cones C= carry two one-parameter families of two-dimensional plane generators, in the second case, they carry a two-parameter family of rectilinear generators, and in the third case, these cones are imaginary. Hypersurfaces V4 of these three kinds are called ultrahyperbolic, hyperbolic, and elliptic, respectively.
5.7
Conformal Structures on a Four-Dimensional Hypersurface
209
In this section, we will study ultrahyperbolic hypersurfaces V4 of the space P5 and the asymptotic CO(2, 2)-structure associated with them in more detail, find conditions of their conformal semiflatness and flatness, investigate some properties of their principal isotropic distributions, and construct examples of hypersurfaces V4 carrying a conformally semiflat and conformally flat CO(2,2)-structure. Let us refer a hypersurface V4 C P5 to the family of isotropic cones relative
to which the equation of the family of asymptotic cones C= takes the form (5.1.1): g = 2(W1w4 - W2W3) = 0.
Consequently the tensor b,j has the following components: 0
(bi3) =
0
0
1
-1 0
-1
1
0
0
0 0
(5.7.2)
Using formulas (4.4.21), (5.1.13), and (5.7.2), we find the components of the tensor of conformal curvature of the CO(2, 2)-structure: ao = (B222BI13 - 3B122B123 + 38112B223 - B111 B224), g
al = ,1 (B222B133 - 3B122B233 + 3B112B234 - B111B244), a2 = 24 (B222B333 - 38122B334 + 3BI12B344 - B111B444),
(5.7.3)
a3 = 1(B224B333 - 3B223B334 +3B123B344 - B113B444), a4 = 18'(B244B333 - 3B234B334 + 3B233B344 - B133B444), bO = 18'(B333B112 - 3B133B123 + 3B113B233 - B111 B334),
b1 = lg(B333B122 - 3B133B223 + 3B113B234 - B111 B344),
b2 = 2(B333B222 - 3B133B224 + 3B113B244 - B111B444),
(5.7.4)
b3 = g (B334 B222 - 3B233B224 + 3B123B244 - B112B444),
b4 =
g
(B344 B222 - 3B234 B224 + 3B223B244 - B122 B444)
are the independent components The quantities a and b, u = 0,0,1,2,3,4, 1, of the tensors C. and Cp, and the quantities B,Jk are the components of the Darboux tensor of the hypersurface V4 defined by equations (4.4.10). Let us consider the (4 x 4)-matrix B112
BI22 B222
B113 B123 B223 B224
B133 B233 B234 B244
B333 B334 B344 B444
(5.7.5)
5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES
210
which is composed of 16 essential components of the Darboux tensor B{jk of the hypersurface V4. Let us denote the ith column-vector of the matrix 13 by Li and its jth row-vector by M,,. In addition we consider the skew-symmetric bilinear form K(X, Y) = KijX'Y' determined by the matrix
(Ki j ) =
0 0 0 1
0 3
-1 0
-3 0
0
0
0
0
0
0
(5.7.6)
In accordance with these notations, formulas (5.7.3) and (5.7.4) for the components of the subtensors Ca and Co take, respectively, the following forms: ao = 1 K(LI, L2),
at = 18K(L1,L3),
a2 = 2K(LI,L4),
(5.7.7)
a3 = 16 K(L2, L4),
a4 = I K(L3, L4) and
bo = ! K(MI, M2),
bI = 18K(MI,M3), b2 = 2K (MI, M4),
(5.7.8)
b3 = 16K(M2,M4
b4 = 8K(M3,M4) Hence the asymptotic pseudoconformal CO(2, 2) -structure on the hypersur-
face V4 is semiflat if and only if one of the following two sets of conditions holds:
K(LI, L2) = 0, K(LI, L3) = 0, K(LI, L4) = 0, K(L2, L4) = 0, K(L3, L4) = 0 or
K(MI, M2) = 0, K(MI, M3) = 0, K(MI, M4) = 0,
(5.7.10)
K(M2, M4) = 0, K(M3, M4) = 0.
This structure is flat if and only if conditions (5.7.9) and (5.7.10) hold simultaneously. 2. We will find the geometric structure of completely isotropic submanifolds
of the CO(2, 2)-structure induced on a ultrahyperbolic hypersurface V4 C P5 by its second fundamental form 0121.
5.7
Conformal Structures on a Four-Dimensional Hypersurface
211
Suppose that on V4 there exists a completely isotropic a-submanifold V2(\) of this CO(2, 2)-structure. This submanifold is determined by the system of differential equations (5.1.2): WI + AW1=0, W2 + AW4 = O.
(5.7.11)
As we noted in Section 5.4, the parameter A satisfies differential equation (5.4.6), which by virtue of (5.4.5) and (4.4.19) takes the following form on V4: da +.\(W4 + W3) - W3 + \2W - I (B112)13 - 3B123\2 + 3B233 \ - B334)W3 (B122A3 - 3B223A2 + 3B234A - B344)W4 = 0.
(5.7.12)
Let x = A0 be a point of the hypersurface V4. We calculate the differentials dAOi d2A0, and d3A0 of this point as it moves along the completely isotropic a-submanifold V2(A). Since equations (5.7.11) hold on the submanifold V2(A), we have dA0 =w0Ao+W3(A3-.\A1)+W4 (A4 - AA2) =W4A0+W3B3+W4B4. (5.7.13) The forms W3 and w4 are linearly independent on the submanifold V2(A), and
the points B3 = A3 - AAl and B4 = A4 - .A2 lie in its tangent 2-plane. Differentiating twice (5.7.13) and applying (5.7.12), we see that d2 A0 = 2(S2(W3)2 +2S3W3W4 +b4( .4)2)A1
I
(w3)2
2
+ 2C2W3W4 + b (w4)2) A2
(mod TAo(V2(A))), (5.7.14)
d3A0 =
2
(S1 (W3)3 +
+313W3(W4)2
(5.7.15)
S4( 4)3)A5
(mod Ti2i(V2(A)))r
where T, (V2 (A)) is the tangent 2-plane to the submanifold V2(A), T(2)(V2(A))
is its osculating subspace at the point x = Ao, and the quantities l;; are computed in the following way: ttl = B111)13 - 3B113A2 + 3B133A - B333,
S2 = B112\3 - 3BI23A2 + 3B233I\ - B334,
(5.7.16)
y3 = B122\3 - 3B223\2 + 3B234A - B344, S4 = B222
A3
- 38224\2 + 3B244A - 8444
If we denote the column-vector with the components t, by {(A), then the last formulas can be written as (A) = L1)3 - 3L2A2 + 3L3A - L4.
(5.7.17)
212
5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES
It follows from relations (5.7.14) and (5.7.15) that the osculating subspace T.(2)(V2(,\)) coincides with or belongs to the four-dimensional tangent subspace T (V4) of the hypersurface V4: T(2)(V2(A)) C T (V4). Relations (5.7.14) and (5.7.15) also imply that the submanifold V2(A) has two second fundamental forms, 1
0(2)(A)
=
0(2)(A)
_ -2(Sl(w3)2
+2e3W3W4 +1;4(W4)2), (5.7.18)
and one third fundamental form, 0(3)(A) =
(5.7.19)
It is easy to verify that the form 1&(3)(A) is proportional to the restriction of the Darboux form P(3) of the hypersurface V4, defined by equation (4.3.12), to the submanifold V2(A). Similar results can be obtained when one considers completely isotropic , 3-submanifolds V2 (µ) determined on V4 by system (5.1.3): wt + µw2 = 0, w3 + µw4 = 0.
(5.7.20)
The 1-forms w2 and w4 are basis forms on V2(p), and the points BZ = A2-pA1 and B4 = A4 - pA3 lie in its tangent 2-plane. The second fundamental forms of such submanifolds have the form 'Y(2)(p) = 2 (rl2(w2)2 + 27)3w2w4 + 174(W4)2)203
(5.7.21)
(1,) = -2(111(W2)2+2172w2W4+113(W4)2),
and their third fundamental form G(3)(µ)
=
1
W4 3 w4 + 3,73W2(W4)2 + l4( ) ) (0l(,d2)3 + 3rl2(W2)2
(5.7.22)
is proportional to the restriction of the Darboux form x+(3) of the hypersurface V4, defined by equation (4.4.12), to the submanifold V2(p). The quantities rl; occurring in equations (5.7.21) are the components of the vector 17(tt) = M1µ3 - 3M2p2 + 3M3p - M4.
(5.7.23)
The following theorem describes the geometrical structure of completely isotropic submanifolds V2 of a hypersurface V4:
Theorem 5.7.1 If a hypersurface V' carries two-dimensional completely isotropic submanifolds V2, then they have one of the following structures: 1. V2 is a two-dimensional plane.
5.7
Conformal Structures on a Four-Dimensional Hypersurface
213
2. V2 is a two-dimensional developable surface, that is, a tangentially degenerate submanifold carrying a family of rectilinear generators, along each of which the tangent subspace to V2 is constant. The osculating subspaces of such submanifolds are three-dimensional.
3. V2 is a tangentially nondegenerate submanifold carrying a family of rectilinear generators. The four-dimensional osculating subspace of such a V2 coincides with the tangent hyperplane of V4. 4. V2 is a tangentially nondegenerate submanifold having four-dimensional osculating subspaces and carrying a conjugate net. (Such submanifolds are called the Carton varieties.) Proof. Let V2(A) be a completely isotropic a-submanifold of a hypersurface V4, and let 0(2) (A), 0(2) (A), and 0(3)(A) be its second and third fundamental forms, respectively. If the forms 0(2) (A) and ¢(2) (A) are identically equal to zero, then by virtue of the relation (5.7.14), V2(A) is a plane generator of the hypersurface V4, so we arrive at the case 1. Suppose that the forms 0(2)(A) and 0(2)(A) are not equal to zero but are proportional. Then from (5.7.18) it follows that l;z S1
=3 = l a 1;2
=k
1;3
By virtue of this expression, we have
0(2)(A) = 2k.i(w3 +kw4)2 and 0(2)(A) = -21:1(w3 +kw4)2, which shows that each of these forms is proportional to the perfect square of the same linear form. Hence V2(A) is a developable surface. Moreover, according to (5.7.14), the osculating subspace of the submanifold V2(A) is three-dimensional, so we arrive at the case 2. If the forms 0('2)(A) and ¢(22) (A) are not proportional but have a common
linear factor, then by virtue of the equation 0(3) (A) =
(A) - w34'(2) (A)
this factor will be a divisor of the third fundamental form 10(3) (A) too. One can prove that this implies that the submanifold V2(A) carries a family of rectilinear generators (e.g., see Akivis and Goldberg [AG 93], p. 235). Thus we arrive at the case 3. Finally, if the forms 0(2)(A) and 4(2)(A) have no linear common factor, then they can be simultaneously reduced to sums of squares, which means that the
submanifold V2(A) carries a conjugate net. Thus the osculating subspace of V2(A) is four-dimensional (see Akivis and Goldberg [AG 93], p. 75), so we arrive at the case 4. Similar arguments hold for completely isotropic submanifolds V2(µ).
5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES
214
Corollary 5.7.2 A completely isotropic submanifold V2 of a hypersurface V4
lies in its osculating subspace if and only if V2 is a plane generator of the hypersurface V4.
Proof. If V2(A) lies in its osculating subspace, then, according to (5.7.15), we have 0131(A) = 0. Then r;; = 0, i = 1,2,3,4, and 0(2)(A) = 0'(2)(A) = 0; that is, V2(A) is a 2-plane. Similar arguments hold for completely isotropic submanifolds V2(µ). U
Corollary 5.7.3 Let V4 be a hypersurface of a projective space P5 whose second fundamental form reduces to 4D(2) = 2(wlw4 _ w2w3), and let t:(A) be the polynomial defined by formula (5.7.17). Then the following conditions are equivalent:
1. The parameter A in equation (5.7.11) is a root of the algebraic equation VA) = 0.
(5.7.24)
2. The hypersurface V4 carries a two-parameter family of two-dimensional plane generators, which are determined by the system of differential equations (5.7.11).
3. The restriction of the cubic Darboux form W(3) of the hypersurface V4, defined by equation (4.4.12), to the two-dimensional distribution (5.7.11) is the null form. Proof. Each of three conditions of this theorem is equivalent to the fact that on the submanifold V2 (A) the quantities r;;, i = 1,2,3,4, defined by (5.7.16), vanish.
Corollary 5.7.4 If through each point of a tangentially nondegenerate hypersurface V4 C P5 there passes a two-dimensional plane generator, then one of the polynomials C, (A) or C,(p) of the asymptotic CO(2,2)-structure on V4 has a triple root. Proof. Suppose that the family of two-dimensional plane generators on the hypersurface V4 is determined by the system of equations (5.7.11). We consider an admissible transformation of the adapted frame, determined by the relations: to)
-4 W' + \W3, W2 --1 W2 + .\W4, W3 -1 W3, W4 - W4.
In the new frame the parameter A vanishes, and condition (5.7.24) becomes L4 = 0. Taking this equation into account, we deduce from formulas (5.7.7) that a2 = a3 = a4 = 0. Therefore the equation (5.4.7) has a triple root A = 0.
5.7
Conformal Structures on a Four-Dimensional Hypersurface
215
One can easily see that a hypersurface of the kind described in Corollary 5.7.4 can be given by the parametric equation: M(u1, u2, u3, u4) = Mo(ul, u2) + u3M1 (ul, u2) + u4M2(ul, u2),
(5.7.25)
where M, M0, M1 and M2 are points of the projective space P5. The twodimensional plane generator u1 = c1, u2 = c2 , where cl and c2 are constants, pass through every point of this hypersurface. 3. Next we will prove the following result:
Theorem 5.7.5 If through each point of a tangentially nondegenerate hypersurface V4 C P5 there pass two two-dimensional plane generators that are in general position, then the semiflat asymptotic CO(2,2) -structure is induced on the hypersurface V4.
Proof. Since the hypersurface V4 is tangentially nondegenerate and carries a family of two-dimensional plane generators, the second fundamental tensor of V4 has signature (2, 2). Moreover, since through any point of V4 there pass two of its plane generators that are in general position, both generators belong to the same family of isotropic planes of the isotropic cone C.; both of them are either a-planes or Q-planes. But then by Corollary 5.7.4, one of the polynomials Ca(\) or Cf(p) must have two distinct triple roots. Thus, since these polynomials are of fourth degree, one of them is identically equal to 0. Therefore, if these plane generators are a-planes, then Ca(J1) = 0, and the asymptotic CO(2, 2)-structure on V4 is anti-self-dual. If these plane generators are
13-planes, then C0(µ) = 0, and the asymptotic CO(2, 2)-structure on V4 is self-dual. Hence the hypersurface V4 carries a semiflat asymptotic CO(2, 2)structure. As we have proved in Theorem 5.4.2, a four-dimensional manifold with a semiflat CO(2, 2)-structure carries a three-parameter family of completely isotropic two-dimensional submanifolds. These submanifolds are a-submanifolds if the CO(2, 2)-structure is anti-self-dual, and they are Q-submanifolds if the CO(2, 2)-structure is self-dual. Thus, through any point of the hypersurface V4 described in Theorem 5.7.5, there passes a one-parameter family of completely isotropic two-dimensional submanifolds. The plane generators of the hypersurface V4 belong to this family. It follows from Theorem 5.7.1 that other completely isotropic two-dimensional submanifolds are developable surfaces or ruled surfaces or two-dimensional Cartan varieties. If all isotropic a-submanifolds are planes, then t (A) - 0, and by (5.7.16), all components B,Jk = 0; consequently the hypersurface V4 is a hyperquadric. Then all 13-submanifolds are planes, and the CO(2, 2)-structure on V4 is conformally flat. Theorem 5.7.5 allows us to describe some classes of hypersurfaces in the space P5 that carry a semiflat conformal CO(2, 2)-structure. We will find now the closed form equations of hypersurfaces V4 C P5 carrying two families of two-dimensional plane generators that are in general posi-
216
5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES
tion. According to Theorem 5.7.5, such hypersurfaces carry semiflat conformal CO(2,2)-structures.
Let us consider in the space P5 nine points Mfg, f,g = 0,1,2, in general position, and let (u°, u1, u2) and (v°, VI, v2) be two sets of variables, each of which is defined up to a common factor. The points Mfg determine a hypersurface
pM = ufvgMfg,
f, g = 0,1, 2.
(5.7.26)
This equation can be written in two ways: pM = v°(u°Moo + u' M10 + u2M2o)
+v'(u°MD1 +u1M11 +u2M21) +v2(u°Mo2 + u'M12 +U2 M22), pM = u°(v°M00 + v' Mlo + v2M2o) +u' (v°Moi + v' Mi i + v2M21) +u2(v°M02 + v' M12 + v2M22).
(5.7.27)
This means that the hypersurface V4 carries two families of plane generators
in general position: of = cft and vg = c9s where cf and cg are arbitrary 2 2 1
1
constants. Since these generators have only one common point, they belong to the same family of the isotropic cone C.. Thus the hypersurface V4 carries a semiflat asymptotic pseudoconformal CO(2, 2)-structure. If we assume that the nine points Mfg occurring in equations (5.7.26) lie in the space Ps and are linearly independent, then equation (5.7.26) determines a Segre variety in Ps which is the embedding of the direct product p2 X P2 of two projective planes. The hypersurface V4 constructed above is a central projection of the Segre variety p2 X P2 C Ps onto a five-dimensional space p5 from a two-dimensional projection center Z. In addition, the points Mfg and a two-dimensional center Z of projection must be in general position. 4. Theorem 5.7.5 implies another important result.
Theorem 5.7.6 If through each point of a tangentially nondegenerate hypersurface V4 C P5 there pass four two-dimensional plane generators, and two of them belong to one of the families of isotropic planes of the asymptotic cone of V4, while the other two belong to its second family of isotropic planes, then the flat asymptotic CO(2,2) -structure is induced on V4. 0 Applying this theorem, we will construct an example of a hypersurface with a flat CO(2, 2)-structure. The equation of such hypersurfaces is obtained from (5.7.26) in the case where the terms containing the points MI, and M22 are absent. Indeed, in this case pM = u°v°Moo + u°v1Mo1 + u°v2M02 + u'v°M1o +ui v2M12 + u2v°M20 + u2v' M21,
(5.7.28)
Notes
217
and each of the four pairs of equations ul : u° = Cl, u2 : u° = c2i V1 : V 0 = c3, v2 : v° = C4; ut : u° = cl, vi : v° = 62; and u2 : u° = c3,v2 : v° = E4, where ci and c';, i = 1, 2, 3, 4, are constants, determines a family of two-dimensional plane generators on V4.
The hypersurface (5.7.28) can also be obtained from the Segre variety P2 X P2 C Ps by projecting onto a five-dimensional space P5 if the twodimensional projection center Z contains the points M1I and M22. Let us note that the conformally flat hypersurface (5.7.28) is not a second order envelope of a one-parameter family of hyperquadrics considered in Subsection 4.4.3, and is not a hyperquadric. 5. Finally let us consider the hyperquadric defined in a projective coordinate system by the equation
X°X5+X'X4+X2X3=0.
(5.7.29)
This hyperquadric can also be obtained from the Segre variety p2 X P2 C P8 by projecting onto a five-dimensional space P5 if the two-dimensional projection center Z contains the points M11 and M22 and the straight line M11 A M22 intersects the center Z. As mentioned above, a flat asymptotic pseudoconformal CO(2, 2)-structure can be realized on the hyperquadric (5.7.29). The hyperquadric (5.7.29) is the unique conformally flat hypersurface, all completely isotropic submanifolds of which are two-dimensional plane generators.
NOTES 5.1-5.2. The study of conformal structures on a four-dimensional manifold is of special interest because of their close relation with the theory of gravitation. Spacetime in general relativity is a four-dimensional Riemannian manifold of signature (1, 3). Since many features of general relativity are of a conformally invariant nature, it is interesting to study pseudoconformal structures of signature (1,3). Along with these kinds of conformal structures, one also can consider conformal structures of signatures (4, 0) and (2, 2). By means of complexification of the manifold M, all
these structures can be reduced to one of them, for example, to the structure of signature (4, 0) or (2, 2). This has been done in many investigations (e.g., see Atiyah, Hitchin, and Singer (AHS 78], Gindikin [Gin 82, 83], Manin [Man 84], and Penrose [P 66, 77]).
The contents of Sections 5.1 and 5.2 are connected with the contents of the well-known paper by Atiyah, Hitchin, and Singer [AHS 78] who considered the decomposition of the Weyl tensor into its self-dual and anti-self-dual parts.
There is also a close relation between these sections and the twistor theory. Twistors were introduced in Penrose (P 67, 68a] (see also Penrose [P 77]). The twistor approach is based on associating a complex manifold Z ("the space of twistors") with a real manifold M endowed with a certain geometric structure and reformulating geometric problems posed for M in terms of the holomorphic geometry of the space Z. This approach has been effectively applied to the solution of a number of problems in
218
5. THE FOUR-DIMENSIONAL CONFORMAL STRUCTURES
geometry and mathematical physics (e.g., see Atiyah, Hitchin, and Singer (AHS 781 and Manin [Ma 84]). There are different ways to define the space of twistors. R. Penrose defined twistors by developing a rather complicated algebraic apparatus in which he used a system of abstract indices, the spinor calculus, and other objects (see Penrose (P 68b, 76, 77) and Penrose and Rindler (PR 86]). Another method is to define twistors using the theory of C-structures (e.g., see Alekseevskii and Graev [AGr 92a, b; 93], Rawnsley (Raw 871, and Wells [Wet 79]). The isotropic fibrations which we consider are essentially twistor fibrations. How-
ever, we refrain from using this term since it is used in geometry and physics with many different meanings (see the references in the above paragraph) while on fourdimensional conformal and pseudoconformal structures, the term "the isotropic fibrations" has a unique meaning. In twistor theory the a- and /3-planes are defined as the proportionality class of a twistor. If one defines the projective twistor space PZ as the projective version of the twistor space Z, then the points of PZ correspond to a-planes. In a similar manner the points of PZ' correspond to 3-planes (e.g., see the book Huggett and Tod [HD 85], p. 56). In physics the adapted frames are called light tetrads or Newman-Penrose tetrads (see Newman and Penrose (NP 62], Penrose [P 68], Penrose and Rindler (PR 86], or Chandrasekhar (Cha 83]). Note one more time that while for the CO(2, 2)- and CO(4)-structures the tensor C of conformal curvature splits into two subtensors C. and Cs, for the
C(1, 3)-structures, there is no such splitting since for these structures we have Co = C,,, and the subtensors Co and Co are two different complex representations of the same real tensor of conformal curvature. Note also that the real theory of four-dimensional Riemannian and pseudo-Riemannian metrics of different signatures and its applications to general relativity and the theory of superstrings were considered in the recent paper Barrett, Gibbons, Perry, Pope, and Ruback [BGPPR 94]. However, in this book we do not consider systematically physical applications of the geometric theories we have constructed. We only note that such applications are possible and supply some references on this matter. The results of these sections are due to Akivis and Konnov [AK 93], §3, and Akivis (A 96).
5.3. The Hodge operator, or the *-operator, is well-known in the theory of harmonic functions on Riemannian manifolds (see Hodge [Hod 41] and also de Rham [Rh 55], §§24--25; Wells [Wel 80], Ch. 5, §1). The Hodge operator on four-dimensional Riemannian manifolds was studied in detail by Atiyah, Hitchin, and Singer [AHS 78)
who introduced the notions of self-dual and anti-self-dual subspaces defined by the Hodge operator. The fact that the Hodge operator is conformally invariant was noted in LeBrun [LeB 82]. In the last paper as well as in Atiyah, Hitchin, and Singer [AHS 781, the Hodge operator was considered in an even-dimensional complex Riemannian manifold whose metric tensor can be reduced to the form g(e;, e,) = 5,,. To our knowledge, the theory of Hodge operator on four-dimensional conformal structures of different real types was considered in this book for the first time (see the forthcoming paper Akivis [A 96]). In particular, it is true for properties of the spectrum of the Hodge operator for conformal structures of different real types and also
for the fact that unlike the CO(2, 2, )- and CO(4)-structure, the CO(1, 3)-structure
Notes
219
cannot be self-dual or anti-self-dual, that is, semiflat without being conformally flat; this fact is of great importance for general relativity. 5.4. The results of this section are due to Akivis and Konnov (AK 93], §2, for CO(2, 2)-structures (see also Akivis [A 83a] and Konnov [Kon 92a]) and Akivis and Zayatuev [AZ 95]) for CO(1, 3)- and CO(4, 0)-structures. See also the forthcoming paper Akivis [A 96]. The classification of the Einstein spaces, which was first found by Petrov (Pe 54] (see also Pirani [Pir 57]), is given in detail in many books in general relativity (e.g., see the books Petrov [Pe 69], §§18-20, Chandrasekhar (Cha 831, Ch. 1, §9, and Penrose
and Rindler [PR 86], Ch. 8). Note that Petrov included types I, D, and 0 (for the last one all coefficients ai = 0) in type 1 of his classification, types II and N in type 2, and type III in type 3. The isotropic directions, which following Chandrasekhar [Cha 83] we called prin-
cipal, were already considered by It. Cartan (Ca 22b] who called them the optical directions. Cartan also indicated that for the Schwarzschild metric these four optical directions are reduced to two double directions. The importance of these directions was recognized in the 1950s after F. Pirani [Pir 57) indicated the physical significance of the Petrov classification. The connection of principal isotropic directions with the Petrov classification was also noted in the book Penrose and Rindler (PR 86], Ch. 8. 5.5. A CO(2, 2)-structure is associated with a three-web W(3, 2, 2) of codimension two given on a four-dimensional manifold. This fact was noted by Akivis in [A 83a] who also considered the CO(2, 2)-structure associated with the set of nondegenerate null-pairs in the real projective plane RP2. G. A. Klekovkin applied the properties of the CO(2, 2)-structure to study webs W(3, 2, 2) (see Klekovkin [Klk 81b, 83, 841). In Goldberg [Go 85, 86, 87, 88], the CO(2, 2)-structures connected with threewebs were used in the construction of maximum rank four-webs W(4, 2, 2) on a fourdimensional manifold obtained as an extension of a three-web W(3, 2, 2). 5.6. As we noted in the text, the Kerr metric is a solution of Einstein's equation. This solution was found by R. P. Kerr (see Kerr [Ke 63] and Kerr and Schild [KS 65])
and was studied in many papers. A significant part of the book Chandrasekhar [Cha 83] is devoted to the investigation of this metric. The Schwarzschild metric is a spherically symmetric solution of the Einstein equation in empty space. A derivation of this solution was first published by K. Schwarzschild [Schw 16a, 16b] shortly after A. Einstein found the fundamental equations (equations (5.4.21)) of general relativity in Einstein [Ein 161. A. Einstein highly praised Schwarzschild's results. The Reissner-Nordstrom metric is a spherically symmetric solution of the system of Maxwell-Einstein equations. It describes a black hole with a mass m and a charge Q. This solution was first presented in the papers Reissner (Re 16] and Nordstrom (No 18] and was independently derived by these two authors. The results of this section are due to Akivis and Zayatuev (AZ 95]. 5.7. The results of this section are due to Akivis and Konnov [AK 93] (§4) (see also Konnov [Kon 92a]).
Chapter 6
Geometry of the Grassmann Manifold The Grassmann manifold, or the Grassmannian, is the set of m-dimensional subspaces of an n-dimensional projective space P". It is denoted by the symbol G(m, n)' . The geometry of Grassmannians was studied in many books and papers (e.g., see Klein [KI 26b] and Hodge and Pedoe [HP 47, 52]). However, the differential geometry of Grassmannians has been considered only in a few papers (see Akivis [A 82b], Leichtweiss [Le 61], and Wong [Won 67]).
The geometry of Grassmannians is closely connected with the geometry of conformal and pseudoconformal spaces. As we have already seen in Section 1.4, the Grassmannian G(1, 3) is endowed with the structure of the pseudoconformal space C. In the same way as one can naturally pass from the geometry of conformal and pseudoconformal spaces to the study of conformal and pseudo-
conformal structures, the geometry of Grassmannians can be extended to the study of almost Grassmann structures. Moreover the same methods that were used in conformal differential geometry can be applied to the study Grassmann and almost Grassmann structures. On the other hand, the theory of Grassmannians is connected with many other branches of geometry and mathematics. This theory finds applications in the theory of multidimensional webs, integral geometry, the theory of hypergeometric functions, mathematical physics, and so on. For this reason we devote a separate chapter to the study of differential geometry of Grassmannians. I Note that sometimes the Grasemannian Gr(m, n) is defined as a set of m-dimensional subspaces of an n-dimensional vector space. The Graasmannian G(m, n) we have defined can
be obtained by the projectivization of that Grassmannian: G(m,n) = PGr(m+1,n+1). The only difference between these two Grassmannians is that the dimensions of their generating elements differ by one. 221
222
6. GEOMETRY OF THE GRASSMANNIAN
6.1
Analytic Geometry of the Grassmannian and the Grassmann Mapping
1. Consider a set of m-dimensional subspaces Pm of an n-dimensional projective space P". We will prove in this section that this set is a differentiable manifold. This is the reason that this set is called the Grossmann manifold or, in short, the Grassmannian. We will denote it by G(m, n). Let us define coordinates on the Grassmann manifold. An element of the Grassmann manifold G(m, n), a subspace Pm, can be given by means of m + 1 linearly independent points xo, x1 , ... , x,,,. We will assume that a projective frame {Ao, A1, ... , A. } is fixed in the space P". In Chapters 6 and 7 we will use the following index ranges:
0 < a, A, 7, d, e, a < m;
m+1 2m + 1, dimp = dimq = m, and dim e' = dim q' = n - m - 1. The subspaces p and q determine in Pn a subspace r of dimension 2m + 1 -the linear span of p and q, and the subspaces p' and q* intersect one another along a subspace s of dimension n - 2m - 2. All transversals of the m-pairs (p, p') and (q, q*) belong to the subspace r, and the cross-ratio of these m-pairs is equal to the cross-ratio of the m-pairs (p, pl ) and (q, qj) where p1 = p' fl r and ql = q' fl r are also m-dimensional subspaces
of the space r. Thus, by Theorem 6.1.1, the cross-ratio W of m-pairs (p,p') and (q, q') is a scalar matrix if and only if four subspaces p, pl, q, and ql of the spacer belong to the same Segrean S(1,m). Consider further the (n - m - 1)-dimensional subspaces p*1 = p A s and qj* = q A s containing the subspace s; here A denotes the linear span of the corresponding subspaces. They, as well as the subspaces p' and q', belong to the bundle with center at s. If the cross-ratio W = (p, p'; q, q') is a scalar matrix, then the subspaces pj, p', qi , and q' belong to the same Segre cone
232
6. GEOMETRY OF THE GRASSMANN MANIFOLD
whose vertex is the center s of the bundle and whose directrix is the Segre variety S(1,m). This cone has two families of plane generators of dimension n - 2m and n - m - 1, respectively.
6.2
Geometry of the Grassmannian G(1, 4)
1. As an example we consider the Grassmannian G(1,4) of straight lines in a four-dimensional projective space p4. A straight line p of this space is determined by points x and y whose coordinates form the matrix
/ x° 1 y0
x2
x1
y2
yI
x3 y3
x4 y4
(6.2.1)
The minors of this matrix xi
P$j = ly i
xj yj
,
i,j = 0,1,2,3,4,
are the Grassmann (Plucker) coordinates of the straight line p. Since (z) = 10, these coordinates define a point p in a projective space P9. But the dimension of the manifold G(1, 4) is equal to six: p = 2.3 = 6. Thus the algebraic variety 11(1, 4) C P9 also has dimension six.
By equations (6.1.2), the variety fl(1,4) is determined by the equations p01p23 + p°2p31 + p°3p12 = 0, p01p24 + p02p41 + p04p12
-0,
p01p34 + p03p41 + p04p13 = 0,
(6.2.2)
pO2p34 + p°3p42 + p04p23 = 0, p12p34 + p13p42 + p14p23 = 0.
However, since p = 6, only three of equations (6.2.2) are independent. It follows from formula (6.1.3) that the variety 1(1, 4) is of degree five. The set of all straight lines intersecting a fixed straight line p C P4 is a four-
dimensional manifold with a singular straight line p. On the variety f2(1,3), there corresponds to this set a cone C,, with vertex at the point corresponding to the straight line p under the Grassmann mapping G(1, 3) -a Ps. The cone Cp carries a one-parameter family of three-dimensional plane generators, corresponding to the bundles of straight lines whose centers lie on the straight line p, and a two-parameter family of two-dimensional plane generators that correspond to the plane fields of straight lines belonging to 2-planes passing through p. Thus the cone Cy is a Segre cone C,,(2,3), and its projectivization PCp(2,3) is a Segre variety S(1,2) = P' X P2 C P6 = PT(')(f2(1,4)). 2. Next, in the space P9, we consider a three-dimensional subspace P3 that is in general position with the variety 11(1,4). The subspace p3 meets
6.2
Geometry of the Grassmannian G(1,4)
233
12(1, 4) in five points. But since the subspace p3 itself is defined in P9 by four points in general position, the four common points of P3 and 12(1,4) uniquely
determine the fifth point of intersection of p3 and 12(1,4). Thus, any four straight lines of the space p4 uniquely determine the fifth straight line of this space. This configuration of five straight lines in P4 corresponding to five points
of intersection of a subspace p3 with the variety 12(1,4) is called the Vlasov configuration (see Vlasov [Vl 10] and Karapetyan [Kar 62a]). Let us study the Vlasov configuration in P4 and some geometric objects connected with this configuration in more detail. Let p l, p2, p3, and p4 be four straight lines in P4 in general position. Each pair of these straight lines determines a three-dimensional subspace (a hyperplane) in p4. There are six such hyperplanes, and we can take five of them as coordinate hyperplanes a', i = 0, 1, 2, 3, 4, and the sixth one as the unit hyperplane
e=a°+a'+a2+a3+a4. We will assume that
piAP2=a°, PiAP3=a',
PiAP4=a2,
P2AP3 =a3, P2AP4 =a4, P3AP4 =e In these formulas pa A pp denotes the linear span of the points Pa and pp, a,/3 = 1,2,3,4. In the space P4, consider the point frame {A1} which is dual to the tangential frame (a'). Then we have
(a',At)=5, and the straight lines p, a = 1, 2, 3, 4, are represented as follows:
p1= a0Aa'Aa2=A3AA4i P2=a°A a3 A a4 =A1 A A2,
P3=a'Aa3Ae=a'Aa3A(a°+a2+a4)=(A2-Ao)A(A4-Ao), p4 =a2Aa4Ae=a2Aa4A(a°+al+a3)=(A, -Ao)A(A3-Ao). (6.2.3)
Every triple of straight lines pa has a unique secant. Consider, for example, the straight lines pl,p2i and p3. In pairs, they define three hyperplanes a°, a',
and a3 that meet in the straight line a° A al A a3. Hence the straight lines p1, p2, and p3 determine a unique secant
q4=a°Aa'Aa3=A2AA4.
(6.2.4)
Similarly {Pi, P2, P4 }
q3
= a° A a2 A a4 = A, A A3,
{Pi,Ps,P4}q2=a'Aa2Ae=(A3-Ao)A(A4-Ao), {P2,P3,P4} qi =a3Aa4Ac=(A, -Ao)A(A2-A°).
(6.2.5)
6. GEOMETRY OF THE GRASSMANN MANIFOLD
234
In pairs, the straight lines pa and qa define four hyperplanes 0°. Let us prove that these hyperplanes meet in the straight line p5. We have
0'=pIAq,=A3AA4A(A,-Ao)A(A2-Ao), Q2=P2Ag2=AlAA2A(A3-Ao)A(A4-A0), Q3=P3Ag3=AlAA3A(A2-Ao)A(A4-Ao), p4
=P4Ag4 =A2AA4A(A, - Ao) A (A3 - Ao).
In point coordinates, with respect to the frame {A;}, the equations of these hyperplanes can be written as r x0 + XI + x2 = 0, x° + x3 + x4 = 0,
Sl x°+x2+x4 =0, x°+x' +x3 =0. Equations of this system are linearly dependent, and a basis of the solution space of this system is
(-1,0,1,1,0), (-1,1,0,0,1). Thus the hyperplanes (i° meet in the straight line
p5 =(A2+A3-A0)A(A,+A4-A0).
(6.2.6)
We will prove now that the straight line p5 is the fifth straight line of the Vlasov configuration defined by the straight lines p, , p2i P3 and p4. To this end, in the space P9 we will find equations of the subspace p3 defined by the points
pa, a = 1,2,3,4. We will look for these equations in the form aiiP'I = 0.
Since the coordinates of the straight lines pa must satisfy these equations, by (6.2.3), we obtain the following six equations: P14 = 0,
P23 = 0,
P01 - p13 = 0,
P02 - p24 = 0, P°3 + p'3 = 0, P°4 + p24 = 0,
which determine the subspace p3 in P9. However, it is easy to check that the Grassmann coordinates of the straight line ps also satisfy these equations. Thus this straight line p5 along with the straight lines pi, p2i p3, and p4 form the Vlasov configuration. Moreover each of the straight lines PI,P2,P3,P4, and Ps have equal status in the Vlasov configuration, and every four of them determine the fifth one. Note that the straight lines qa and p5 also form the Vlasov configuration defined by the subspace P3 C P9. The subspace P3 is defined by the following system of equations: P14 = 0,
P23 = 0,
001 - P12 = 0,
p03 - p34 = 0, p02 + p12 = 0, p04 + p34 = 0.
6.2
Geometry of the Grassmannian G(1, 4)
235
The point ps is the only common point of the subspaces p3 and 3. Consider two-dimensional planes intersecting all five straight lines of the Vlasov configuration. We will represent such 2-planes in the form P.
a=AAp= \ipja'nay = Iaija'Aaj, where A = \iai and p = pia' are hyperplanes in P4 and pi
a,j -
I
Ai
pj
(6.2.7)
are their Grassmann coordinates. The condition that the 2-plane a and the straight line p with coordinates p'3 intersect one another can be written as
(a, P) = aii p'j = 0.
(6.2.8)
To "see" the Grassmann coordinates of the straight lines composing the Vlasov configuration, we will rewrite equations (6.2.3) and (6.2.6) in the form Pi = A3 A A4, P2 = Al A A2,
p3=AoAA2-A0AA4+A2AA4i p4 = AoAA, -AoAA3+Al AA3, ps = -AoAA, + AoAA2+AoAA3 -AoAA4 -Al AA2-A,AA3+A2AA4+A3AA4. By (6.2.8), the condition that the 2-plane a and the straight line p intersect
one another can be written as a34 = 0, a12 = 0,
Opt - a04 + a24 = 0, a0l - a03 + a13 = 0,
(6.2.9)
-a01 + 0`02 + a03 - a04 - a12 - a13 + a24 + a34 = 0.
As one can expect, the last equation is a linear combination of the first four equations, and as a result the 2-planes a are defined by the first four equations of this system. Consider the Grassmannian G(2, 4) of two-dimensional planes in the space P4. As in the case of the Grassmannian G(1,4), its dimension is equal to six. By means of the Grassmann coordinates aii, the Grassmannian G(2, 4) can be mapped onto the algebraic variety 11(2,4) of the projective space (P9)' which is dual to the space P9 containing the variety 11(1,4). As is the case for the variety 12(1,4), the degree of the variety 11(2,4) is equal to five. Equations (6.2.9) define in (P9)' a subspace of dimension five that meets the variety Q(2,4) in a two-dimensional surface. In view of this, in the space P4 there exists a two-parameter family of 2-planes a intersecting all five straight lines
6. GEOMETRY OF THE GRASSMANN MANIFOLD
236
of the Vlasou configuration. The set of these 2-planes form a congruence called the Vlasov congruence. Let us find the equation of the Vlasov congruence in tangential coordinates.
To this end, we substitute for the Grassmann coordinates a, in equations (6.2.9) their values (6.2.7). We find that A3p4 - A4A3 = 0, A1 /p2 - 1\2/41 = 0, AO/12 - 1\2/10
1\oµ4 + A4/W + 1\2/44 - 1\4/12 = 0,
(6.2.10)
A0141 - AI/10 -A0143 + A3p0 + \1/13 - 1\301 = 0.
All 2-planes determined by the system of equations (6.2.9) intersect the straight lines of the Vlasov configuration. Hence the hyperplane is can be chosen in such
a way that it passes through the straight line p1. Then, by (6.2.3), we obtain 143 = p4 = 0. As a result the system of equations (6.2.10) takes the form AI/12-A2111=0, 1\0112 - A2/W + A4p0 -,\4/12 = 0,
Ao/h - AI/1o + Alpo - A3p = 0. Excluding the quantities µo, p1 and P2 from these equations, we arrive at the following cubic equation: AI (A0 - A3)(A2 - A4) - A2(AO - A4)(A1 -- A3) = 0.
(6.2.11)
This equation determines the Vlasov congruence in tangential coordinates in the space P4: any hyperplane A whose coordinates satisfy equation (6.2.11) contains at least one 2-plane of the Vlasov congruence. Since this equation is of the third degree, the Vlasov congruence is a congruence of the third class. In other words, every pencil of hyperplanes of the space p4 contains three hyperplanes whose coordinates satisfy equation (6.2.11). Note that Karapetyan (Kar 62a] considered the dual Vlasov configuration formed by five 2-planes of the space p4 corresponding to five points of intersection of the variety l(2, 4) with the subspace (P3)' of the space (P9)' which is dual to the space P9 and in general position with f2(2, 4). A two-parameter family of straight lines intersecting five 2-planes of the dual Vlasov configuration forms a hypersurface called the Vlasov hypersurface. It is a hypersurface of third order.
6.3
Differential Geometry of the Grassmannian
1. We will pass now to the study of differential geometry of the Grassmannian. In the space P" we consider the family R(P") of projective point frames {A(}
6.3
Differential Geometry of the Grassmannian
237
and the family R'(P") of tangential frames {af} formed by the hyperplanes at. The condition of duality of these frames has the form (A(, a") = 6k,
(6.3.1)
where the parentheses denote the convolution of the corresponding elements, and the condition (A, a) = 0 is the incidence condition of the point A and the hyperplane a. Let us write the equations of infinitesimal displacement of the point and tangential frames of the space P" (cf. Section 4.3): dAt = w" A,,, dat = 86a",
where " and 0 are 1-forms. By differentiating (6.3.1), we can easily find that 6n + wt = 0. In view of this, the equations of infinitesimal displacement take the form: dAt = w' A,r, dac = (6.3.2) Relations (6.3.1) also imply that the forms WE are related by the condition
Wp +Wj +...+wn =0.
Thus, each of the families of frames R(P") and 7Z* (P") depends on (n + 1)Z - 1 = nZ + 2n parameters. In addition the forms w, satisfy the structure equations of the space P" (cf. Section 4.4): (6.3.3) dw(=wnAWC. Let p be an m-dimensional subspace in P", or briefly an m-plane. We will assume that m < i (n - 1), since the case m > (n - 1) is dual to the first one z and can be studied in a similar manner. Moreover we will assume that m > I
and n - m > 2, since if m = 0 and m = n - 1, the Grassmannian G(m, n) becomes a projective space. We associate with the plane p a subfamily R(p) of projective frames of the space P" such that the vertices A" of its frames lie in the plane p. We denote by R' (p) the subfamily of projective frames that are dual to the frames of the subbundle RZ(p). The hyperplanes a' of the frames of R' (p) pass through the m-plane p. By virtue of this, the plane p can be determined in two ways:
p=AO AA, A...AAm= am+' A...Aa",
(6.3.4)
where as earlier, the expression Ao A A, A ... A A,,, is the linear span of the points A,, and a'+1 A ... A an is the intersection of the hyperplanes a'. The manifold of m-planes p generates the two fiberings in the frame man-
ifolds R(P") and R'(P") whose fibers are the subfamilies R(p) and 1'(p). These fiberings are determined as the projections
n : R(P") -a G(m, n), a' : R' (P") - G(m, n),
(6.3.5)
6. GEOMETRY OF THE GRASSMANN MANIFOLD
238
where 7t-1(p) = R(p), (n')-1(p) = R'(p). From equations (6.3.2) it follows A an that the condition for the plane p = A0 A Al A ... A Am = am+' A to be fixed can be written as wa = 0. So the 1-forms w, are horizontal (base) forms for the fiberings (6.3.5). These forms are linearly independent on the Grassmannian G(m, n), and their number is equal to the dimension p = (m + 1) (n - m) of G(m,n). 0, equations (6.3.2) take the form For Gap = -1rpa'V - apa{,
6A,, = rrpAp,
(6.3.6)
6Ai = 7r°A° + 7r? Aj, 6a' _
where 6 = dI,< =0 and of = we(d). The 1-forms 7rO, 7ri and ni are invariant forms of the stationary subgroup of the m-plane p in the space Pn. They are the fiber forms of the fiberings (6.3.5). 2. We consider now the Grassmann mapping y of the Grassmannian
G(m, n) onto the algebraic variety f l(m, n) of the space P", where +1 y : G(m,n) -a fl(m,n). Suppose that TM (0) is the tangent bundle of the variety It (m, n). Let us find the differential of a point p E fl(m, n) that corresponds to an m-plane p in Pn. By differentiating (6.3.4), we find that dp = wp + w'pi,
(6.3.7)
where w = wo + ... + wm and
p° = AOA...AA°_1 AA, AA0+1 A...AAm = am+' A...Aa'-' Aa° Aa'+l Aan are linearly independent points in P^' that also belong to the variety f2(m, n). The points p° together with the point p determine the tangent subspace TP1) to
the variety fl(m, n), and the dimension of this subspace
is
equal to
pi = p = (m + 1)(n - m). Formulas (6.3.7) show that the space TP1)(fl) is isomorphic to the vector space of rectangular (m + 1) x (n - m) matrices. Next we will find the tangent bundle Ty2)(fl) of second order of 1(m, n) whose element is an osculating subspace of the variety Sl(m, n). To this end, we calculate the second differential of the point p: d2p =
a determined by the points p and cap°. The dimension of the subspace Dy3)(a) is dim Dy3)(a) = 2(n - m). The subspaces Dy31(a) form the first family of plane generators of the cone
242
6. GEOMETRY OF THE GRASSMANN MANIFOLD
C. In the same manner, if we set TI = ci\ and vary entries of the matrix (oa ), then a point q E CP3) will describe a subspace Op3> ((3) in the tangent subspace T11 determined by the points p and cap°. The dimension of the subspace Ay31(Q) is dim Op3)(Ji) = 2(m + 1). The subspaces 43j(3) form the second family of plane generators of the cone Cp31 The cone Cp3l is connected also with the following construction on the Grassmannian G(m, n). Let p be a fixed m-dimensional subspace of the space
P". Consider the set of m-dimensional subspaces intersecting the subspace p at a subspace q of dimension m - 2. This set is a submanifold of the Grassmannian G(m, n), and the subspace p is singular on UD31. On the variety 1(m, n), to the submanifold UP31 there corresponds a submanifold with a singular point p which we will denote also by U. The asymptotic cone CD3' C Tp) is a tangent cone to the submanifold UP(3) C fl(m, n) at the point P.
The submanifold Up3l in G(m, n) is stratified into two families of Grassmann manifolds. In fact, if we fix an (m - 2)-subspace q C p, then the set of rn-dimensional subspaces p passing through q is equivalent to the Grassmannian G(1, n - m + 1), since every subspace p' meets the subspace q*, which is complementary to q, in a straight line. On the other hand, if we fix an (m + 2)-dimensional subspace r passing through p, then the set of all m-dimensional subspaces p` lying in r and intersecting p at an (m - 2)-subspace forms the Grassmannian G(m,m + 2). Under the Grassmann mapping, to the Grassmannians G(1, n - m + 1) and G(m, m + 2) there correspond the algebraic varieties fl(1, n - m + 1) and UP(3) fl(m, m + 2) belonging to and passing through the point p. The dimensions of these varieties are dim Q(1, n - m + 1) = dim G(1, n - m + 1) = 2(n - m) and dim fl(m, m + 2) = dim G(m, m + 2) = 2(m + 1). The plane generators 43)(a) and Di,3)(f3) of the cone Cn3) are the tangent subspaces to the varieties It(1, n - m + 1) and !i(m, m + 2), respectively.
On the Grassmannian G(m, n), to the intersection of the varieties S?(1,n. - m + 1) and S1(m,m + 2) there corresponds a set of m-subspaces p passing through the (m - 2)-subspace q and belonging to the (m + 2)-subspace r. This set is equivalent to the Grassmannian G(1, 3) and is of dimension four. Hence two generators A ,3) (a) and Op3) (A) of the cone Cp3' intersect one another at a four-dimensional subspace of the space TP'I.
The structure of the asymptotic cone CLkl of order k < m + 1 can be investigated in a similar manner. The cone CD k) is a determinantal variety of dimension (n - k + 2)(k - 1) and carries two families of plane generators D,,ki(a) and A ($) of dimension (n - m)(k - 1) and (m + 1)(k - 1), respectively. Moreover two generators belonging to different families have in common a subspace of dimension (k -1)2. We will call the generators of the first family the a-subspaces of the asymptotic cone and the generators of the second
6.3
Differential Geometry of the Grassmannian
243
family its f3-subspaces.
The projectivization PCP(k) of the cone Cpk) is an algebraic variety in the space Tell (fl). The latter space is a collection of (k-1)-secants of the Segre vari-
ety S(m, n-m - 1) = PC,21. Thus all asymptotic cones Cpkl, k = 2.... ,m+1, are determined by the cone C,2) of second order. In the same way as above, we can prove that the submanifold PCP(k) is the set of singular points of the submanifold Cpk+>
Note one more time that for k = 2 the asymptotic cone CP21 lies on the variety fl(m,n), while for k > 2 the asymptotic cones Cpki lie in the subspace Tpll(fl) but not in fl(m,n) itself. 4. The plane generators of each of two families of asymptotic cones Cnk) form a fiber bundle over the variety fl(m, n). Let us consider, for example, the fiber bundle E( k) of generators 4Dk)(a) of dimension (n - m)(k - 1). Its fiber has the dimension (m - k + 2)(k - 1). An integral submanifold of the fiber bundle E,(, k) is called a manifold VQkI
of dimension (n - m)(k - 1) whose all tangent subspaces belong to E. The integral submanifolds are asymptotic a-submanifolds of order k of the variety 11(m, n).
If k = 2, then the asymptotic a-submanifolds are a-subspaces of dimension
n - m of the variety fl(m, n). On the Grassmannian G(m, n), to these asubmanifolds there correspond bundles of m-subspaces with an (m - 1)-dimensional center.
If k = 3, then the asymptotic a-submanifolds are submanifolds VQ31 of dimension 2(n - m) to which on the Grassmannian G(m, n) there correspond bundles of m-subspaces with (m - 2)-dimensional centers. The asymptotic a-submanifolds of any order k can be defined in a similar manner. Consider next the fiber bundle Eokl of generators OP(k) (/3) of the asymptotic cones Cpk) and its integral submanifolds Vpk), dim VAk) = (m+ 1)(k- 1). These integral submanifolds are asymptotic 13-submanifolds of order k of the variety f] (m, n).
If k = 2, then these asymptotic f3-submanifolds are f3-subspaces of dimen-
sion m + I of the variety f2(m, n). On the Grassmannian G(m, n), to these f3-submanifolds there correspond bundles of m-subspaces belonging to a subspace of dimension m + 1.
If k = 3, then the asymptotic f3-submanifolds are submanifolds V(3) of dimension 2(m + 1). On the Grassmannian G(m, n), to these f3-submanifolds there correspond bundles of m-subspaces belonging to a subspace of dimension
m+2. Thus, for each k = 2, ... , m + 1, the variety f? (m, n) carries two families
of plane generators z$ (a) and o, (j3) of dimension (n - m)(k - 1) and (m + 1)(k - 1), respectively. If asymptotic submanifolds of different families intersect each other, then the dimension of their intersection is equal to (k-1)2.
244
6. GEOMETRY OF THE GRASSMANN MANIFOLD
Finally, consider the asymptotic lines of order k on Sl(m, n). Any such line at any of its points is tangent to an asymptotic direction of order k. In the space P", to these asymptotic lines there correspond one-parameter families of enplanes in each of which any two infinitesimally close planes have an (m-k+1)plane in common and belong to an (m+k-1)-plane. Such families of m-planes
are called (m - k + 1)-focal. In particular, in the space P", to the asymptotic lines of second order there correspond (m - 1)-focal (or torsal) families of enplanes, and to the asymptotic lines of order in + 1 there correspond 0-focal families of m-planes.
6.4
Submanifolds of the Grassmannian G(m, n)
1. Families of m-dimensional subspaces in a projective space P" were studied
rather intensively. This started in the nineteenth century from the study of different families of straight lines in a three-dimensional projective space Ps (see Section 3.4). Later these investigations were generalized to the projective
space P". We will consider only smooth families F of subspaces. If an m-dimensional
subspace p of the family 7 depends on r parameters, we will denote such a
family by P. If r < n - m, then the family P forms a point submanifold of dimension m + r in the space P". However, this submanifold can have singular points (e.g., see Akivis [A 57, 87]). If r = n - m, then a family Tr is called a congruence. A congruence of subspaces is characterized by the fact
that there passes a finite number of subspaces of Fr through any point of general position in P". If r > n - m, then a family P is called a complex. If r = (m + 1)(n - m) - 1, then a family .7'' is called a hypercomplex. The latter family is a submanifold of codimension one on the Grassmannian G(m, n). Different types of families of subspaces were investigated by R. M. Geidelman, Moscow, S. E. Karapetyan, Erevan, K. I. Grincevicius, Vilnius, L. Z. Kruglya-
kov, and R. N. Shcherbakov, both Tomsk, and their students and colleagues (see the book Kruglyakov [Kru 80) and the survey papers Geidelman [Ge 67a) and Shcherbakov [Sh 67]).
However, in most of these works the Grassmann mapping of families of m-dimensional subspaces onto the algebraic variety !l(m, n) C PN, N = ( +j) - 1 was not used at all or used occasionally; there are some exceptions; see, for example, the papers Karapetyan [Kar 62a, b, c] and the recent book Mizin, Chupakhin, and Shcherbakov [MCS 911 in which the Grassmann mapping was used as the main tool of investigation. The application of the Grassmann mapping allows one to see many facts of the geometry of families of m-dimensional subspaces from another point of view and to find many new results. We will conduct such an investigation for families of two-dimensional subspaces (2-subspaces) in the space P5. The choice of dimensions 2 and 5 is motivated by the fact that in the space P5, a complementary subspace to a
6.4
Submanifolds of the Grassmannian G(m, n)
245
2-subspace again is a 2-subspace (as in the space p3 a complementary subspace to a straight line is again a straight line).
We will consider families F? depending on r = 2, 3, and 5 parameters. In the first case, a family F2 constitutes a planar hypersurface in the space P5, in the second, a family F3 is a congruence of 2-subspaces, and in the third case, a family F5 is a complex of 2-subspaces. 2. A family of 2-subspaces in the space P5 is the Grassmannian G(2, 5). Its dimension is p = 9, and the Grassmann mapping sends it bijectively onto the
algebraic variety f)(2,5) C PN, N = (3) - 1 = 19. The degree of the variety 11(2, 5) is calculated by formula (6.1.3) and is equal to 42.
We will write for the Grassmannian G(2,5) some formulas from Section 6.3. Since m = 2, then the variety 0(2,5) has at any of its points only two asymptotic cones Cpl) and CP31, CP21 C CP3). In notations of Section 6.2, the equations of the asymptotic cone CP21 of the variety 11(2,5) can be written in the form: rank (m,`,) = 1, a= 0, 1, 2; i = 3A5, 5, (6.4.1)
and the equations of the asymptotic cone CP3) in the form
det (w,) = 0.
(6.4.2)
It follows that the cone CP3> is a hypercone of third order in the tangent subspace TDII(fl) whose dimension is equal to 9. The cone f'-(2) is the Segre cone of dimension 5 whose projectivization is the Segre variety S(2,2) C Ps. The degree of this Segre variety S(2, 2) is equal to (a) = 6. The Segre cone carries two two-parameter families of three-dimensional plane generators Opal(a) and Op21(f3), and the cubic cone -P( 2) carries two two-parameter families of six-dimensional plane generators A(3) (a) and AP(3) Consider next a two-parameter family F of 2-subspaces in the space P5. On the variety f2(2, 5) to such a family there corresponds a two-dimensional smooth submanifold V2. The classification of points of the submanifold V2
depends on the mutual location of its tangent subspaces Tyll (V2) and the cones Cpl) and Cp3).
In the general case, the subspace T,')(V2) has no common straight lines with the cone CP2) and meets the cone C,? in three straight lines. Thus, on V2, there are no asymptotic directions of first order, and there are asymptotic directions of second order. If V2 is a submanifold of general type, then the described situation holds at any point of V2. By virtue of this, a three-web, formed by three families of asymptotic lines of second order, arises on V2. This relationship between the geometry of families and the theory of twodimensional three-webs was considered in Zhogova [Zh 78, 79). In the space P5, to the asymptotic lines of second order of the submanifold V2 there correspond one-parameter subfamilies F1 of 2-subspaces, and these
246
6. GEOMETRY OF THE GRASSMANN MANIFOLD
subfamilies have one-dimensional envelopes. In any 2-subspace of the family .P2, there are three focal points of second order in which this subspace is tangent to the envelopes of three subfamilies passing through this 2-subspace. Note also that a two-parameter family F2 of 2-subspaces in the space P5 can be considered as a hypersurface with two-dimensional plane generators
in P5. In this case, three focal points in a 2-subspace are singular points of this hypersurface-in these points the tangent subspace to a hypersurface is of dimension three.
3. Consider a three-parameter family F3 of 2-subspaces in the space P5. It is a congruence. On the variety fl(2, 5), to such a family there corresponds a three-dimensional submanifold V3. The tangent subspace Tpll(V3) meets the asymptotic cone '"P(3) of third order in a two-dimensional cone of third order. Each generator of this cone determines asymptotic directions of third order at a point p E V3 that are tangent to the asymptotic lines of third order of V3 passing through the point p. In the space P5, to these asymptotic lines there correspond 0-focal families of 2-subspaces containing the subspace p. This implies that the focal points of the subspace p, that is, the points at which 0-focal families are tangent to their envelopes, form a cubic curve W3 in the 2-subspace p. To find an equation of the curve W3, we will write the equation of the congruence .F3 in the form wa = pupBa
(6.4.3)
where a, ,C = 0, 1, 2; i = 3, 4, 5, and the I-forms 00 are linearly independent basis forms on the congruence y3. In the tangent subspace TPII(f1), equations (6.4.3) represent parametric equations of the three-dimensional subspace Tptl(V3). Suppose that x = xaA,, is an arbitrary point of the subspace p C P5. The differential of this point has the form dx = (dxa + x0w, )Aa + xawQA;.
The focal points of the subspace p and the focal directions on the congruence are determined by the conditions xaWa = 0.
By (6.4.3), these conditions can be written as
xap.pBa = 0. Since at the focal points this system must have a nontrivial solution with respect to the forms 0a, the locus of focal points is determined by the following equation: det (xap;,Q) = 0. The latter equation is of third degree with respect to xa, that is, it determines a cubic curve W3 in the subspace p.
6.4
Submanifolds of the Grassmannian G(m, n)
247
On the other hand, the equations of the cone Ty')(V3) n C(3) of focal directions have the form det (paaB0) = 0,
which also immediately follow from equation (6.4.2) of the cone C;31
Consider now the projectivization PTp' 1(12) of the variety D = Q(2,5) and the projectivization PTnll (V3) of the submanifold V3. Denote by 03 the cubic curve in which the projectivizations PTpll (V3) and PC,3) meet: PT$1)(V3) n PCP(3) = 03- If the 2-subspace PTp')(V3) does not meet the Segrean PCP(2) = S(2, 2), then the above cubic curve does not carry singular points. This corresponds to a congruence F3 of general type. If the 2-subspace PTyll (V3) does meet the Segrean S(2, 2), then to each intersection point there corresponds a 1-focal direction on a congruence F3. Then through a 2-subspace p there passes a torse (i.e. a developable submanifold), formed by 2-subspaces of F3. Furthermore, the 2-subspace p contains the characteristic line I of this torse, and the 2-subspace p itself belongs to a three-dimensional subspace L that is tangent to the torse along p. If the curve 03 has one singular point, then the focal cubic W3 decomposes
into a straight line I and a conic C (see Figure 6.4.1). If the curve 03 has two singular points, it necessarily has a third one and thus the focal cubic W3 decomposes into three straight lines that are in general position (see Figure 6.4.2). Congruences F3 of this kind are called totally focal. We will consider the latter case in more detail. Let us choose a moving frame in such a way that the points A0, A,, and A2 are the intersection points of three characteristic straight lines of a subspace p, and that the points A3, A4,
Figure 6.4.1
Figure 6.4.2
248
6. GEOMETRY OF THE GRASSMANN MANIFOLD
and A5 are located in the subspaces Lo, LI and L2, where L. is a 3-subspace tangent to the torse whose characteristic in the subspace p is the straight line 1,, (see Figure 6.4.2). Let 90, 91, and 92 be basis forms of the congruences J chosen in such a way that they are basis forms of the torses indicated above. Then the torse To is determined by equations 91 = 92 = 0. Since the straight line 10 = Al A A2 is the characteristic straight line of the subspace p, then NO C p, from which we obtain w1 w2 0 (mod 91,92).
Since the 3-subspace Lo = p A A3 is tangent to the torse described by the subspace p, we also have wo, wo = 0
(mod 91,92).
In a similar manner we obtain wa, w2 M 0
(mod 90,92),
wi, wi =- 0
(mod 90, 92)
wo, wi
(mod 9°, 91), (mod 90, 9').
and
0
W232 wz = 0
We recall that in these equations i = 3,4,5. Comparing all these relations, we find that w30 =
Wp90r
wi = 0, wz = 0,
w4 0
= 0+
5
w0 =
0+
,) = go" wl = 0, w2 = 0,
w25 =
(6.4.4) r02.
The coefficients p, q and r in these equations can be reduced to 1 by normalizing
the points A3, A4, and A5. Taking exterior derivatives of equations (6.4.4), where p = q = r = 1, we arrive at the following exterior quadratic equations:
won91 -w3n9°=0, won92-wyA9°=0, w; A 9° - w3 A W =0, w2 A 0° - w5 A 92 = 0,
wIA92-w5A0'=0,
w2
(6.4.5)
A0'-w4A92=0.
Consider the submanifold described by the point A0 of intersection of the characteristic straight lines 11 and 12. By (6.4.4), the differential of the point A0 can be written as dAo = woA0 +woA, +w0A2 +w0A3.
(6.4.6)
6.4
Submonifolds of the Grassmannian G(m, n)
249
Thus the subspace L° is tangent to the three-dimensional submanifold (Ao). The 1-forms w0 and wo occurring in formula (6.4.6) can be found from equations (6.4.5) by means of Cartan's lemma: W02 =10262
100 =10101 + 101000,
+ lone°,
(6.4.7)
-w3 =10082 -1308°, -w3 =10082 -13000.
As to the 1-forms w03, this form can be found from equations (6.4.4) by taking p = 1. Substituting the values of all these forms into equation (6.4.6), we find
that dAo = woAo + 1018' Al + 10202A2 + (100A1 + 100A2 + A3)9°.
We further specialize our moving frame by placing the point A3 on the tangent to the line 91 = 92 = 0 of the submanifold (Ao). Then we obtain
100=loo=0 and
dAo =woAo+1018'A1 +10292A2+8°A3. Equations (6.4.7) now become w0 =1010'+
42
= 10202,
(6.4.8)
w3 = 1300°, w3 = 1308°
We need to prove that the coordinate lines on the submanifold (Ao) form a conjugate net. To this end, we find d2Ao = [130(0°)2 + 1011(01)2 ]A4 + [153 0(90)2 + 12 02 (82)2JA5
(mod Lo).
It follows that the second fundamental forms of the submanifold (Ao) are 4(2) = 130()2 + 101(0')2, '
(2)
= 130(90)2 + 102(82)2
The fact that these forms are sums of squares proves that the coordinate lines on the submanifold (Ao) form a conjugate net (see Section 3.2, and for more detail, see Akivis and Goldberg [AG 931, Ch. 3). Moreover the three-dimensional submanifold (Ao) is stratified into three families of two-dimensional surfaces carrying conjugate nets. In fact, taking the exterior derivatives of the basis forms 8° and applying equations (6.4.4), we find that
2+aA0" dO' = PaQ - w2+o)
where there is no summation with respect to a. By the Frobenius theorem, each of the equations 00 = 0 is completely integrable, and this proves that the submanifold (Ao) is stratified as indicated above. Similar conclusions are true for the submanifolds described by the points Al and A2.
6. GEOMETRY OF THE GRASSMANN MANIFOLD
250
Since the totally focal congruences considered above are congruences of special kind, we must prove their existence. This can be done by applying the Cartan test to the system of equations (6.4.4) in the same way as this was done in Section 3.2 for submanifolds carrying a net of curvature lines. It turns out that a totally focal congruence of 2-subspaces in the space P5 exists, and the solution of the system defining such a congruence depends on six functions of two variables.
4. Consider finally a five-parameter family (a complex) F5 of 2-subspaces
in the space P5. The number of parameters, on which a 2-subspace of Y5 depends, coincides with the dimension of the space P5. This is the reason that such complexes are of interest for integral geometry in the sense of 1. M. Gelfand (see Gelfand and Graev [GG 68]).
Under the Grassmann mapping, to a complex FS there corresponds a fivedimensional submanifold V5 C 0(2, 5). At any point p, the submanifold V5 has the five-dimensional subspace Tpll(VS) whose projectivization is the fourdimensional projective space PTP11(V5). To different cases of mutual location of the subspace Tpl1(V5) and the asymptotic cones ($2) and Cp3) there correspond different classes of complexes P. It is convenient to conduct the investigation of these cases by considering the projectivization of the tangent subspace Tpl) (52) under which PCP(') = S(2,2) and PC P31 = VT C Ps, where VT is a cubic hypersurface in Ps. In the general case, the subspace PTp11(V5) meets the variety S(2, 2) at six points which we will call the characteristic points. On the submanifold V5, to these points there correspond six fields of asymptotic directions of first order. They are tangent to six families of asymptotic lines of first
order on V5. In the space P5, to these families there correspond six families of torses (developable surfaces) on the complex P. Through any 2-subspace p C .PS, there pass six torses, one from each family. Thus six characteristic straight lines arise on the 2-subspace p, and six three-dimensional subspaces tangent to these torses pass through the 2-subspace p. Moreover the subspace PTp'1(VS) meets the hypercubic V3 in a threedimensional cubic submanifold. Since the hypercubic V3 C P8 carries two two-parameter families of five-dimensional plane generators, the submanifold
V3 = V3 n PT, (V5) carries two two-parameter families of rectilinear a-generators and fl-generators. In view of this, the submanifold- V5 carries two fiber bundles El?) and F(2) of two-dimensional asymptotic directions of second order. The asymptotic lines of second order on V5 are tangent to twodimensional directions belonging to these fiber bundles. In the space P5, to these lines there correspond 0-focal one-parameter families of 2-subspaces belonging to the complex P. If we fix a point m in a 2-subspace p C .PS, then through this point there passes a two-parameter family of 2-subspaces of the complex Y$ which contains
the 2-subspace p. This family is a four-dimensional cone C,,, with vertex at a point m. On the submanifold V5 C 52(2,5), to the cone C,,, there corre-
6.4
Submanifolds of the Grassmannian G(m, n)
251
sponds an integral submanifold of the distribution E.( 2). On the other hand, if we consider a four-dimensional subspace p passing through a 2-subspace p, p D p, then a two-parameter family of 2-subspaces of the complex F5 belongs to this subspace p. On the submanifold V5 C Q(2,5), to the subspace p there corresponds an integral submanifold of the distribution Ep . A projectivization PCm of the cone C,,, is a two-parameter family of straight lines in a four-dimensional projective space p4. If the vertex m of this cone does not belong to any of six characteristic straight lines, then the family of straight lines indicated above constitutes a ruled hypersurface without singular points. If the vertex m lies on one of the characteristic straight lines, then there is one singular point on each straight line of the family PC,,,, and the family PCm is semifocal. If the vertex m is the intersection point of two characteristic straight lines, then there are two singular points on each straight line of the family PC,,,, and the family PCm is focal. In this case the ruled hypersurface indicated above is tangentially degenerate (see Subsection 3.3.3). In the general case six characteristic straight lines in the 2-subspace p and six three-dimensional tangent subspaces to the torses passing through p are in general position. If they are in a special (not general) position, then the corresponding complexes F belong to special classes of complexes. Such special classes of complexes were considered in Bubyakin [Bub 90, 91]. We will describe briefly results of these papers.
Theorem 6.4.1 If three characteristic straight lines of each 2-subspace p c F5 belong to a pencil with center at a point m, and the corresponding threedimensional characteristic subspaces are in general position, then the point m describes a hypersurface in the space P5 to which the 2-subspaces of the complex .F5 are tangent (see Figure 6.4.3). The tangent three-dimensional subspaces of three other torses belong to the tangent subspace of this hypersurface.
Figure 6.4.3
Figure 6.4.4
6. GEOMETRY OF THE GRASSMANN MANIFOLD
252
Figure 6.4.5
Figure 6.4.6
Theorem 6.4.2 If two triplets of characteristic straight lines of each 2-subspace p C J belong to two pencils with centers at points m1 and m2, then the straight line m1m2 is characteristic, and the points m1 and m2 describe hypersurfaces in the space P5 to which the 2-subspaces of the complex 25 are tangent (see Figure 6.4.4).
In addition to the two cases indicated in Theorems 6.4.1 and 6.4.2, there are only two more configurations of characteristic straight lines of a complex .FS (see Figures 6.4.5 and 6.4.6). For these additional configurations the points m1 i m2 and m3 (or m1, m2, m3, and m4) describe hypersurfaces to which the 2-subspaces of the complex F5 are tangent. As it follows from Theorem 6.4.1, all four classes of complexes indicated above are self-dual; that is, each of them is transformed into itself under any correlative transformation of the space P5. One can also consider special classes of complexes .F5 characterized by a pairwise congruence of characteristic straight lines or characteristic three-dimensional characteristic subspaces of 2-subspaces
of the complex P.
6.5
Normalization of the Grassmann Manifold
1. In this section on the Grassmann manifold G(m,n) of m-dimensional subspaces of an n-dimensional projective space P", we consider a certain supplementary construction called the normalization. By means of this normalization, one can construct the structure of a Riemannian or semi-Riemannian manifold or an affine connection on G(m, n). Let U be an open domain of the Grassmann manifold G(m, n) of dimension p = (m + 1)(n - in) coinciding with the dimension of G(m, n). This domain
6.5
Normalization of the Grassmann Manifold
253
can coincide with the entire manifold G(m, n) or can be its proper subset. The domain U is said to be normalized if to each its m-dimensional subspaces p there corresponds a chosen subspace p' of dimension n - m - 1 in the projective space P", such that p' does not have common points with p. The subspace p' is called the normalizing subspace for the subspace p. We will denote a normalized domain U by U". If U = G(m, n), then we will denote the normalized Grassmann manifold G(m, n) by G" (m, n).
Since the subspace p' belongs to the Grassmannian G(n - in - 1, n), a normalization of the manifold G(m, n) is defined by a normalizing mapping v : G(m, n)
G(n - in - 1, n)
(6.5.1)
given in the domain U C G(m, n) and having a submanifold U' of the Grassmannian G(n - in - 1, n) as its image. Of course we assume that the mapping v is differentiable. Let r be the dimension of the submanifold U'. The number r coincides with the rank of the mapping v. Since dim G(n - in - 1, n) = p = (m + 1)(n - m),
we have 0 < r < p. If r = p, then U' is an open domain of the manifold G(n-m-1, n). If 0 < r < p, then U' is a proper submanifold of G(n-m-1, n). If r = 0, then U' consists of one fixed subspace p' of dimension n - in - 1 in the projective space P". If r = p, the normalization is called nondegenerate. In this case there is a one-to-one differentiable correspondence between the domains U and U'. If 0 < r < p, then the complete preimage v-1(p') of the normalizing subspace p' is a differentiable submanifold of dimension p-r on the Grassmannian G(m, n). If r = 0, then the complete preimage v-' (p') coincides with the entire domain U.
Consider also the case in = 0. In this case the manifold G(0, n) coincides with the projective space P", and the manifold G(n - 1, n) coincides with the dual projective space (P")'. Thus, to a point x E U C P", the normalization v sets in correspondence a hyperplane t; not passing through the point x. 2. Let us write the equations of the normalizing mapping v using differential forms. To this end, with the pair of subspaces p and p' we associate a family of point frames (A(} in such a way that A. E p and A, E p'. For each frame of this family, we have
dAQ=w Ap+w,A,, dAi=w°A,+wAj.
(6.5.2)
As in Section 6.3, the 1-forms w , are basis forms of the frame bundle associated
with the Grassmannian G(m, n). As to the I-forms w° defining displacements of the subspace p', they are no longer fiber forms. They are expressed in terms of the basis forms w' by relations w°
(6.5.3)
These relations are differential equations of the normalizing mapping (6.5.1). The coefficients VP form a square matrix of order p = (m + 1)(n - m), whose rank r is equal to the rank of the mapping v: rank (a A) = r.
254
6. GEOMETRY OF THE GRASSMANN MANIFOLD
The 1-forms wo and w; are fiber forms of the frame bundle associated with the normalized Grassmannian G"(m, n). As in Section 8.3, we set 7rg = wQ(6) and ir; = w? (6), where 6 = dl,,i =o. For the frame bundle in question, the forms aQ are invariant forms of the group GL(m + 1), and the forms 7r are invariant
forms of the group GL(n - m). The quotients of these two groups modulo the subgroup H of homotheties are isomorphic to the groups of projective transformations of the subspaces p and p' of the space P^, respectively. The quotient of group G = GL(m + 1) x GL(n - m) modulo the subgroup H is the stationary subgroup of the m-pair (p, p*). Taking the exterior derivatives of the basis forms wa by means of the structure equations (6.3.3), we find that
This implies the relations (6.5.4)
6W,° = -("J. Tr + wp7rQ,
which describe the law of transformation of the basis forms w, under admissible
transformations of frames associated with a nondegenerate m-pair (p, p') by means of differential forms. These formulas are analogous to formulas (2.1.17). Next, making use of equations (6.3.3), we take exterior derivatives of equations (6.5.3). This leads to the equations (dA,p
- A,k W - \QOWk +
A, WQ +
A,Pw7) A A
0,
or
VapAwp=0,
(6.5.5)
where, as usual, VA ?P denotes the expression occurring in parentheses in the previous formula. Applying Cartan's lemma to equation (6.5.5), we find that DA°a =
k-,Wy,
(6.5.6)
where \"3' - ) 113 If we fix an m-pair (p, p' ), then equations (6.5.6) take the form
Oaa'p = 0,
where as in Subsection 2.1.2, V5A f = VA,"a(6). The last relations show that the coefficients) a form a tensor, which is called the fundamental tensor of the normalized domain U" C G(m, n). This tensor is connected with a first-order differential neighborhood of the m-pair (p, Since the stationary subgroup of an m-pair is the product of the general linear groups GL(m+ 1) and GL(n-m), any geometric object of the normalized Grassmann manifold is a tensor. In particular, the object )'ijk occurring in
6.5
Normalization of the Grassmann Manifold
255
equations (6.5.6) is a tensor. This tensor is connected with a second order differential neighborhood of the normalized Grassmann manifold. In the normalized domain U" C G(m,n), we consider the quadratic differential form
9=w;wa Substituting the values (6.5.3) of the forms w° into this form, we obtain (6.5.7)
9=
siwhere the coefficients are obtained if one symmetrizes the tensor aQ multaneously with respect to both vertical pairs of indices:
g?P = 2(a A + ap°).
Hence the quantities g,? themselves form a tensor that is symmetric with respect to these pairs of indices. In view of this, the quadratic differential form g is invariant in the domain U". Denote the rank of the matrix of coefficients of the quadratic form g by F. If i= = p, then the quadratic form g is nondegenerate and defines a Riemannian
(or pseudo-Riemannian metric) in the domain U". If r < p, then the form g defines a semi-Riemannian metric in the domain U" for which the equation g°pw'a = 0
defines an isotropic distribution of dimension p-r. If the rank r of the mapping v vanishes, then r` = 0, and the form g vanishes. The normalization v is said to be harmonic if the coefficients in equations (6.5.3) are symmetric with respect to the vertical pairs of indices:
a°a = A.
(6.5.8)
VP and r = r. If r < p, and the normalization v If this is the case, then is harmonic, then the isotropic iistribution defined by the form g is integrable, and its integral manifolds coincide with the complete preimages v-1(p') of the normalizing subspaces p'. 3. Now we will establish a geometric meaning for the quadratic form (6.5.7). To this end, we find the matrix coordinates X and Y of the subspaces p = Ao A A, A ... A A, and p' = (Ao + dAo) A (Al + dA1) A ... A (Am + dAm )
with respect to the frame R = {Ao, Al , ... , A,,). We have
X = 1
1
0
...
0%
0
1
...
0
0 0
0
...
1
0
...
0
0..0.......0
_
Im+l
( 0(n-m) x (m+1)
6. GEOMETRY OF THE GRASSMANN MANIFOLD
256
where I,,,+l is the identity matrix of order m + 1 and °(n_,n) x (,n+1) is the zero (n - m) x (m + 1) matrix. By (6.5.2), we also find that
Yap+Wp As was indicated in Subsection 6.1.4, the matrix coordinate of an mdimensional subspace is determined up to multiplication from the right by a nondegenerate square matrix of order m + 1. As such a matrix, we take the matrix (b; +woo)-1 - (6000 -wp).
(6.5.9)
In the last and the following formulas, we assume that two matrices are equiva-
lent if they differ by second-order terms with respect to the elements of the matrix (wi ). Multiplying the matrix Y from the right by matrix (6.5.9), we find that WI WI
(6.5.10)
.
P,
Consider further the normalizing subspaces p' = An,+1 A ... A An and p" = (Am+1 + dAm+1) A ... A (An + dAn). The tangential matrix coordinate U of the first of these subspaces can be easily found: 1
...
0
0
...
0
1
0
...
0
O(m+1)x(n-m)) , 0
where O(m+l)x(n_m) is the zero (m + 1) x (n - m) matrix. The tangential matrix coordinate V of the subspace p" can be found from condition (6.1.6) which in the case in question can be written in the following form:
v°(A;+dA;)=v7(d +w;)+vpwp=0.
(6.5.11)
To find a basis of the solution space of this system, we take vp = dp. Multiplying (6.5.11) from the right by the matrix
(6 + Wk)-1 ' (a - 4), we find that
ilk ^. - 4"(6L -wk) - -Wk. Thus the tangential matrix coordinate V of the subspace p'' has the form
V=(do, -wk).
(6.5.12)
Let us find the cross-ratio W of two such m-pairs (p,p*) and (p', p"') by applying formula (6.1.12). To this end first we compute the products of the matrices occurring in formula (6.1.12): Ux = UY = (ap), VY = (6000 - Wi wp.
(6.5.13)
6.5
Normalization of the Grassmann Manifold
257
Thus we have (VY)-1
=K
+ w°wp)
and
W=
bR + w°wp
(6.5.14)
- Wk l
O(n-m) x (n+1)
/
.
(6.5.15)
In expressions (6.5.13)-(6.5.15), we retain the terms of second order with respect to the elements of the matrix (w{ ). Since such terms are principal, we discard the terms of order higher than two. The formula (6.5.15) determines the matrix W which is the cross-ratio of m-pairs (p, p') and (p', p"). To compute the quadratic form (6.5.7), we find the trace of the matrix W:
tr W =m+l+w°w,, Since for small x we have log(1 + x)
x,
it follows that
log(1+m+lw°w°) ^ m+lw'w°' and as a result we find that
g=w°w,
(m+l)log (l+m+ltr W).
(6.5.16)
The last formula gives the expression of the quadratic form g in terms of the cross-ratio of two infinitesimally close m-pairs (p, p') and (p', p"). 4. A normalization of the Grassmann manifold G(m, n) defines an affine connection on it. In fact, taking the exterior derivatives of the basis forms 4 of the manifold G(rn, n) and applying structure equations (6.3.3), we obtain (6.5.17)
Consider the 1-forms w'Q =bQw;
-54.
(6.5.18)
These forms are expressed in terms of the fiber forms wQ and w. of the frame bundle associated with a normalized Grassmann manifold G"(m,n) (or a domain U" of this manifold). In the tangent space Tp(fl) to the manifold fl(m, n), which is the image of the manifold G(m, n) under the Grassmann mapping, these forms define a subgroup of the general linear group whose transformations preserve the cone CD determined by equations (6.3.11). Exterior differentiation of equations (6.5.18) leads to the following exterior quadratic equations:
°j
k°
dw'p = b° AY°wl °j - bpwk n w' + b{wry n wR ry a n wl, jl e n w'ry - b'Aa`wk 7 kl
(6.5.19)
It is essential that the right-hand sides of equations (6.5.19) are expressed only in terms of the basis forms w, of the normalized Grassmann manifold G" (m, n).
258
6. GEOMETRY OF THE GRASSMANN MANIFOLD
By the facts from the general theory of spaces with of ine connection (e.g., see Kobayashi and Nomizu [KN 63], Ch. III, or Lichnerowicz [Lie 55], Ch. III, or Laptev [Lap 66]), these equations show that the forms wa define an affine
connection on G"(m,n), and the forms occurring in the right-hand sides of equations (6.5.19) are the curvature forms of this connection. Denote this affine connection by r". The connection r" is uniquely determined by the normalization v. Let us write the curvature forms of the connection 1' in the form WQ
= (64 , i + d;b Aki )w' A wk.
(6.5.20)
The alternated coefficients occurring in the right-hand sides of the last equations form the curvature tensor of the constructed connection. Equations (6.5.20) imply that this tensor has the following form:
Ra kl = 2 (baokAj"l + ba 5 ), - b J Ajt -
(6.5.21)
namely this tensor is expressed only in terms of the components of the fundamental tensor of the normalized Grassmann manifold G" (m, n). Equations (6.5.17) show that the affine connection r, is torsion-free. In view of this, the following theorem holds:
Theorem 6.5.1 The normalization v of a normalized domain U" C G(m, n) uniquely determines a torsion-free affine connection r,, with the connection forms (6.5.18) on it. The curvature tensor of this connection is expressed in terms of the fundamental tensor of the normalization v according to formulas (6.5.21).
We will find also the Ricci tensor of the connection r,. Contracting the tensor (6.5.20) with respect to the indices i, l and a, e, we obtain the following expression for the Ricci tensor of the connection F":
Rjk = R"jk; =
2
Ajk + Akj - (n + 1)Ajk
).
(6.5.22)
From these relations it follows immediately that the Ricci tensor of the connection r, is symmetric if and only if the normalization v of the Grassmann manifold G"(m,n) is harmonic. 5. Suppose that the normalization v of the Grassmann manifold G" (m, n) is harmonic (i.e., conditions (6.5.8) hold) and that its fundamental tensor A, is of maximal rank. Then the quadratic form g can be expressed as 9 = A°aw' ij aw'p,
and it defines a Riemannian (or pseudo-Riemannian) metric on the normalized Grassmann manifold G"(m,n).
6.5
Normalization of the Grassmann Manifold
259
Since the left-hand side of relation (6.5.6) is the covariant differential of the tensor A°Q with respect to the connection r, this relation shows that the connection r" is not the Levi-Civita connection of the metric defined by the form g. Nevertheless, we still can construct from the curvature tensor of the connection r" defined by formulas (6.5.21) a tensor that is covariant with respect to the indices i, j, k, and 1. This tensor is defined by the following formula:
Rap'Ye - 1 /A"Rmo
2l im
ijkl
pjkl
- App jm
pikt
Substituting the expression (6.5.21) for the tensor Rpfki c in the above formula, we obtain
R 3kl -
4
(Aiijk -
ik
Aji + A jf Alk - Aij Aki
( 6.5.23 )
Af3'YAac ik + ApaA'c jk it - ApcAar ji lk ,+ ji kl )'
-ApaAc7
it
One can immediately verify that this tensor satisfies the following relations: (6.5.24) R jkic = -R ik-y _ -Rj klc - Rktijp which are the standard relations for the curvature tensor of a Itiemannian
manifold. Next we define the following tensor: (6.5.25)
9ljkl c - AO Aji - A tcARk
Since, in the case in question, the fundamental tensor of the normalization v satisfies the symmetry relations (6.5.8), the tensor (6.5.25) also satisfies the conditions of type (6.5.24). The tensors ski c and g ski c allow us to define a sectional curvature on the normalized Grassmann manifold G"(m,n). To this end, in the tangent space T. (G(m, n) ), we consider two vectors { _ ({Q) and i = (711) and the bivector p = A rl defined by and rl. The coordinates of the bivector p are ij
pap =
1
i
j
i
The sectional curvature of the manifold G"(m, n) at a point x is defined as the ratio
K(p) = R(p,p)
(6.5.26)
9(p, p)
of two quadratic forms defined by the tensors (6.5.21) and (6.5.23): R
p) -
apryc
ij
kl
apryc
ij
ki
- Rijkl papp'tc and 9(p p) = 9ijkt It is possible to find the principal bivectors of the space TT(G(m,n)) for which the sectional curvature takes stationary values. However, for the general case this involves tedious calculations.
papp7e.
6. GEOMETRY OF THE GRASSMANN MANIFOLD
260
6.6
Homogeneous Normalization of the Grassmann Manifold
1. As was indicated earlier, a Grassmann manifold G(m, n) is a homogeneous space. However, in general, a normalized Grassmann manifold G"(m,n) is
not a homogeneous space. In fact even two m-pairs (p, p') and (q, q') have a matrix invariant W-their cross-ratio. In view of this, in general, there is no projective transformation superposing two neighborhoods U(p, p') and &(p, j r) of two m-pairs belonging to a normalized Grassmann manifold GM (m, n) (or its open domain U°). On the other hand, if a normalized Grassmann manifold G° (m, n) is homogeneous, then its fundamental tensor determining the location of an m-pair (p', p" ), which is infinitesimally close to the m-pair (p, p' ), must be covariantly constant; that is, it must satisfy the condition
V) A=0,
(6.6.1)
where V is the operator of covariant differentiation with respect to the affine connection F. Taking the exterior derivatives of the system of equations (6.6.1) by means of structure equations (6.3.3) of the projective space and excluding the differentials dA Q, we arrive at the system of relations
a ij kl jl + kj f it + ij k1 + 7!j 0/7 7 a, Q7 _ c(3 my ik - iii xij \Ik = 0.
op ye ik
a0 Y
aQ 7
1
ary
Oe
(6.6.2)
\Ik
Conditions (6.6.1) and (6.6.2) are necessary and sufficient for the normalization v of the normalized Grassmann manifold G°(m, n) with the fundamental tensor .\,°Q to be homogeneous.
2. Let us find some solutions of the system of equations (6.6.1) and (6.6.2). To this end, first we consider a polar normalization, namely a normalization of the Grassmann manifold G(m, n) by means of a nondegenerate hyperquadric
Q of the space P". Let po be an m-dimensional subspace of the space P" which is not tangent to the hyperquadric Q, and let po be an (n - m - 1)-dimensional subspace of P" which is polar-conjugate to po with respect to this hyperquadric. The subspaces po and po form a nondegenerate m-pair (po,po). The set of subspaces p, located in the same manner with respect to the hyperquadric Q as po (we will clarify below the meaning of the expression "in the same manner"), form an open domain U, and the subspaces p' polar-conjugate to the subspaces p with respect to the hyperquadric Q define the polar normalization of this domain. If the hyperquadric Q is imaginary, then the domain U coincides with the entire Grassmann manifold G(m, n). Let us associate a family of projective frames {AC} with an m-pair (p, p')
in such a way that the points A. E p and Ai E p'. As we did in Chapter 1,
6.6
Homogeneous Normalization of the Grassmann Manifold
261
we denote by (AF, An) the scalar product of the points A( and A, with respect to the hyperquadric Q. Since the points A. and Ai are polar-conjugate with respect to this hyperquadric, we have
gi. = (Ai, A.) = 0.
(6.6.3)
(Ai, Aj) = gij and (Ao, A0) = gall
(6.6.4)
The scalar products
form nondegenerate symmetric matrices (goo) and (gij). With respect to any chosen frame, the equation of the hyperquadric Q can be written as
900xox0 + 9ijxixj = 0.
(6.6.5)
Moreover the signature of each of the quadratic forms g0pxox0 and gijxixj is not changed when the subspace p moves in the domain U" C G(m, n). This condition clarifies the meaning of the expression "in the same manner" which we used above to characterize the domain U. Taking derivatives of equations (6.6.3) and (6.6.4) by means of equations (6.5.2), we find that
9ijwj = 0, dgij = 9ikLJ + 9kj(
,
d900 = 9oywp + 970w0
The first relation implies that
w° = -go0gij 4,
(6.6.6)
and the last two relations can be written as
Vgij = 0, VgoO = 0,
(6.6.7)
where, as earlier, V is the symbol of covariant differentiation with respect to the connection r l. Comparing equations (6.6.6) and (6.5.3), we obtain the following expression for the fundamental tensor of the polar normalization:
00 - a0 gij - -9 gij.
(6.6.8)
Since the tensors g°0 and gij are symmetric, this fundamental tensor satisfies condition (6.5.8), and the polar normalization is harmonic. Since the tensors goo and gij are nondegenerate, the fundamental tensor of the polar normalization is also nondegenerate. From relations (6.6.7) it follows that for the polar normalization we have V.1,oF = 0,
(6.6.9)
262
6. GEOMETRY OF THE GRASSMANN MANIFOLD
which shows that its fundamental tensor is covariantly constant with respect to the connection [''. This can also be immediately verified by checking that by (6.6.8), conditions (6.6.2) are satisfied identically. Hence the polar normalization of the Grassmann manifold is homogeneous. We will give another geometric proof of this. The projective space P", in which a fixed hyperquadric Q is given, is a pseudoelliptic space SQ of signature
q, where the number q is one unit less than the number of minuses in the canonical form of the quadratic form occurring in the left-hand side of equation (6.6.5). But the motions of this space transfer the normalized domain U" into itself. Thus this domain is a homogeneous domain of the space Sa .
For the polar normalization, the quadratic form (6.5.7) can be written as follows-
g = -9°p9ijwuwp. Thus it is nondegenerate and defines a Riemannian (or pseudo-Riemannian) metric on the Grassmann manifold G" with a polar normalization v. By relation (6.6.9), the connection F" is the Levi-Civita connection defined by this metric.
Substituting values (6.6.8) of the fundamental tensor of the polar normalization into expressions (6.5.21), we obtain the following expression for the curvature tensor: Raki f =
2
(9009"`(9ii9jk - gikgjl) + (9Q`g0" -
9Q"g0')9ij9ki).
(6.6.10)
It is also not so difficult to find the expression for the Ricci tensor of the connection F" defined by the polar normalization v. Substituting the values (6.6.8) of the components of the fundamental tensor of the polar normalization v into (6.5.22), we find that ROk = n
1
2
0"9jk;
9
in other words, the Ricci tensor of a polar-normalized Grassmann manifold G"(m,n) is proportional to its metric tensor. But this means that such a polarnormalized Grossmann manifold is an Einstein space (see Subsection 5.4.6 and also Petrov [Pe 69]).
Now we calculate the sectional curvature of the polar-normalized Grassmann manifold G"(m,n). To this end, we compute first the components of the tensor (6.5.25) for the polar normalization: Qo"e = Q" ae
9ijki
9
9
9ik9ji - 9Q"90" 9i:9,k
Hence the quadratic form g(p,p) can be written as 9(p, p) = (E,0 (77,17) - (C' 17)"
(6.6.11)
where we denote by (£, q) the scalar product of the vectors l: and p with respect
to the metric tensor g = (-g'Ogij):
6.6
Homogeneous Normalization of the Grassmann Manifold
263
The expression for the quadratic form R(p,p) is more complicated. By (6.6.10), we have
R(p,p) = 2[g"'g7`r1li) +gijgkt(W,?lt)W,
(6.6.12)
) - (,i,Ck)(,J,rlt))],
where (t., Y1#) = gijtor?p and
tjt) =
The sectional curvature K(p) of the polar-normalized Grassmann manifold G"(m,n) can be found by formula (6.5.26) where the quadratic forms R(p,p) and g(p,p) are expressed by formulas (6.6.12) and (6.6.11), respectively. From
the above formulas it follows that for m > 0 and n - m > 1, the sectional curvature K(p) is variable at a fixed point x E G(m, n), and that for m = 0 or n-rn = 1, the sectional curvature K(p) is constant at a fixed point x E G(m, n). This is natural because G(0, n) = S9" and G(n - 1, n) = (Sq)*. 3. In conclusion, we consider the case when the normalizing mapping v
has zero rank: r = 0. Then the set of normalizing subspaces consists of a single subspace p' of dimension n - m - 1, and the normalized domain U" of the Grassmann manifold G(m, n) consists of the m-dimensional subspaces p not intersecting the normalizing subspace p'. It follows from subsection 6.1.2 that the domain U" is diffeomorphic to the affine space A" of dimension
p = (m + 1)(n - m). A projective space P", in which a subspace p' of dimension n - m - 1 is fixed, is called the m-quasiafine space (see Dobromyslov [Dob 88]). The reason for this name is that if m = 0, this space reduces to an n-dimensional affine space An. We will denote the m-quasiaffine space by A. If we take an m-dimensional subspace p E Ate, as the basic element of the space AM,, then the
space Am coincides with the domain U" of the Grassmann manifold G(m, n) which we considered above.
If we associate a family of point frames with the subspace p E U" in the manner indicated in Subsection 6.5.2, then since the normalizing subspace p' is fixed, we find that dA,, = wOA0 +w' Ai, dAi = w; Aj.
(6.6.13)
Thus equations (6.5.3) take the form W° = 0,
(6.6.14)
and the tensor at for the case in question vanishes:
aQ = 0.
(6.6.15)
Hence the quadratic form g defined by equation (6.5.7) also vanishes, and it defines no metric in the domain U".
264
6. GEOMETRY OF THE GRASSMANN MANIFOLD
The forms w'Q defined by formulas (6.5.18) determine the affine connection r' in the domain U". But by (6.5.21) and (6.6.15), the curvature tensor pi ow of this connection vanishes, and the connection r" is flat. Thus, the domain U" is endowed with the structure of the affine space AP of dimension p = (m + 1)(n - m). However, since the forms w'Q are not linearly independent and expressed by formulas (6.5.18) in terms of the forms w. and w.0, the isotropy group (the stationary group) H of this space is the direct product
H = GL(m + 1) x GL(n - m).
(6.6.16)
This group preserves the Segre cone C; (m + 1, n - m) with vertex at the point x. Moreover all these cones are parallel with respect to the connection r,,
that is, with respect to a parallel transport in the space AP. The common generatrix of all these Segre cones is the Segre variety S(m, n - m - 1) lying in the hyperplane at infinity of the space AP. The variety S(m, n - m - 1) is the embedding of the direct product PI X Pn-m-1 of the projective spaces P"' and Pn-m-1 into the hyperplane PAP = Pp-1. The variety S(m, n - m - 1) carries two families of plane generators of dimensions m and n - m - 1. The
equations of the variety S(m,n - m - 1) in the space PAP can be written in the form (6.1.4) where a = 0, 1,-, m and p = m + 1, ... , n. The affine space AP with the absolute S(m, n - m - 1) and the isotropy group (6.6.16) is said to be the Segre-qlline space and is denoted by SAP. Thus we have proved the following result:
Theorem 6.6.1 Let U" be the domain of the Grassmann manifold G(m,n) formed by its m-dimensional subspaces p not having common points with a fixed subspace p' of dimension n - m - 1 (the normalizing subspace). Then the domain U" admits a mapping onto a Segre-affine space SAP that preserves the, structure of U".
The Segre-affine space SAP is a homogeneous space, and its fundamental group
G == GL(m + 1) x GL(n - m) x T(p), where T(p) is the group of parallel translations of this space, and the symbol x as earlier is the symbol of the semidirect product. Therefore the normalization
of the domain U" C G(m, n) by means of a fixed subspace p' of dimension n - m - 1 is really a homogeneous normalization. This matches the fact that by relations (6.6.15), equations (6.6.1) and (6.6.2), which are conditions for a normalization v to be homogeneous, are satisfied identically. The mapping s: U" -4 SAP described in Theorem 6.6.1 is called the stereographic projection of the Grassmann manifold G(m, n). This mapping is analogous to the stereographic projection of the conformal space C" onto the Euclidean space R" and of the pseudoconformal space Ce onto the pseudoEuclidean space Rq described in Chapter 1. Since the Grassmann manifold
Notes
265
G(1, 3) is equivalent to the pseudoconformal space C2, it admits the stereographic projection onto the pseudo-Euclidean space RZ which is equivalent to the Segre-affine space SA4.
NOTES 6.1. The projective matrix coordinates were introduced in Hua Lo-gen and Rosenfeld [HR 57] (for a more detailed and systematic treatment, see Rosenfeld [Ro 58]). The cross-ratio of two m-pairs was defined in Fuhrman [Fuhr 55]. For formula (6.1.12) for the cross-ratio of two m-pairs and its derivation, see Rosenfeld [Ro 96], §2.4.4.
The cross-ratio of four m-dimensional subspaces in a space of dimension n = 2m + I was considered in Kaplenko and Ponomarev (KP 81). The cross-ratio of four m-dimensional subspaces in (2m)-dimensional space was considered in Goldberg [Go 77, 80) (see also the book Goldberg [Go 88], p. 303). 6.2. The Vlasov configuration was introduced by Vlasov [VI 10] and was studied by Karapetyan [Kar 62a]). 6.3. Algebraic geometry of the Grassmannian was studied in detail starting from the paper Severi [Sev 15] (see also the books Hodge and Pedoe [HP 47, 52], Chapters VII and XIV). The asymptotic cones of second order on the Grassmannian G(m, n) were considered in Karapetyan [Kar 63a, b]. The determinantal varieties were studied in detail in the book Room (Roo 38). On the Segre varieties, see Griffiths and Harris [GH 791, and on the Segre cones, see Akivis [A 80]. The focal families of m-planes were studied by Korovin [Kor 50] and Geidelman (see the paper Geidelman [Ge 67a] and the book Finikov [Fin 561, Ch. 25). However, as far as the authors know, prior to the paper Akivis [A 82b] there was no detailed study of the differential geometry of the Grassmannian. In our exposition we follow that paper. 6.4. More details on the theory of two-parameter families and complexes of twodimensional subspaces in the space P5 can be found in Zhogova [Zh 78, 79] and Bubyakin [Bub 90, 91).
6.5. The normalization v which to a point r E P" sets in correspondence a hyperplane t not passing through the point x was considered in the book Norden [N 50a], §60.
For more detail on a semi-Riemannian metric, see the paper Akivis and Chebysheva [AC 811 in which with an invariant framing of a semi-Riemannian manifold was constructed. 6.6. The polar normalization was considered in the book Norden [N 50a], §572-73.
Essentially, the case where the hyperquadric Q is imaginary and consequently the domain U coincides with the entire Grassmann manifold G(m, n), was studied in detail in the paper Leichtweiss [Le 611. In this paper the Riemannian geometry of the Grassmann manifold of subspaces of an Euclidean space was mainly investigated. The quasiaf lne spaces were introduced by Rosenfeld [Ro 59] (see also Rosenfeld [Ro 96], §5.2.2). The Segre-affine spaces were first introduced in Dobromyslov [Dob 88] and were studied in detail in Rosenfeld, Kostrikina, Stepanova, and Yuchtina [RKSYu 90]. The stereographic projection of the Grassmann manifold G(1, 3) was considered in
the book Semple and Roth [SR 85), and for the general Grassmann manifold G(m, n), it was considered in the paper Semple [Sem 321.
266
6. GEOMETRY OF THE GRASSMANN MANIFOLD
The stereographic projection, the m-quasiaffine space and the Segre-affine space were studied in Dobromyslov [Dob 88].
Chapter 7
Manifolds Endowed with Almost Grassmann Structures 7.1 1.
Almost Grassmann Structures on a Differentiable Manifold
As we saw in Chapter 6, the Grassmannian G(m, n) of m-dimensional
subspaces of a projective space P^ admits a bijective mapping on the algebraic variety 1(m, n) of dimension p = (m+1)(n-m) belonging to a projective space
P", where N = (n) - 1. This mapping was called the Grassmann mapping. Under this mapping, to the family of m-dimensional subspaces of the space P"
intersecting a fixed subspace Pm = x in a subspace of dimension m - 1, on the variety 1 = f2(m, n), there corresponds a cone with vertex at the point x. (Note that we denoted the subspace P" and the corresponding point on fl (m, n) by the letter x not p as in Chapter 6.) This cone is a Segre cone, and it carries two families of plane generators f and 0 of dimensions p = m + 1 and q = n - m, respectively. We denote this cone by SC., (p, q). It is located in the tangent subspace T,, (n) and is the intersection T. (n) fl f2. Now we define the notion of an almost Grassmann structure on an arbitrary differentiable manifold M of dimension pq.
Definition 7.1.1 Let M be a differentiable manifold of dimension pq, and let SC(p, q) be a differentiable field of Segre cones with the base M such that SC,, (p, q) C Tt(M), x E M. The pair (M,SC(p,q)) is said to be an almost Grassmann structure and is denoted by AG(p - 1, p + q - 1). The manifold M endowed with such a structure is said to be an almost Grassmann manifold.
267
268
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
It follows from our previous considerations that the algebraic variety SZ(m, n),
onto which the Grassmannian G(m, n) is mapped and along with it the Grass-
mannian G(m, n) itself are endowed with an almost Grassmann structure AG(m, n), since on the variety 11(m, n), the field SC(p, q), where p = m+ 1 and q = n - m, a field of Segre cones is defined naturally. As was the case for Grassmann structures, the almost Grassmann structure
AG(p - 1, p + q - 1) is equivalent to the structure AG(q - 1, p + q - 1), since both of these structures are generated on the manifold M by a differentiable field of Segre cones SC. (p, q). The structural group of the almost Grassmann structure is a subgroup of the general linear group GL(pq) of transformations of the space T= (M), which leave the cone SC: (p, q) C T=(M) invariant. We denote this group by G = GL(p, q).
To clarify the structure of this group, in the tangent space T=(M) we consider a family of frames {e° }, a= 1,...,p; i =p+ 1, ... , p + q, such that for any fixed i, the vectors e; belong to a p-dimensional generator of the Segre cone SC =(p, q), and for any fixed a, the vectors e; belong to a q-dimensional generator q of SC= (p, q). In such a frame the equations of the cone SC= (p, q) can be written as follows:
za=tas',
a=1,...,p; i=p+1,...,p+q,
(7.1.1)
where zQ are the coordinates of a vector z = z ea belonging to the space T,(M), and to and s' are parameters on which a vector z E SCA(M) depends. The family of frames {e°} attached to the cone SC= (p, q) C T=(M) admits a transformation of the form 'e° = AapA;ea,
(7.1.2)
where (Aa) and (A') are nonsingular square matrices of orders p and q, respectively. These matrices are not defined uniquely since they admit a multiplication by reciprocal scalars. However, they can be made unique by restricting
to unimodular matrices (Ap) or (A;): det(A') = 1 or det(A;) = 1. Thus the structural group of the almost Grassmann structure defined by equations (7.1.2) can be represented in the form
G = SL(p) x GL(q)
GL(p) x SL(q),
(7.1.3)
where SL(p) and SL(q) are special linear groups of dimension p and q, respectively. Such representation has been used by T. Hangan [Han 66, 68, 80], V. V. Goldberg [Go 75a) (see also the book [Go 88], Ch. 2), and Yu. I. Mikhailov [Mi 78]. Unlike this approach, we will assume that both matrices (Ap) and (A;) are unimodular though the right-hand side of equation (7.1.2) admits a multiplication by a scalar factor. As a result we obtain the more symmetric representation of the group G:
G = SL(p) x SL(q) x H,
(7.1.4)
7.1
Almost Grassmann Structures on a Differentiable Manifold
269
where H = R' ® Id is the group of homotheties of the T=(M), and R' is the multiplicative group of real numbers. It follows from condition (7.1.1) that the p-dimensional plane generators of the Segre cone SC1 , (p, q) are determined by values of the parameters s' and
that tQ are coordinates of points of a generator . But a plane generator is not changed if we multiply the parameters s' by the same number. Thus the family of plane generators l; depends on q - 1 parameters. Similarly, q-dimensional plane generators rl of the Segre cone SC= (p, q) are
determined by values of the parameters tQ, and s' are coordinates of points of a generator q. But a plane generator rl is not changed if we multiply the parameters tQ by the same number. Thus the family of plane generators r) depends on p - 1 parameters. The p-dimensional subspaces l: form a fiber bundle on the manifold M. The base of this bundle is the manifold M, and its fiber attached to a point x E M is the set of all p-dimensional plane generators f of the Segre cone SC= (p, q). The dimension of a fiber is q - 1, and it is parameterized by means of a projective space Pa, dim P. = q - 1. We will denote this fiber bundle of p-subspaces by
Ea = (M, Pa)
In a similar manner q-dimensional plane generators r) of the Segre cone
SCy (p, q) form on M the fiber bundle EE = (M, Pp) with the base M and fibers
of dimension p - I = dim P. The fibers are q-dimensional plane generators r/ of the Segre cone SC,(p,q). Consider the manifold M. = M x P. of dimension pq + q - 1. The fiber bundle E0 induces on MQ the distribution Da of plane elements e of dimension p (see Figure 7.1.1). In a similar manner, on the manifold Mp = M x Pp, the fiber bundle Ep induces the distribution Op of plane elements q, of dimension q (see Figure 7.1.2).
Figure 7.1.1
Figure 7.1.2
270
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
Definition 7.1.2 An almost Grassmann structure AG(p-1,p+q-1) Is said to be a-semiintegrable if the distribution A. is integrable on this structure. Similarly an almost Grassmann structure AG(p - 1, p + q - 1) is said to be p-semiintegrable if the distribution Op is integrable on this structure. A structure AG(p-1, p+q-1) is called integrable if it is both a- and /3-semiintegrable.
Integral manifolds V of the distribution A, of an a-semiintegrable almost Grassmann structure are of dimension p. They are projected on the original manifold M in the form of a submanifold V, of the same dimension p, which, at any of its points, is tangent to the p-subspace {o of the fiber bundle E0. Through each point x E M, there passes a (q - 1)-parameter family of submanifolds Va. Similarly integral manifolds f/0 of the distribution Op of a p-semiint.egrable
almost Grassmann structure are of dimension q. They are projected on the original manifold M in the form of a subtanifold Vi3 of the same dimension q, which, at any of its points, is tangent to the q-subspace tlp of the fiber bundle E0. Through each point x E M, there passes a (p - 1)-parameter family of submanifolds Vp.
If an almost Grassmann structure on Af is integrable, then through each point x E M, there pass a (q - 1)-parameter family of submanifolds 1". and a (p - 1)-parameter family of submanifolds Vp which were described above. The Grassmann manifold G(m, n) is an integrable almost Grassmann structure AG(m,n), since it admits a bijective mapping onto the manifold 0(m, n) of dimension pq, p = m + 1, q = n - m, through every point x of which there pass a (q - 1)-parameter family of p-dimensional plane generators (which are and a (p - 1)-parameter family of q-dimensional plane the submanifolds generators (which are the submanifolds Vp). In the projective space P", to submanifolds V. there corresponds a family of m-dimensional subspaces belonging to a subspace of dimension m + 1, and to submanifolds Vp there corresponds a family of rn-dimensional subspaces passing through a subspace of dimension
m - 1. 2. Let us consider some examples. First, we consider a pseudoconformal CO(2, 2)-structure on a four-dimensional manifold Af (see Section 5.1). The isotropic cones C. of this structure carry two families of plane generators. Therefore a pseudoconformal Hence these cones are Segre cones CO(2, 2)-structure is equivalent to an almost Grassmann structure AG(1, 3). If we complexify the four-dimensional tangent subspace T=(M) and consider Segre cones with complex generators, then conformal C0(1, 3)- and CO(4, 0)structures can also be considered as complex almost Grassmann structures of the same type AG(1, 3). However, in this book we will consider only real almost Grassmann structures. Almost Grassmann structures arise also in the study of multidimensional webs (see Akivis and Shelekhov (AS 92J and Goldberg [G 88]). We consider first a three-web W(3, 2, q) formed on a manifold AP 9 of dimension 2q by three
foliations A o = 1, 2, 3, of codimension q that are in general position.
7.1
Almost Grassmann Structures on a Differentiable Manifold
271
Through any point x E M29, there pass three leaves F, belonging to the foliations a,. In the tangent subspace T= (M2q), we consider three subspaces T. (.F.,) that are tangent to F, at the point x. If we take the projectivization of this configuration with center at the point x, then we obtain a projective space P2q-1 of dimension 2q - 1 containing three subspaces of dimension q - 1 that are in general position. As we saw in Subsection 6.1.6, these three subspaces determine a Segre variety S(l,q - 1), and the latter variety is the directrix for a Segre cone SC.(2,q) C T.(M2q). Thus on M2q a field of Segre cones arises, and this field determines an almost Grassmann structure on M2q. However, the structural group of the three-web in question is smaller than that of the induced almost Grassmann structure, since transformations of this group must keep invariant the subspaces TT(F,). Thus the structural group of the three-web is the group GL(q). In the same manner we can prove that an (p+ 1)-web W (p+ 1, p, q), formed on a manifold M of dimension pq by p + 1 foliations a or = 1, ... 'P+ 1, of codimension q which are in general position, generates an almost Grassmann structure on M. The structural group of the web W (p + 1, p, q) is the same group GL(q), and this group does not depend on p. 3. Let us reduce the structure equations of the Grassmannian G(m, n) that have been already considered in Chapter 6 (see Section 6.5) to a form convenient for a further generalization. As earlier, we denote the points of a moving frame of the space P" by AE and write the equations of infinitesimal displacement of this frame in the form dAf = 0"A,,,
0,
. .
. ,nn..
(7.1.5)
Since the fundamental group of the space P" is locally isomorphic to the group
SL(n + 1), the forms 0 are connected by the relation tE = 0.
(7.1.6)
As was indicated in Section 6.3, the structure equations of the space P" have the form
dln=9 A9.
(7.1.7)
By (7.1.7), the exterior differential of the left-hand side of equations (7.1.6) is identically equal to 0, and hence this equation is completely integrable.
Let a subspace P' = x be an element of the Grassmannian G(m, n). We place the points Ao, A1, ... , A,,, of the frame into this subspace. Since by (7.1.5), we have
dA,,=8 A0+0'Ai,
(7.1.8)
where here and in what follows in this subsection, a,# = 0, ... , m and i = m + 1, ... , n, the 1-forms 9 , are basis forms of G(m, n). These forms are linearly independent, and their number is equal to p = (m + 1)(n - m) = p q, where p = m + 1, q = n - m; that is, it equals the dimension of the Grassmannian G(m, n). We will assume that the integers p and q satisfy the inequalities
272
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
p > 2 and q > 2, since for p = 1, we have m = 0, and the Grassmannian G(0, n) is the projective space P", and for q = 1, we have m = n - 1, and the Grassmannian G(n - 1, n) is isomorphic to the dual projective space (P")'. Let us rename the basis forms by setting 90 = wi and find their exterior differentials: (7.1.9)
dwp = Ba n wp +WJ- A Off.
Define the trace-free forms
wa = 6a - pda97, wj = B
- Qk
(7.1.10)
satisfying the conditions
wa = 0, w; = 0.
(7.1.11)
Excluding the forms 9 and 9 from equations (7.1.9), we find that
dwawanwp+wfanw;+wnwa
(7.1.12)
where
w = pe7 - qek. But since, by (7.1.6), we have
ek = -e7
y,
A:
then the expression for the 1-form w can be written in the form
(1+1)e;. P
(7.1.13)
4
Taking the exterior derivatives of equations (7.1.10) and eliminating the
forms 9 and t from the equations obtained, we find that dwa =wQnwA+wkn (aa k - I 0a8k p
(7.1.14)
dw = w; A wk + (ake; - look) n wQ. Finally, taking the exterior derivative of equation (7.1.13), we find that
dw=
(1+1)waA9°.
p
(7.1.15)
q
If we set
p+q)Ba,
(7.1.16)
then equation (7.2.15) takes the form
dw=w°nwQ,
(7.1.17)
7.1
Almost Grassmann Structures on a Differentiable Manifold
273
and equations (7.1.14) become dwa = wa A wA +
wjkA wk +
p
+ q (ba wk A ,, _Y - pw Q A wk) , q
p+q
Awry
- qw
(7.1.18)
wy) .
Taking the exterior derivatives of equations (7.1.16) and applying equations (7.1.7) and previous relations between the forms BE and wf, we find that
Awl +Awp°+w°Aw.
(7.1.19)
Finally, exterior differentiation of equations (7.1.19) leads to identities. Thus the structure equations of the Grassmannian G(m, n) take the form (7.1.12), (7.1.17), (7.1.18) and (7.1.19). This system of differential equations is closed in the sense that its further exterior differentiation leads to identities.
If we fix a subspace x = PI C P" (an element of the Grassmannian G(m, n)), then we obtain wQ = 0, and equations (7.1.17) and (7.1.18) become d7r13 = nQ A rrA, drr = n A Irk, d7r = 0,
(7.1.20)
where, as in previous chapters, it = w(b), as = w13 (b), it = w (b), and b is the operator of differentiation with respect to the fiber parameters of the frame bundle associated with the Grassmannian G(m, n) (see Subsection 6.3.1). Moreover the forms xa and 7r satisfy equations similar to equations (7.1.11), so these forms are trace-free. The forms 7rO are invariant forms of the group SL(p) which is locally isomorphic to the group of projective transformations
of the subspace P"'. The forms ar are invariant forms of the group SL(q) which is locally isomorphic to the group of projective transformations of the bundle of (m + 1)-dimensional subspaces of the space P" containing P"`. We
will denote this bundle by P"/P'". The form it is an invariant form of the group H = R' ® Id of homotheties of the space P" with center at P'". The direct product of these three groups is the structural group G of the Grassmann manifold G(m, n):
G = SL(p) x SL(q) x H.
(7.1.21)
Finally, the forms ir° = w° (b), which by (7.1.19) satisfy the structure equations dir° = rr, A rra + 7rR A ir; + a° Air, (7.1.22) are also fiber forms on the Grassmannian G(m, n), but unlike the forms Ira, ir; , and it, they determine admissible transformations of second-order frames associated with the Grassmannian G(m, n).
It follows from Subsection 6.6.3 that a projective space P", in which an m-dimensional subspace PI is fixed, is an (n-m-1)-quasiaffine space An_m_1 Its dimension coincides with the dimension of the Grassmannian G(n-m-1, n), and this dimension is the same as the dimension of the Grassmannian G(m, n):
274
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
p = (m + 1)(n - m). The forms a° determine the parallel translation of the element P"-m-1 = Am+1 A ... A An of the space An_,,,_,. All together, the forms ire, 1T , it, and n°, satisfying the structure equations (7.1.20) and (7.1.22),
are invariant forms of the fundamental group G' of the space An-m-1 which is the semidirect product of the groups G and T(pq):
G' = G a T(pq),
(7.1.23)
where T(pq) is the group of parallel translations of the space The group G' coincides with the stationary subgroup H. of the element x = P" of the Grassmannian G(m, n). Ann_
7.2
Structure Equations and Torsion Tensor of an Almost Grassmann Manifold
1. Consider a differentiable manifold M of dimension pq endowed with an almost Grassmann structure AG(p-1, p+q -1). Suppose that x E M, T.(M) is the tangent space of the manifold M at the point x and that {e°) is an adapted frame of the structure AG(p - 1,p + q - 1). The decomposition of a vector z E Tt(M) with respect to this basis can be written in the form z = wa(z)e°,
where w, are 1-forms making up the co-frame in the space T=(M). If z = dx is the differential of a point x E M, then the forms wa(dx) are differential forms defined on a first-order frame bundle associated with the almost Grassmann structure. These forms constitute a completely integrable system of forms. As a result we have (7.2.1)
dw' = w'p A w'Q .
The forms wa are called also the basis forms of the manifold M. As earlier, we set rr'a°. = w'', (b), where b is the operator of differentiation with respect to the fiber parameters of the frame bundle. These forms determine an infinitesimal transformation of the adapted frames: be'? = it ea.
(7.2.2)
On the other hand, the admissible transformations of adapted frames can be written as closed form equations (7.1.2). Solving equations (7.1.2), we obtain e? = A°A 'ep
(7.2.3)
where (Ap) and (Ai) are the inverse matrices of the matrices (Ap) and (A'), respectively:
Ay AF = ApA° = 6, AkA = AJAk = 5
.
(7.2.4)
Structure Equations of an Almost Grossmann Manifold
7.2
275
It follows from (7.2.4) that
Ap . 6Ay = -AI bAp, A; bA1 = -A3 dAk,
(7.2.5)
Suppose now that (x, 'e°) is a fixed frame, d('e;) = 0. Then, differentiating (7.2.3) for a fixed x E M and using (7.2.4) and (7.2.5), we obtain Sea = (dp°ir; - d, rp )eA,
(7.2.6)
1r = Ak dA, , ira = Aa bAp.
(7.2.7)
where
Comparing formulas (7.2.2) and (7.2.6), we find that = bairi - b; lrpa.
7r'
(7.2.8)
In these formulas the forms lri are invariant forms of the group GL(q), the forms ira are invariant forms of the group GL(p), and the forms ap' are invariant forms of the structural group G of the almost Grassmann structure AG(p - 1,p + q - 1). If a point x E M is variable, then from equations (7.2.8) we find that wAi = by°
- b w+
uQ kWy0
(7.2.9)
,
where uQik are certain functions defined on the first order frame bundle. Substituting for WQa in (7.2.1) their values taken from (7.2.9), we obtain duly = Wa Awp +wa AWE +u'a kwp Au)
(7.2.10)
where ua k denotes the result of alternation of the quantities ii' O" occurring in
(7.2.9) with respect to the pairs of indices (Q) and (k): u` k = -u«' . If we set
lbiWk, aQWry, W! =WT + p q then it is easy to see that wQ = 0 and wk = 0, and the above structure equations take the form wa ==Wa +
dwQ = w A
V,' A wa + W A wa' + uQ kwp A Wy,k,
where w = I_W7 - QWk. If we suppress -, then the structure equations take the form: du,a =Wa AWj +W , AWR +W AWc,
c
kWp AWk
(7.2.11)
where i7A io-V TEajk = -uakf
(7.2.12)
276
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
and
W7 = 0, Wk = 0.
(7.2.13)
Conditions (7.2.13) mean that the subgroups GL(p) and GL(q) of the structural group G of the almost Grassmann structure AG(p-1, p+q- 1) are reduced to the groups SL(p) and SL(p), respectively, and that the group G itself is represented in the form (7.1.4). As for the Grassmannian G(p - 1, p+ q - 1) (see Subsection 7.1.3), for the almost Grassmann manifold the forms wa, w; , and w are fiber forms defined on the second-order frame bundle associated with the
almost Grassmann manifold AG(p - 1,p+ q -1). The structure equations (7.2.11) differ from the structure equations (7.1.9) of the Grassmannian G(p - 1, p + q - 1) only by the last term. 2. We obtain the remaining structure equations of the almost Grassmann manifold M by exterior differentiation of (7.2.11). This gives
nflAwp- Q' A wQ+(Vu'Q k+u'Q."W)AwpAwk
(7.2.14)
+dw Aw, + 2uaic-fmkum,Rwfs A WQ Awy = 0, where Q0 = dwQ - wa A wA, fl' = dw' - wk A wk, VuQ,k = duopjk - u6 kWQ
- tiafkwj - uo'Iwk +
uIQ k-f
wf + uokc
i
+ ua kwe .
To solve equations (7.2.14), we represent the forms and dw from the left-hand side of equations (7.2.14) as a sum of terms containing the basis forms and the terms not containing these forms:
1l =w7Awak+4i«, fly=wokAwk+dw=w°AWQ+4',
(7.2.15)
where 41,a, I , 4 and wak,w4k are certain 2- and 1-forms not expressed in terms of the basis forms wi, only. By (7.2.13), we have fly = 0 and ftk = 0, which implies that (7.2.16)
4K7 = 0, 4kk = 0. and
wak=0,
;k=0
(7.2.17)
Substituting (7.2.15) into equations (7.2.14), we obtain 16 A 4 A wry (b 4ip - 6131'. + 606' fl A w! + 2uaml0l-OU
+(f5(,Wlolk1 +
6Qry61jlw,) + Vua k +
Awp AWy = 0. (7.2.18)
7.2
Structure Equations of an Almost Grassmann Manifold
277
The first term in the left-hand side of (7.2.18) does not have similar terms among other terms of this side. Thus this term vanishes. But since the first factor of this term does not contain the basis forms, this factor itself vanishes:
+b 54' = 0.
(7.2.19)
Contracting (7.2.19) with respect to the indices a and Q, applying (7.2.16), and dividing by p, we find that
-4i +bj4' = 0. Contracting this equation with respect to the indices i and j, we obtain 4' = 0, and consequently 1i = 0. Finally, by (7.2.19), we find that 4 = 0. Now equation (7.2.18) contains only the last two terms. It follows that the 1-form which is multiplied by wR A wk is expressed only in terms of the basis forms. Therefore, if the principal parameters are fixed (i.e., if wi = 0), then we obtain R7 r ak
- b'
k
7R + *5ait,k- ba77r'Rkj
aj
u' + u' aQjk7 + ba7bk' 7rp - 60b a ` ak7 + 2 (V 6 k aR7j
it) = 0. (7.2.20)
It follows from equation (7.2.20) that the quantities u'a k form a geometric object that is defined in a second-order differential neighborhood of the almost Grassmann structure AG(p - 1, p + q - 1). Consider the quantities iR7
R7
ia7
i7
(7.2.21)
link - uaik, ujk = U01 jk.
If we contract equations (7.2.20) with respect to the indices i and j, then after some calculations we find that uR7 + uR7ir = _ 2 r X07 q( ok 6 ak ak
by it
- aitk) - ,rak - a ( 5
-YO
ki
- aRk) ]
'
( 7.2.22)
Similarly, contracting equations (7.2.20) with respect to the indices a and fl, we obtain VJujki-f
2[p(xjik-Miak)-
i7
k j- bk (Ira7aj
- sj7)]
(7.2.23)
Formulas (7.2.22) and (7.2.23) show that each of the quantities uQk and u"k form a geometric object that is defined in a second-order differential neighborhood of the almost Grassmann structure AG(p - 1,p+ q - 1). Let us prove that if we make a specialization of second-order frames, then we can reduce these geometric objects to 0. In our proof we will apply the same method that we used in constructing the tensor of conformal curvature in Chapter 4 (see Subsection 4.1.3). We will prove this for the geometric object uo". To this end, we must show that the 1-forms in the right-hand sides of equations (7.2.22) are linearly
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
278
independent. First, we note that the forms 7rak are linearly independent in the set of second-order frames. Let us equate to 0 the right-hand sides of equations (7.2.22): -Yo
0
9(7rok - balk) - yak - 6-V (71j, - k)
(7.2.24)
If we contract equations (7.2.24) first with respect to the indices a and 0 and second with respect to the indices a and -y, we arrive at the system pq)ir
1ak + 7rk;
(7.2.25)
gook - PAki = (q - p)ir If we solve this system, we find the quantities 7r"ak and irki:
rya -- 7rak
q(p2 - 1)
P(q2 - 1) " Irk.
,. = p+q k+ ski - - p+q If
(7.2.26)
Substituting these values of 7rk7 into equations (7.2.24), after some calculations, we reduce the equations obtained to the following form: (7.2.27)
-yo = 0, 9Wak - yak
where
irek
- 7rak - p
q (6.07rk
- p8,prk )
.
+ Interchanging in (7.2.27) the indices 0 and y, we obtain Ary
ryR
- lrak + gook -
(7.2.28)
O.
Since the determinant of the system of equations (7.2.27)-(7.2.28) is equal to q2 - 1 0 0, the system has only the trivial solution. But the forms *Qk as the forms 7rak are linearly independent. Thus the
forms q;rak - a.k are linearly independent too. But, up to the factor -, a the latter forms coincide with the right-hand sides of equations (7.2.22).
Hence the geometric object uek = u'Q k can be reduced to 0. Similarly the geometric object ufk = can be reduced to 0. This operation leads to a reduction of the set of second order frames of the almost Grassmann structure AG(p - 1, p + q - 1). Before this reduction, the set of second-order frames depended on pq(p2 + q2) parameters equal to the number of linearly independent forms among the forms 7rak and 7r"k. After the reduction, the forms WCkk3" and vanish, and the forms 7rak and 7r,k are expressed in terms of the 1-forms irk: 7rak - p
q (b°7rk
+
- pbaak)
,
7r;k - p
q
+
(bj7rk
- gbk7r,7).
(7.2.29)
7.2
Structure Equations of an Almost Grassmann Manifold
279
Since there are pq forms Irk , and they are linearly independent, the reduced family of second-order frames depends on pq parameters. The 1-forms Irk define
admissible transformations of frames in this reduced family of second-order frames.
Denote by a.k the quantities ua k after the specialization indicated above. Then the quantities a'Q k satisfy the conditions iory
iRry
aajk = 0, aaik = 0
(7.2.30)
and
as k = -
(7.2.31)
Qk,.
The last relations follow from conditions (7.2.12). Substituting expressions (7.2.29) into equations (7.2.20), we find that
a, klr = 0.
V6
(7.2.32)
This implies the following theorem:
Theorem 7.2.1 The quantities a`o k, defined in a second-order neighborhood by the reduction of second-order frames indicated above, form a relative tensor of weight -1 and satisfy conditions (7.2.30) and (7.2.31).
Definition 7.2.2 The tensor {aQ } is said to be the first structure tensor, or the torsion tensor, of an almost Grassmann manifold AG(p - 1, p + q - 1). After the specialization of second-order frames has been made, the first structure equations (7.2.11) become
dwa=wQnwj' +wQAwp+wAw.' +aakwRnw1
(7.2.33)
3. We will now find the expression for the tensor a'Q k in terms of the quantities
uQ k occurring in equations (7.2.11). We assume that the specialization of second-order frames indicated above has not been made and that the quantities uQ k satisfy equations (7.2.20), which we write in the form iRry iRry 06uojk + uojk7r =
1
2
i
i
iR rya Qkl - tSi7rak + koj ry
Rry
R
iry
(7.2.34)
+a og7rk -
We will eliminate the fiber forms irak, and Irk from equations (7.2.34). To this end, we construct the following three objects: iRry
_-
yolk iRry _
2
,(
q2 - 16 i 2
k
(aryl
IoIkJ
I7R) 1 + ulolkll,
Ikil(((7.2.35)
b(R ( ulilryl + ulilryll
yOjk - -p2 -1 o zQ
.
q likkri
(p2 - 1)(q2 - 1) L(pq - 1) \aljalalullIkl + blkoIQlujliryllJ +(p - q) (a[i Jalukit + a(kbJai 111ji 9] 1
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
280
where the quantities uak and u,k are defined by formulas (7.2.21). A straightforward calculation with help of equations (7.2.22) and (7.2.23) gives the following differential equations for the objects x'a k, y;k, and zQ k x'RY vdxi0Y Lj ajk + ajk ir = 2 [6
a k) 6t-( irYo kja - 6Yxo)] a3
607rY
ak
66a> 2. Hence this system has only the trivial solution: b4 ml = 0. As a result the homogeneous system in question has only the trivial solution: bajp,,, = 0 provided that q > 2; thus the original nonhomogeneous
The Complete Structure Object of an Almost Grassmann Manifold 289
7.3
system has a unique solution expressing the quantities bolo, in terms of the components a'Q k of the tensor a and their Pfaffian derivatives. In a similar manner we can prove that if p > 2, then the quantities are expressed in terms of the components ask of the tensor a and their Pfafflan derivatives. Note that the condition q > 2 is required only for finding of beo ) = b`('") and the condition p > 2 for finding of bj(k (see Lemma 7.4.1, p. 292).
Now we can see that the tensor a satisfies certain differential equations. These equations can be obtained if we substitute for the components of bl and b2 in equations (7.3.43) their values found in the way indicated above. The conditions obtained in this manner are analogues of the Bianchi equations in the theory of spaces with affine connection. 8. Next we will find new closed form equations and differential equations in equathat the components of c satisfy. If we substitute for the 1-forms tions (7.3.9) and (7.3.10) their values taken from (7.3.32) and apply (7.3.34), we arrive at the following exterior cubic equations: Abo-Y6 A wl6 akl A Wk Y
-
she)w,- /1 wY p pq + q6' aa°me°Y6 ski +2b O-Yu aks aaim
k
1
/1 w6
0 (7.3.45)
_
2b sm oki J )we"' / wk / w6 = 0,
ob kt n wk /1 wa + ( p +
(7.3.46)
where Ob#-ry6 aki = VbQy6 aki + 2b/iY6 akl w+
ObiY6 ,lkl + jkl = Obi" Jkl+ 2b'ry6w
p-Y +q(s
R m sY6) we mA6 -6e6a aaklwml
6a6kaCal
(&o mai
ej
+ ppq
oM
amo61we c55' o k ejl 11 m
It follows from equations (7.3.45) and (7.3.46) that
Qb"' = bY6e wm QbJYd - bf7de m jklm e , 1M akl aklmwe
(7.3.47)
where bjtktm and bakim are the Pfaffian derivatives of 6akl and bake respectively. Substituting (7.3.47) into equations (7.3.45) and (7.3.46), we find the following differential equations for the components of b: bplY6e1
a(klm]
_
P9 6(ecIAh6)
p + q a lmkil
- 2bp(Yleasl6e) a(kle olim) -- 0 (7.3.48)
6 (klml
p + g6lmcljl M) +
0.
Equations (7.3.48) can be written in the form 6[f Clmkll]
-
BO-The
(7.3.49)
290
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
and
(7.3.50)
b1-clilk) = Bjki-I
where the quantities Bakem and Bjkl,`,+ are skew-symmetric with respect to the last three pairs of indices and are expressed in terms of the components of the subobjects (a, b') and (a, b2), respectively, and their Pfaffian derivatives. We will now prove that if p > 2, then the components of c are expressed in terms of the components of the subobject (a, b') and their Pfafan derivatives,
and that if q > 2, then the components of c are expressed in terms of the components of the subobject (a, b2) and their Pfaffian derivatives. We will prove only the first part of this statement. The proof of the second part is similar. The components of c satisfy equations (7.3.49) that are a nonhomogeneous system of linear equations with respect to c . Consider the homogeneous
system corresponding to this nonhomogeneous system; that is, set a = b' = 0 in this nonhomogeneous system. This gives a;ckld + batik + ba
j
kf = 0.
Contracting this equation with respect to the indices a and e, a, and S, and a and ry, we obtain
k jkl + Ije"Y
+ klj =
pcas7
76 _
-a-Y6
jkl +
0,
Ijk + klf - 0+
fOZ, + co" + l _c'klj - 0. jkl
(7.3.51)
If we symmetrize and alternate equations (7.3.51) with respect to the indices
'y and d, we obtain two homogeneous systems of equations with respect to 6) andkl ai with different order of lower indices (cf. Subsection 7.3.5). Cak1 '1 he determinants of the matrices of coefficients of these systems are equal to (p - 1)2(p + 2) and (p + 1)2(p - 2), respectively. They do not vanish if p > 2. Hence these systems have only the trivial solution. As a result the homogeneous system in question has only the trivial solution c0,216 = 0 provided that p > 2; thus the original nonhomogeneous system has a unique solution expressing the components of c in terms of the components of the subobject (a, b') and their Pfaffian derivatives. Now we can see that the object (a, b) satisfies certain differential equations.
These equations can be obtained if we substitute for the components of c in equations (7.3.49) and (7.3.50) their values found in the way indicated above. The conditions obtained are other analogues of the Bianchi equations in the theory of spaces with affine connection. 7. An almost Grassmann structure AG(p - 1, p + q - 1) is said to be locally Grassmann (or locally flat) if it is locally equivalent to a Grassmann structure. This means that a locally flat almost Grassmann structure AG(p - 1, p + q - 1) admits a mapping onto an open domain of the algebraic variety fl(m, n) of a
7.3
The Complete Structure Object of an Almost Grassmann Manifold 291
projective space PI, where N = (m+i) - 1, m = p - 1, n = p + q - 1, under which the Segre cones of the structure AG(p - 1, p + q - 1) correspond to the asymptotic cones of the variety 1(m, n). From the equivalence theorem of E. Cartan (see Cartan [Ca 08) or Gard-
ner (Gar 891), it follows that in order for an almost Grassmann structure AG(p -1, p + q - 1) to be locally Grassmann, it is necessary and sufficient that its structure equations have the form (7.1.9), (7.1.14), (7.1.15), and (7.1.17).
Comparing these equations with equations (7.3.35), we see that an almost Grassmann structure AG(p - 1, p + q - 1) is locally Grassmann if and only if its complete structure object S = (a, b, c) vanishes. However, we established in this section that if p > 2 and q > 2, the components of b are expressed in terms of the components of the tensor a and their Pfaffian derivatives, and the components of c are expressed in terms of the components of the subobject (a, b) and their Pfaffian derivatives. Moreover it follows from our considerations that the vanishing of the tensor a on a manifold M carrying an almost Grassmann structure implies the vanishing of the components of b and c. Thus we have proved the following result:
Theorem 7.3.2 For p > 2 and q > 2, an almost Grassmann structure AG(p - l,p + q - 1) is locally Grassmann if and only if its first structure tensor a vanishes. 8. We will now write the structure equations (7.3.35) in index-free notation. To this end, we denote the matrix 1-form (w.), defined in a first-order frame bundle, by w and write equation (7.2.33) in the form
dw=r. Aw-w A8 -90Aw+fl,
(7.3.52)
where 0. = (90*) and Bp = (9) are the matrix 1-forms defined in a second-order frame bundle for which
tr 0o = 0, tr Bp = 0; by the letter k we denote the scalar form w occurring in equation (7.2.33) and also defined in a second-order frame bundle. Note that in the exterior products of 1-forms, occurring in equations (7.3.52) and in further structure equations of this subsection, multiplication is performed according to the regular rules of matrix multiplication-row by column (see Subsection 4.1.6). The 2-form H = (f)a) is the torsion form with the components (7.3.53)
292
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
The remaining structure equations (7.3.35) can be written in the form dea + 9a, A 8a =
[-Ia tr (co A w) + pcp A w] + 9a,
p+ q p d00 + Oa A 90 = P q [-Io tr (cp A w) + qw A,p]
+00, (7.3.54)
dx=tr(,pAw), dip + ea A ,p + W A Bo +K AV = - (ap) A w + +,
where V = (w°) is a matrix 1-form defined in a third-order frame bundle; In = (ba) and to = (b;) are the unit tensors of orders p and q, respectively; and 2-forms 9a = (9p), 90 = (9'), and 4' = ($i) are the curvature 2-forms of the AG(p - 1, p + q - 1)-structure whose components are 9a = baryfwk Awl
7.4
9' = b'`kl6wk Awl
4'° = ca7dwl A wk
(7.3.55)
Manifolds Endowed with Semiintegrable Almost Grassmann Structures
1. In this section we will establish geometric conditions for an almost Grassmann structure AG(p - 1, p + q - 1) defined on a manifold M to be semiintegrable. The conditions will be expressed in terms of the structure object S of the almost Grassmann structure AG(p - 1, p + q - 1) and its subobjects Sa and So which will be defined in this section. In what follows, we will often encounter quantities satisfying the conditions similar to conditions (7.2.12) for the quantities uapk. For calculations with quantities of this kind, the following lemma is very useful:
Lemma 7.4.1 If a system of quantities T:'.vo is skew-symmetric with respect to the pairs of indices (a.) and (0), namely satisfies the conditions :..00 = -T :'. °,
(7.4.1)
then the following identities hold:
j
T...]ao] = T....a0
T ..
l
T..1
T...(a0) ..*J
= T...no lif]+
-T'jao) = T..(ll )$ T..li l = -T..° On = T...lijl .
] = 0,
(7.4.2)
T.. gyp) = 0.
In these relations the symmetrization and the alternation are carried separately over the lower indices and the upper indices. In addition the following decompositions take place: .. no =T...iao) +T...no
T".: ro
=
T..I;aal
+T...no.
(7.4.3)
Manifolds with Semiintegrable Almost Grassmann Structures
7.4
293
Proof. All these identities can be proved by direct calculation with help of (7.4.1).
In addition, in the proof of the main theorem, we will use the following lemma:
Lemma 7.4.2 The condition Tlikl7l
-0
(7.4.4)
where the alternation is carried over three vertical pairs of indices, implies the condition (7.4.5) T(ijk) l - 0 where the alternation and symmetrization are carried separately over the upper triple and the lower triple of indices. and collect Proof. To prove this, one should write down 36 terms of T(la°.p71 k) from them 6 groups of 6 terms to each of which the hypothesis (7.4.4) can be
applied. 0 Next we will prove the following important result on the decomposition of the torsion tensor of an almost Grassmann structure AG(p - I, p + q - 1):
Theorem 7.4.3 The torsion tensor a =
} of the almost Grassmann struck ture AG(p - 1, p + q - 1) decomposes into two subtensors: aiO'Y
a=as-f ao, where
(7.4.6)
i(o7) }. io7 na = {aCr(jk)}, ao = {aOjk
Proof. Since the tensor ak is skew-symmetric with respect to the pairs of indices (o) and (k), then, by Lemma 7.4.1, the decomposition (7.4.6) is equivalent to the obvious decomposition io7 io7 07 aajk - aa(jk) + aaijkl'
U Note that by Lemma 7.4.1, the subtensors a,, and ao can be also represented in the form
a,, = (a`a,,)), ao = {aiA7 } Q[jkl
Note also that like the tensor a, its subtensors aQ and ao are skew-symmetric with respect to the pairs of indices (Q) and (k): iW7 i7d ' iA7 a«(jk) = -aa(kj)' a.(jk) = -aO(kj),
294
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
and they are also trace-free, since it follows from (7.2.30) that
ao(jk) - 0, a«k - 0+ aelk = 0, aaukl = 0.
(7.4.7)
Theorem 7.4.4 If p = 2, then as = 0, and if q = 2, then ap = 0. Proof. Suppose that p = 2. Then a,#, ry = 1, 2. Since, by definition and Lemma 7.4.1, the tensor as is skew-symmetric with respect to the indices A and ry, we have aill
a(lk)
ai22 = a7k) = 0.
But the first condition of (7.4.7) gives i2l i12 i22 al(ik) + a2 ()k) - 0, al (1k) + a2(Jk) = 0.
It follows from these relations that a2(fk) = aI(ik) = 0+
that is all components of the tensor as vanish. For the case q = 2, the proof is similar. 0 2. Now we will prove the following necessary and sufficient conditions for an almost Grassmann structure AG(p - 1,p + q - 1) to be a-semiintegrable or p-semiintegrable.
Theorem 7.4.5 (i) If p > 2 and q > 2, then for an almost Grassmann structure AG(p - 1, p + q - 1) to be a-semiintegrable, it is necessary and sufficient that the following condition holds: as = 0.
(ii) If p > 2 and q >
2,
then for an almost Grassmann structure
AG(p - 1, p + q - 1) to be A-semiintegrable, it is necessary and sufficient that the following condition holds: ap = 0.
Proof. We will prove part (i) of theorem. The proof of part (ii) is similar. Suppose that 0a, a = 1, . . . , p, are basis forms of the integral submanifolds Va, dim Va = p, of the distribution Aa appearing in Definition 7.1.2. Then w4 = s`ea,
a = 1,...,p; i = p+ 1,...,p+q.
(7.4.8)
For the structure AG(p - 1, p + q - 1) to be a-semiintegrable, it is necessary and sufficient that system (7.4.8) be completely integrable. Taking the exterior derivatives of equations (7.4.8) by means of structure equations (7.2.11), we find that (ds' + sjw - siw) A 0a + si(dOa - wQ A 00) = a`o ks3sk0s A 0.y.
(7.4.9)
7.4
Manifolds with Semiintegrable Almost Grassmann Structures
295
It follows from these equations that
dOa - won 8p =
(7.4.10)
A 80,
where vg is an 1-form that is not expressed in terms of the basis forms O. For brevity, we set gyp' = ds' + s'wj - s'w. (7.4.11) Then the exterior quadratic equation (7.4.9) takes the form (6arP' + 8V O.) A 8Q
(7.4.12)
= a'a ksisk8p A 8y.
From (7.4.12) it follows that for 00, = 0, the 1-form
s'W« vanishes:
69(p'(6) + s'cpa(6) = 0.
(7.4.13)
Contracting equation (7.4.13) with respect to the indices a and Q, we find that gyp' = -s' 'P(6),
= 6!W(6),
(7.4.14)
where we set w(6) = It follows from (7.4.14) that on the subvariety V0, the 1-forms W' and WO can be written as follows: 'P' _ -s'W + SiO00, woo = 6aW + 80."0-,.
(7.4.15)
Substituting these expressions into equations (7.4.10) and (7.4.11), we find that d8,, - w« A 8Q = P A Ba + say 8y A Bp
(7.4.16)
where say = AC OP) and
ds' + sjw - s'w = -s'W + sio0o.
(7.4.17)
Substituting (7.4.16) and (7.4.17) into equation (7.4.9), we obtain
- s'soy - 6QRslilyl = aapk lsjsk.
(7.4.18)
Contracting equation (7.4.18) with respect to the indices a and Q, we obtain
-2s'say - ps'y + 8'y = 0, from which it follows that
s'y = s'sy,
(7.4.19)
where we set sy = -P2 sa'r. Substituting (7.4.19) into (7.4.18), we find that s'(6yso - dasy - 2,,O-f) =
2a'lRklsisk.
(7.4.20)
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
296
It follows that basa
- 60s'y - 2sQ' =
(7.4.21)
where so ctj = -s"A. a.? Substituting (7.4.21) into (7.4.20), we arrive at the equation = a'"07
sa(j a7 d$
(7.4.22)
a(jk),
k)
where the alternation sign in the right-hand side is dropped by Lemma 7.4.1.
Contracting (7.4.22) with respect to the indices i and j and taking into account equations (7.2.30) and (7.2.31), we obtain s a, ak = O ,
(7 4 23) .
.
from which, by (7.4.22), it follows that aia, a(jk)
= 0.
(7.4.24)
This proves that if an almost Grassmann structure AG(p - 1, p + q - 1) is a-semiintegrable, then its torsion tensor satisfies the condition (7.4.24), as = 0. Since, by Theorem 7.4.4, for p = 2 the subtensor as = 0, condition (7.4.24) is identically satisfied. Hence, while proving sufficiency of this condition for a-semiintegrability, we must assume that p > 2. Let us return to equations (7.4.16) and (7.4.17). Substitute into equation (7.4.17) the values s'a taken from (7.4.19) and set
P = V - saga.
(7.4.25)
In addition, by (7.4.23), relations (7.4.21) imply that sa, = altisal a
a
Then equations (7.4.16) and (7.4.17) take the form d9a - (wQ + 6.00) A Bp = 0
(7.4.26)
ds'+sjwj' -s'(w-W-)=0.
(7.4.27)
and
Taking the exterior derivatives of (7.4.27), we obtain the following exterior quadratic equation: s'4i +
A 9Q = 0,
(7.4.28)
where 4
= dip - (p+1)g s kwk A9 ry
p+q
Next, taking the exterior derivatives of (7.4.26), we find that
A06=0.
(7.4.29)
7.4
297
Manifolds with Semiintegruble Almost Grassmann Structures
Equation (7.4.28) shows that the 2-form 4i can be written as 4i = A- sks'O,, A 06,
(7.4.30)
where the coefficients Ak6 are symmetric with respect to the lower indices and
skew-symmetric with respect to the upper indices. Substituting this value of the form t into equations (7.4.28) and (7.4.29), we arrive at the conditions b(k,)
0
(7.4.31)
blaktj + a[a Aki 1 - 0.
(7.4.32)
and
Contracting equation (7.4.31) with respect to the indices i and j and equation (7.4.32) with respect to the indices a and Q, we find that 2(q + 2)Aki + bk1i + bk;l + bi k + b,4k; = 0
(7.4.33)
and
2(p
- 2)Aki + bOk + boik + baki + balk = 0.
(7.4.34)
Note that for p = 2 equation (7.4.32) becomes an identity, and we will not obtain equation (7.4.34). If we add equations (7.4.33) and (7.4.34) and apply condition (7.3.29), we find that Aki = 0. (7.4.35) As a result equations (7.4.31) and (7.4.32) take the form
0kl)
= 0+ 6[a ki) = 0.
(7.4.36)
By Lemma 7.4.1, conditions (7.4.36) are equivalent to the conditions b(jkt) = 0+ b[Oki 61 - 0.
(7.4.37)
It follows from equations (7.4.35) and (7.4.30) that d 4p =
(p + 1)g s k Wk n By.
p+q
(
7 4 38) .
.
Finally, taking the exterior derivatives of equations (7.4.38) and applying (7.4.26), (7.4.27), and (7.3.35), we obtain the condition [QQyI - 0. c(ijk)
(7.4.39)
These equations will not be trivial only if p > 2. But, by Lemma 7.4.2, conditions (7.4.39) follow from integrability conditions (7.3.34).
298
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
Thus the system of Pfaf flan equations (7.4.8), defining integral submanifolds of an a-semiintegrable almost Grassmann structure, together with Pfaffian equations (7.4.17) and (7.4.38) following from (7.4.8), is completely integrable if and only if conditions (7.4.24) and (7.4.37) are satisfied. But as we showed
earlier, for p > 2, conditions (7.4.37) follow from condition (7.4.24). Hence only condition (7.4.24) is necessary and sufficient for complete integrability of the system of equations (7.4.8), (7.4.17), and (7.4.38), that is, for the almost Grassmann structure to be a-semiintegrable for p > 2. This proves part (i). As we noted in the beginning, the proof of part (ii) is similar. We note only that the equations of integral submanifolds Vo, dim Vp = q, of the distribution G10 appearing in Definition 7.1.2 can be written in the form
a=1,...,P; i=p+I....,P+q,
wQ=s09$,
where the 1-forms 9' are linearly independent on the submanifold V0. We introduce the following notations: ba
=
{b{ikt)}+ ba = 1 okl l}, Ca =
6p = 1
[jkll},
ICI- j01 11
b0 = {baki6)}, co =
It follows from our previous considerations that
1. for p = 2 we have ba, = O and ca = 0;
2. for q = 2 we have by = 0 and co = 0;
3. forp > 2 we have ca = 0; and
4. forq>2wehave c0=0. The last two results follow from conditions (7.3.34) and Lemma 7.4.2. These
results combined with equations (7.3.27) and (7.3.28) imply that the tensors a,, ao and the quantities b,,bQ,ba,by form the following geometric objects: (aa, bI), a
z (aa, ba), S. = (/aa, bia, b2a),
(a0, bb),
(ao,bo), So = (ao,bp,b2),
which are subobjects of the second structural object and the complete structural object of the almost Grassmann structure. From the proof of Theorem 7.4.5 it follows that for p > 2 the condition as = 0 implies the conditions b l = bQ = 0. Similarly for q > 2 the condition ao = 0 implies the conditions bQ = bQ = 0. Now we consider the cases p = 2 and q = 2. For definiteness we take the case p = 2. As we have already seen, for p = 2, the tensor as as well as the quantities bQ and ca vanish (aa = bQ = ca = 0), and the object bQ becomes
7.4
Manifolds with Semiintegrable Almost Grassmann Structures
299
a tensor. Thus the vanishing of this tensor is necessary and sufficient for the almost Grassmann structure AG(1, q + 1) to be a-semiintegrable. Hence we have proved the following result:
Theorem 7.4.6 (i) If p = 2, then the structure subobject Sa consists only of the tensor b'', and the vanishing of this tensor is necessary and sufficient for the almost Grassmann structure AG(1, q + 1) to be a-semiintegrable.
(ii) If q = 2, then the structure subobject So consists only of the tensor bQ, and the vanishing of this tensor is necessary and sufficient for the almost Grassmann structure AG(p-1, p+ 1) (which is equivalent to the structure AG(l,p+ 1)) to be p-semiintegrable.
(iii) If p = q = 2, then the complete structural object S consists only of the tensors b1, and b2,, and the vanishing of one of these tensors is necessary
and sufficient for the almost Grossmann structure AG(1,3) to be a- or (3-semiintegrable, respectively.
We will make two more remarks:
1. The tensors ba and bQ are defined in a third-order differential neighborhood of the almost Grassmann structure.
2. For p = q = 2, as was indicated earlier (see Subsection 7.1.2), the almost Grassmann structure AG(1,3) is equivalent to the conformal CO(2, 2)-structure. Thus by results of Subsection 5.1.3, we have the following decomposition of its complete structural object: S = b,4-b2 . This matches the splitting of the tensor of conformal curvature of the CO(2, 2)-structure: C = C. 4-Co. 3. Now we can compare the differential geometry of conformal and pseudoconformal structures with that of almost Grassmann structures. A conformal or pseudoconformal structure CO(p, q) is defined on a differentiable manifold M of dimension n = p + q by a differentiable field of second-order cones Cz(p, q) of signature (p, q) lying in the tangent space T,, (M). A cone Cx(p,q) is invariant under transformations of the group G °_w SO(p,q) x H, where SO(p,q) is the pseudoorthogonal group of signature (p, q) and H is the group of homotheties.
When we derive the structure equations (4.1.31)-(4.1.35) of a conformal (SO (p, q) x H) a T(n) structure, we prolong the group G to the group G' which is the group of motions in the compactified space T2(M) enlarged to an n-dimensional quadric Qy of index q. This quadric can be embedded into a projective space P"+l of dimension n + 1 and is determined in it by a homogeneous equation of second order whose left-hand side is a quadratic form of signature (p + 1, q + 1). Since a point x E Q, at which Qx is tangent to M, is fixed, the geometry of Qy is equivalent to that of a pseudo-Euclidean space RQ .
The group G' is isomorphic to the group of motions of this space, and T(n) is its group of translations.
300
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
The first structure tensor appearing in the structure equations (4.1.34) is the tensor of conformal curvature which is determined in a third order differential neighborhood. In general, if n > 4, this tensor does not vanish. The vanishing of this tensor leads to a local conformally flat structure.
In the case of a four-dimensional conformal structure CO(p, q), where p + q = 4, the tensor of conformal curvature splits into two subtensors which are the curvature tensors of two fiber bundles Ea and E0 associated with this conformal structure. The vanishing of one of these subtensors leads to a semiintegrable conformal structure. An almost Grassmann structure AG(p - l, p + q - 1) is defined on a differentiable manifold M of dimension n = pq by a differentiable field of algebraic Segre cones SC,. (p, q) C T., (M) whose projectivizations are the Segre varieties S(p - 1, q - 1). Each of these cones carries two families of plane generators of dimensions p and q that form two fiber bundles Ea, and Ep on the manifold M. A cone SC,(p, q) is invariant under transformations of the group G °-' SL(p) x SL(q) x H where SL(p) and SL(q) are the special linear groups of orders p and q, respectively. When we derive structure equations (7.3.35) of an almost Grassmann structure, we prolong the group G to the group G' °-r (SL(p) x SL(q) x H) of T(pq). The group G' is the group of motions in the compactified space T=(M) which is obtained by joining to the space T=(M) the Segre cone SC,,. (M) with vertex at the point at infinity of the space T,T(M). The compactified space Ts(M) is equivalent to the algebraic variety fl(p-1, p+q-1) with a fixed point x at which this space is "glued" to the almost Grassmann manifold AG(p - 1, p + q - 1). The variety l(p - 1, p + q - 1) itself is the image of the Grassmannian in the projective space PN, where N = (Dq°) - 1. The variety 0 (p-1, p+q-1) with a fixed point x is equivalent to the Segreaffine space SAD" (see Subsection 6.6.3) of dimension pq. The latter space is a stereographic projection of the variety fl(p - 1,p+ q - 1) from the point x onto a flat space of dimension pq. The group G' is the group of motions of the space SAD", and the group T(pq) is the group of translations of this space. Unlike the CO(p, q)-structure, the first structure tensor (the torsion tensor) of the almost Grassmann structure AG(p - l,p + q - 1) is determined in its differential neighborhood of second order. If p > 2 and q > 2, then just like for CO(2, 2)-structure, this tensor splits into two subtensors that are the first
structure tensors of two fiber bundles E, and E. The vanishing of one of these subtensors leads to a semiintegrable almost Grassmann structure.
On the other hand, if p = 2 or q = 2, then the corresponding torsion tensor vanishes, and the condition of semiintegrability of an almost Grassmann structure will be connected with the vanishing of the second structure tensorthe curvature tensor of the corresponding fiber bundle. Finally, if p = q = 2 (note that this is the only positive integer solution to the equation p + q = pq), then the almost Grassmann structure AG(1, 3) becomes
7.5
Multidimensional (p + 1)-Webs and Almost Grassmann Structures
301
Table 7.4.1
#
Property
CO(p, q)
1.
dimM
n=p+q
2.
Invariant construction inT=(M)
2nd-order cone
Segre cone
C.(p,q)
SC.(p,q)
Order of
s = 1
s = 1
3.
AG(m,n)
p=m+l, q=n-m
G-structure 4.
Structure group
5.
Prolonged structure group
6.
Type of
SO(p, q) x H G'
G x T(p + q)
G'-° SL(p) x SL(q) x H G' '-5 G x T (p q)
t=2
t=2
Torsion-free
With torsion
(b, c)
(a, b', V, c)
G-structure 7.
Existence of
torsion 8.
Complete
structure object 9.
Local space
(q).
(G(m,n))=
10.
Locally flat
Cq
G(m, n)
structure
11.
Existence of isotropic bundles
`dp & q : p = q = 2: E,, (M, SL(2)) and & (M, SL (p)) and Ea(M, SL(q)) E,(M, SL(2))
12.
p+q=pq
CO(2,2)
AG(1, 3)
p=q=2 the CO(2, 2)-structure. Its torsion tensor vanishes, and the role of its curvature tensor was described earlier (see Sections 5.1 and 5.4). The preceding comparison of the conformal CO(p, q)-structures and almost Grassmann AG(m, n)-structures is summarized in Table 7.4.1.
7.5
Multidimensional (p + 1)-Webs and Almost Grassmann Structures Associated with Them
1. We will define the notion of a d-web of codimension q given on a differentiable manifold of dimension pq.
Definition 7.5.1 Let M be a C'-manifold of dimension pq, p > 2, q > 1, s > 3. We say that a d-web W (d, p, q) of codimension q is given in M if
302
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
1. d foliations A, a = 1,... , d, of codimension q are given in M; and 2. d leaves (of the foliations A,) passing through a point x E M are in general position; namely any p of the d tangent subspaces to the leaves at the point x have in common only the point x. There exists a neighborhood of each point x of the web W (d, p, q), t where the foliations A, are fibrations. Therefore, from a local point of view, a d-web
can be considered as formed by d fibrations. We denote the bases of these fibrations by X,. Example 7.5.2 Consider in an affine space AP4 of dimension pq d families of parallel (p - 1)q-dimensional planes that are in general position. They form a d-web called a parallel d-web.
Example 7.5.3 Let X o = 1, ... , d, be d smooth submanifolds of dimension q in a projective space pP+q-1, and let L be an (p - 1)-plane that intersects each X, at the points x,. A d-web arises in a neighborhood of the (p-1)-plane L on the Grassmannian G(p - 1,p + q - 1) of all (p - 1)-planes of the space pp+q-t (dimG(p- l,p+q-1) = pq (see Section 6.1)). The leaves of this web are bundles of (p - 1)-planes with vertices located on the submanifolds X,. Such a web is called the Grassmann d-web and is denoted by GW (d, p, q). Our definition of the Grassmann d-web is essentially of local nature, since, only for (p - 1)-planes sufficiently close to x, can we assert that they intersect each of
the submanifolds X, only at one point, as it was for the (p - 1)-plane L.
Example 7.5.4 A Grassmann d-web is called algebraic and is denoted by AW (d, p, q) if the submanifolds X, defining it belong to the same algebraic q-dimensional submanifold Vd of degree d. There are many special cases of algebraic d-webs. They are characterized by the fact that the submanifold Vd is decomposed into two or more submanifolds Vdk, 0 < dk < d, Ek dk = d, and each X, belongs to one of V9. For example, all submanifolds X, can be q-planes. In this case the submanifold Vd is decomposed into those q-planes.
Definition 7.5.5 Let M and M be two manifolds of the same dimension pq. Two webs W (d, p, q) and W (d, p, q) defined in M and M are said to be equivalent if there exists a local diffeomorphism w: M -4 M that transfers the foliations of the first web W (d, p, q) into the foliations of the second web W (d, p, q).
In particular, d-webs equivalent to the parallel, Grassmann, and algebraic d-webs, which were considered above, are called parallelizable, Grassmannizable, and algebraizable, respectively. Thus a parallelizable web W (d, p, q) is equivalent to a web consisting of d families of parallel planes of codimension q. If d < p, a web W (d, p, q) is always For brevity, we will use these words instead of the words "each point x of a manifold M carrying a d-web W".
7.5
Multidimensional (p + 1)-Webs and Almost Grassmann Structures 303
parallelizable. Because of this we will assume that d > p + 1. In Sections 7.6 and 7.8 we will study Grassmann and Grassmannizable d-webs in more detail. 2. Consider a (p + 1.)-web W (p + 1, p, q) defined on a manifold M. The foliations .10, a = 0,1, ... , p, forming this web can be defined by the following completely integrable systems of Pfaffian equations: W4 = 0, c
o = O,l,...,p; i = p + 1,...,p + q.
(7.5.1)
Since the number of 1-forms on the left-hand sides of equations (7.5.1) is (p+l)q
and dim M = pq, the forms w{ are connected by linear equations. It can be proved (see Goldberg [Go 73, 74a] or Goldberg [Go 88], §1.2) that these equations can be reduced to the following form:
w'+w'+...+w'=0. 0 p
(7.5.2)
1
Relations (7.5.2) remain invariant under the transformations
AJw', det(AJ) # 0,
(7.5.3)
forming the group G = GL(q)-the structure group of a web W (p + 1, p, q) (see Subsection 7.1.2). By conditions (7.5.2), the structure equations of a web W (p + 1, p, q) can be reduced to the form
dw'=wiA ,'+E 003k, a R#a
p
o
o
where a,,0 = 1, ... ,p, and a'.k is the torsion tensor of the web satisfying the conditions p,k
Rakj, E a ik
(7.5.5)
0
CO
(see Goldberg [Go 73, 74a] or Goldberg [Go 881, §2.1). In addition, we suppose that 0. The forms ww satisfy the structure equations
as k =
dwf = w A wk +
b
klwk A wl,
(7.5.6)
Ck,
where b
ki
is the curvature tensor of the web, and define an affine connection
t on the manifold MPQ (see Goldberg [Go 88], §1.3). The tensors a
and
no
bp';ki are the torsion tensor and the curvature tensor of this connection.
a
Equations (7.5.6) are differential prolongations of equations (7.5.4). In addition, as another result of exterior differentiation of equations (7.5.4), we obtain the Pfaffian equations p
VClip a ka ikt +apya ,,k noa i + a ma a kl)w1, ya 7_1
R7
'Y
a
(7.5.7)
304
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
and the closed form relations (7.5.8)
(7.5.9)
p(ikll = 0,
b,';k+2bbkll=0.
a«
ao
(7.5.10)
Relation (7.5.8) shows that if p > 2, then the curvature tensor of a web W (p + 1, p, q) is completely determined by the covariant derivatives of the torsion tensor of this web. This implies the following result:
Theorem 7.5.6 For a web W (p + 1, p, q), p > 2, to be parallelizable, it is necessary and sufficient that its torsion tensor vanishes, ask = 0. Proof. By (7.5.7) and (7.5.8), the condition «a k = 0 implies that b '
= 0.
As a result the structure equations of the web under consideration take the
form dw' 0
0
dw' =wh Awk,
o =0,1,...,p; i,j,k=p+1,...,p+q.
But these equations determine a (p + 1)-web in an afiine space APQ of dimension pq formed by foliations of parallel planes of codimension q (see Goldberg [Go 73, 74a] or 88), §1.5). The converse follows immediately from the previous equations. Note that for a multidimensional three-web W(3, 2, q) the symmetric part of the curvature tensor cannot be expressed in terms of the covariant derivatives of the torsion tensor (see equations (1.31) and (1.33) in Akivis and Shelekhov [AS 92]). The parallelizability condition for webs W(3, 2, q) is expressed in
terms of both the torsion and curvature tensors (see Akivis and Shelekhov [AS 92), §1.5).
3. We will now show that an almost Grassmann structure AG(p-1, p+q-1) is associated with each (p + 1)-web W (p + 1, p, q).
Let T,,(M) be the tangent space to M at the point z. The co-basis forms p; i = p + 1, ... , p + q, of the (p + 1)-web introduced above can be taken as coordinates in the space TT(M). Then the equations of the subspaces T, of this space that are tangent to the leaves of the web passing through the point x can be written in the form (7.5.1). By virtue of (7.5.2), we can see that the relations
hold. Equations (7.5.1) and (7.5.2) are invariant under transformations of the group GL(q) of the web W(p+ 1,p,q).
Grassmann (p+ 1) -Webs
7.6
305
Let (r, K, 01, ... , o _ 1) be a permutation of the indices (0, 1, ... , p). In T=(M), we consider the intersection of the subspaces T,,, k = 1, . . . , p - 1. Denote this intersection by TT,. Its dimension is q, and it is defined by the equations w' = 0. The number of such subspaces is (PZ1) = P(P'). z If p = 2, o. this number is equal to 3, and the subspaces TTK coincide with the subspaces T. tangent to the leaves of the web passing through the point x. In the space T=(M) there exists a unique Segre cone SC=(p,q) containing all subspaces TT,,. This cone can be defined by parametric equations (7.1.1) where zo = w'. By (7.5.2), it follows from these equations that Q
W` _ -o'rb' o
where rro = - EP.=1 j7,,. The subspaces TTK belonging to the Segre cone can be given on this cone by the equations q,. = 0, where the indices ok take the values indicated above. These subspaces belong to the family of the q-dimensional plane generators q4 of the Segre cone SC, (p, q). Since the family of Segre cones SC= (p, q) given in the tangent spaces T. (M)
defines an almost Grassmann structure AG(p -1, p+ q - 1) in the manifold M, the following theorem holds:
Theorem 7.5.7 An almost Grassmann structure AG(p-1, p+q-1) is invariantly connected with an (p + 1) -web W (p + 1, p, q) given on a smooth manifold
M of dimension pq. The structure group of this web is a normal subgroup of the structure group of the almost Grassmann structure.
Note that the last statement of Theorem 7.5.7 follows from the fact that the structural group of a (p + 1)-web is the group GL(q) and the structure group of the almost Grassmann structure is either of two following isomorphic groups: SL(p) x GL(q) SL(p) x SL(q) x H. The q-dimensional plane generators r)q of the Segre cones SC, (p, q) associated with a web W (p + 1, p, q) are called its isoclinic subspaces. In addition to them, the Segre cones SC., (p, q) carry the p-dimensional plane generators cP. They are called the transversal subspaces of this web.
7.6
Grassmann (p + 1)-Webs
1. In Section 7.5 we constructed an important example of a web W (p+ 1, p, q)-
the so-called Grassmann (p+1)-webs in the Grassmannian G(p-1,p+q-1) of (p-1)-planes of a projective space PP+q-1 of dimension p+q-1 (see Example 7.5.3). In this section we will study Grassmann (p + 1)-webs in more detail. First, note that dim G(p -1, p + q - 1) = pq. Next, consider a submanifold in G(p - 1, p+ q - 1) formed by (p - 1)-planes passing through a fixed point
306
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
z E PP+q-I. We will call this manifold a bundle of (p-1)-planes and denote it by S. It is easy to see that the bundle S. is isomorphic to the Grassmannian G(p - 2, p + q - 2) and that dim S= _ (p - 1)q and codim S., = q. Using bundles of (p - 1)-planes, we can construct foliations and webs of codimension q in the Grassmannian G(p - 1, p + q - 1). In fact, in the space pP+q-1, let us consider a smooth manifold X of dimension q and the set of the bundles S,T with vertices belonging to X: x E X. If we exclude the (p - 1)planes tangent to the manifold X and the (p-1)-planes intersecting X at more than one point from each of the bundles St, the remaining parts Sx of S. form a foliation in an open domain D of the Grassmannian G(p - 1, p + q - 1). In the space PP+q-1, we further consider submanifolds X a = 0,1, ... , p, of dimension q in general position. Each of these submanifolds generates a
foliation in an open domain D. C G(p - 1, p + q - 1) described above. All foliations constructed in this manner generate a (p + 1)-web of codimension q in the domain D = ns=o D. In Section 7.5 the (p + 1)-webs described above were called Grassmann (p + 1)-webs and were denoted by GW(p + 1,p, q). Next, denote by L a moving (p - 1)-plane of a Grassmann (p + 1)-web and
by A, the points of intersection of L and the submanifolds X,. Since the submanifolds X. are in general position, p of those points, for example, the points A1,.. . , AP, can be taken as the vertices of a projective frame of the (p - 1)-plane L. We also take the vertex A0 as the unit point of this frame. Thus A0 = Al + A2 +... + AP. Let us take the points A,, i = p + I.... ,p+ q, that supplement the points Aa, a = 1,. .. , p, to a complete frame of the space PP+q-1 As usual, the equations of infinitesimal displacement of this moving frame can be written in the form dAE = wf An,
,n = 1,...,p,p + I,...,p + q,
(7.6.1)
and the structure equations which the forms wf satisfy, in the form: d w E = wf A t,
f,i,C = 1,...,p,p + 1,...,p + q.
(7.6.2)
Since the (p-1)-plane L is not tangent to any of the submanifolds X,,, the 1-forms w,, can be taken as co-basis forms on these submanifolds. Since the points Aa are fixed when the subspace L is fixed, we must have wa = Aoiwa,
0 # a; a,Q = 1,...,p; i = p+ 1,...,p+q,
(7.6.3)
and
dAa = w°A,, +wa 1 Ai +
\
AaiAp f .
(7.6.4)
p#a
Here and in what follows, the summation is carried over the indices i, j, k according to the usual rule, while the summation is carried over the indices a, j3, ry only if there is the summation sign.
7.6
Grassmann (p + 1)-Webs
307
Let us locate the points A; in the space TA,, tangent to the manifold Xe generated by the point A0. Then we have dAo = wAo + Ai
wQ
(7.6.5)
where (7.6.6) Q
We define
wQ=-wo.
(7.6.7)
Since the point AO generates a q-dimensional manifold X0, the forms wo are linearly independent. In the frame that we have constructed the equations
wa =0
w, = O, ... , wyi= 0,
determine p + 1 foliations in the Grassmannian G(p - 1, p + q - 1), and these foliations form a Grassmann (p+ 1)-web GW(p+ 1,p, q). The forms w. are the co-basis forms of this web, and equations (7.6.3) and (7.6.7) are its fundamental equations. 2. Let us find the torsion and curvature tensors of a Grassmann web GW (p + 1, p, q). To this end, we first prolong equations (7.6.3); that is, we take the exterior derivatives of these equations and apply the Cartan lemma to the exterior quadratic equations obtained as the result of exterior differentiation. As a result we obtain
Oxa; + \n;Aojw0 + E (Aai
- Aoi)(A'3 - )'3 )w,, + wR = Aa;jwJo+
(7.6.8)
7¢a,Q
where a,$ and -y are distinct and Vij = AQji. In equations (7.6.8) we used the notations
Vap;=dAp;-aaj9;, 9 =w -b; w,
(7.6.9)
and the forms wo are determined by formula (7.6.7). Exterior differentiation of relation (7.6.6) leads to the equation
Awo=0. The solution of this equation can be represented in the form w° = w° +
.?
wo
where
P,=Pji,
EP,a.=0, a
(7.6.10)
308
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
and
(7.6.11)
P
are fiber forms, so they are not linear combinations of the co-basis forms wQ.
Let us fix the (p - 1)-plane L of the web under consideration, that is, set w`o = 0. Then equations (7.6.8) take the form
VsAoi + np = 0,
(7.6.12)
where as usual b denotes the differentiation symbol with respect to the fiber parameters and aR = wo(b). We define the quantities
A,=
1
R
p(P-.00 "
(7.6.13)
It follows from equations (7.6.12) that when a point x is fixed, the quantities Ai satisfy the equations
Daai+rro=0. On the other hand, equations (7.6.1) and (7.6.11) imply that
bAi = iA; + r°Ao. The last two relations lead to the equations: b(Ai + AiAo) = 7r, (Ai + A,Ao),
which show that the plane L spanned by the points A, + A,A0 is invariant. Let us locate the points A, in L. Then the quantities A, vanish, and (7.6.13) gives
E 1oi = 0.
(7.6.14)
o#p
The forms ir? become zero, and the forms w° become linear combinations of the co-basis forms wi : gAwA.
(7.6.15)
A
Equations (7.6.15), (7.6.10), and (7.6.7) imply that (qQ
- p,)wa.
(7.6.16)
We can now find the torsion and curvature tensors of the web GW(p + 1, p, q). Applying exterior differentiation to its co-basis forms wa, we get dwQ=w1Awj' +w.0 Awa.
7.7 Transversally Geodesic and Isoclinic (p + 1) -Webs
309
Next from equation (7.6.6) we obtain
Wa =w- wp. 00a
Substituting these expressions into the previous formulas and applying relations (7.6.3), we obtain
dw,, =wi A0 +
Aw,a,
kAO
(7.6.17)
p#a
where the 1-form B is determined by formula (7.6.9). By (7.6.14), the expressions bkAO + 6IAak satisfy equations (7.5.5). Therefore equations (7.6.17) are the structure equations of a Grassmann web GW(p + 1, p, q), and the torsion tensor of this web has the form (7.6.18)
Qik = bkA«j + 6 apk.
Since for p > 2 the torsion tensor completely defines the geometry of a web W (p + 1, p, q) (see Section 7.5), we arrive at the following result:
Theorem 7.6.1 For a web W (p + 1, p, q), p > 2, to be Grassmannizable, that is, to be equivalent to a Grassmann web GW (p + 1, p, q), it is necessary and sufficient that its torsion tensor has the form (7.6.18). The forms B determined by equations (7.6.9) define an affine connection 1' on a web GW (p + 1, p, q). It follows from the previous equations that
dw, - w n wk
w n w'a,
dw = -wok A w°t.
a
By virtue of formulas (7.6.9), (7.6.15) and (7.6.16), we find from this equation
that
d9i - ej kA eki =
is
i s
(btgjk + bjglk
i s wak A w0. 1 bkPjl)
04 This gives the following expression for the curvature tensor of the Grassmann (p + 1)-web under consideration:
b'jki = 2
7.7
bjgik + bkipp) -
2
(6'
'3, +
5 pQk).
(7.6.19)
Transversally Geodesic and Isoclinic (p + 1)-Webs
1. A web W (p+ 1, p, q) is called transversally geodesic if the almost Grassmann structure AG(p - 1, p + q - 1) associated with this web is a-semiintegrable. A
310
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
web W (p + 1, p, q) is called isoclinic if the structure AG(p - 1, p + q - 1) is p-semiintegrable. We first consider transversally geodesic (p + 1)-webs W and find analytic conditions characterizing them. The semiintegrability of the almost Grassmann structure associated with a web means the existence of a family of subvarieties
VP on the manifold M that are tangent to the p-planes c". The equations of these subvarieties have the form w' = '9a,
a = 1,...,p; i = p + 1,...,p + q,
(7.7.1)
Cr
where 9a are 1-forms independent on VP, and ' are the coordinates of a vector
determining the location of the transversal subspace of the web. By means of formulas (7.5.4), exterior differentiation of equations (7.7.1) leads to the following exterior quadratic equations:
a' 8,) A 0a =
(7.7.2)
p#a where we used the notations
e'w, ,
V'=
a"= aap.kf'k. ap
(7.7.3)
The quantities a' satisfy the equations aA
a = Ra a', > a'aA= 0.
a13
(7.7.4)
a.A
If we add up equations (7.7.2) written for all a = 1,. .. ,p, and use conditions (7.7.4), we find that
V ' A (o) = -'d (> 90)
.
(7.7.5)
01
Ck
Equations (7.7.5) show that
d(1: where 0 is an 1-form. Substituting the last expression into equations (7.7.5), we obtain the equation
(Vt' - e'9) A
1>
9a) = o.
By Cartan's lemma, we find that V{' = {`9 +
0a.
a
(7.7.6)
7.7 Transversally Geodesic and Isoclinic (p + 1)- Webs
311
On a submanifold VP, the foliations of a web W (p + 1, p, q) cut out a (p + 1)-web W (p + 1, p, 1) of codimension 1. The forms 0. are the basis forms of this web, and the leaves of its (p - 1)-dimensional foliations are determined by the following systems of Pfaffian equations: 0« = 0,
0a = 0.
By (7.5.4), the structure equations of a web W(p+ 1,p,1) have the form
a0aA00,
(7.7.7)
096000
where
Ea=0. -'000
Substituting expressions (7.7.6) and (7.7.7) into equations (7.7.2), we find that
w=0and
a'+a0a'=f'a. a0
Summing up all these equations in a and 3, we obtain a' = 0, a0' = f' R.
(7.7.8)
By (7.7.8), equations (7.7.6) and (7.7.3) take the form Vf' = f'0,
(7.7.9)
and
aaaXfk = a0f'.
(7.7.10)
The following theorem gives the geometric meaning of relation (7.7.9):
Theorem 7.7.1 The subvarieties VP defined on M by equations (7.7.1) are totally geodesic in the connection 1' induced by a web W (p + 1, p, q).
Proof. Denote by lei) the frame that is dual to the co-frame 10) Q consisting of the co-basis forms of the web W (p + 1, p, q), and consider the vectors fa = f'e1. By (7.7.1), these vectors are tangent to VP. By (7.7.9), they, as a well as all vectors of the form cafes where ca are constants, can be parallel translated along VP. Therefore, VP is a totally geodesic submanifold. It follows that the submanifolds VP cut out the leaves of the (p + 1)-web (which are themselves totally geodesic submanifolds on M) along geodesic lines.
This is the reason that the submanifolds VP are called transversally geodesic submanifolds of a web W (p + 1, p, q), and the web itself is called transversally geodesic.
312
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
Next, we will study equations (7.7.10). First of all, note that if p = 2, the left-hand side of (7.7.10) is identically zero, since by (7.5.5) the torsion tensor a' -k of a web W(3, 2, q) is skew-symmetric in the indices j and k. We can see 123
from relations (7.7.3) and (7.7.8) that in this case la' = 0 and equation (7.7.10)
becomes an identity. Therefore, if p = 2, the property of a three-web to be transversally geodesic can be expressed in terms of the curvature tensor (see Akivis and Shelekhov [AS 92], §3.1).
Suppose further that p > 2. Since equation (7.7.10) must be identically satisfied with respect to C', the expression p on its right-hand side is linear in {'; namely a = a kek.
ao
(7.7.11)
no
hold. Substituting (7.7.11) into (7.7.10), we obtain the equation
bk
( .03 k - .1Q)(3e
0,
pk).
(7.7.12)
Theorem 7.7.2 For a web W (p + 1, p, q), p > 2, to be transversally geodesic, it is necessary and sufficient that the symmetric part of its torsion tensor has the form (7.7.12). Proof. The necessity of the condition of Theorem 7.7.1 was proved above. Its sufficiency follows from the fact that by (7.7.12), the system of Pfaffian equations (7.7.1) and (7.7.9) defining the transversally geodesic submanifolds of a web W (p + 1, p, q) is completely integrable. 0 Since the almost Grassmann structure associated with a transversally geodesic (p + 1)-web is a-semi integrable, the condition (7.7.12) of transversal geodesicity we have obtained is equivalent to the condition aQ = 0 of semiintegrability of this structure. It follows that the vanishing of the tensor as is equivalent to the condition (7.7.12), and conversely. In view of this, the tensor as must be expressed in terms of the tensor ai(ik), and conversely. This was proved analytically in Goldberg [Go 75a] (see also Goldberg [Go 88], §2.4). To this end, the following formula derived in Goldberg [Go 75a] was used: aQ(,kl = 2 a (jk)
- ,.
2
,
6(i (a k)i + a irlk) J ' p
E (a 1.)r + a.rizIk)) + (q + 1)(p - 1) &(j6*y
-
6 tjk) 6#Y
(a not summed). (7.7.13)
2. We now consider isoclinic webs W (p + 1, p, q). The almost Grassmann structure associated with such a web must be a-semiintegrable. Therefore
7.7 Transversally Geodesic and Isoclinic (p + 1) -Webs
313
on the manifold M, there exists a family of submanifolds Va tangent to the isoclinic subspaces rlo. The equations of these submanifolds can be written in the form
a=1,...,p; i=p+1,...,p+q,
w'=r1a9', a
(7.7.14)
where the 9i are 1-forms, which are linearly independent on VQ, and qa are parameters determining the location of the isoclinic subspace of the web.
Theorem 7.7.3 For a web W (p + 1, p, q), p > 3, q > 2, to be isoclinic, it is necessary and sufficient that the skew-symmetric part of its torsion tensor has the form: p[jkl = b [i6kl. (7.7.15) 00
The proof of this theorem is similar to that of Theorem 7.7.2. Note that the condition of Theorem 7.7.2 is equivalent to the condition for an almost Grassmann structure associated with a web W (p + 1, p, q) to be /3-semiintegrable. As was proved in Section 7.4, this condition has the form ao = 0. It follows that the tensor ao must be expressed in terms of the tensor aa and, conversely.
These expressions were also found in Goldberg [Go 75a] (see also Goldberg [Go 88], §2.4):
aa(jkl =
2
2
[jk] +/
q
2 1
a(j a7l1Ikl
2
(a
blj
\4 - 1)(p + 1)
p+
d#7
1
6#y
a7(jkl
(7.7.16)
(a not summed).
a 67
d7
The following important theorem follows from Theorems 7.7.2 and 7.7.3;
Theorem 7.7.4 A web W (p + 1, p, q), p > 2, q > 2, is Grassmannizable if and only if it is both isoclinic and transversally geodesic.
Proof. Suppose that a web W (p + 1, p, q) is Grassmannizable. Then it is equivalent to a Grassmann web GW (p + 1, p, q). As was proved in Section 7.6 (see formula (7.6.18)), the torsion tensor of a Grassmann web has the form ajjk = bkaaj
a
ki
This implies the relations
p[jkl -
Qfi -
Xolj)akl'
aR(jk) = (\a(j +'\R(j)ak);
clearly the torsion tensor of the web under consideration satisfies the condition of Theorems 7.7.2 and 7.7.3. Thus a Grassmannizable web is both isoclinic an transversally geodesic.
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
314
Conversely, suppose that a web W (p+ 1, p, q) is both isoclinic and transver-
sally geodesic. Then relations (7.7.12) and (7.7.15) hold on this web. So we have
Qaik = as
+ 0ljkl = aa(jbk) + b 1,j 00
l = (a0j + pj)bk + ( k - k)bj a
Thus the torsion tensor of this web has the form (7.6.18), and therefore the web is Grassmannizable. 0 Hence relations (7.7.12) and (7.7.15) are analytic conditions of the Grassmannizability of a (p + I)-web W (p + 1, p, q). These conditions are conditions for an almost Grassmann structure associated with an (p+ 1)-web to be locally integrable, and hence, for p > 2 and q > 2, they equivalent to the vanishing of the torsion tensor a = of the almost Grassmann structure. If p = 2, then a web becomes a three-web, and its condition of Grassmannizability is expressed not only in terms of the torsion tensor but also in terms of the curvature tensor of the three-web. Clearly this condition is connected with a differential neighborhood of third order (see Akivis and Shelekhov [AS 921, §3.4).
A similar situation occurs for q = 2. If p = q = 2, then we have a three-web on a four-dimensional manifold M. This web induces on M a CO(2, 2) -structure which is torsion-free, and all main properties of this structure are expressed in terms of its tensor of conformal curvature.
7.8
Grassmannizable d-Webs
1. In Sections 7.5-7.7 we studied the webs W (p + 1, p, q) formed by p + I foliations of codimension q on a manifold M of dimension pq. In this section
we consider the webs W (d, p, q), d > p + 1 on a manifold M of the same dimension pq.
As in Section 7.5 we define the foliations as and A a = p+q+ 1,. .., q+d, forming the web W (d, p, q) on the manifold M by the following completely integrable systems of equations:
WI=0, W'=0, P O
(7.8.1)
where i = p + 1 , . . . , p + q; a = 1 , . . . , p ; o = p + q + 1,...,q + d. Since the foliations as and a, are in general position, each of the subsystems of system (7.8.1) corresponding to p values of indexes a and a is linearly independent.
We take the forms w', or = 1,. .. , p, as co-basis forms of the manifold M. a Then other forms of system (7.8.1) are their linear combinations:
a=p+q+1,...,q+d,
315
Grassmannizable d- Webs
7.8
where all matrices (A ) are nonsingular. By a change of the co-bases in the foliations Aa and A we can reduce the last equations to the form go.?
P+q+1
"P-1Jp-1 +A'wJ+...+ . a2J 2
a1J I
a
(7.8.2)
p
2
1
J.
(7.8-3) P
where a = p + q + 2, ... , q + d. Now all the forms w' and w' admit only the a o concordant transformations of the form 'w' =
Ja
a
Jo
o
These transformations form the structural group GL(q) of the web W (d, p, q), and the matrices (A ) become tensors with respect to these transformations. as We denote these tensors by A and note that Af = b). as ap Besides being nonsingular, the tensors A satisfy an additional condition: as their differences A - A as well as some other of their combinations must be as ap nonsingular. These conditions follow from the fact that the foliations Af are in general position.
The structure equations of a web W(d,p,q) consist of equations (7.5.4) (which are the conditions of integrability of the systems of equations w' = 0, a 0), and the conditions of integrability of the systems a = 1, ... , p, and w p+q+1
of equations w' = 0, a = p + q + 2,...,q + d, which contain the differentials of a
the tensors A . We will not write the general form of all these equations. as 2. Let us consider a four-web W (4, 2, q) in more detail. For this web, equations (7.8.2) and (7.8.3) take the form
-w' = w' + w', -w' = 3
1
2
4
J1
w' 2
where the tensors A' and b - M must be nonsingular. The tensor M is called the basis afnor of the web W (4, 2, q) and is denoted by A. Let us clarify its geometric meaning.
Let x be an arbitrary point of a manifold M of dimension 2q carrying a web W (4, 2, q), and let T=(M) be the tangent space to M at this point. Denote
by T a = 1,2,3,4, the subspaces of TT(M) tangent to the leaves 1, of the web W passing through the point x. Under the projectivization of the tangent space TT(M) with center at the point x, the projectivizations of the subspaces T. are subspaces P, C P2q-1. Consider in the space T. (M) the frame {e;, e;} which is dual to the co-frame 1 2 {w',w'} and such that any vector l; E Ts(M) can be written in the form 1
2
t = w'(t)ej - w(t)e+. 2 2 1
1
316
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
Then e; E T1, e; E T2, + e; E T3, and e; E T4, and each of these e, systems of vectors form a basis in the corresponding subspace T,. The triples of subspaces (TI, T2, T3 ) and {T1, T2, T4 } define in the space T,, (M) two systems of transversal bivectors, and the triples {PI, P2, P3} and
{PI,P2iP4} define in the space p2,7-1 two Segre varieties, S(1, q - 1) and S(1, q - 1), which are the projectivizations of these systems of bivectors. Let
1 = E'e; be an arbitrary vector from T1. The bivector H = £I A 6, where 1:2 = ej, passes through ti, and this bivector H is transversal to the first Iel, triple of subspaces. Similarly the bivector H = E .j 2, where SI = passes through t;2 = ty'e;, and this bivector H is transversal to the second 2
triple of subspaces. This defines the linear transformation A: T1 -+ TI which can be written in the form £' = A'tyj (see Figure 6.1.2, p. 228). Thus, as was proved in Subsection 6.1.5, the operator A is the cross-ratio of the quadruple of subspaces P1, P2, P3, and P4 which also can be considered as the cross-ratio of four subspaces T1,T2iT3, and T4. Let rl be the eigenvector of the operator A: T1 -4 T, corresponding to an eigenvalue A. Since it = Ail, the transversal bivectors H and k, defined by the eigenvector rl, belong to the common transversal subspace of the quadruple of subspaces T. We arrive at the following theorem:
Theorem 7.8.1 At each point x E M, the basis affinor A =
of a fourweb W (4, 2, q) is the cross-ratio of four subspaces T1, T2, T3 and T4 which are tangent to the leaves of the web passing through the point x: A = (T1, T2; 2'3, T4 ). To the eigenvectors of the operator A, there corresponds the common transversal subspace of the quadruple of subspaces T.
3. Now we return to the study of the general webs W (d, p, q). Each subsystem of foliations Ar, , ... Arp+, , where r = {a, o) is the combined index taking the d values, 1 , ... , p, p+ q + 1, ... , q + d, forms a (p+ 1)-subweb on the manifold M. We denote this subweb by ITI, ... , Ty+I ]. The total number In the tangent space TT(M) each of of such subwebs is (p+1) = e these subwebs determines (see Section 7.5) the Segre cone SCz(p, q) and consequently the almost Grassmann structure AG(p - 1, p + q - 1) in the manifold M. Thus a system of almost Grassmann structures arises in the manifold Al. However, the most interesting case is indicated in the following definition (cf. Section 7.5):
Definition 7.8.2 A web W (d, p, q) is said to be almost Grassmannizable if all almost Grassmann structures defined by its (p + 1)-subwebs coincide. Theorem 7.8.1, proved in Subsection 7.8.2, implies that the web W (4, 2, q) is almost Grassmannizable if and only if its basis affinor is scalar: A = AI. In fact in this case all transversal bivectors of the subweb [1, 2,31 are also
7.8
Grassmannizable d- Webs
317
transversal bivectors of the subweb [1, 2, 4], and consequently for all of its other three-subwebs, [1, 3,41 and 12,3,4]. But the transversal bivectors constitute one of the families of the plane generators of the Segre cones SC(2, q) associated with the web W. Therefore, if A = AI, then the Segre cones defined by different subwebs of the web W(4, 2, q) coincide. The converse is obvious. In the general case we have the following result:
Theorem 7.8.3 For a web W (d, p, q) to be almost Grassmannizable, it is necessary and sufficient that all its basis affinors A be scalar, that is, proportional as
to the identity afnor I =
Proof. The almost Grassmann structures, determined on the manifold M by two (p + 1)-subwebs of a web W (d, p, q), coincide if and only if at each point
x E M the Segre cones located in the tangent space T=(M) and determined by the tangent subspaces to the leaves of these subwebs coincide. Consider the subwebs [1, ... , p, p + 1] and [1, ... , p, a] on M. As shown in Section 7.5, the Segre cone determined in T=(M) by the first subweb can be given by the equations (7.7.1). In a similar way we can show that the Segre cone determined in T=(M) by the second subweb can be given by the equations
zo=?.(asafli).
(7.8.4)
A; = ao a dj
(7.8.5)
If ao
then equations (7.8.4) take the form
zi = 111X an
(7.8.6)
and determine the same Segre cone as equations (7.7.1). Conversely, if equations (7.8.4) define the same Segre cone as equations (7.7.1), the tensors ao
have the form (7.8.5); that is, they are proportional to the identity tensor. It follows from Theorem 7.8.3 that for an almost Grassmannizable web W (d, p, q), equations (7.8.3) take the form
-w`=\w'+...+ A all a,p-1P-I
+w' P
a
Since the foliations ar are in general position, in the matrix I
...
1
1
A
...
A
1
P+q+2,1
p+q+2,p-l
..........................
A
q+d,l
...
A
q+d,p-l
1
I
(7.8.7)
318
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
composed of the coefficients on the right-hand sides of equations (7.8.2) and (7.8.7), all the minors of any order are different from zero. Denote an almost Grassmannizable web W (d, p, q) by AGW (d, p, q) and consider the almost Grassmann structure AG(p - 1, p + q - 1) associated with this web. If this structure is a-semiintegrable, then the web AGW (d, p, q) is called transversally geodesic (cf. Section 7.7). If the almost Grassmann structure AG(p - I, p + q - 1) is 0-semiintegrable, then the web AGW (d, p, q) is called isoclinic. A web AGW (d, p, q) is called Grassmannizable if it is equivalent to a Grass-
mann web GW (d, p, q) formed on the Grassmannian G(p - 1, p + q - 1) by d foliations AE whose structure has been described in Section 7.6. It follows from Theorem 7.7.4 that a web AGW (d, p, q) is Grassmannizable if and only if it is both isoclinic and transversally geodesic. However, these conditions of Grassmannizability can be weakened, since the following theorem holds:
Theorem 7.8.4 If d > p + 2 and q > 3, an almost Grassmannizable web AGW(d, p, q) is isoclinic.
Proof. We write the system of Pfaffian forms defining the foliation Ap+q+2 on the web AGW (d, p, q) in the form (7.8.8)
p+q+2
pP
II
In equation (7.8.8) we omitted the index p + q + 2 in the coefficients
A
p+q+2,a
and assumed that A is not necessarily equal to one. By the Fobenius theorem, P
the condition of complete integrability of system (7.8.8) can be written in the form
d w
w J A O'. 3 p+q+2
p+q+2
(7.8.9)
By virtue of formulas (7.5.4), exterior differentiation of (7.8.8) leads to the exterior quadratic equations
-d w '_- w jAw'+EdAAw'+EAa""'{{,twiAWk. p+q+2 j a a a
p+q+2
a
ZOO. (f'
(7.8.10)
Q
In these equations the coefficients A are relative invariants. This implies that a
dA = AA9+E Afp . Q
Substituting these expansions into (7.8.10), we obtain
-d w
p+q+2
w
p+q+2
a,Q
Aw k. A dk+Aa'k aaQ )wj a Q
a82
Notes
319
From condition (7.8.9) it follows that the second term on the right-hand side of the last equation must have the form - w A o`, where a` _ E µ'k wk. P+q+2
>
>
a a3a
Equating these two expressions and applying (7.8.8), we get
(A jbk + A a'k)wj A wk = E a,0aj3 aaft a 0 ap ap
a
A wk. R
Comparing the alternated coefficients and applying relations (7.5.5), we arrive at the equations
(a -
Rkb - jbk' + 'jk - pUkj.
aa
(7.8.11)
Setting a: = A in (7.8.11), we find that ki
aQU6k) a
By virtue of these equations, the alternation of relations (7.8.11) with respect to the indices j and k gives
ljbkl,
(7.8.12)
where we used the notation 1
AQ k
a Ak
From relations (7.8.12) and Theorem 7.7.3 it follows that if q > 3, then the (p + 1)-subweb [1,. .. , p, p + 1) of the web AGW (d, p, q) is isoclinic. This immediately implies the isoclinicity of the web AGW (d, p, q). Theorems 7.7.4 and 7.8.4 give another result:
Theorem 7.8.5 If d > p + 2 and q > 3 and a web W (d, p, q) is almost Grassmannizable and transversally geodesic, then it is Grassmannizable.
NOTES 7.1. Almost Grassmann manifolds were introduced in Hangan [Han 66) as a generalization of the Grassmannian G(m, n). Hangan [Han 66, 681 and T. Ishihara [I 721 studied mostly some special almost Grassmann manifolds, especially locally Grassmann manifolds. A. B. Goncharov [Gon 871 considered the almost Grassmann manifolds as generalized conformal structures. R. J. Baston [Bas 91a1 constructed a theory of a general class of structures, called almost Hermitian symmetric (AHS) structures,
320
7. MANIFOLDS ENDOWED WITH ALMOST GRASSMANN STRUCTURES
which include conformal, projective, almost Grassmann, and quaternionic structures and for which the construction of the Cartan normal connection is possible. He constructed a tensor invariant for them and proved that its vanishing is equivalent to the structure being locally that of a Hermitian symmetric space. Subsequently Baston [Bas 91b) computed an algebra of differential invariants of the AHS structures. In Goncharov [Con 87] the AHS structures were studied from the point of view of cone structures (see Baston [Bas 91a, b] and Concharov [Gon 87] for further references on generalized conformal structures and their invariants). The local twistor theory of almost Grassmann structures was constructed in the recent paper by Bailey and Eastwood [BE 91] where the almost Grassmann structures were called paraconformal structures. In the another recent paper, Dhooghe [Dh 94] (see also Dhooghe (Dh 93]) considered the almost Grassmann structures (he called them Grassmannian structures) as subbundles of the second-order frame bundle and constructed a canonical normal connection for these structures. As we noted in Subsection 7.2.2, a pseudoconformal CO(2, 2)-structure is equivalent to an almost Grassmann structure AG(1,3). Since, as we saw in Chapter 5, fourdimensional conformal structures play an important role in general relativity, this provides a physical justification for studying the general almost Grassmann structures AG(m, n). In our exposition we defined the almost Grassmann structures geometrically following Akivis [A 80, 82a] (see also the paper Mikhailov (Mi 78] and the books Akivis and Shelekhov [AS 92], §8.3, and Goldberg [Go 88], §§2.1 and 2.2). 7.2-7.4. In Goldberg [Go 75a] (see also Goldberg (Go 881, §2.2, Eq. (2.2.38)) the expression (7.2.38) of the torsion tensor a'o k in a general (not specialized) frame was constructed for the first time. Using another method, Hangan [Han 80] deduced this expression again. The theorem similar to Theorem 7.3.1 was proved in Hangan [Han 80] in terms of Lie algebras. Mikhailov (Mi 72, 74, 77, 81) considered almost Grassmann structures and found their realizations in the frame of theory of two-webs.
Our structure equations (7.3.35) are very close to the structure equations in Dhooghe [Dh 94].
7.5. The basic equations of the theory of (p + 1)-webs W (p + 1, p, q) as well as the connection r were obtained in Goldberg [Go 73, 74a) (see also the book Goldberg (Go 88], Chapter 1). Theorem 7.5.6 can also be found in these papers and the book. 7.6. Grassmann webs GW (p + 1, p, q) for p = 2 were considered in Akivis [A 731, for p = 3 in Akivis and Goldberg [AG 74], and for any p in Goldberg [Go 75b]. For examples of Grassmann (and algebraic) webs GW (4, 2, q), see Goldberg [Go 82b]. 7.7. Transversally geodesic webs W (p + 1, p, q) were introduced in Goldberg [Go 73, 74a], and isoclinic webs W (p + 1, p, q) were introduced in Goldberg [Go 74b]. In connection with the theory of almost Grassmann structures, these webs were considered in Goldberg (Go 75a) and in Akivis [A 80, 82a]. The Grassmannizability problem was solved for webs W(3, 2, q) in Akivis [A 74] and for webs W (p + 1, p, q) (Theorem 7.7.5) in Akivis (A 80, 82a] and Goldberg [Go 82a].
7.8. The theory of webs W(4, 2, q) was constructed in Goldberg [Go 77, 80]. The geometric meaning of the basis affinor a' (Theorem 7.8.1) was also established there. A geometric definition of almost Grassmannizability for webs W (d, p, q) in the
case d > p + I was introduced in Akivis [A 83b]. But actually this kind of webs
Notes
321
was considered by Akivis (A 81) who gave the analytical characterization of these webs. Analytically a definition of almost Grassmannizable webs was given in Goldberg
[Go 84) (in this paper they were called scalar webs). Theorem 7.8.4 supplements Theorem 8.1.10 on almost Grassmann webs AGW (d, 2, q) from Goldberg [Go 881. It has appeared that the almost Grassmannizable webs are related to webs W (d, p, q) of maximum q-rank. S. S. Chern and P. A. Griffiths [CG 78] proved a geometric theorem which, in terms of almost Grassmannizable webs, can be formulated as follows: for d > 2p + 1 and p > 3, a web W (d, p, 2) of maximum 2-rank is almost Grassmannizable. J. B. Lit-
tle [Lit 89] extended this result to webs W (d, p, q). He proved that if q > 2 and d > q(p - 1) + 2, then every web W (d, p, q) of maximum q-rank is almost Grassmannizable. The last result (partially) gives an affirmative answer to a problem posed by V. V. Goldberg whether every web W (d, p, q) of maximum q-rank is almost Grassmannizable. It follows from Little's result mentioned above that a web W (d, 2, q) of maximum q-rank is almost Grassmannizable if q > 2 and d > q+2. Since the last two inequalities imply that d > 4, the case d = 4 should be studied separately. This case was considered earlier by V. V. Goldberg [Go 85) (see also Goldberg [Go 88), §8.3) who showed that if q > 2 and a web W(4, 2, q) admits at least one abelian equation, then the web is almost Grassmannizable. Recently V. V. Goldberg [Go 92) gave a description of almost Grassmannizable 6-webs AGW (6, 3, 2) of maximum 2-rank.
Bibliography [AdM 671
Adati, T., and T. Miyazawa, On a Riemannian space with recur. rent conformal curvature, Tensor 18 (1967), no. 3, 348-354. (MR' 35 #60936; Zbl 152, 391.)
[A 481
Akivis, M. A., Pairs of T-complexes, Dokl. Akad. Nauk SSSR 81 (1948), no. 1, 181-184 (Russian). (MR 10, 400; Zbl 38, 340.)
[A 501
Akivis, M. A., Pairs of T-complexes, Mat. Sb. (N.S.) 27 (69) (1950), no. 3, 365-378 (Russian). (MR 13, 152; Zb1 38, 340-341.)
[A 52a]
Akivis, M. A., Invariant construction of the geometry of a hypersurface of a conformal space, Dokl. Akad. Nauk SSSR 82 (1952), no. 3, 325328 (Russian). (MR 13, 777; ZbI 47, 152.)
[A 52b)
Akivis, M. A., Invariant construction of the geometry of a hypersurface
of a conformal space, Mat. Sb. (N.S.) 31 (73) (1952), no. 1, (Russian). (MR 14, 318; Zbl 48, 397.) [A 571
Akivis, M. A., Focal images of a surface of rank r, Izv. Vyssh. Uchebn. Zaved. Mat. 1957, no. 1, 9-19 (Russian). (MR 25 #498; Zbl 94, 186.)
[A 61a]
Akivis, M. A., On the conformal differential geometry of multidimensional surfaces, Mat. Sb. (N.S.) 53 (95) (1958), no. 4, 53-72 (Russian). (MR 23 #A2813; Zbl 103, 148.)
[A 61b]
Akivis, M. A., On multidimensional surfaces carrying a net of conjugate lines, Dokl. Akad. Nauk SSSR 139 (1961), no. 6, 1279-1282 (Russian); English transl. in Soviet Math. Dokl. 2 (1961), no. 4, 1065-1068. (MR 24 #A2908; Zbl 134, 169.)
[A 63a]
Akivis, M. A., On the structure of multidimensional surfaces carrying a net of curvature lines, Dokl. Akad. Nauk SSSR 149 (1963), no. 6, 1247-1249 (Russian); English transl. in Soviet Math. DokI. 4 (1963), no. 2, 529 -531. (MR 27 #668; Zbl 129, 140.)
'In the bibliography we will use the following abbreviations for the review journals:.lbuch for Jahrbuch fur die Fortschritte der Mathematik, MR for Mathematical Reviews, and Zbl for Zentratblatt fur Mathematik and ihren Grenzgebiete. 323
BIBLIOGRAPHY
324 [A 63b]
Akivis, M. A., On the structure of surfaces carrying a net of conjugate
lines, Moskov. Gos. Ped. Inst. Uchen. Zap. No. 208 (1963), 31-47 (Russian). [A 64]
Akivis, M. A., Conformal differential geometry, Geometry 1963, pp. 108-137. Akad. Nauk SSSR Inst. Nauchn. Informatsii, Moscow, 1965 (Russian). (MR 33 #3217.)
[A 65]
Akivis, M. A., On an invariant differential-geometric characterization of the Dupin cyclide, Uspekhi Mat. Nauk 20 (1965), no. 1, 177-180 (Russian). (MR 30 #4212; Zbl 137, 411.)
[A 69]
Akivis, M. A., Three-webs of multidimensional surfaces, Trudy Geom. Sem. Inst. Nauchn. Inform., Akad. Nauk SSSR 2 (1969), 7-31 (Russian). (MR 40 #7967; Zbl 244:53014.)
[A 73]
Akivis, M. A., The local differentiable quasigroups and three-webs that are determined by a triple of hypersurfaces, Sibirsk. Mat. Zh. 14 (1973),
no. 3, 467-474 (Russian); English transl. in Siberian Math. J. 14 (1973), no. 3, 319-324. (MR 48 #2911; Zbl 267:53005 & 281:53002.) [A 74]
Akivis, M. A., Isoclinic three-webs and their interpretation in a ruled space of projective connection, Sibirsk. Mat. Zh. 15 (1974), no. 1, 3-15 (Russian); English transl. in Siberian Math. J. 15 (1974), no. 1, 1-9. (MR 50 #3129; Zbl 288:53021 & 289:53020.)
[A 80]
Akivis, M. A., Webs and almost-Grassmann structures, Dokl. Akad. Nauk SSSR 252 (1980), no. 2, 267-270 (Russian); English transl. in Soviet Math. Dokl. 21 (1980), no. 3, 707-709. (MR 82a:53016; Zbl 479:53015.)
[A 81]
Akivis, M. A., A geometric condition of isoclinity of a multidimensional web, Webs and Quasigroups, Kalinin. Gos. Univ., Kalinin, 1981, 3-7 (Russian). (MR 83e:53010; Zbl 497:53026.)
[A 82a]
Akivis, M. A., Webs and almost-Grassmann structures, Sibirsk. Mat.
Zh. 23 (1982), no. 6, 6-15 (Russian); English transl. in Siberian Math. J. 23 (1982), no. 6, 763-770. (MR 84b:53018; Zbl 505:53004 & 516:53013.) [A 82b]
Akivis, M. A., On the differential geometry of a Grassmann manifold, Tensor (N.S.) 38 (1982), 273-282 (Russian). (MR 87e:53021; Zbl 504:53010.)
[A 83a]
Akivis, M. A., Completely isotropic submanifolds of a four-dimensional pseudoconformal structure, Izv. Vyssh. Uchebn. Zaved. Mat. 1983, no. 1 (248), 3-11 (Russian); English transi. in Soviet Math. (Iz. VUZ) 27 (1983), no. 1, 1-11. (MR 841:53016; Zbl 512:53056 & 526:53054.)
[A 83b]
Akivis, M. A., The local algebrnizability condition for a system of sub-
manifolds of a real projective space, Dokl. Akad. Nauk SSSR 272
BIBLIOGRAPHY
325
(1983), no. 6, 1289-1291 (Russian); English transi. in Soviet Math. Dokl. 28 (1983), no. 2, 507-509. (MR 85c:53018; ZbI 547:53006.) [A 851
Akivis, M. A., On the theory of conformal structures, Geom. Sb. Vyp. 26, 44-52, Tomsk. Univ., Tomsk, 1985 (Russian).
[A 87]
Akivis, M. A., On multidimensional strongly parabolic surfaces, Izv. Vyssh. Uchebn. Zaved. Mat. 1987, no. 5 (311), 3-10 (Russian); English transl. in Soviet Math. (Iz. VUZ) 31 (1987), no. 5, 1-11. (MR 89g:53016; Zbl 632:53012.)
[A 961
Akivis, M. A., On the real theory of four-dimensional conformal structures, J. Geom. Phys. 384 (1996), 1-28.
[AC 811
Akivis, M. A., and B. P. Chebysheva, Invariant framing of a semiRiemannian manifold, Sibirsk. Mat. Zh. 22 (1981), no. 6, 7-14 (Russian); English transl. in Siberian Math. J. 22 (1981), no. 6, 809-815. (MR 83d:53018; Zbl 491:53015.)
(AG 74]
Akivis, M. A., and V. V. Goldberg, The four-web and the local differentiable ternary quasigroup that are determined by a quadruple of surfaces of codimension two, Izv. Vyssh. Uchebn. Zaved. Mat. 1974, no. 5 (144), 12-24 (Russian); English transi. in Soviet Math. (Iz. VUZ) 18 (1974), no. 5, 9-19. (MR 50 #8321; ZbI 297:53037.)
[AG 93]
Akivis, M. A., and V. V. Goldberg, Projective differential geometry of submanifolds, North-Holland, Amsterdam, 1993, xi+364 pp. (MR 941:53001.)
[AK 93]
Akivis, M. A., and V. V. Konnov, Local aspects in conformal structure theory, Uspekhi Mat. Nauk 48 (1993), no. 1, 3-40 (Russian); English transl. in Russian Math. Surveys 48 (1993), no. 1, 1-35. (MR 94g:53009; Zbl 804:53022.)
[AS 92]
Akivis, M. A., and A. M. Shelekhov, Geometry and algebra of multidimensional three-webs, Kluwer Academic Publishers, Dordrecht, 1992, xvii+358 pp. (MR 93k:53021; Zbl 771:53001.)
(AZ 95]
Akivis, M. A., and B. V. Zayatuev, Geometry of isotropic bundles on a four-dimensional pseudoconformal structure CO(1,3), Webs and Quasigroups, Tver Gos. Univ., Tver, 1995, 44-61.
[AGr 92a]
Alekseevskii, D. V., and M. I. Graev, Twistors and G-structures, Izv. Ross. Akad. Nauk. Ser. Mat. 56 (1992), no. 1, 3-37 (Russian); English transl. in Izv. Ross. Acad. Sci. Izv. Math. 40 (1993), no. 1, 1-31. (MR 93e:53036; Zbl 764:53021.)
[AGr 92b]
Alekseevskii, D. V., and M. I. Graev, Twistors of a Riemannian manifold and CR-structures, Izv. Vyssh. Uchebn. Zaved. Mat. 1992, no. 5 (360), 3-19 (Russian); English transl. in Russian Math. (Iz. VUZ) 36 (1992), no. 5, 1-16. (MR 94b:32052; Zbl 779:53042.)
BIBLIOGRAPHY
326
(AGr 931
Alekseevskii, D. V., and M. I. Graev, G-structures of twistor types and their twistor spaces, J. Geom. Phys. 10 (1993), no. 3, 203-229. (MR 94e:53026; Zbl 779:53020.)
[AVS 881
Alekseevskii, D. V., E. B. Vinberg, and A. S. Solodovnikov, Geometry of spaces of constant curvature, pp. 5-146. In: Itogi Nauki i Tekhniki, Sovremennye Problemy Matem., Fundam. Napravleniya, vol. 29, Ge-
ometriya 2, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tehn. Inform., Moscow, 1988, 264 pp. (Zbl 699:53001); English transl. in Encyclopaedia of Math. Sci., vol. 29: Geometry II. Spaces of constant curvature, ed. E.B. Vinberg, pp. 1-138, Springer-Verlag, Berlin, 1993, viii+254 pp. (MR 95b:53042; Zbl 787:53001.) [AHS 78]
Atiyah, M. F., N. L. Hitchin, and I. Singer, Self-duality in four. dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425-461. (MR 80d:53023; Zbl 389:53011.)
[Ba 501
Backes, F., Sur une figure de rdfdrence mobile conatitude par cinq spheres non ndcessarement orthogonales, C. R. Acad. Sci. Paris 230 (1950), 1569-1571. (MR 12, 130; Zbl 41, 92.)
[Ba 51a)
Backes, F., La mdthode du pentasph2re oblique mobile et sea applications, Colloque de Gdometrie Difftrentielle, Louvain, 1951, 183-190, George Thone, Liege & Masson, Paris, 1951. (MR 13, 686; Zbl 44, 181.)
[Ba 51b)
Backes, F., La methode du pentasphdre oblique mobile et quelques-unes de sea applications, Acad. Roy. BeIg. Cl. Sci. Mem. Collect. in 8°. (2) 26, no. 1613, 87 pp. (MR 13, 686; Zbl 45, 428.)
[Ba 56]
Backes, F., Sur lea spheres d deux parametres dont lea points caractdristiques sont repartis sur en cercle, Acad. Roy. Belg. Bull. Cl. Sci. (5) 42 (1956), no. 2, 153-162. (MR 17, 884; Zbl 75, 311.)
[Ba 61]
Backes, F., Sur les congruences R en gdomdtrie anallagmatique, Acad. Roy. Belg. Bull. Cl. Sci. (5) 47 (1961), no. 5, 318-327. (Zbl 104, 163.)
[BE 91)
Bailey, T. N., and M. G. Eastwood, Complex paraconformal manifolds: their differential geometry and twistor theory, Forum Math. 3 (1991), no. 1, 61-103. (MR 92a:32038; Zbl 728:53005.)
[Bar 61]
Barner, M., Zur Mobius-Geometrie: Die Inversionsgeometrie ebener Kurven, J. Reine Angew. Math. 208 (1961), 192-220. (MR 26 #5459; Zbl 104, 162.)
[BGPPR 94)
Barrett, J. W., G. W. Gibbons, M. J. Perry, C. N. Pope, and P. Ruback, Kleinian geometry and the N = 2 superstring, Internat. J. Modern Phys. A 9 (1994), no. 9, 1457-1493. (MR 95a:81198.)
[Bas 91a)
Baston, R. J., Almost Hermitian symmetric manifolds. I. Local turistor theory, Duke Math. J. 63 (1991), no. 1, 81-112. (MR 93d:53064; Zbl 724:53019.)
327
BIBLIOGRAPHY
[Bas 91b)
Baston, R. J., Almost Hermitian symmetric manifolds. II. Differential invariants, Duke Math. J. 63 (1991), no. 1, 113-138. (MR 93d:53065; Zbl 724:53020.)
[Berw 27)
Berwald, L., Konforme Differentialgeometrie, Enzyklopadie der Mathematischen Wissenschaften, Bd. III, 3. Teil, Kap. 11, 1927, 118-120. (Jbuch 52, 680.)
[Bla 82]
Blair, D. E., On conformal images of flat submanifolds, Geom. Dedicata 12 (1982), no. 2, 205-208. (MR 83i:53074; Zbl 506:53004.)
(BI 21)
Blaschke,
W.,
Vorlesungen iiber Differentralgeometrie
and ge.
ometrische Grundlagen von Einstein Relativitatstheorie, Band 1, Springer-Verlag, Berlin, 1921, xii+230 pp. (Jbuch 48, 1305-1306); 2d ed., 1924, xii+242 pp. (Jbuch 50, 452-453); republished by Dover, New York, 1945, xiv + 322 pp. (MR 7, 391; 3d ed., 1930, x+311 pp. (Jbuch 58, 588); Zbl 63.I, A85); 4th ed., 1945, x+311 pp. [BI 25]
Blaschke, W., Uber konforme Geometric. III. Kreisgeometrie rechtwin-
kliger Kurvennetze auf der Kugel, Abh. Math. Sem. Univ. Hamburg 4 (1925), 148-163. (Jbuch 51, 586.) [BI 29]
W., Vorlesungen iiber Differentialgeometrie and geometrische Grundlagen von Einsteins Relativitiitstheorie, vol. 3; Differentialgeometrie der Kreise and Kugeln, Springer-Verlag, Berlin, 1929, x+474 pp. (Jbuch 55, 422-427.)
[BI 55]
Blaschke, W., Einfiihrung in die Geometric der Waben, Birkhii.userVerlag, Basel-Stuttgart, 1955, 108 pp. (MR 17, 780; Zbl 68, 365); Russian transl., GITTL, Moskva, 1959, 144 pp. (MR 22 #2942.)
[Bo 35)
Bol, G., Uber 3-Gewebe in vierdimensionalen Raum, Math. Ann. 110 (1935), 431--463. (Zbl 10, 222.)
[Bo 50]
Bol, G., Projektive Differentialgeometrie, Vandenhoeck & Ruprecht, G6ttingen, vol. 1, 1950, vii+365 pp. (MR 11, 539; Zbl 35, 234); vol. 2, 1954, v+372 pp. (MR 16, 1150; Zbl 59, 155); vol. 3, 1967, viii+527 pp. (MR 37 #840; Zbl 173, 233.)
[Bom 12]
Bompiani, E., Sull'equazione di Laplace, Rend. Circ. Mat. Palermo 34 (1912), 383-407. (Jbuch 43, 687-688.)
[Br 88]
Bryant, R. L., Surfaces in conformal geometry, The mathematical
Blaschke,
heritage of Hermann Weyl (Durham, NC, 1987), 227-240, Proc. Sym-
pos. Pure Math., 48, Amer. Math. Soc., Providence, RI, 1988. (MR 89m:53090; Zbl 645:53010.) [BCGGG 91]
Bryant, R. L., S. S. Chern, R. B. Gardner, H. L. Goldsmith, and P. A. Griffiths, Exterior differential systems, Springer-Verlag, New York, 1991, vii+475 pp. (MR 92h:58007; Zbl 726:58002.)
BIBLIOGRAPHY
328
[Bub 90]
Bubyakin, I. V., On some properties of five-dimensional complexes of two-dimensional planes in the projective space P5, Differentsial'naya
Geom. Mnogoobraz. Figur No. 21 (1990), 12-16 (Russian). (Zbl 823:53014.) [Bub 91]
Bubyakin, 1. V., On the geometry of five-dimensional complexes of two-dimensional planes in a projective space P5, Functional Anal.
i Prilozhen. 25 (1991), no. 3, 73-76 (Russian); English transl. in Functional Analysis and Its Appl. 25 (1991), no. 3, 223-224. (MR 92i:53014; Zbl 736:53005.) [Bur 121
Burali-Forti, C., Fondamenti per la geometria differenziale su di una superficie cot metodo vettorioale generate, Rend. Circ. Mat. Palermo 33 (1912), 1-40. (Jbuch 43, 680.)
[BN 701
Bushmanova, C. V., and A. P. Norden, A polar normalization of surfaces and a congruence of circles in a conformal plane, Kazan. Cos. Univ. Uchen. Zap. 129 (1970), kn. 6, 22-32 (Russian). (MR 44 #4666; Zbl 224:53014.)
[CD 87]
do Carmo, M., and M. Dajczer, Conformal rigidity, Amer. J. Math. 109 (1987), no. 5, 963-985. (MR 89e:53016; Zbl 631:53043.)
[CDM 85)
do Carmo, M., M. Dajczer, and F. Mercuri, Compact conformally flat hypersurfaces, Trans. Amer. Math. Soc. 288 (1985), no. 1, 189-203. (MR 86b:53052; Zbl 537:53050 & 554:53040.)
[Ca 08]
Cartan, E., Les sous-groupes des grouper continus de transformations, Ann. Sci. Ecole Norm. (3) 25 (1908), 57-194; (Jbuch 39, 206-207); cRuvres completes: Partie II, Algebre. Formes dtfferentielles, systemes differentielles, vols. 1-2, Gauthier-Villars, Paris, 1953, pp. 719-856. (MR 15, 383; Zbl 15, 83.)
[Ca 17]
Cartan, E., La deformation des hypersurfaces dans l'espace conforme reel a n > 5 dimensions, Bull. Soc. Math. France 45 (1917), 57-121 (Jbuch 46, 1129); tBuvres completes: Partie III, Divers, geometrie, differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 221-285. (MR 17, 697; Zb1 15, 83.)
[Ca 19]
Cartan, E., Sur les varietes de courbure constante dun espace euclidien ou non-euclidien, Bull. Soc. Math. France 47 (1919), 125-160 (Jbuch 47, 692-693); (uvres completes: Partie III, Divers, geometrie, differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 321-359. (MR 17, 697; Zbl 15, 83.)
[Ca 20a]
Cartan, It., Sur les varietes de courbure constante d'un espace euclidien ou non-euclidien, Bull. Soc. Math. France 48 (1920), 132-208 (Jbuch 47, 692-693); Wuvres completes: Partie III, Divers, geometrie,
differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 360-432. (MR 17, 697; Zbl 15, 83.)
BIBLIOGRAPHY
[Ca 20b]
329
Cartan, E., Sur to deformation projective des surfaces, Ann. Sci. Ecole Norm. Sup. 37 (1920), 259-356 (Jbuch 47, 656-657); Buvres completes: Partie III, Divers, geometrie, differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 441-538. (MR 17, 697; ZbI 15, 83.)
[Ca 20c]
Cartan, It., Sur le probleme general de la deformation, C. R. Congrbs Internat. Math. Strasbourg 1920, 397-406 (Jbuch 48, 817); Buvres completes: Partie III, Divers, geometrie, differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 539-548. (MR 17, 697; Zbl 15, 83.)
[Ca 22a]
Cartan, It., Sur les equations de la gravitation d' Einstein, J. Math. Pures Appl. 1 (1922), 141-203 (Jbuch 48, 993); Separate: GauthierVillars, Paris, 65 pp.; (Buvres completes: Partie 111, Divers, geometrie,
differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 549-611. (MR 17, 697; Zbl 15, 83.) [Ca 22b]
Cartan, E., Sur les espaces conformes generalises et I'Univers optique, C. R. Acad. Sci. Paris 174, 857-859; Buvres completes: Partie 111,
Divers, geometrie, differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 622-624 (MR 17, 697; Zbl 15, 83); English transi. in On generalized conformal spaces and the optical Universe in On manifolds with an affine connection and the theory of general relativity, Bibliopolis, Naples, 1986. (MR 88b:01071; Zbl 657:53001.) (Ca 231
Cartan, It., Les espaces a connexion conforme, Ann. Soc. Polon. Math. 2 (1923), 171-221 (Jbuch 50, 493); Buvres completes: Partie 111,
Divers, geometrie, differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 747. 797. (MR 17, 697; ZbI 15, 83.) [Ca 31]
Cartan, E., Sur les developpantes dune surface reglee, Bull Sect. Sci. Acad. Roumain. 14 (1931), 167-174 (Zbl 3, 130); Buvres completes: Partie III, Divers, geometree, differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1187-1194. (MR 17, 697; Zbl 15, 83.)
[Ca 37a]
Cartan, It., La theorie de groupes fins et continus ei la geometrie differentielle traitees par la methode di repere mobile, Gauthier-Villars, Paris, 1937, vi+269 pp. (Zbl 18, 298); 2d ed., 1951 (Zbl 54, 14-16.).
(Ca 37b)
Cartan, It., L'extension du calcul tensorial aux geometries non affines, Ann. of Math. (2) 38 (1937), 1-13 (Zbl 15, 416); Buvres completes: Partie III, Divers, geometrie, differentielle, vols. 1-2, Gauthier-Villars, Paris, 1955, pp. 1411-1423. (MR 17, 697; Zbl 15, 83.)
(CaH 67]
Cartan, H., Formes diffenntielles. Application elementaires au calcul des variations at d la theorie des courbes et surfaces, Hermann, Paris, 1967, 186 pp. (MR 37 #6358; ZbI 184, 127); English transl. in Differential forms, Houghton Mifflin, Boston, 1970, 166 pp. (MR 42 #2379; Zbl 213, 370.)
[Cay 59)
Cayley, A., A sixth memoir on quantres, Phil. Trans. of the Royal Soc. London 149 (1859), no. 1, 61-90; see also Cayley, A., Collected Mathematical papers, Cambridge University Press, Cambridge, 1889, vol. 2, p. 561. (Jbuch 20, 24-25.)
330 (Ce 891
BIBLIOGRAPHY
Cecil, T. E., Reducible Dupin submanifolds, Geom. Dedicata 32 (1989), no. 3, 281-300. (MR 91g:53062; Zbl 697:53056.)
(Ce 91)
Cecil, T. E., Lie sphere geometry and Dupin submanifolds, Geometry and topology of submanifolds, III (Leeds, 1990), 90-107, eds. L. Verstraelen and A. West, World Sci. Publishing, River Edge, NJ, 1991. (Zbl 773:53003.)
(Ce 92]
Cecil, T. E., Lie sphere geometry with applications to submanifolds, Springer-Verlag, New York, 1992, xii+207 pp. (MR 94m:53076; Zbl 752:53003.)
[CC 891
Cecil, T. E., and S. S. Chern, Dupin submanifolds in Lie sphere geometry, Differential geometry and topology (Tianjin, 1986-87), 1-48, Lecture Notes in Math 1369, Springer, Berlin, 1989. (MR 901:53079; Zbl 678:53003.)
[CR 78)
Cecil, T. E., and P. Ryan, Focal sets, taut embeddings and the cyclides of Dupin, Math. Ann. 236 (1978), no. 2, 177-190. (MR 80a:53003; Zbl 365:53004 & 379:53002.)
(CR 80)
Cecil, T. E., and P. Ryan, Conformal geometry and cyclides of Dupin,
Canad. J. Math. 32 (1980), no. 4, 767-782. (MR 82f:53004; Zbl 483:53050.) [CR 85]
Cecil, T. E., and P. Ryan, Tight and taut immersions of submanifolds, Research Notes in Mathematics, 107. Pitman (Advanced Publishing Program), Boston, 1985, vi+366 pp. (MR 87b:53089; Zbl 596:53002.)
(Cha 831
Chandrasekhar, S., The mathematical theory of black holes, Clarendon Press, Oxford & Oxford University Press, New York, 1983, xxi+646 pp. (MR 85c:83002, Zbl 511:53076.)
[Ch 73a]
Chen, B. Y., Geometry of submanifolds, Marcel Dekker, New York, 1973, vii+278 pp. (MR 50 #5697; Zb1 262:53026.)
[Ch 73b]
Chen, B. Y., An invariant of conformal mappings, Proc. Amer. Math. Soc. 40 (1973), 563-564. (MR 47 #9489; Zbl 266:53020.)
[Ch 74]
Chen, B. Y., Some conformal invariants of submanifolds and their applications, Boll. Un. Mat. Ital. (4) 10 (1974), 380-385. (MR 51 #6663; Zbl 321:53042.)
[CY 73)
Chen, B. Y., and K. Yano, Special conformally flat spaces and canal
hypersurfaces, Ti hoku Math. J. (2) 25 (1973), 177-184. (MR 48 #12351; Zbl 266:53043.) [C 86]
Chern, S. S., On a conformal invariant of three-dimensional manifolds, Aspects of mathematics and its applications, 245-252, North-Holland
Mathematical Library, 34, North-Holland, Amsterdam, 1986. (MR 87h:53051; Zbl 589:53011.)
BIBLIOGRAPHY [C 911
331
Chern, S. S., An introduction to Dupin submanifolds, Differential geometry, A symposium in honour of M. do Carmo, Proc. Int. Conferences, Rio cle Janeiro/Brasil, 1988, 95-102, Pitman Monograph Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, 1991. (MR 93g:53084; Zbl 721:53052.)
(CG 78]
Chern, S. S., and P. A. Griffiths, An inequality for the rank of a web and webs of maximum rank, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 78-83. (MR 80b:53009; Zbl 402:57001.)
[Dar 17)
Darboux, G., Principles de gdomdtrie analytique, Paris, 1917, vi+520 pp. (Jbuch 46, 877.)
[Dar 73]
Darboux, C., Sur une classe remarquable de courbes et des surfaces algdbriques et sur la thdorie des imaginaires, Mem. de Bordeaux (1873), VIII, pp. 292-350; IX, pp. 1-280. (Jbuch 5, 323, 371, 399.)
(Del 27]
Delens, P. C., Mdthodes et problemes de gdomdtrie diffdrentielles euclidiennes et conforme, These, Gauthier-Villars, Paris, 1927, x+184 pp. (Jbuch 53, 659.)
[Demo 05]
Demoulin, A., Principes de geometric anallagmatique ei de gdomdtrie
rdglde intrinseque, C. R. Acad. Sci. Paris 140 (1905), 1526-1529. (Jbuch 36, 682.) [Demo 19]
Demoulin, A., Sur lea congruences de spheres cycliques et sur lea systemes triples orthogonaux d lignes de courbure planes ou spheriques dons un systeme, Bull. Cl. Sci. (Bruxelles) 1919, 339-359. (Jbuch 47, 667.)
[Demo 21)
Demoulin, A., Recherches sur les systemes triples orthogonaux, Mem. Soc. Roy. Sci. Liege in 8° (3) 11 (1921), 98 pp. (Jbuch 48, 813-814.)
[Demo 26]
Demoulin, A., Sur la gdomdtrie conforme et des systemes triples orthogonaux, C. R. Acad. Sci. Paris 182 (1922), 1008-1010. (Jbuch 52, 764.)
[De 891
Deszcz, R., Notes on totally umbilical submanifolds, Geometry and topology of submanifolds (Marceille, 1987), 89-97, World. Sci. Publishing, Teaneck, NJ, 1989. (MR 92f:53060; Zbl 735:53042.)
(De 90)
Deszcz, R., On conformally flat Riemannian manifolds satisfying certain curvature conditions, Tensor (N.S.) 49 (1990), no. 2, 134-145. (MR 92b:53048; Zbl 742:53006.)
[Dh 93]
Dhooghe, P. F., Grassmannianlike manifolds. Geometry and topology of submanifolds, V (Leuven/Brussels, 1992), 147-160, World Sci. Publishing, River Edge, NJ, 1993. (MR 96e:53033.)
[Dh 941
Dhooghe, P. F., Grassmannian structures on manifolds. Bull. Belg. Math. Soc. Simon Stevin 1 (1994), no. 1, 597- 622. (MR 95m:53034.)
BIBLIOGRAPHY
332
[D 64)
J., Alghbre lineaire et geometrie elementaire, Hermann, Paris, 1964, 223 pp. (MR 30 #2015; Zbl 185, 488); English transi., Linear algebra and geometry, Houghton Mifflin, Boston, 1969, 207 pp.
(MR 42 #6004; Zbl 185, 488.) [Dob 88]
Dobromyslov, V. A., On the geometry of the k-quasiaffine space, Webs and Quasigroups, Kalinin. Gos. Univ., Kalinin, 1988, 147-155. (MR 89h:53046; Zbl 477:53008.)
[DFN 921
Dubrovin, B. A., A. T. Fomenko, and S. P. Novikov, Modern geometry-methods and applications, part 1, 2d ed. Springer-Verlag, New York, 1992, xvi+468 pp. (MR 92h:53001; Zbl 751:53001.)
[Du 22]
Dupin, C., Applications de geometrie et de mecanique, Paris, 1822.
[Ein 05)
Einstein, A., Zur Elektrodynamik bewegter Korper, Ann. Physik (4) 17 (1905), 891-921 (Jbuch 36, 920-921); see also H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The principle of relativity, Dover, New York, 1923, pp. 35-65.
[Ein 16)
Einstein, A., Die Grundlage der allgemeine Relativitntstheorie, Ann. Physik (4) 49 (1916), 769-822; reprinted by J. A. Barth, Leipzig, 1916, 64 pp.; English transl., The principle of relativity, Univ. of Calcutta, Calcutta, 1920, xxiii+186 pp.
[Ei 23]
Eisenhart, L. P., Transformation of surfaces, Princeton Univ. Press, Princeton, 1923, ix+379 pp. (Jbuch 49, 501-503.)
[Ei 26]
Eisenhart, L. P., Riemannian geometry, Princeton Univ. Press, Princeton, NJ, 1926, vii+262 pp. (Jbuch 52, 721); 2d printing, 1949, vii+306 pp. (MR 11, 687; Zbl 41, 294); 6th printing, 1966, vii+306 pp. (Zbl 174, 533.)
[Eu 11)
Euler, L., Opera omnia (1). Opera mathematica, t. 1-29, LeipzigBerlin-Zurich, 1911-1956. (MR 15, 89; 770; 16, 1; 17, 2; 18, 709; 19, 826; 20 #3769, 3770, 6970.)
[Eu 69]
[Eu 77a]
Euler, L., Considerationes de trajectoriis orthogonalibus, Novi commentarii academiae scientiarum Petropolitanae, vol. 14 (1769), part 1, Petersburg, 1770, pp. 104-128; see also [Eu 11], Ser. I, vol. 28 (1955), pp. 99-119. Euler, L., De repruesentatione superficiei sphaericae super piano, Acta
academiae scientiarum Petropolitanae, vol. 1 (1777), part 1, Petersburg, 1778, pp. 107-132; see also [Eu 11], Ser. I, vol. 28 (1955), pp. 248-275. [Eu 77b]
Euler, L., De projectione geographica superficiei sphaericae, Acta academiae scientiarum Petropolitanae, vol. 1 (1777), part 1, Petersburg, 1778, pp. 133-142; see also [Eu 11], Ser. I, vol. 28 (1955), pp. 276-287.
BIBLIOGRAPHY
333
[Fia 42]
Fialkow, A., The conformal theory of curves, Bull. Amer. Math. Soc. 51 (1942), 435-501. (MR 3, 307; Zb1 62, A275.)
[Fia 451
Fialkow, A., Conformal classes of surfaces, Amer. J. Math. 87 (1945), 583-616. (MR 7, 175; Zb1 63, A275.)
[Fin 50]
Finikov, S. P., Theory of congruences, Gostekhizdat, Moscow, 1950, 528 pp. (Russian). (MR 12, 744); German transl. by G. Bol, Akademie Verlag, Berlin, 1959, xvi+491 pp. (MR 21 #5212; Zbl 65, 367.)
[Fin 56]
Finikov, S. P., Theory of pairs of congruences, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956, 443 pp. (Russian). (MR 19, 676; Zbl 72, 168); French transl. by M. Decuyper, U.E.R. Mathematiques Pures et Appliquees, No. 68, Universit des Sciences at Techniques de Lille 1, Villeneuve d'Ascq, n°68, 2 vols, 1976, xxix+616 pp. (MR 55 #4023a & 4023b; Zbl 342:53010.)
[Fi 02]
Finzi, A., L e ipersuperficie a tre dimensioni the si possono rappresent are conformente sullo spazio euclideo, Atti R. Istit. Veneto (VIII) 5 (= 62) (1902), 1049-1062. (Jbuch 34, 668.)
(Fi 21]
Finzi, A., Sulla representabilitd conforme di due varset`a ad n dimensioni Tuna ultra, Atti R. Istit. Veneto 80'I (1921), 777-789.
(Fi 221
Finzi, A., Sulle varietd in rappresentazione conforme con la variet'a euclidea a pid di tre dimensioni, Rend. d. Linc. (V) 31' (1922), 8-12. (Jbuch 48, 854.)
(Fi 231
Finzi, A., Sulla curvature confomme di una variet'a, Rend. d. Linc. (V) 32' (1923), 215-218. (Jbuch 49, 547.)
[Fu 09]
Fubini, G., Suite rappresentazione the conservano le spersfere, Ann. di Matem. (III) 16 (1909), 141-160. (Jbuch 40, 718.)
(Fu 16]
Fubini, G., Applicabititd proiettiva di due superficie, Rend. Circ. Mat. Palermo 41 (1916), 135-162. (Jbuch 46, 1098-1099.)
[Fu 18a]
Fubini, G., Studi relativi all' elemento lineare proiettivo di una ipersuperficie, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (5) 27 (1918), 99-106. (Jbuch 46, 1095.)
(Fu 18b)
Fubini, G., It problema delta deformazione proiettivo delle ipersuperficie. Le varieta a un qualsiasi numero di dimensioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (5) 27 (1918), 147-155. (Jbuch 46, 1095-1097.)
[Fu 20]
Fubini, G., Sur les surfaces projectivement applicables, C. R. Acad. Sc. 171 (1920), 27-29. (Jbuch 47, 656.)
(PC 26]
Fubini, G., and E. Cech, Geometria proiettiva dsferenziale, Zanichelli, Bologna, vol. 1, 1926, 394 pp., vol. 2, 1927, 400 pp. (Jbuch 52, 751752.)
BIBLIOGRAPHY
334 [Fuhr 551
Fuhrman, A., Klasse ahnlicher Matrizen als verallgemeinerte Doppelverh6ltnisse, Math. Z. 62 (1955), 211-240. (MR 17, 1122; Zbl 68, 340.)
[Ga 531
Gantmacher, F. R., Theory of matrices, Gosud. Izdat. Tehn.-Teor. Lit., Moscow, 1953, 491 pp. (Russian). (MR 16, 438; Zbl 50, 248249); 2d ed., 1966, 576 pp. (MR 34 #2585; Zbl 145, 36-37); English transi. of 1st ed., Chelsea Publishing, New York, 1959, vol. 1, x+374 pp.; vol. 2, ix+286 pp. (MR 21 #6372c.)
(Gar 891
Gardner, R., The method of equivalence and its applications, CBMSNSF Regional conference Series in Applied Mathematics, 58, SIAM, Philadelphia, PA, 1989, vii+127 pp. (MR 91j:58007; Zbl 694:53027.)
(Ge 491
Geidelman, R. M., On congruences of circles which is decomposed into canal surfaces, Dokl. Akad. Nauk SSSR 66 (1949), 145-147 (Russian). (MR 13, 491; Zbl 40, 90.)
(Ge 50a)
Geidelman, R. M., On congruences of circles having a single family of canal surfaces, Dokl. Akad. Nauk SSSR 70 (1950), 369-372 (Russian). (MR 11, 540; Zbl 41, 294.)
[Ge 50b1
Geidelman, R. M., The conformal deformation of congruences of circles having two families of canal surfaces, Dokl. Akad. Nauk SSSR 72 (1950), 829-832 (Russian). (MR 12, 532; Zbl 41, 295.)
[Ge 571
Geidelman, R. M., A metric characterization of congruences of circles with families of canal-surfaces, Uspekhi Mat. Nauk 12 (1957), no. 4, 281-284 (Russian). (MR 19, 676; Zbl 81, 158.)
[Ge 60)
Geidelman, R. M., Conformal theory of two-parameter families of spheres, Dokl. Akad. Nauk SSSR 134 (1960), no. 4, 753-756 (Russian); English transl. in Soviet Math. Dokl. 1 (1960), 1130-1132. (MR 23 #A2812; Zbl 100, 178.)
[Ge 67a)
Geidelman, R. M., Differential geometry of families of subspaces in multidimensional homogeneous spaces, Itogi Nauki; Algebra, Topology, Geometry 1965, pp. 323-374. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tehn. Inform., Moscow, 1967 (Russian). (MR 35 #7224; Zbl 189, 2 & 198, 539.)
[Ge 67b)
Geidelman, R. M., The fundamentals of the conformal theory of families of spheres, An. 5ti. Univ. "Al. I. Cuza" Ia§i Sect, I Mat. (N.S.) 13 (1967), 309-328 (Russian). (MR 38 #3785; Zbl 172, 472.)
[GG 681
Gelfand, 1. M., and M. I. Graev, Complexes of straight lines in the space C Functional Anal. i Prilozhen. 2 (1968), no. 3, 39-52 (Russian); English transl. in Functional Analysis and Its Appl. 2 (1968), no. 3, 219-229 (1969). (MR 38 #6522; Zbl 179, 509.)
[GVV 81)
Gheysens, L., P. Verheyen, and L. Verstraelen, Sur les surfaces A ou les surfaces de Chen, C. R. Acad. Sci. Paris Sdr. I Math. 292 (1981), no. 19, 913-916. (MR 82f:53064; Zbl 474:53052.)
BIBLIOGRAPHY
[GVV 83)
335
Gheysens, L., P. Verheyen, and L. Verstraelen, Characterization and examples of Chen submanifolds, J. Geom. 20 (1983), no. 1, 47-62. (MR 84j:53029; Zbl 518:53023.)
[Gin 82]
Gindikin, S. G., Pencils of differential forms and Einstein's equations, Yadernaya Fiz. 36 (1982), no. 2, 537-548 (Russian); English transl. in Soviet J. Nuclear Phys. 36 (1982), no. 2, 313-319. (MR 851:32042; Zbl 588:53049.)
[Gin 83)
Gindikin, S. C., The complex universe of Roger Penrose, Math. Intelligencer 5 (1983), no. 1, 27-35. (MR 89g:01036; Zbl 527:14003.)
[Go 73)
Goldberg, V. V., (n + 1)-webs of multidimensional surfaces. Dokl. Akad. Nauk SSSR 210 (1973), no. 4, 756-759 (Russian); English
transl. in Soviet Math. Dokl. 14 (1973), no. 3, 795-799. (MR 48 #2919; Zbl 304:53017.) (Go 74a)
Goldberg, V. V., (n + 1)-webs of multidimensional surfaces, Bulgar. Akad. Nauk Izv. Mat. Inst. 15 (1974), 405-424 (Russian). (MR 51 #13889; Zbl 346:53010.)
[Go 74b]
Goldberg, V. V., Isoclinic (n + 1)-webs of multidimensional surfaces, Dokl. Akad. Nauk SSSR 218 (1974), no. 5, 1005-1008 (Russian); English transi. in Soviet Math. Dokl. 15 (1974), no. 5, 1437-1441. (MR 52 #11763; Zbl 314:53012.)
[Go 75a)
Goldberg, V. V., The almost Grassmann manifold that is connected with an (n + 1) -web of multidimensional surfaces, Izv. Vyssh. Uchebn.
Zaved. Mat. 1975, no. 8 (159), 29-38 (Russian); English transl. in Soviet Math. (Iz. VUZ) 19 (1975), no. 8, 23-31. (MR 54 #8318; Zbl 315:20056 & 349:20026.) [Go 75b)
Goldberg, V. V., The (n + 1)-web determined by n + I surfaces of codimension n - 1, Problems in Geometry, vol. 7, 173-195, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1975 (Russian). (MR 57 #17537; Zbl 548:53013.)
[Go 77]
Goldberg, V. V., On the theory of four-webs of multidimensional surfaces on a differentiable manifold X2,, Izv. Vyssh. Uchebn. Zaved. Mat. 1977, no. 11(186), 15-22 (Russian); English transl. in Soviet Math. (Iz. VUZ) 21 (1977), no. 11, 97-100. (MR 58 #30859; Zbi 398:53009 & 453:53010.)
[Go 80)
Goldberg, V. V., On the theory of four-webs of multidimensional surfaces on a differentiable manifold X2,, Serdica 6 (1980), no. 2, 105-119 (Russian). (MR 82f:53023.)
[Go 82a]
Goldberg, V. V., The solutions of the Grassmannization and algebraization problems for (n+1)-webs of multidimensional surfaces, Tensor (N.S.) 36 (1982), no. 1, 9-21. (MR 87a:53027; Zbl 479:53014.)
BIBLIOGRAPHY
336
[Go 82b]
Goldberg, V. V., Grossmann and algebraic four-webs in a projective space, Tensor (N.S.) 38 (1982), 179-197. (MR 87e:53024; Zbl 513:53009.)
[Go 841
Goldberg, V. V., An inequality for the 1-rank of a scalar web SW(d, 2, r) and scalar webs of maximum 1-rank, Geom. Dedicata 17 (1984), no. 2, 109-129. (MR 86f:53014; Zbl 554:53015.)
[Go 85)
Goldberg, V. V., 4-tissus isoclines exceptionnels de codimension deux et de 2-rang maximal, C. R. Acad. Sci. Paris Mr. I Math. 301 (1985), no. 11, 593-596. (MR 87b:53025; Zb1 579:53015.)
[Go 861
Goldberg, V. V., Isoclinic webs W(4, 2, 2) of maximum 2-rank, Differential Geometry, Peniscola 1985, 168-183. Lecture Notes in Math., 1209, Springer-Verlag, Berlin, 1986. (MR 88h:53021 & 88m, 6477; Zbl 607:53008.)
[Go 87]
Goldberg, V. V., Nonisochnic 2-codimensional 4-webs of maximum
2-rank, Proc. Amer. Math. Soc. 100 (1987), no. 4, 701-708. (MR 88i:53037; Zbl 628:53018.) [Go 881
Goldberg, V. V., Theory of multicodimensional (n + 1)-webs, Kluwer Academic Publishers, Dordrecht, 1988, xxii+466 pp. (MR 89h:53021; Zbl 668:53001.)
[Go 92]
Goldberg, V. V., Maximum 2-rank webs AGW (6, 3, 2), Differential Geometry and Its Applications 2 (1992), no. 2, 133-165. (MR 941:53005; Zbl 735:53011.)
[Con 871
Goncharov, A. B., Generalized conformal structures on manifolds, Selecta Math. Soviet. 6 (1987), 306-340. (MR 89e:53050; Zbl 632:53038.)
[Gra 441
Grassmann, H., Die Wissenschaft der extensiven Griisse oder die Ausdehnungslehre, Theil 1: Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, Verlag von Otto Wigand, Leipzig, 1844; see also in: Gesammelte mathematische and physicalische Werke, Bd. 1, Teil 1, B. G. Teubner, Leipzig, 1894, 1-319 (Jbuch 25, 27-29); republished by Chelsea Publishing, Bronx, NY, 1969, xii+435 pp. (MR 39 #6727.)
[Gra 621
Grassmann, H., Die Ausdehnungslehre, Verlag von Otto Wigand, Leipzig, 1862; see also in: Gesammelte mathematische and physicalische Werke, Bd. 1, Teil 2, B. G. Teubner, Leipzig, 1896, 1-506 (Jbuch 25, 27-29); republished by Chelsea Publishing, Bronx, NY, 1969.
[Gr 741
Griffiths, P. A., On Carton's method of Lie groups and moving frames as applied to uniqueness and existence question in differential geome-
try, Duke Math. J. 41 (1974), no. 4, 775-814. (MR 53 #14355; Zbl 294:53034.) [GH 79]
Griffiths, P. A., and J. Harris, Algebraic geometry and local differential geometry, Ann. Sci. Ecole Norm. Sup. (4) 12 (1979), 355-452. (MR 81k:53004; Zbl 426:14019.)
BIBLIOGRAPHY
337
[Haa 371
Haantjes, J., Conformal representations of an n-dimensional euclidean space with a nondefinite fundamental form on itself, Nederl. Akad. Wetensh. Proc. Ser. A 40 (1937), 700-705. (Zbl 17, 422.)
[Haa 411
Haantjes, J., Conformal differential geometry I. Curves in conformal euclidean spaces, Nederl. Akad. Wetensh. Proc. Ser. A 44 (1941), 814824. (MR 3, 189; Zbl 25, 365.)
[Haa 42a]
Haantjes, J., Conformal differential geometry II. Curves in conformal two-dimensional spaces, Nederl. Akad. Wetensh. Proc. Ser. A 45 (1942), 249-255. (MR 6 #21; Zbl 26, 353.)
[Haa 42b]
Haantjes, J., Conformal differential geometry III. Curves in threedimensional space, Nederl. Akad. Wetensh. Proc. Ser. A 45 (1942), 836-841. (MR 6, 21; Zbl 27, 348.)
[Haa 42c)
Haantjes, J., Conformal differential geometry IV. Surfaces in threedimensional space, Nederl. Akad. Wetensh. Proc. Ser. A 45 (1942), 918-923. (MR 6, 21; Zbl 27, 348.)
[Haa 43]
Haantjes, J., Conformal differential geometry V. Special surfaces, Nederl. Akad. Wetensh. Verslagen, Afd. Naturwiskunde 52 (1943), 322331. (MR 7, 394; Zbl 63, A372.)
[Hai 37)
Haimovici, A., Directions concourantes le long dune courbe sur une surface dun espace conforme, C. R. Acad. Sci. Roumanie 1 (1937), 296-301. (Jbuch 63, 241.)
[Hai 39)
Haimovici, A., Directions concourantes et directions paralldles sur une variete dun espace conforme, Ann. Sci. Univ. Jassy I. Math. 25 (1939), 153-222. (Zbl 21, 64.)
[Han 66]
Hangan, Th., Ceometrie dfferentielle grassmannienne, Rev. Roumaine Math. Pures Appl. 11 (1966), no. 5, 519-531. (MR 34 #744; Zbl 163, 434.)
[Han 68]
Hangan, T., Tensor-product tangent bundles, Arch. Math. (Basel) 19 (1968), no. 4, 436-440. (MR 38 #3795; Zbl 172, 470.)
[Han 801
Hangan, Th., Sur l'integrabilite des structures tangentes produits tensoriels reels, Ann. Mat. Pura Appl. (4) 126 (1980), 149-185. (MR 82e:53051; Zbl 457:53016.)
[Har 92]
Harris, J., Algebraic geometry: A first course, Springer-Verlag, New York, 1992, xx+328 pp. (MR 93j:14001; Zbl 779:14001.)
[HI 36]
V., Systeme de connexions de M. Weyl, Bull. Acad. Sci. Boh. 37 (1936), 181-184. (Zbl 19, 45.)
[HI 45]
Hlavaty, V., Differentielle Liniengeometrie (Tcheque), P. Noordhoff-
Groningen, 1945, xii+568 pp. (MR 8, 346; Zbl 63, A407); English transl., Differential line geometry, P. Noordhoff-Groningen, 1953, x+495 pp. (MR 15, 252; Zbl 51, 391.)
BIBLIOGRAPHY
338 [Hod 41]
Hodge, W. V. D., The theory and applications of harmonic integrals, Cambridge Univ. Press, Cambridge & Macmillan, New York, 1941, ix+281 pp. (MR 2, 296-297; Zbl 24, p. 397); 2d ed., 1952, x+282 pp. (MR 14, 500; Zbl 48, 157.)
[HP 47]
Hodge, W. V. D., and D. Pedoe, Methods of algebraic geometry, vol. 1, Cambridge Univ. Press, Cambridge & Macmillan, New York, 1947, viii+440 pp. (MR 10, 396; Zb1 157, 275.)
[HP 521
Hodge, W. V. D., and D. Pedoe, Methods of algebraic geometry, vol. 2, Cambridge Univ. Press, Cambridge & Macmillan, 1947, x+394 pp. (MR 13, 972; Zbl 48, 145.)
[Hou 741
Houh, C. S., On spherical A-submanifolds, Chinese J. Math. (1) 2 (1974), 128-135. (MR 52 #6597; Zbl 363:53009.)
[HM 79]
Hsiung, C. C., and L. R. Mugridge, Euclidean and conformal invariants of submanifolds, Geom. Dedicata 8 (1979), 31-38. (MR 80i:53027; Zbl 406:53010.)
[HR 571
Hua, L. K., and B. A. Rosenfeld, The geometry of rectangular matrices and its applications to real projective and non-Euclidean geometry, Sci.
Sinica 6 (1957), no. 6, 995-1011 (Russian). = Izv. Vyssh. Uchebn. Zaved. Mat. 1957, no. 1, 233-247. (MR 22 # 8421; 28 #1791; Zbl 91, 329.) (HD 85]
Huggett, S. A., and K. P. Tod, An introduction to twistor theory, Cambridge Univ. Press, Cambridge, 1985, vii+145 pp. (MR 87i:32042; Zbl 573:53001); 2d ed., 1994, xii+178 pp. (Zbl 809:53001.)
[I 72]
Ishihara, T., On tensor-product structures and Crassmannian structures, J. Math. Tokushima Univ. 1972, no. 4, 1-17. (MR 45 #1064; Zbl 218, 342.)
(JM 94]
Jensen, G. R., and E. Musso, Rigidity of hypersurfaces in complex projective space, Ann. Sci. Ecole Norm. Sup. (4) 27 (1994), 227-248. (MR 95a:53020; Zbl 829:57021.)
[KP 81]
Kaplenko, A. F., and V. A. Ponomarev, Cross-ratio of a nondegenerate quadruple of subspaces, Functional Anal. i Prilozhen. 15 (1981), no. 1, 76-77 (Russian); English transl. in Functional Analysis and Its Appl. 15 (1981), no. 1, 61-62. (MR 82d:15018; Zbl 453:51004 & 464:51002.)
[Kar 62a]
Karapetyan, S. E., Linear manifolds of lines and planes in a projective 4-space, Izv. Akad. Nauk Armyan. SSR Ser. Fiz.-Mat. 15 (1962), no. 1, 53-72 (Russian). (MR 27 #1895; Zbl 115, 253.)
[Kar 62b]
Karapetyan, S. E., Projective differential geometry of two-parameter families of lines and planes in a 4-space, I, Izv. Akad. Nauk Armyan. SSR Ser. Fiz.-Mat. 15 (1962), no. 2, 25-43 (Russian). (MR 27 #1896; Zbl 115, 159.)
BIBLIOGRAPHY
339
(Kar 62c]
Karapetyan, S. E., Projective differential geometry of two-parameter families of lines and planes in a 4-space, II, Izv. Akad. Nauk Armyan. SSR Ser. Fiz.-Mat. 15 (1962), no. 3, 17-28 (Russian). (MR 27 #1897; Zbl 121, 385.)
[Kar 63a]
Karapetyan, S. E., Projective differential geometry of families of multidimensional planes. I, Izv. Akad. Nauk Armyan. SSR Ser. Fiz.-Mat. 16 (1963), no. 3, 3-22 (Russian). (MR 27 #666; Zbl 134, 391.)
[Kar 63b]
Karapetyan, S. E., Projective differential geometry of families of multi. dimensional planes. II, Izv. Akad. Nauk Armyan. SSR Ser. Fiz.-Mat. 18 (1963), no. 5, 3-22 (Russian). (MR 29 #539; Zb1 124, 143.)
(Kar 641
Karapetyan, S. E., Projective differential geometry of families of multidimensional planes. III, Izv. Akad. Nauk Armyan. SSR Ser. Fiz.-Mat.
17 (1964), no. 1, 3-21 (Russian). (MR 29 #540; Zbl 129, 140.) [KC 41]
Kasner, E., and J. de Cicco, Families of curves conformally equivalent to circles, 'Trans. Amer. Math. Soc. 49 (1941), 378-391. (MR 2, 298; Zbl 24, 423.)
[Ke 63]
Kerr, R. P., Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11 (1963), 237-238. (MR 27 #6594; Zbl 112, 219.)
(KS 65)
Kerr, R. P., and A. Schild, Some algebraically degenerate solutions of Einstein's gravitational field equation, Proc. Symp. Appl. Math., vol. 17, pp. 199-209. Amer. Math. Soc., Providence, RI, 1965. (MR 35 #7675.)
[KI 26a]
Klein, F., Vorlesungen uber hohere Geometric, 3. Aufl. SpringerVerlag, Berlin, 1926, viii+405 pp. (Jbuch 52, 624), reprinted by Springer-Verlag, Berlin, 1968 and by Chelsea Publishing, New York, 1949. (MR 17, 445; Zbl 41, 81.)
[KI 26b]
Klein, F., Vorlesungen fiber die Entwicklung der Mathematik im 19. Jahrhundert, Springer-Verlag, Berlin, vol. 1, 1926, xiv+385 pp.; vol.
2, 1927, xiii+208 pp. (Jbuch 52, pp. 22-24); 2 vols. reprinted as one by Springer-Verlag in 1979, (MR 82c:01027; Zbl 398:01006) and
by Chelsea Publishing, New York, in 1950, 1956, and 1967; English transl., Development of Mathematics in the 19th century, Math.
Sci. Press, Brookline, MA, 1979, ix+630 pp. (MR 81c:01023; Zbl 411:01009.) [KI 281
Klein, F., Vorlesungen fiber nicht-euclidische Geometric, SpringerVerlag, Berlin, 1928, xii+326 pp.; reprinted by Springer-Verlag, Berlin, 1968. (Jbuch 52, 624.)
[Kl 72a]
Klein, F., Vergleichende Betrachtungen uber neuere geometrische
Forschungen (Programm zum Eintritt in die philosophische Fakultiit and den Senat der k. Friedrich-Alexanders-Universitat zu Erlangen), Verlag von A. Deichert, Erlangen, 1872, 48 pp. (Jbuch 4, 229-231);
340
BIBLIOGRAPHY
see also Math. Ann. 43 (1893), 63-100, and in Klein. F., Gesammelte Mathematische Abhandlungen, vol. 1, Springer-Verlag, Berlin, 1973, xii+612 pp. 460-497 (MR 52 #10349; Zbl 269:01015); English transl. in Bull. NY Math. Soc. 2 (1892), p. 215. [KI 72b]
Klein, F., Uber Liniengeometrie and metrische Geometric, Math. Ann. 5 (1872), 257-277. (Jbuch 4, 411-412.)
[Klk 81a]
Klekovkin, G. A., A pencil of Weyl connections and a normal conformal connection on a manifold with relatively invariant quadratic form, Webs and Quasigroups, Kalinin. Gos. Univ., Kalinin, 1981, 4755 (Russian). (MR 83h:53035; Zbl 497:53028.)
[Klk 81b]
Klekovkin, G. A., A pencil of Weyl connections associated with a fourdimensional three-web, Geometry of Imbedded Manifolds, Moskov. Cos. Ped. Inst., Moscow, 1981, 59-62 (Russian).
[Klk 831
Klekovkin, G. A., Weyl geometries generated by a four-dimensional
three-web, Ukrain. Geom. Sb. 26 (1983), 56-63 (Russian). (MR 85h:53017; Zbl 525:53020.) [Klk 84]
Klekovkin, G. A., Four-dimensional three-webs with a covariantly con-
stant curvature tensor, Webs and Quasigroups, Kalinin. Gos. Univ., Kalinin, 1984, 56-63 (Russian). (MR 88c:53001; Zbl 558:53010.) [Ko 721
Kobayashi, S.,
Transformation groups in differential geometry,
Springer-Verlag, Berlin, 1972, viii+182 pp. (MR 50 #8360; Zbl 246:53031.) [KN 63]
Kobayashi, S., and K. Nomizu, Foundations of differential geometry, 2 vols., Wiley-Interscience, New York, vol. 1, 1963, xi+329 pp., (MR 27 #2945; Zbl 119, 375), vol. 2, 1969, xv+470 pp. (MR 38 #6501; Zbl 175, 465.)
(KNS 91]
Kolgf, I., P. W. Michor, and J. Slovak, Natural operations in differential geometry, Springer-Verlag, Berlin, 1991, vi+434 pp. (MR 94a:58004; Zbl 782:53013.)
[Ko 95]
Konigs, G., La gdomitrie regle a et ses applications, Gauthier-Villars, Paris, 1895, 14 pp. (Jbuch 26, 758-759.)
[Kon 92a]
Konnov, V. V., Asymptotic pseudoconformal structure on a fourdimensional hypersurface and its completely isotropic two-dimensional submanifolds, Izv. Vyssh. Uchebn. Zaved. Mat. 1992, no. 6 (361), 7179 (Russian); English transl. in Russian Math. 36 (1992), no. 6, 67-74. (MR 94d:53019; Zbl 777:53025.)
[Kon 92b]
Konnov, V. V., Asymptotic conformal structure on a hypersurface, in Ryzhkov, V.V. et al. (eds.), Algebraic methods in geometry, Collection of scientific works, Izdat. Rossijskogo Univ. Druzhby Narodov, Moskva, pp. 14-19 (Russian). (Zbl 806:53055.)
BIBLIOGRAPHY
341
[Kor 50)
Korovin, V. I: Stratification of pairs of complexes of two-dimensional planes in five-dimensional projective space, Dokl. Akad. Nauk SSSR 72 (1950), 837-840 (Russian). (MR 12, 281; Zbl 39, 174.)
(Kos 89]
Kossowski, M. The intrinsic conformal curvature and Gauss map of a light-like hypersurface in Minkowski space, Trans. Amer. Math. Soc. 316 (1989), no. 1, 369-383. (MR 90b:53076; Zbl 691:53046.)
[Kov 63)
Kovantsov, N. I., Theory of complexes, Izdat. Kiev Univ., Kiev, 1963, 292 pp. (Russian). (MR 33 # 4816.)
[Kow 731
Kowalski, 0., Partial curvature structures and conformal geometry of submonifolds, J. Differential Geom. 8 (1973), 53-70. (MR 50 #11068; ZbI 273:530:[2.)
(Kr 62)
Krivonosov, L. N., Families of spheres with an indefinite metric in a conformal space, Gor'kov. Pedag. Inst. Uchen. Zap., 1962, vyp. 41, 20-62 (Russian).
[Kru 80)
Kruglyakov, L. Z., Foundations of projective differential geometry of families of multidimensional planes, Izdat. Tomsk. Univ., Tomsk, 1980, 111 pp. (Russian). (MR 83h:53003; ZbI 499:53005.)
[Kul 70)
Kulkarni, R. S., Curvature structures and conformal connections, J. Differential Geom. 4 (1970), 425-452. (MR 44 #2173; ZbI 192, 586.)
(Kul 881
Kulkarni, R. S., Conformal structures and Mobius structures, Conformal geometry, Semin., MPI, Bonn, 1985-86, 1-39, Aspects Math., E, 12, Vieweg, Braunschweig, 1988, vii+236 pp. (MR 901:53026; Zbl 659:53014.)
(LagR 41 a]
Lagrange, R., Sur les invariants conformes dune courbe, C. R. Acad. Sci. Paris 242 (1941), 1123-1126. (MR 5, 77; Zbl 25, 265.)
[LagR 41b]
Lagrange, R., Propridtds diffdrentielles des courbes des l'espace conforme a n dimensions, C. R. Acad. Sci. Paris 213 (1941), 551-553. (MR 5, 77; Zbl 26, 353.)
[LagR 50]
Lagrange, R., Les courbes dons l'espace anallagmatique, Acta Math. (3) 82 (1950), 327-355. (MR 14, 204; Zbl 35, 379.)
[Lap 49]
Laptev, G. F., An invariant construction of the projective differential geometry of a hypersurface, Dokl. Akad. Nauk SSSR 73 (1950), no. 1, 17-20 (Russian). (MR 11, 53; Zbl 41, 59.)
(Lap 50)
Laptev, G. F., On manifolds of geometric elements with a differential connection, Dokl. Akad. Nauk SSSR 73 (1950), no. 1, 17-20 (Russian). (MR 12, 443; ZbI 40, 246.)
[Lap 53]
Laptev, G. F., Differential geometry of imbedded manifolds. Grouptheoretic method of differential geometry investigations, Trudy Moskov. Mat. Obshch. 2 (1953), 275--382 (Russian). (MR 15, 254; Zbl 53, 428.)
342 [Lap 58a]
BIBLIOGRAPHY
Laptev, G. F., Group-theoretic method of differential geometry investigations, Trudy III Vsesoyuzn. Mat. S'ezda, Moscow, 1958, 409-418 (Russian). (ZbI 93, 353); English transi. in Amer. Math. Soc. Transl.
IT, Ser. 37, 337-350, Trans]. of Trudy III Vsesoyuzn. Mat S'ezda, Moskva, Iun'-Iul' 1956,3, 409-418 (1958). (Zbl 139, 157.) [Lap 58b)
Laptev, G. F., A hypersurface in a space with an projective connection, Dokl. Akad. Nauk SSSR 121 (1958), no. 1, 41-44 (Russian). (MR 20 #6137; Zbl 85, 164.)
[Lap 65]
Laptev, G. F., Differential geometry of multidimensional surfaces, Geometry 1963, pp. 5-64. Akad. Nauk SSSR Inst. Nauchn. Informatsii, Moscow, 1965 (Russian). (MR 33 #4817.)
[Lap 661
Laptev, G. F., The main infinitesimal structures of higher orders on a differentiable manifold, Trudy Geom. Sem. 1 (1966), 139-190 (Russian). (MR 34 #6681; ZbI 171, 423.)
(LeB 821
LeBrun, C., H-spaces with a cosmological constant, Proc. Roy. Soc. London Ser. A 380 (1982), 171-185. (MR 83d:53019; Zbl 154, 214.)
[Le 61]
Leichtweiss, K., Zur Riemannschen Geometric in Grassmannschen Mannigfaltigkeiten, Math. Z. 76 (1961), 334-336. (MR 23 #A4102; Zb1 113, 371.)
[Lic 55]
Lichnerowicz,
A.,
Thdorie globate des connexions et groupes
d'holonomie, Edizioni cremonese, Rome, 1955, xv+282 pp. (MR 19, 453; Zb1 116, 391); English transl., Global theory of connections and holonomy groups, Noordhoff International Publishing, 1976, xiv+250 pp. (MR 54 #1121; ZbI 337:53031.) [LS 96]
Lie, S., and G. Scheffers, Geometric der Beriihrungstransformationen. I, B. G. Teubner, Leipzig, 1896, vol. 1, xi+694 pp. (Jbuch 27, pp. 547556); 2d corrected ed., Chelsea Publishing, Bronx, NY, 1977, xii+694 pp. (MR 57 #45; Zbl 406:01015.)
[Lieb 23]
Liebmann, H., Beitrdge zur Inversionsgeometrie der Kurven, Munch. Ber. 1923, 79-94. (Jbuch 49, 531.)
(Lio 501
Liouville, J., Extension an cas de trois dimensions de to question du trace 6th appendix to the book: G. Monge, Application de l'analyse d la gComdtrie, cingieme edition, revue corrigLe et annotte par J. Liouville. Bachelier, Paris, 1850, 608-616; reprinted by University Microfilms International, Ann Arbor, MI, 1979.
[Lit 891
Little, J. B., On webs of maximum rank, Geom. Dedicata 31 (1986), no. 19-35. (MR 90g:53023; Zbl 677:53017.)
[Lu 75]
Lumiste, U, G., Differential geometry of submanifolds, Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, vol. 13, 273-340, Akad. Nauk SSSR Inst. Nauchn. Informatsii, Moscow, 1975 (Russian); En-
glish. transl. in J. Soviet Math. 7 (1977), no. 4, 654-677. (MR 55 #4008; Zbl 421:53036.)
BIBLIOGRAPHY
343
[Ma 42]
Maeda, J., Differential Mobius geometry of plane curves, Japan J. Math. 18 (1942), 67-260. (MR 7, p. 265; Zbl 63, p. A580.)
(Man 84]
Manin, Yu. I., Gauge field theory and complex geometry, "Nauka", Moscow, 1984, 336 pp. (Russian). (MR 86m:32001; Zbl 576:53002); English transi., Springer-Verlag, Berlin, 1988, x+297 pp. (MR 89d:32001; Zbl 641:53001.)
[Mat 55]
Matsumoto, M., A theorem for hypersurfaces of conformally fiat space,
Mem. Coll. Sci. Univ. Kyoto A29 (1955), no. 3, 219-233. (MR 20 #4685; Zbl 68, 359.) (Mi 721
Mikhailov, Yu. I., On multidimensional two-webs of type T,m,n, V All-
Union Conf. on Contemporary Problems of Geometry, Abstracts of Talks, 1972, p. 141 (Russian). [Mi 74)
Mikhailov, Yu. I., On multidimensional two-webs of the type T,, Trudy Geom. Sem. 5 (1974), 335-344, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (Russian). (MR 53 #3916; Zbl 304:53009.)
(Mi 77)
Mikhailov, Yu. I., Classification of two-webs of the type T.7",, Differential Geometry, Kalinin. Gos. Univ., Kalinin, 1977, 88-94 (Russian). (MR 82j:53028.)
[Mi 78]
Mikhailov, Yu. I., On the structure of almost Grassmannian manifolds, Izv. Vyssh. Uchebn. Zaved. Mat. 1978, no. 2, 62-72 (Russian); English trans). in Soviet Math. (Iz. VUZ) 22 (1978), no. 2, 54-63. (MR 81e:53031; Zbl 398:53006.)
[Mi 811
Mikhailov, Yu. I., Some two-webs of type Tm +a, Webs and Quasigroups, Kalinin. Gos. Univ., Kalinin, 1981, 76-83 (Russian). (MR 831:53034; Zbl 497:53019.)
(Min 09]
Minkowski, H., Raum and Zest (Address delivered at the 80th Assembly of German Natural Scientists and Physicians at Cologne, September 21, 1908, Leipzig, 1909), Jahresber. Deutsch. Math.-Vereign. 18 (1909), 75-88; it was published as a book by B. G. Teubner, Leipzig,
1909, 14 pp. (Jbuch 40, 745); see also H. Minkowski, Gesammelte Abhandlungen, 2. Band, pp. 431-444, Chelsea Publishing, 1967, or H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The principle of relativity, Dover, New York, 1923, pp. 73-91. [Miy 84]
Miyaoka, R., Compact Dupin hypersurfaces with three principal curvatures, Math. Z. 187 (1984), no. 4, 433-452. (MR 851s:53041; Zbl 529:53045 & 545:53047.)
[Miy 89a]
Miyaoka, R., Dupin hypersurfaces and a Lie invariant, Kodai. Math. J. 12 (1989), no. 2, 228-256. (MR 90i:53083; Zbl 687:53053.)
(Miy 89b]
Miyaoka, R., Dupin hypersurfaces with six principal curvatures, Kodai. Math. J. 12 (1989), no. 3, 308-315. (MR 90k:53092; 711:53049.)
BIBLIOGRAPHY
344 [MO 891
Miyaoka, R., and T. Ozawa, Construction of taut embeddings and the Cecil-Ryan conjecture, In: Geometry of manifolds (Matsumoto, 1988), 181-189, Perspect. Math, 8, Academic Press, Boston, 1989. (MR 92f:53071; Zbl 687:53055.)
[MCS 91]
Mizin, A. G., N. P. Chupakhin, and N. R. Shcherbakov, Introduction to the projective differential geometry of manifolds of straight lines, Izdatel'stvo Tomskogo Univ., Tomsk, 1991, 150 pp. (Russian). (Zbl 787:53012.)
[Mu 40a]
Muto, Y., On some properties of umbilical points of hypersurfaces, Proc. Imp. Acad. Tokyo 16 (1940), 79-82. (MR 1, 272; ZbI 23, 167.)
(Mu 40b]
Muto, Y., On some properties of subspaces in a cnformally connected manifold, Proc. Phys.-Math. Soc. Japan 22 (1940), 621-636. (MR 2, 166; Zbl 24, 82.)
(Mu 42]
Muto, Y., Theory of subspaces in a space with a conformal connection, Tensor 5 (1942), 31-46 (Japanese). (MR 2, 203; Zbl 63, A656.)
[NP 62]
Newman, E. T., and R. Penrose, An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. 3 (1962), 566-578;
(MR 25 #4904; Zbl 108, p. 409); Errata 4 (1963), p. 998. (MR 27 #3412.) (Ni 741
Nishikawa, S., Conformally flat hypersurfaces in a Euclidean space,
Tohoku Math. J. (2) 26 (1974), 563-572. (MR 50 #14605; ZbI 299:53037.) [NM 74]
Nishikawa, S., and Y. Maeda, Conformally flat hypersurfaces in a con-
formally flat Riemannian manifold, Tohoku Math. J. (2) 26 (1974), 159-168. (MR 49 #3730; Zbl 278:53018.) IN 47]
Norden, A. P., La connexion affine sur les surfaces de l'espace projectif, Mat. Sb. (N.S.) 20 (60) (1947), 263-281. (MR 9, 67; ZbI 41, 306.)
IN 481
Norden, A. P., On normalized surfaces of the Mobius space, Dokl. Akad. Nauk SSSR 61 (1948), no. 2, 207-210 (Russian). (MR 10, 67; Zbl 41, 494.)
IN 49]
Norden, A. P., Conformal interpretation of Weyl's spaces, Mat. Sb. (N.S.) 24 (66) (1949), 75-85 (Russian). (MR 11, 55; Zb1 35, 240.)
IN 50a]
Norden, A. P., Affinely connected spaces, Gosudarstv. Izdat. Tehn: Teor. Lit., Moscow, 463 pp. (Russian). (MR 12, 44; ZbI 41, 502.) 2d ed., Izdat. "Nauka", Moscow, 1976, 432 pp. (MR 57 #7421.)
IN 50b]
Norden, A. P., On normalized surfaces of a conformal space, Izv. Akad.
Nauk SSSR Ser. Mat. 14 (1950), no. 2, 105-122 (Russian). (MR 12, 54; Zbl 41, 307.) [No 18]
Nordstrom, G., On the energy of the gravitational field of Einstein's theory, Proc. Kon. Ned. Acad. Wet. 20 (1918), 1238-1245 = Amst. Akad. Versl. 26, 1201-1208. (Jbuch 46, 1346-1347.)
BIBLIOGRAPHY
345
[Og 671
Ogiue, K., Theory of conformal connections, Ki dai Math. Sem. Rep. 19 (1967), 193-224. (MR 36 #812; Zbl 163, 165.)
(Pen 76]
Pendl, A., Zur Mdbiusgeometrie der Kurventheorie, Monatsh. Math. 81 (1976), no. 2, 141-148. (MR 54 #11199; Zbl 323:53006.)
[P 671
Penrose, R., Twistor algebra J. Math. Phys. 8 (1967), 345-366. (MR 35 #7567; Zbl 174, 559.)
[P 68a]
Penrose, R., Twistor quantization and curved space-time, Internat. J. Theoret. Phys. 1 (1968), no. 1, 61-99.
[P 68b)
Penrose, R: Structure of space-time, in Battelle rencontres: 1967 lectures in mathematics and physics, Chapter VII, eds. C. M. DeWitt and J. A. Wheeler, Benjamin, New York, 1968 (MR 38 #955; ZbI 174, 559); Russian trans)., Structura prostransivo-vremeni, Mir, Moscow, 1972, 183 pp. (MR 50 #6418.)
[P 76]
Penrose, R., Nonlinear gravitons and curved twistor theory. The riddle of gravitation, General Relativity and Gravitation 7 (1976), no. 1, 3152. (MR 55 #11905; Zbl 354:53025.)
[P 77]
Penrose, R., The twistor programme, Rep. Math. Phys. 12 (1977), no. 1, 65--76. (MR 57 #4948.)
[PR 86]
Penrose, R., and W. Rindler, Spinors and space-time, vol. 2: Spinor and twistor methods in space-time geometry, Cambridge Univ. Press, Cambridge, 1986, x+501 pp. (MR 81d:83010; Zbl 591:53002); Russian transi., Spinory i prostranstvo-vremya, Mir, Moscow, 1988, 576 pp. (MR 90f:83003.)
[Per 35]
Perepelkine, D., Sur la transformation conforme et la courbure riemannienne normale dune V,,, dons V,,, C. R. Acad. Sci. Paris 200 (1935), 513-515. (Zbl 10, 419.)
[Pet 46]
Petrescu, S., Quelques proprigtEs conformer des sons espaces V1, dans un V. de V,,, Bull. Sci. Ecole Polytech. Timi§oara 12 (1946), 167-174. (MR 9, 66; Zbl 63, A719.)
[Pet 48]
Petrescu, S., Sur quelques proprietds conformes des espaces non holonomes V,m, Mathematica Timi§oara 23 (1948), 108-122. (MR 10, 448; Zbl 31, 78.)
[Pe 54)
Petrov, A. Z., Classification of spaces defined by gravitational fields, Kazan. Gos. Univ. Uchen. Zap. 114 (1954), no. 8, 55-69 (Russian); English transi. in Trans. No. 29, Jet Propulsion Lab, California Inst. Tech., Pasadena, 1963. (MR 17, 892.)
(Pe 69]
Petrov, A. Z., Einstein spaces, Pergamon Press, Oxford,
1969,
xiii+411 pp. (MR 39 #6225; Zbl 174, 183); revised, corrected, and modified edition of the Russian original, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961, 463 pp. (MR 25 #4897; Zbl 98, 189.)
346
BIBLIOGRAPHY
[Pi 85a]
Pinkall, U., Dupin hypersurfaces, Math. Ann. 270 (1985), no. 3, 427440. (MR 86e:53044; Zbl 538:53004.)
[Pi 85b]
Pinkall, U., Dupinsche Hyperfi8chen in E4, Manuscripta Math. 51 (1985), no. 1-3, 89-119. (MR 86m:53010; Zbl 572:53028.)
[PT 891
Pinkall, U., and G. Thorbergsson, Deformation of Dupin hypersurfaces, Proc. Amer. Math. Soc. 107 (1989), no. 4, 1037-1043. (MR 90c:53145; Zbl 682:53061.)
[Pir 57]
Pirani, F. A. E., Invariant formulation of gravitational radiation theory, Phys. Rev. (2) 105 (1957), 1089-1099. (MR 20 #3020; Zbl 77, 419.)
(P1 46]
PIucker, J., System der Geometric des Raumes in newer analytischer Behandlungsweise, Schaubsche Buchh., Diisseldorf, 1846.
[PI 68]
Pliicker, J., Neue Geometric des Raumes, gegriindet auf die Betrachlung der geraden Linie als Raumenelement, B. G. Teubner, Leipzig, Abt. 1 & Abt. 2, 1868-1869. (Jbuch 1, 198-205; 2, 601.)
[Po 061
Poincar6, H., Sur les dynamique de l'electron, Rend. Circ. Mat. Palermo 21 (1906), 129-175 (Jbuch 37, 886); see H. Poincare, (Buvres,
vol. 9, Gauthier-Villars, Paris, 1954, pp. 494-550. (MR 18, 435; Zbl 59, 1.) [Raw 87]
Rawnsley, J. H., Turistor methods, In: Differential Geometry (Lyngsby, 1985), 97-133, Lecture Notes in Math. 1263, Springer-Verlag, Berlin, 1987. (MR 88j:53046; Zbl 652:53026.)
[Re 16]
Reissner, H., Die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie, Ann. Physik 50 (1916), 106-120.
(Rh 55)
de Rham, G., Variltds diffdrentiables. Formes, courants, formes har-
moniques, Paris, Hermann, 1955, vii+196 pp. (MR 16, 957-959; Zbl 65, 324); English transl., Differentiable manifolds: forms, currents, harmonic forms, Springer-Verlag, Berlin, 1984, x+167 pp. (MR 85m:58005.) (Roo 38)
Room, T. G., The geometry of determinantal loci, Cambridge Univ. Press, Cambridge, 1938, xxviii+483 pp. (Zbl 20, 54.)
(RB 79]
Rosca, R., and Buchner, K: Spatial submanifolds structured by conformally parallel flat connection in an even-dimensional Minkowski space, Tensor (N.S.) 33 (1979), no. 3, 300--306. (MR 81g:53047; Zbl 417:53011.)
(Ro 47]
Rosenfeld, B. A., The metric and affine connection in spaces of planes, spheres or quadrics, Dokl. Akad. Nauk SSSR 57 (1947), no. 6, 543-546 (Russian). (MR 9, 249; ZbI 39, 179.)
[Ro 48aJ
Rosenfeld, B. A., Differential geometry of symmetry figures, Dokl. Akad. Nauk SSSR 59 (1948), no. 6, 1057-1060 (Russian). (MR 10, 66; Zbl 31, 416.)
BIBLIOGRAPHY
347
[Ro 48b)
Rosenfeld, B. A., Conformal differential geometry of families Cm and Cn, Mat. Sb. (N.S.) 23 (65) (1948), 297-313 (Russian). (MR 10, 403; ZbI 41, 494.)
[Ro 491
Rosenfeld, B. A., Projective differential geometry of families of pairs
Pm + P"-'"-t in P", Mat. Sb. 24 (66) (1949), 405-428 (Russian). (MR 11, 133-134; ZbI 41, 494); English transl. in Amer. Math. Soc. Translations no. 77, 1952, 32 pp. (MR 14, 498.) [Ro 58]
Rosenfeld, B. A., Rectangular matrices and non-Euclidean geometries, Uspekhi Mat. Nauk 13 (1958), no. 6 (84), 21-48 (Russian). (MR 22 # 8423; Zbl 121, 377.)
[Ro 59]
Rosenfeld, B. A., Quasielliptic spaces, Trudy Moskov. Mat. Obshch. 8 (1959), 49-70 (Russian). (MR 21 #5987; ZbI 196, 230.)
[Ro 96]
Rosenfeld, B. A., Geometry of Lie groups, Kluwer Academic Publishers, Dordrecht, 1996 (to appear).
[RKSYu 90]
Rosenfeld, B. A., L. P. Kostrikina, G. V. Stepanova, and T. I. Yuchtina, Focally affine spaces, Izv. Vyssh. Uchebn. Zaved. Mat. 1992, no. 5 (336), 60-68 (Russian); English transl. in Sov. Math. (Iz. VUZ) 34 (1990), no. 5, 70-78. (MR 92c:53008; ZbI 711:53009 & 726:53009.)
[RZT 88]
Rosenfeld, B. A., M. P. Zamakhovskii, and T. A. Timoshenko, Quasielliptic spaces, Algebra, Topologiya, Geometriya vol. 26, 125160 (Russian); Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuzn. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988; English transl. in J. Soviet Math. 42 (1990), no. 3, 1014-1034. (MR 90c:22050; Zbl 677:53015 & 699:53017.)
[Rou 741
Rouxel, B., Sur certaines families de spheres et d'hypersphe res dun espace euclidien quadridimensionnel E4, Acad. Roy. Belg. Cl. Sci. (5) 60 (1974), no. 2, 170-182. (MR 52 #9086; Zbl 283:53008.)
[Rou 79]
Rouxel, B., Sur les courbes isotropes, pseudo-isotropes et les surfaces isotropes dun espace-temps de Minkowski M4, Rend. Sem. Fac. Sci. Univ. Cagliari 49 (1979), 571-584. (MR 81j:53017; Zbl 447:53021.)
[Rou 80]
Rouxel, B., Ruled A-surfaces in Euclidean space E", Soochow J. Math. 6 (1980), 117-121. (MR 82j:53011; Zbl 465:53004.)
[Rou 81a)
Rouxel, B., Sur une famille de A-surfaces dun espace euclidien E4, 10` Osterreichischer Math. Kongress, Insbriich, 1981, p. 185.
[Rou 81b]
Rouxel, B., A-submanifolds in Euclidean space, Kodai Math. J. 4 (1981), no. 1, 181-188. (MR 82g:53028; Zbl 467:53004.)
[Rou 82]
Rouxel, B., Sur les A-surfaces dun espace-temps de Minkowski M4, Riv. Mat. Univ. Parma (4) 8 (1982), 309-315. (MR 85e:53071; Zb1 514:53016.)
348 [S 62]
BIBLIOGRAPHY
Sacksteder, R., The rigidity of hypersurfaces, J. Math. Mekh. 11 (1962), 929-939. (MR 26 #1833; ZbI 108, 347.)
[SaS 39)
Sasaki, S., On the theory of curves in a curved conformal space, Sci. Rep. Tohoku Imp. Univ. Ser. 1, 27 (1939), 392-409. (MR 1, 175; Zbl 20, 260.)
[SaS 40]
Sasaki, S., On the theory of surfaces in a curved conformal space, Sci. Rep. Tohoku Imp. Univ. Ser. 1, 28 (1940), 261-285. (MR 1, 273; Zbl 23, 75.)
(SaS 48)
Sasaki, S., Geometry of conformal connection, Kawade-shobb, Tokyo, 1948, 3+3+265 pp. (Japanese). (MR 12, 442.)
[SaT 88]
Sasaki, T., On the projective geometry of hypersurfaces, Equations differentielles dans le champ complexe, vol. 3 (Strasbourg, 1985), 115-
161, Publ. Inst. Rech. Math. Av., Univ. Louis Pasteur, Strasbourg, 1988. (MR 93d:53021; Zbl 790:53011.) [SSu 80]
Schiemankgk, C., and R. Sulanke, Submanifolds of the Mobius space, Math. Nachr. 96 (1980), 165-183. (MR 82d:53017; ZbI 484:53008.)
[S 181
Schouten, J. A., Die direkte Analysis zur neueren Relativitatstheorie, Amsterdam Akad. Verh. 12 (1918), no. 6, 95 pp. (Jbuch 46, 11271128.)
[S 21)
Schouten, J. A., Uber die konforme Abbildung n-dimensionaler Man. nigfaltigkeiten mit quadratischer Maflbestimmung auf eine Mannigfaltigkeit mit euklidischer Maflbestimmung, Math. Z. 11 (1921), 55-88. (Jbuch 48, 857-858.)
[S 24]
Schouten, J. A., Der Ricci Kalkiil, Springer-Verlag, Berlin, 1924, 312 pp.; English trans]. in Ricci calculus. An introduction to tensor anal-
ysis and its geometrical applications (Jbuch 50, 588-589); 2d ed., Springer-Verlag, Berlin, 1954, xx+516 pp. (MR 16, 521; Zbl 57, 378380.) [S 27]
Schouten, J. A., Uber n-fache Orthogonalsysteme in V,,, Math. Z. 26 (1927), 706-730. (Jbuch 53, 685.)
[SH 36]
Schouten, J. A., and J. Haantjes, Beitrnge zur allgemeinen (gekriimmten) konformen Differentialgeometrie. 1, 11 Math. Ann. 112 (1936), 594-629; 113 (1936), 568-583. (Zbl 13, 367.)
[SS 38)
Schouten, J. A., and D. J. Struik, Einfiihrung in der neueren Methoden der Differentialgeometrie II, 2. Aufl., 2. Band, Geometrie von D.J. Struik, P. Noordhoff N.V., Groningen/Batavia, 1938, xii+338 pp. (Zbl 19, 183); Russian trans]., Gosudarstv. Izd. lnostr. Lit., Moscow, 1948, 348 pp. (MR 12, 128.)
[Schu 261
Schubarth, E., Sur les courbes admettant un groupe de transformations de Moebius, Enseign. Math. 25 (1926), 234-239. (Jbuch 53, 648.)
BIBLIOGRAPHY [Sehw 16a]
349
Schwarzschild, K., Ober das Gravitationsfeld eines Maflenpunktes nach
der Einsteinschen Theorie, Berliner Sitzungsberichte (Phys. Math. Classe) 3 Feb. 1916 (Mitt. Jan. 13), 189-196. (Jbuch 46, pp. 12961297.) [Sehw 16b]
Schwarzschild, K., Ober das Gravitationsfeld einer Kugel aus incompressibler Fliissigkeit nach der Einsteinschen Theorie, Berliner Sitzungsberichte (Phys., Math. Classe) 23 Mar. 1916 (Mitt. Feb. 24), 424-434. (Jbuch 46, pp. 1297-1298.)
[Seg 85]
Segre, C., Sulla geometria delta recta e delle sue sene quadratiche, Mem. Accad. Sci. Torino (2) 36 (1885), 87-157. (Jbuch 18, 691-693.)
[Seg 07]
Segre, C., Su una classes di superficie degli iperspazi legate colle equazioni tineari alle derivate parziali di 2° ordine, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 42 (1907), 1047-1049. (Jbuch 38, 671-673.)
[Sem 32)
Semple, J. G., On representatson of the Sk's of Sn and of the Grassmann manifolds G(k, n), Proc. London Math. Soc. (2) 32 (1931), 200221. (Zbl 1, 157.)
[SR 85)
Semple, J. G., and L. Roth, Introduction to algebraic geometry, Clarendon Press & Oxford Univ. Press, New York, 1985, xviii+454 pp. (MR 86m:14001; Zbl 576:14001.)
[Sev 15)
Severi, F., Suite varietd the rappresenta gli spazi subordinati di data dimensione, immersi in uno spazio lineare, Ann. Mat. (3) 24 (1915), 89-120. (Jbuch 45, 915 & 1379-1380.)
[Sh 67]
Shcherbakov, R. N., Differential line geometry of three-dimensional space, Itogi Nauki; Algebra, Topology, Geometry, 1965, pp. 265-321. Akad. Nauk SSSR Inst. Nauchn. Tekhn. Informatsii, Moscow, 1967 (Russian). (MR 35 #7233; Zbl 189, 2); English transl. in Progress Math. 6 (1970), 53-111. (Zbl 202, 206.)
[St 64)
Sternberg, S., Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964, 390 pp. (MR 23 #1797; Zbl 211, 535); 2d ed., Chelsea Publishing, New York, 1983, xi+442 pp. (MR 88f:58001; Zbl 518:53001.)
[Su 81]
Sulanke, R., Submanifolds of the Mobius space. II. Frenet formulas and curves of constant curvatures, Math. Nachr. 100 (1981), 235-247. (MR 83d:53046; Zbl 484:53009.)
[Su 82)
Sulanke, R., Submanifolds of the Mobius space. III. The analogue of 0. Bonnet's theorem for hypersurfaces, Tensor (N. S.) 38 (1982), 311317. (MR 87m:53066; Zbl 511:53060.)
[Su 84)
Sulanke, R., Submanifolds of the Mobius space. IV. Conformal invariants of immersions into spaces of constant curvature, Potsdamer Forsch. B 43 (1984), 21-26. (Zbl 643:53013.)
BIBLIOGRAPHY
350
[Su 88]
Sulanke, R., Mobius geometry. V. Homogeneous surfaces in the Mobius space S3, Topics in differential geometry, Vols 1-2 (Debrecen, 1984), 1141-1154, Colloq. Math. Soc. Janos Bolyai, 46, North-Holland, Amsterdam, 1988. (MR 90e:53067; ZbI 643:53014.)
[Su 92]
Sulanke, R., Mobius geometry. VII. On channel surfaces, Proceedings of the 3rd Congress of Geometry (Thessaloniki, 1991), 410-419 Aristotle Univ. Thessaloniki, Thessaloniki, 1992. (MR 93m:53049; Zbl 759:53037.)
[Ta 38]
Takasu, T., Differentialgeometrien in der Kugelraumen, I. Konforme Differentialgeometrie von Liouville and Mobius, Maruzen Comp., Tokyo, 1938, xviii+458 pp. (Zbl 19, 44.)
[Ta 391
Takasu, T., Differentialgeometrien in der Kugelraumen, II. Laguerrische Differentialkugelgeometrie, Maruzen Comp., Tokyo, 1939, xx+444 pp. (MR 1, 286; Zbl 22, 265); 2d printing, Taigado Publishing, Kyoto & Hofner Publishing, New York, 1950, xxi+444p. (MR 14, 279.)
[ThJ 26]
Thomas, J. M., Conformal invariants, Proc. Nat. Acad. Sci. U.S.A. 12 (1926), 389-393. (Jbuch 52, 736.)
[ThT 341
Thomas, T. Y., The differential invariants of generalized spaces, Cambridge Univ., Cambridge, 1934, x+241 pp. (Zbl 9, p. 85.)
[Tho 23)
Thomsen, G., Ober konforme Geometrie I. Grundlagen der konformen
Flachentheorie, Abh. Math. Sem. Univ. Hamburg 3 (1923), 31-56. (Jbuch 49, 530.) [Tho 25)
Thomsen, G., Uber konforme Geometrie II. Uber Kreisscharen and Kurven in der Ebene and Uber Kugelscharen and Kurven in Raum, Abh. Math. Sem. Univ. Hamburg 4 (1925), 117-147. (Jbuch 51, 585586.)
(Thor 83]
Thorbergsson, G., Dupin hypersurfaces, Bull. London. Math. Soc. 15 (1983), no. 5, 493-498. (MR 85b:53066; Zbl 592:53044.)
[Ti 61]
Tikhonov, V. A., The Ribaucour transformation in conformal geometry, Izv. Vyssh. Uchebn. Zaved. Mat. 1961, no. 3 (22), 136-147 (Russian). (MR 28 #5486.)
[Ti 631
Tikhonov, V. A., On a type of Ribaucour transformations, Sibirsk. Mat. Zh. 4 (1963), no. 3, 683-688 (Russian). (MR 27 #6199; Zbl 129, 140.)
[Ti 64]
Tikhonov, V. A., On degenerate Ribaucour transformations, Izv. Vyssh. Uchebn. Zaved. Mat. 1964, no. 5 (42), 104-108 (Russian). (MR 30 #2413; ZbI 136, 172.)
[T r 94]
Tresse, A., Sur le invariants dsfft rentiels des groupes continus de trans-
formations, Acta Math. 18 (1894), 1-88. (Jbuch 25, 641-642.)
BIBLIOGRAPHY
351
(Va 481
Vasilyev, A. M., Involutive systems of line complexes, Doki. Akad. Nauk SSSR 61 (1948, 189-191 (Russian). (MR 10, 64; Zbl 35, 378.)
(Ved 50a]
Vedernikov, V. I., The conformal applicability of surfaces, Dokl. Akad. Nauk SSSR 73 (1950), 437-440 (Russian). (MR 12, 442; ZbI 41, 495.)
(Ved 50b]
Vedernikov, V. I., Conformal applicability of surfaces, Kazan. Univ. Uchen. Zap. 110 (1950), no. 3, 35-55 (Russian).
(Ved 54]
Vedernikov, V. I., Conformal applicability of surfaces with preservation of conjugate geometries, Trudy Voronezh. Gos. Univ. Fiz.-Mat. Sb. 33 (1954), 37-42 (Russian). (MR 18, 448.)
[Ved 571
Vedernikov, V. I., Surfaces enveloping a family of hyperspheres, Izv. Vyssh. Uchebn. Zaved. Mat. 1957, no. 1, 89-97 (Russian). (MR 25 #517; Zbl 94, 166.)
[Ved 581
Vedernikov, V. I., On a multidimensional generalization of the Dupin cyclides, Izv. Vyssh. Uchebn. Zaved. Mat. 1958, no. 6 (7), 58-72 (Russian). (MR 23 #A489; Zb1 119, 171.)
[Ved 621
Vedernikov, V. I., (n - 1)-parameter families of hyperspheres in Mn, Izv. Vyssh. Uchebn. Zaved. Mat. 1962, no. 2 (27), 35-43 (Russian). (MR 26 #2967; Zbl 135, 220.)
[Ved 63]
Vedernikov, V. I., Conformal applicability of surfaces in the space Mn, Izv. Vyssh. Uchebn. Zaved. Mat. 1963, no. 1 (32), 33-41 (Russian). (MR 28 #2496; Zbl 134, 174.)
(VT 541
Vedernikov, V. I., and V. A. Tikhonov, A metrical characterization of the fundamental forms and quantities of the conformal theory of surfaces, Trudy Voronezh. Gos. Univ. Fiz.-Mat. Sb. 33 (1954), 43-52 (Russian). (MR 18 #148.)
(Ver 521
Verbitsky, L. L., Geometry of conformal Euclidean spaces of class 1, Trudy Sem. Vektor. Tenzor. Anal. 9 (1952), 146-182 (Russian). (MR 14, 795; Zbl 49, 234.)
[Ver 59]
Verbitsky, L. L., Fundamentals of curve theory in conformal space of n dimensions, Izv. Vyssh. Uchebn. Zaved. Mat. 1959, no. 6 (13), 26-37 (Russian). (MR 24 #A498; Zbl 102, 372.)
(VV 801
Verheyen, P., and L. Verstraelen, Conformally flat totally real submanifolds of complex projective spaces, Soochow J. Math. 6 (1980), 137-143. (MR 82j:53098; ZbI 456:53035.)
[Vers 78]
Verstraelen, L., A-surfaces with flat normal connection, J. Korean
Math. Soc. (1) 15 (1978/79), no. 1, 1-7. (MR 58 #12972; ZbI 385:53013.) [Ves 26a]
Vessiot, E., Contribution d to giometrie conforme. Theorie des surfaces. I, II, Bull. Soc. Math. France 54 (1926), 139-179. (Jbuch 53, 700-701.)
352
BIBLIOGRAPHY
(Ves 26b]
Vessiot, E., Sur la gdomdtrie conforme des surfaces, C. R. Acad. Sci. Paris 182 (1926), 752-754. (Jbuch 52, 764.)
[Ves 271
Vessiot, E., Contribution d la gdomdtrie conforme. Thdorie des surfaces, Bull. Soc. Math. France 55 (1926), 39-79. (Jbuch 53, 700-701.)
(VI 10]
Vlasov, A. K., Polar systems of higher orders in the first-order form. Investigation of a construction of geometric theory which corresponds to the theory of algebraic equations and forms, Abb. Univ. Moscow, Phys. Math. Abt. R, 1910, Moscow, xii+186 pp. (Russian). (Jbuch 41, 608-609.)
[Vo 80]
Voss, A., Zur Theorie der T ansformation quadratischer Differentialausdrucke and der Kri mmung ht herer Mannigfaltigkeiten, Math. Ann. 16 (1880), 129-178. (Jbuch 12, 570-572.)
(Vr 40)
Vranceanu, G., Sur lea espaces a connexion conforme, Disquisit. Math. Phys. 1 (1940), 63-81. (MR 8, 603; Zbl 23, 271.)
(Vr 43)
Vranceanu, G., Sur la thdorie des espaces d connexion conforme, Bull. Math. Soc. Roumaine Sci. 45 (1943), 3-31. (MR 7, 34; Zbl 63, A997.)
[Vr 51]
Vranceanu, G., Lecjii de geometric differenliala, vol. 2, Editura Academies Republicii Populace Romine, 1951, 398 pp. (MR 18, 1049; Zbl 45, 428.)
[Wel 79]
Wells, R. 0., Jr., Complex manifolds and mathematical physics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 2, 296-313. (MR 80h:32001; Zbl 444:32014.)
[Wel 80]
Wells, R. 0., Jr., Differential analysis on complex manifolds, 2d ed. Springer-Verlag, New York, 1980, x+260 pp. (MR 83f:53001; Zbl 435.32014.)
(We 181
Weyl, H., Reine Infinitesimalgeometrie, Math. Z. 2 (1918), 384-411. (Jbuch 46, 1301.)
(We 211
Weyl, H., Zur Infinitesimalgeometrie. Etnordnung der projektiven and konformen Auffassung, Gi ttinger Nachr., 1921, 99-112. (Jbuch 48, 844.)
[Wol 77]
Wolf, J. A., Spaces of constant curvature, 4th ed., Publish or Perish, Berkeley, 1977, xvi+408 pp. (MR 49 #7958.)
[Won 43]
Wong, Y. C., Family of totally umbilical hypersurfaces in an Einstein
space, Ann. of Math. (2) 44 (1943), 271-297. (MR 4, 258; Zbl 60, 387.) [Won 67]
Wong, Y. C., Differential geometry of Grassmann manifolds, Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 589-594. (MR 35 #4825; Zbl 154, 214.)
BIBLIOGRAPHY
353
[Wou 48J
van der Woude, W., On conformal differential geometry. Theory of plane curves, Nederl. Akad. Wetensh. Proc. Ser. A 51 (1948), 16-24 = Indagationes Math. 10 (1948), 3-11. (MR 9, 467; Zbl 29, 164.)
[Y 39a)
Yano, K., Sur la connexion conforme de Weyl-Hlavaty et la ge omdtrie conforme, Proc. Imp. Acad. Japan 15 (1939), 116-120. (Zbl 21, 427.)
[Y 39b]
Yano, K., Sur les equation de Gauss dons la gEomdtrie conforme des espaces de Riemann, Proc. Imp. Acad. Tokyo 15 (1939), 247-252. (MR 1, 175; Zbl 22, 398.)
(Y 39c)
Yano, K., Sur les equations de Codazzi dons la gdomdtrie conforme des espaces de Riemann, Proc. Imp. Acad. Tokyo 15 (1939), 340-344. (MR 1, 175; Zb1 22, 398.)
[Y 39d]
Yano, K., Sur la thdorie des espaces it connexion conforme, J. Fac. Sci., Imp. Univ. Tokyo Sect. 1 4 (1939), 1-59. Thesis, Univ. of Tokyo. (MR 1, 88; Zbl 22, 170.)
[Y 40a]
Yano, K., Sur quelques propridtds conformes de V% dons V,,, dons V,,, Proc. Imp. Acad. Tokyo 18 (1940), 173-177. (MR 2, 21; Zbl 23, 271.)
[Y 40b]
Yano, K., Concircular geometry III, Theory of curves, Proc. Imp. Acad. Tokyo 16 (1940), 442-448. (MR 2, 303; Zbl 25, 85.)
(Y 40c]
Yano, K., Concircular geometry IV, Theory of subspaces, Proc. Imp. Acad. Tokyo 16 (1940), 505-511. (MR 2, 303; Zbl 25, 85.)
(Y 42]
Yano, K., On the fundamental theorem of conformal geometry, Tensor 5 (1942), 51-59 (Japanese). (MR 9, 209; Zbl 63/II, p. A1045.)
[Y 43a]
Yano, K., Sur les Equations fondamentales dons la gComCtrie conforme
des sous-espaces, Proc. Imp. Acad. Tokyo 19 (1943), 326-334. (MR 7, 332; Zb1 60, 389.) [Y 43b]
Yano, K., Sur une application du tenseur conforme C,k et du scalaire conforme C, Proc. Imp. Acad. Tokyo 19 (1943), 335-340. (MR 7, 332; Zbl 60, 389.)
[Y 47]
Yano, K., Geometry of connections, Kawade-shobo, Tokyo, 1947, 2+4+2+185 pp. (Japanese). (MR 12, 282.)
[Y 741
Yano, K., On complex conformal connections, Kodai Math. Sem. Rep. 28 (1974/1975), 137-151. (MR 51 #13905; Zb1 302:53013.)
[Y 76]
Yano, K., On contact conformal connections, Kodai Math. Sem. Rep. 28 (1976), 90-103. (MR 57 #7455; ZbI 341:53031.)
[YC 71a]
Yano, K., and B. Y. Chen, On the concurrent vector fields of immersed
submanifolds, Kodai Math. Sem. Rep. 23 (1971), 343-350. (MR 45 #5922; Zbl 221:53049.)
354 [YC 71b]
BIBLIOGRAPHY
Yano, K., and B. Y. Chen, Minimal submanifolds of a higher dimensional sphere, Tensor (N.S.) 22 (1971), 370-373. (MR 44 #4699; Zbl 218, 350.)
[YC 73]
Yano, K., and B. Y. Chen, Some results on conformally flat submanifolds, Tamkang. J. Math. 4 (1973), no. 2, 167-174. (MR 50 #5667; Zbl 283:53026.)
[YI 69)
Yano, K., and S. Ishihara, Pseudo-umbilical submanifolds of codimension 2, Kodai Math. Sem. Rep. 21 (1969), 365-382. (MR 40 #1947; Zbl 197, 182.)
[YM 38]
Yano, K., and Y. Mutb, Sur la determination dune connexion conforme, Proc. Physico-Math. Soc. Japan 20 (1938), 267-279. (Zbl 18, 376.)
(YM 391
Yano, K., and Y. Muto, A projective treatment of a conformally connected manifold, Proc. Physico-Math. Soc. Japan 21 (1939), 270-286. (ZbI 21, 427.)
(YM 41a]
Yano, K., and Y. Muto, Sur la thdorie des hypersurfaces dans un espace a connexion conforme, Japan J. Math. 17 (1941), 229-288. (MR 7, 331; Zbl 60, 389.)
[YM 41b]
Yano, K., and Y. Muto, Sur la thdorie des espaces d connexion conforme normale et la gComdtrie conforme des espaces de Riemann, J. Fac. Sci. Imp. Univ. Tokyo Sect. 14 (1941), 117-169. (MR 3, 192; Zbl 63, A1045.)
[YM 42a]
Yano, K., and Y. Muto, Sur le thdorbme fondamental dons la geomdtrie conforme des soul-espaces riemanniens, Proc. PhysicoMath. Soc. Japan 24 (1942), 437-449. (MR 7, 332; Zbl 60, 389.)
[YM 42b]
Yano, K., and Y. Muto, On the fundamental theorem of conformal geometry, Tensor 5 (1942), 51-59 (Japanese). (MR 9, 203; Zbl 63, A 1045.)
[Z 78]
Zhogova, T. B., On the focal three-web of a two-parameter family of two-dimensional planes in P5, The Geometry of Imbedded Manifolds, Moskov. Gos. Ped. Inst., Moscow, 1978, 40-46 (Russian). (MR 82a:53010; Zbl 444:53009.)
(Z 79]
Zhogova, T. B., On a class of two-parameter families of twodimensional planes in Ps with a hexagonal focal three-web, The Geometry of Imbedded Manifolds, Moscow, 1979, 44-50 (Russian). (MR 82f:53026; Zbl 484:53004.)
(Zi 02]
Zindler, K., Liniengeometrie mit Anwendungen, G. J. Goschen, Leipzig, 1. Band, 1902, viii+380 pp. (Jbuch 33, 682-683); 2. Band, 1906, vii+248 pp. (Jbuch 37, 673-674.)
Symbols Frequently Used The list below contains many of the symbols whose meaning is usually fixed throughout the book.
I. Groups
G': G2(n):
structure group of CO(p, q)-structure and AG(p - 1, p + q - 1)-structure, 120, 166, 268, 273, 275,286,299-301 prolonged group of G, 128, 166, 274, 286, 299-301 group of admissible transformations of second-order
GL(n):
frames, 121 n-dimensional general linear group, 34, 120, 166, 254
G:
SU(n): T(n):
structural group of AG(p - 1, p + q - 1), 268 group of homotheties, 34, 120, 166, 254, 268 stationary subgroup (isotropy group) of a point x E V, 2, 34, 41, 44, 75, 80 n-dimensional orthogonal group, 2, 175 n-dimensional pseudoorthogonal group of signature (p, q), n = p + q, 7, 17, 120 group of conformal transformations of C", 7, 75 group of conformal transformations of Ca, 17, 127 fundamental group of projective transformations of the space P"+1, 150 multiplicative group of reals, 269 special n-dimensional linear group, 151, 166, 268 special n-dimensional orthogonal group, 120, 172 Lorentz group, 169 special n-dimensional pseudoorthogonal group of signature (p, q), n = p + q, 6, 7, 17, 120, 299 special n-dimensional unitary group, 172 n-dimensional group of parallel translations, 2, 34,
Z2:
127, 166, 264, 274 cyclic group of second order, 7
GL(p, q): H:
H, H= (V): O(n): O(p, q):
PO(n + 2, 1): PO(n + 2, q + 1): PSL(n + 2):
R': SL(n): SO(n): SO(1,3): SO(p,q):
355
SYMBOLS FREQUENTLY USED
356
II. Manifolds, Submanifolds, Spaces and Structures
AG(p-1,p+q- 1): A":
A: m AGW(d,p, q):
almost Grassmann structure on Ma9, 267 affine space of dimension n, 263, 302 m-quasiaffine space of dimension n, 263, 273 almost Grassmannizable d-web of codimension q on MDO: 318
algebraic d-web of codimension q on MDQ: 302 field of complex numbers, 170
AW(d,p,q): C:
C".:
isotropic cone with vertex at oo, 16 conformal space of dimension n, 3, 89 n-dimensional pseudoconformal space of index q,
C': C"v (C")Z and
(C,").
:
CO(p, q):
CPa and CPg: 'ipk). CT=(M): C=:
Cr"":
0(a) and o(p): O(k)(a) and E« and E0: G(m, n): G" (m, n): G(1, 3): GW (d, p, q):
H": G:
AI:
f2(m,n): fl (1, 3):
PN: (PN)*: papa,... a-: p`j: Q": QQ
R:
R":
yk)(f3)
16, 31, 100, 127, 141, 169, 221, 264 local conformal and pseudoconformal space at point x, 126, 132 pseudoconformal structure of signature (p, q), 120, 163 complex projective lines, 171 asymptotic cone of order k of fl(m, n) at p, 240 complexified tangent space at x E M, 170 isotropic cone with vertex at x, 6, 13, 100, 141, 164 osculating (m + ml)-sphere of VI, 85 distributions of two-dimensional elements on EQ and E0, 186 plane generators of Cpk), 241-243 isotropic fiber bundles and fiber bundles of plane generators of Segre cones: 165, 269 Grassmannian of m-dimensional subspaces in P", 221, 267 normalized Grassmann manifold, 253
Pliicker manifold of straight lines in P3, 19, 108, 221 Grassmann d-web of codimension q on M": 302, 306 n-dimensional hyperbolic (Lobachevsky) space, 66 Lie hyperquadric, 25 differentiable manifold, 119 image of the Grassmannian G(m, n), 223 Pliicker hyperquadric, 20 projective space of dimension N, 3, 7 projective space dual to pN, 224 Grassmann coordinates of P"' C P", 222 Pliicker coordinates of a straight line in p3, 20, 232 hyperquadric in Pn+1, 6, 66, 87 hyperquadric of index q in Pn+i, 16, 131, 141 field of real numbers, 19, 230 vector space of all ordered n-tuples of real numbers,
86,198,224
SYMBOLS FREQUENTLY USED
357
RPo and RP0: RP2: Rk(X):
n-dimensional Euclidean space, 1, 34, 66, 115, 264 n-dimensional pseudo-Euclidean space of index q, 14, 127, 166, 264 real projective lines, 165 real projective plane, 193 bundle of frames of order k over X, 31, 40, 44, 49, 75,
Sk:
120, 141, 145 k-dimensional elliptic space or k-dimensional sphere
R": R' n:
S"'1: SQ :
SA': SC. (p, q):
in C", 17, 66, 72, 145 hypersphere in C", 72 n-dimensional pseudoelliptic space of signature q, 262 Segre-affine space of dimension p, 264, 300 Segre cone of G(m, n) or AG(m, n) with vertex at x, 267
S(k, 1):
T=(M): T.kl (M): U°: V'": V'": VQ
V"-1: W (d, p, q):
n-dimensional Segre variety (Segrean), n = k + 1, 225 tangent space of M at x, 100, 120, 126, 268 osculating subspace of order k of M at x, 211, 239 normalized domain of the Grassmannian, 253 submanifold of dimension in, 73 tangentially degenerate submanifold of dimension m and rank r, 68, 108 n-dimensional Riemannian manifold of signature q, 140 hypersurface in a n-dimensional space, 31 d-web of codimension q on MDQ, 195, 270, 301
III. Tensors and Geometric Objects alk: aq or b..: a
a= {a' k}: (a, b): B;,,k:
b = (CJkg): CQ and CO:
filzl: 4iZi g:
torsion tensor of a web W (d, n, q), 303
second fundamental tensor of V"-1 C P", 38, 152, 208 torsion tensor of AG(p - 1, p + q - 1), 279 second structure tensor of AG(p - 1, p + q - 1): 286 Darboux tensor of a hypersurface, 152, 209 curvature tensor of a web W(d, n, q), 303 tensor of conformal curvature (Weyl tensor), 125, 142 subtensors of the tensor of conformal curvature of CO(2, 2)-structure, 168
second fundamental form of V"` in C" (or P"), 37 second fundamental forms of V'" in C", 78 fundamental form of CQ , CO(p, q)-structure and AG(p - 1, p + q - 1)-structure, and also first fundamental form of V'" C C" and V'" C CQ, 13, 18, 37, 74, 100, 119, 141, 148, 163
9;j: K(!; A ri):
fundamental (metric) tensor, 9, 73 conformal sectional curvature, 183
SYMBOLS FREQUENTLY USED
358
scalar curvature, 52, 133, 190 Ricci tensor, 51, 133, 190 torsion tensor of an affine connection, 147 curvature tensor of affine connection or Riemannian manifold, 51, 130, 133, 147
R R;i : R2 k:
Rj'kI: S = (a,6,c):
third (complete) structure tensor of AG(p - I, p + q - 1), 287
S and S0: Tjk:
structure subobjects of the structure object S of AG(p - I, p + q - 1), 298 deformation tensor, 47, 137
IV. Other Symbols d: 6:
6ij,6;i:
I'°: ry:
Id:
a: V: Va: v:
PT: a': ®: ®:
x: A:
oo:
(, ):
exterior differential, 11 symbol of differentiation with respect to fiber parameters, 122 Kronecker symbol, 11, 54, 91 affine connection on G°(m,n), 258 affine connection, 136, 147, 196 identity operator, 115, 179, 231 semidirect product, 2, 41, 128, 173, 264, 274, 286 operator of covariant differentiation, 76, 123 operator of covariant differentiation with respect to fiber parameters, 35, 75 normalizing mapping, 253 projectivization of T, 23, 221, 232 symbol of local isomorphism of groups, 2, 299 direct sum, 180 tensor product, 170 direct (Cartesian) product, 44, 166, 216, 225, 264, 273 symbol of equivalence of matrices or structures or spaces, 17, 256 exterior multiplication, 11 point at infinity, 3, 16 scalar product, 1, 32, 105, 126, 261
Author Index Adati, T., 135, 187, 323
Darboux, G., ix, 28, 70, 72, 331 Decuyper, M., 333 Delens, P. C., 70, 331 Demoulin, A., 28, 70, 72, 331 Deszcz, R., 115, 331 Dhooghe, P. F., 320, 331 Dieudonne, J., 1, 19, 332 Dobromyslov, V. A., 263, 265, 266, 332 Dubrovin, B. A., 28, 130, 134, 332 Dupin, C., 72, 332
Akivis, M. A., x, xiii, 6,23, 24, 33, 38, 67, 68, 71, 72, 87, 90, 97, 99, 107, 110, 111, 116, 117, 152, 153, 156, 161, 195, 196, 213, 218,
219, 221, 239, 244, 249, 265, 270, 304, 312, 314, 320, 321,
323-325
Alekseevskii, D. V., 2, 218, 325, 326 Atiyah, M. F., 163, 180, 217, 218, 326
Backes, F., 28, 72, 326 Bailey, T. N., 320, 326 Garner, M., 115, 326 Barrett, J. W., 120, 163, 218, 326 Baston, R. J., 319, 320, 326, 327 Berwald, L., ix, 70, 327 Blair, D. F., 115, 327
Eastwood, M. G., 320, 326 Einstein, A., 15, 219, 332, 343 Eisenhart, L. P., 70, 129, 130, 134, 136, 150, 160, 161, 177, 332
Euler, L., 28, 332 Fialkow, A., 115, 116, 333 Finikov, S. P., 112, 113, 117, 265, 333 Finzi, A., 71, 161, 333 Fomenko, A. T., 28, 130, 134, 332 Fubini, G., ix, 70, 72, 160, 333 Fuhrman, A., 265, 334
Blaschke, W., ix, 28, 29, 63, 70, 72, 94,
115, 327
Bol, G., 117, 195, 327, 333 Bompiani, E., 111, 117, 327 Bryant, R. L., xiii, 33, 115, 124, 327 Bubyakin, I. V., 251, 265, 328
Buchner, K., 116, 346 Burali-Forti, C., 52, 328 Bushmanova, G. V., 115, 328
Gantmacher, F. R., 63, 334 Gardner, R. B., xiii, 14, 33, 124, 131, 291, 327, 334
Carmo, do M., 71, 72, 115, 162, 328 Cartan, It., ix, 14, 24, 28, 72, 98, 128, 150, 160, 161, 219, 291, 328, 329 Cartan, H., 35, 329 Cayley, A., 28, 329 tech, E., ix, 72, 333 Cecil, T. E., 29, 72, 330 Chandrasekhar, S., 140, 170, 189, 191,
Geidelman, R. M., x, 72, 244, 265, 334 Gelfand, 1. M., 250, 334 Gheysens, L., 116, 334, 335 Gibbons, G. W., 120, 163, 218, 326 Gindikin, S. C., 163, 217, 335
Goldberg, V. V., xiii, 6, 23, 33, 38, 67, 68, 72, 90, 94, 97, 99, 107, 110,
117, 152, 153, 156, 187, 198202, 213, 219, 239, 249, 265, 268, 270, 281, 303, 304, 312,
202, 206, 207, 218, 219, 330 Chebysheva, B. P., 265, 325
Chen, B. Y., 70, 72, 115, 116, 330, 353,
313, 320, 321, 325, 335, 336
Goldsmith, H. 1.., xiii, 33, 124, 327 Goncharov, A. B., 319, 320, 336
354
Chern, S. S., xiii, 33, 70, 72, 124, 321, 327,
Graev, M. 1., 218, 250, 325, 326, 334 Grassmann, H., 28, 29, 336
330, 331
Chupakhin, N. P., 244, 344 Cicco, de J., 115, 339
Grifiths, Ph. A., xiii, 33, 72, 124, 161, 239, 265, 321, 327, 331, 336 Grincevicius, K. 1., 244
Dajczer, M., 71, 72, 115, 162, 328 359
AUTHOR INDEX
360 Haantjes, J., 28, 71, 115, 160, 337, 348 Haimovici, A., 115, 337
Hangan, T., 268, 319, 320, 337 Harris, J., 161, 224, 239, 265, 336, 337 Hitchin, N. L., 163, 180, 217, 218, 326 Hlavaty, V., ix, 117, 161, 337 Hodge, W. V. D., 218, 221, 223, 265, 338 Houh, C. S., 116, 338
Hsiung, S. S., 70, 338 Hua, L. K., 265, 338 Huggett, S. A., 218, 338
Matsumoto, M., 71, 72, 343 Mercuri, F., 71, 162, 328 Michor, P., V. xiii, 340 Mikhailov, Yu. 1., 268, 320, 343 Minkowski, H., 28, 332, 343 Miyaoka, R., 72, 343, 344 Miyazawa, T., 135, 187, 323 Mizin, A. G., 244, 344 Monge, G., 28, 342 Mugridge, L. R., 70, 338 Musso, E., 72, 338 MOto, Y., 71, 115, 160, 344, 354
Ishihara, S., 115, 354 Ishihara, T., 319, 338
Jensen, C., 72, 338
Newman, E. T., 170, 218, 344 Nishikawa, S., 71, 344 Nomizu, K., xiii, 46, 94, 131, 147, 258, 340
Karapetyan, S. E., 233, 236, 244, 265,
Norden, A. P., x, 115, 132, 136, 137, 149, 161, 265, 328, 344
338, 339 Kasner, E., 115, 339 Kerr, R. P., 219, 339 Klein, F., x, 7, 19, 20, 28, 29, 54, 221, 230, 339, 340
Ogiue, K., 160, 345 Ozawa, T., 72, 344
Kaplenko, A. F., 265, 338
Klekovkin, G. A., 134, 219, 340 Kobayashi, S., xiii, 46, 94, 120, 121, 128, 131, 147, 160, 258, 340
K61at, I., xiii, 340 Kiinigs, G., 117, 340 Konnov, V. V., 161, 162, 218, 219, 325, 340
Korovin, V. 1., 265, 341 Kossowski, M., 101, 341 Koetrikina, L. P., 265, 347 Kovantsov, N. 1., 113, 117, 341 Kowalski, 0., 115, 341 Krivonosov, L. N., 72, 341 Kruglyakov, L. Z., 244, 341 Kulkarni, R., 160, 341
Lagrange, R., 115, 341 Laptev, G. F., 36, 71, 77, 116, 121, 124, 147, 153, 156, 258, 341, 342
LeBrun, C., 218, 342 Leichtweiss, K., 221, 265, 342 Lichnerowicz, A., 147, 258, 342 Lie, S., 27, 342
Liebmann, H., 115, 342 Liouville, J., 2, 28, 342 Little, J. B., 321, 342 Lorentz, H. A., 332, 343 Lumiste, Yu. G., 71, 116, 342
Nordstrom, G., 219, 344 Novikov, S. P., 28, 130, 134, 332
Pedoe, D., 221, 223, 265, 338 Pendl, A., 115, 345 Penrose, R., 163, 165, 170, 189, 191, 217-219, 344, 345 Perepelkine, D., 115, 345
Perry, M. J., 120, 163, 218, 326 Petrescu, S., 115, 345 Petrov, A. Z., 191, 219, 262, 345 Pinkall, U., 72, 346 Pirani, F. A. E., 191, 219, 346 Pliicker, J., 19, 28, 29, 346 Poincark, H., 28, 346 Ponomarev, V. A., 265, 338 Pope, C. N., 120, 163, 218, 326 Rawnsley, J. H., 218, 346 Reissner, H., 219, 346 Rham, G. de, 218, 346 Ribaucour, A., ix, 70, 72 Rindler, W., 165, 189, 191, 218, 219, 345 Room, T. G., 265, 346 Rosca, R., 116, 346 Rosenfeld, B. A., x, 1, 7, 28, 115-117, 151, 226, 229, 265, 338, 346, 347 Roth, L., 265, 349 Rouxel, B., 116, 347 Rubak, P., 120, 163, 218, 326 Ryan, P., 72, 330
Maeda, J., 115, 343 Maeda, Y., 71, 344
Saksteder, R., 72, 348
Manin, Yu., 1. 217, 218, 343
Sasaki, S., ix, x, 115, 160, 161, 348
AUTHOR INDEX Sasaki, T., 162, 348 Scheffers, G., 27, 342
Schiemankgk, C., 49, 52, 115, 348 Schild, A., 219, 339 Schouten, J. A., ix, x, 28, 52, 160, 161,
348
Schubarth, E., 115, 348 Schwarzschild, K., 219, 349 Segre, C., 28, 349 Semple, J. G., 265, 349 Seven, F., 224, 265, 349 Shcherbakov, N. R., 244, 344 Shcherbakov, R. N., 244, 349 Shelekhov, A. M., 195, 196, 270, 304, 312, 314, 320, 325 Singer, 1., 163, 180, 217, 218, 326
SiovSk, J., xiii, 340 Solodovnikov, A. S., 2, 326 Stepanova, G. B., 265, 347
Sternberg, S., xiii, 125, 160, 72, 349 Struik, D. J., 28, 52, 160, 161, 348 Sulanke, R., 49, 52, 71, 72, 115, 348-350
Takasu, T., ix, 70, 71, 115, 350 Thomas, J. M., ix, 160, 350 Thomas, T. Y., ix, 160, 350 Thomsen, G., ix, 70, 71, 115, 350 Thorbergsson, G., 72, 346, 350 Tikhonov, V. A., 72, 115, 350, 351 Timoshenko, T. A., 116, 347
361
Tod, K. P., 218, 338 Tresse, A., 70, 350
Vasilyev, A. M., 24, 351 Vedernikov, V. I., x, 72, 115, 116, 161, 351 Verbitsky, L. L., x, 115, 116, 160, 161, 351 Verheyen, P., 115, 116, 334, 335, 351 Verstraelen, L., 115, 116, 334, 335, 351 Vessiot, E., ix, 70, 351, 352
Vinberg, E. B., 2, 326 Vlasov, A. K., 233, 265, 352 Voss, A., ix, 70, 352 Vranceanu, G., 160, 352 Wells, R. 0., Jr., 177, 218, 352 Weyl, H., ix, 160, 161, 332, 343, 352 Wilczynski, E. J., ix
Wolf, J. A., 14, 352 Wong, Y. C., 71, 221, 352 Woude, W. van der, 115, 353 Yano, K., ix, 28, 72, 115, 160, 330, 353, 354
Yuchtina, T. 1., 265, 347
Zamakhovskii, M. P., 116, 347 Zayatuev, B. V., 219, 325 Zhogova, T. B., 245, 265, 354 Zindler, K., 117, 354
Subject Index variety 11(1,3), 109-114, 117, 232 variety 11(1,4), 232, 233
Abelian equation, 321 Absolute, 18, 66, 127, 264
variety 11(2,4), 235, 236
parallelism, 148 tensor, 36, 40
variety n(2,5), 245, 250
Adapted frames, 164, 165, 188, 218, 274
Algebraizable webs, 302 Almost Grassmann manifold, 319
Admissible transformation of
adapted frames, 135, 188, 214, 274 frames of first order, 148 m-pair, 254
Almost Grassmann structure, ix-xii, 221, 267-300, 320 associated with web, 304, 305, 312,
316, 318
integrable, 270 semiintegrable, xii, 270, 292, 300 structure equations of, xii, 274-276, 285, 300, 320
second order, 121, 273, 284
third order, 125, 283, 284
reduced family of second-order frames, 279 Affine connection, 132, 133, 137, 141,
structure group of, 268
torsion tensor of, 274, 279, 287, 293 subtensors of, 293
147-150, 252, 264
associated with web, 196,309, 311 curvature tensor of, 136, 147, 150 connection forms of, 136, 147 P-, 258-264
Almost Grassmannizable web, 316-320, 321 Almost liermitian symmetric structure, 319
o-plane(s), 164, 166, 171, 184, 185, 196, 215, 218
on normalized submanifold, 115,
o-semifiat conformal structure, 183 a-semiintegrable almost Grassmann structure, 270, 294-299, 309,
141
torsion-free, 115, 148, 258 torsion tensor of, 147 coordinate system, ix differential geometry, ix
318
a-semiintegrable CO(2,2)-structure, 186 o-semirecurrent CO(2, 2)-structure, 187
space, 117, 150, 161, 263, 264, 302,
a-submani fold (s), 215
304
a-subspace(s), 242, 243
transformation(s), 147
Alternation, 40, 275, 285, 287, 292, 293,
Weyl connection, 133, 149, 150, 205
319
Affinor, 62
Anti-involutive operator, 180 Anti-self-conjugate subspaces, 182 Anti-self-dual
Burali-Forti, 52, 113 symmetric, 52, 107 AG(m, n)-structure, 267, 301 AG(1,3)-structure, 270, 300, 301 Algebraic d-web, 302 geometry, 224, 225 of Grassmannian, 265
asymptotic CO(2, 2)-structure, 215 CO(1,3)-structure, 218 CO(2, 2)-structure, 197, 199 part of Weyl tensor, 217 structure, 183, 185, 186 subspace, 180, 182, 183, 218
variety A(m,n), 20-23, 28, 223-225, 238-244,267-271,286,290,291,
Apolarity, 51, 78, 85, 92 Apolar tensors, 38, 39, 58, 61, 66 Asymptotic
300
cone on, 225 manifold, 240
cone(s),
363
SUBJECT INDEX
364
of algebraic variety fl(m, n), 240-246, 250, 291 of CO(2, 2)-structure, 208, 209, 214 of Darboux hyperquadric, 104, 106 filtration of, 240 of hyperquadric, 17-18 of hypersurface, 208, 209 of order k, 240, 242, 243
of tangent space, 178 0-plane(s), 164, 166, 171, 184, 185, 196, 215, 218
l-semiflat conformal structure, 183 fl-semiintegrable almost Grassmann structure, 270, 294, 299, 310, 318
#-semi integrable CO(2, 2)-structure, 186, 194, 202
of second order, 240, 241, 243, 245, 250
of submanifold, 106 of third order, 241, 242, 245-247, 250
CO(2,2)-structure, 208, 209, 210 semiflat, 215, 216 conformal connection, 157 conformal structure, 150, 153, 161 flat, 155, 158, 161 direction(s) of second order, 250
of submanifold, 245, 246, 250 of variety f1, 239, 240, 244 form of second order, 152 line(s) of order k, 244 of submanifold, 250 of surface, 110, 111 of third order, 246 of variety fl, 244 Axially symmetric metric, 202, 204, 206, 207
fl-semirecurrent CO(2,2)-structure, 187
$-submanifold, 215 fl-subspace, 243
Bianchi equations, 289, 290 Bijective correspondence, 171 Bijective mapping, 245, 267, 270 Biquadratic algebraic submanifold, 155 Bisecant variety, 241 Bivector(s), 178, 183, 259 indices, 179, 181 isotropic, 184 space, 178, 180
Bundle of central hyperspheres, 85 frame, 11, 176 invariant, of 2nd fundamental forms, 78, 90 of isotropic frames, 204 of m-subspaces, 243, 273, 302, 306 of normal hyperspheres, 82
of second fundamental forms, 79 of straight lines, 22, 114 of subspaces, 224, 231 tangent, 46, 75, 238
Base
of bundle of first-order frames, 75,
of tangent hyperspheres, 78
120
of fibration, 302 forms, 33, 238 of frame fiber bundle, 11, 75, 121 hypersurface, 151 frame fiber bundle, 11 of isotropic fiber bundle, 164, 171,
176
parameters, 122 variables, 121 Basis affinor, 315, 316, 320 Form(s), 74
of affine connection, 136 of almost Grassmann structure, 274 of frame bundle of Grassmannian, 253, 271 of Grassmannian, 238, 253, 254, 257
of hypersurface, 34, 35, 46, 48, 49 of manifold, 177 of subspace, 229
Canal hypersurface, xi, 57, 60, 61 submanifold, 145 surface, ix, 70, 72 Canonical
form, 262 frame, 70 normal connection, 320
Cartan's lemma, 33 normal connection, 128, 320 number, 99 test, 99, 250 variety, 98, 213, 215
Cartesian coordinates, 127, 159 coordinate system, ix, 3, 159 Center of bundle of subspaces, 232, 243 hypersphere, 2, 4 inversion, 3
SUBJECT INDEX pencil, 251, 252 projectivization, 225 Central rn-sphere, 78, 79, 145, 1.16 tangent complex, 112 tangent hypersphere, 40, 41, 45, 47, 61, 66, 78, 79, 81, 84
Character, 99 Characteristic(s), 57, 155, 157 cone, 79 equation, 54 polynomial, 181 straight line of subspace, 247-252
straight line of torse, 247, 248 subspace(s), 251 Classical differential geometry, ix, 70 Closed
form equation(s), 71, 198, 200, 201, 215, 287
365
linear, 21, 107, 109-112 plane, 189, 231 projective line, 171 representation, 173, 218 Riemannian manifold, 218 of straight lines, 108-113, 117 of general type, 109, 111 linear, 21-23, 28, 109 special, 109-111
of subspaces, 244 of tangents, 117 transformation, 170 of 2-subspaces, 245, 250-252, 265
Complexification, xi, 217, 270 Complexified space, 180
Complexified tangent space, 170-172 Complex projective line(s), 171, 172 Cone(s) asymptotic, see Asymptotic cone
system, 99, 273 Co-basis, 35
of asymptotic directions of variety
forms, 304, 306 Co-frame, 120, 274 Commuting of lnors, 62 Compact differentiable manifold, 3 pseudoconformal space, 16 Compactification, 3, 16, 64, 126, 127, 204,
characteristic, 79
286
Compactified tangent space, 127, 300 Complementary subspaces, 223, 226, 242, 245
Complete fundamental object, 71, 77, 87 invariant normalization, 82 structure object, 281, 287, 291, 298, 299, 301
Completely integrable system, 11, 56, 60, 64, 71, 148, 186, 271, 274, 294, 296, 312
isotropic submanifolds, xii, 183, 185, 186, 194, 211-215
Complex, 28, 29 conjugate curvature tensors, 174-176 directions, 188
eigenvalues, 230 fiber bundles, 171 forms, 170 generators, 171
subspaces, 180, 182 transversals, 230
coordinates, 2 form(s), 173, 174 generator(s), 270
fl, 240 degenerate, 54
field of, x imaginary, 6 isotropic, see Isotropic cone of second order, x, 53, 79, 80, 120, 141, 195, 243, 299, 301
plane generator of, 195 structure, 320 Conformal connection, 161, 204 correspondence, 64, 65 deformation, 116, 161 differential geometry, ix, x of submanifolds, x, 71, 115 differential invariant, ix, 70 flatness, 209 geometry, ix, x, 70-72, 115, 121 invariance, 160 mapping, 131, 145, 161 model of H", 66 model of S", 66 moving frame, 8, 9, 66, 87 of CO(p, q)-structure, 126 of hypersurface, 31 of submanifold, 73, 90 rigidity, 46, 72 sectional curvature, 184
semiflatness, 209 space, ix-xi, 3, 8, 25, 28, 31, 33, 45, 87, 116, 141, 144, 150, 161, 221, 264 of Lorentzian signature, xi projective interpretation of, 28
proper, 16
SUBJECT INDEX
366
structure, ix, x, xii, 37, 119, 141,
144, 148, 150, 153, 160, 161,
202, 208, 209, 217, 299, 320
of third class, 236
of 2-subspaces, 245-247
Conjugate
conformally flat, xii, 175 curvature forms of, 126
bundles, 164
CO(p,q), 120, 125, 126, 132, 141,
net, 67, 213, 249 points, 69
142, 146, 284, 301
on four-dimensional manifold, 175 on hypersurface, 153, 161 realization of, xii on submanifold, 90 ultrahyperbolic, 120, 163 theory of spheres, x
transformation(s), 2, 3, 6, 8, 14, 28, 46, 49, 52, 70, 85, 87, 119, 136, 140, 160
group of, 3, 6, 28 of pseudo- Riemannian metric, 136
of Riemannian metric, 136, 137, 140
Conformally connected space, ix
equivalent hypersurfaces, 46, 49 equivalent Riemannian metrics, 132, 189
Euclidean space, 161
flat conformal structure, xii, 175, 300 CO(1,3)-structure, 175, 218 CO(2,2)-structure, 169, 185, 194, 209, 215, 218
hypersurface, 217 submanifold(s), 144 invariant form, ix geodesics, 161 metric, 116 operator, 179, 218 properties, 161 tensor, 116 Lorentzian structure, 120 recurrent structure, 135, 136 semiflat CO(1,3)-structure, 175 semiflat CO(2, 2)-structure, 209 symmetric invariant connection, 136 Congruence(s), 28, 29 of circles, ix
normal, 64 of hyperapheres, 72 of isotropic geodesics, 189, 191-193 pair of, 117 of pairs of points, 115 quadratic, 117
of spheres, 72 of straight lines, 108, 109, 113, 117 of subspaces, 244
directions, 67
subspace(s), 260
Connected
component of identity, 10 conjugate, domain, 86, 87 hypersurface, 45 submanifold, 73 Connection, affine, see Affine connection Connection forms of
affine connection, 136, 147 Riemannian connection, 136-140
Connection r,, 258-264 curvature tensor of, 258 Contact hypersphere, 54, 56 Coordinate(s) of bivector, 184, 259 Cartesian, 127, 259
Grassmann, 222, 223, 232, 234, 235 homogeneous, see Homogeneous coordinates of linear subspace, 28 nonhomogeneous, see Nonhomogeneous coordinates PlOcker, 20
projective of point, 227 simplex, 224
of straight line, 28 tangential, 236 transformation, xi Correlation of projective space, 224 Covariant derivative, 50, 148 of curvature tensor, 187 differential, 259
differentiation, 123, 136, 148, 260 Covariantly constant fundamental tensor, 260, 262 tensor, 135 Covector, 133 form, 131 CO(1,3)-structure, 163, 169-172, 174-176, 180, 183, 188-193, 202-204, 217-219, 270 CO(2, 2)-structure, 163-169, 172, 175, 176, 180,194,195,217-219,270,301 CO(4,0)-structure, 163, 172, 175, 176, 180, 183, 189, 190, 217-219, 270 Cross-ratio of four points, 229
SUBJECT INDEX of four subspaces, 265, 316
of quadruple of plane generators, 231
of two m-pairs, 228-231, 256, 257, 260, 265
of two points and two hyperplanes, 229 Cross-section of bundle of frames, 204 Cubic cone of directions, 245 curve, 246, 247 Darboux form, 153, 212, 214 equation, 236
hypersurface, 250 submanifold, 250 Curvature form(s) of almost Grassmann structure, 292
connection r,, 258 CO(p,q)-structure, 132 isotropic fiber bundle, 169, 182, 183 Weyl connection, 51 Curvature lines, 52, 53, 62-64, 70, 89, 92 isothermic, ix, 70
spherical, ix, 70 Curvature object of CO(p, q)-structure, 126 Curvature tensor, 129 of affine connection, 51, 133, 136, 147, 303
of almost Grassmann structure, 300 of connection C", 258, 259, 262 of CO(1,3)-structure, 205 of CO(2,2)-structure, 301
of empty space, 190
of fiber bundles E. and E8, 168, 174, 183, 191-193, 300 of four-dimensional Riemannian manifold, 190 of Grassmann (p + 1)-web, 307, 309 of isotropic bundle, 168, 174, 183 of (p + 1)-web, 303, 304
of Riemannian manifold, 259 of three-web, 196
367
tensor, 152, 154, 157, 209 Deformation conformal, 116, 161 of submanifolds, 72 tensor, 47, 137 Degenerate cone, 54 congruence, 114
of first kind, 114 of second kind, 114 form, 102
hypersphere, 64 inversion, 66 linear congruence, 22 m-pair, 227 null-pair, 193 Degree of Segre variety, 225, 245 variety fl(1,4), 232 variety 11(2, 5), 245 Derivational equations, 70, 71, 116 Determinantal variety, 240, 242, 265 dimension of, 242 plane generators of, 242
Developable ruled surface, 109
submanifold, 247, see also Torse surface(s), 24, 68, 69, 108, 1 t 1-113, 213, 215, 250 Differentiable correspondence, 253
field of Segre cones, 267, 268, 300 manifold, x, 20, 27, 31, 35, 119, 222-224, 267, 274 compact, 3
structure equations of, 121 mapping(s), 20 submanifold, 253 Differential covariant, 259
of Weyl connection, 51, 133, 150 Curve, 8, 13, 42, 43, 70, 161 cubic, 246, 247 integral, 188, 189 isotropic, 24, 102, 109 Curvilinear coordinates, 86, 119, 120 Curvilinear two-web, 94 Cuspidal edge, 112 Cyclic group, 7
equation(s) of distribution, 202
Darboux form, 212, 214 hyperquadric, 16, 26, 104, 105 asymptotic cone of, 106
form, xiii, 35, 120, 125 exterior, x, 71 -geometric structure, 128 geometry, ix, xiii affine, ix classical, ix
mapping, 6, 10, 13, 17, 18, 24-26, 28, 66, 67, 87-90, 108, 117
geodesics, 138, 139 geometric object, 36 hypersurface, 46 normalizing mapping, 253 relative invariant, 50, 82 submanifold, 210 tensor, 39, 77, 125, 130, 168 three-web, 195
SUBJECT INDEX
368
conformal, ix of Grassmannians, 221, 236, 265 of m-spheres, 115
proper, 253
simply connected, 86, 87 Double congruence of isotropic geodesics,
projective, ix
of submanifold, 77 of submanifold of spheres, ix invariant, 70 operator V, 123 prolongation, 128, 286, 303 Differentiation with respect to fiber parameters, 122, 176 Dilation, 160 Dimension of asymptotic cone of fl, 241 determinantal variety, 242 Grassmannian, 221, 238, 252, 271
kth osculating subspace to fl(m,n), 239 plane generator of cone, 242 plane generator of Segre variety, 264
191-193
family of isotropic lines, 102 line, 229 principal directions, 219
principal distribution, 189, 191-193, 206 quadric, 155, 157 root(s), 189, 191, 192
Dual
frame, 237, 311, 315 projective space, 224, 235, 236, 253, 272
space, 224, 235, 236 Dupin's cyclide, 58, 72 Dupin's submanifold, 72 d-web, 301, 314, 316
quasiaffine space, 273 Segre cone, 240
Eigendirection(s), 54, 108, 181
Segre variety, 225
Eigensubspace, 180-183
tangent subspace to 0(m, n), 238,
Eigenvalue(s), 54, 91, 181, 229-231
245, 246 variety fl(1,4), 232
Direction(s)
Eigenvector of operator, 316
Einstein equation, 190, 206, 219 Einstein space, 163, 190, 218
asymptotic, see Asymptotic direction(s)
of type D,
conjugate, 67 focal, 246, 247
of type N, 192, 218 of type 0, 218
isotropic, see Isotropic direction(s)
of type 1, 191, 218 of type 11, 191, 192, 218 of type 111, 192, 218
optical, 219 principal, see Principal direction(s) Direct product of projective spaces, 225, 264,
Directrices of linear congruence, 22 Directrix of Segre cone, 232, 271 Discriminant of quadratic form, 176 Discriminant tensor, 177, 180 Distribution(s), 94, 251 O(Q) and A (O), 194, 197
double principal, 189, 191-193, 206 holonomic, 94 horizontal, 147 integrable, 198, 200-202 involutive, 94, 95 of plane elements, 269, 298
of two-dimensional elements, 186
Domain connected, 86, 87 external, 102 homogeneous, 262 internal, 102
192, 218
Elation, 160 Elliptic congruence, 113 hypersurface, 208 linear congruence, 22 point, 111 space, 66 submanifold, 111 Embedding, 225, 264 Empty space, 190 curvature tensor of, 190 Energy-momentum tensor, 190 Envelope of family of hyperquadrics, 155-159 of hyperspheres, 57, 58, 60 of spheres, 71, 98, 116, 145 Envelope of 0-focal family, 244, 246 Equation(s), of asymptotic cone, 241, 245 characteristic, 54
normalized, 253, 254, 257, 260-264 open, 253, 260
of cone C:, 195, 196 of embedding, 225
of principal directions, 53
of geodesics, 138, 140, 189
SUBJECT INDEX
369
of hyperquadric, 24, 261 Maurer-Cartan, 151 of Segre cone, 268 of Segre variety, 230
of Grassmannian, 273 of normalized Grassmannian, 254, 257
of second order frame bundle, 276 of frame fiber bundle, I1
of Vlasov congruence, 236 Equiaffine Weyl connection, 134 Ricci tensor of, 134
Equivalent webs, 302 Erlanger program, 28 Euclidean geometry, 72 metric, 150 plane, 28 space, xi, 1-3, 5, 7, 8, 15, 28, 33, 66, 70, 115, 117, 264, 265 multidimensional, 70, 115, 161 three-dimensional, 28, 70, 117
of isotropic fiber bundle, 164, 171 parameters, 77, 80, 143, 176, 204, 274, 308
variables, 121 Fibering(s), 237
Fibration(s), 302
isotropic, 218 twistor, 218
Field of asymptotic directions, 250 of cones, x
of second order, 120, 299 of geometric objects, 71
Existence of geodesics, 138
totally focal congruence, 250
Exterior derivative, 272, 273 differential forms method, x, 71 differentiation, 45, 50, 64, 75-77, 82, 99,121,143,151,153,198,257, 281, 303, 307, 318
of Segre cones, 271, 300
tensor, xiii theory, xi vector, xiii, 148 Filtration of asymptotic cones, 240 Finite type C-structure, 125, 128
First fundamental tensor of hypersurface,
product, 19, 177
46, 51
quadratic form(s), 169, 178 External domain, 102
integral(s), 12 order frames, 59, 66 bundle of, 120, 121, 126, 128, 131 prolongation of group, 173 structure tensor of almost Grassmann
Family of a-planes, 166 #-planes, 166
frames, 14, 41, 44, 268 of Segre cone, 268 hyperquadrics, 155-159 hyperspheres, 26, 45 hypereurfaces, 57, 58,
plane generators, 168, 224, 242, 267 planes, 23
structure, 279, 286, 291, 300
structure tensors of bundles E. and
Ep, 306 Five-dimensional projective space, 20, 169, 216, 217 submanifold, 2,50 Five-parameter family of 2-subspaces, 250
Flat affine connection 1", 264 asymptotic conformal structure, 155,
point frames, 253, 263 projective frames, 150, 236, 237, 260
158, 161, 162
spheres, 71
CO(2, 2)-structure, 186, 210, 216, 217
straight lines, 22, 244 submanifolds, 240, 241 tangential frames, 237 torses, 250
metric conformal structure, 162 Flatness, 161
2-subspaces, 244-252
Focal
Fiber bundle(s), 243 of normalized Grassmannian, 254, 257
of plane generators of Segre cone, 269, 300 form(s), 121, 279, 308
of fourth order frame bundle, 285
isotropic distribution, 208
direction(s), 246, 247 family, 68-70 of m-planes, 265 of 2-subspaces, 251 point(s), 246 submanifold, 108 Focus
of generator, 68
370
SUBJECT INDEX
of straight line, 68 Foliation(s), 94, 188, 195-198, 202, 270, 271, 302, 304, 306, 307, 310, 314
isotropic, 186-188, 202 leaf of, 271
one-parameter, 94 Form(s) cubic Darboux, 153, 212, 214 fiber, see Fiber form(s) horizontal, 11, 238 linear, 213 principal, 135 third fundamental, 212, 213 Four-dimensional cone, 250 conformal structure, 163, 218, 219,
320
CO(1,3)-structure, 163, 174-176, 169-173 CO(2, 2)-structure, 163, 164, 166, 175, 176, 270, 314 CO(4, 0)-structure, 163, 175, 176 hypersurface,208-217 manifold, 195, 217, 232, 272, 314 projective space, 250
pseudoconformal structure, xi, xii quadric, 20
Riemannian manifold, xi, 140, 177, 190
curvature tensor of, 190 three-web, 194, 195 Fourth order geometric object, 37, 39
Four-web, 197, 219, 315, 320 of maximum rank, 219 Frame fiber bundle, 11, 176
of almost Grassmann structure of first order, 274, 286, 291 of fourth order, 282 of second order, 286, 291, 320 of third order, 285, 286, 292 base forms of, 11 base of, 11
in C.1, 128 fiber of, 11
of Grassmannian, 253 basis forms of, 253 R1 (M), 120, 121, 126, 128, 147 R2(M), 121, 123, 128, 147, 204 R3(M), 121, 124
RI(Vm), 75, 80, 141, 145 7Z2(Vm), 80, 145 7Z1(V"-1), 31, 33, 40, 41, 151, J55
R2(V"-1), 41 7Z3(Vn-1),44,49 Frenet equations, x, 71, 116
FYobenius theorem, 12, 249, 318 Fundamental form(s), 169
of pseudo- Euclidean space, 14 of variety 11(m, n), 239, 240 group, ix of conformal space, 7 of Lie sphere geometry, 27 of projective space, 19, 24, 271 of Segre-aflne space, 264
of space An274
geometric object(s) of first order, 37 fourth order, 37 kth order, 76 second order, 37, 76, 79, 81 third order, 37, 45, 86 sequence of objects, 77 tensor, 124, 125, 133, 160
of normalized Grassmannian, 258-260 of polar normalization, 261 theorem, xi, 49
Gauss equation, 52 Generalized conformal structure, 319 Segre theorem, 97 General linear group, 34-36, 120, 121 General principal distributions, 192, 192 General relativity, ix, xi, xii, 102, 108, 140, 189, 202, 217, 218, 219, 320
Generating element, 71, 221 Generatrix of Segre cone, 264 Geodesics, 137-139, 161, 189, 191, 192, 310
Geometric object, 36, 37, 277, 278, 283
Geometry algebraic, 224, 225, 265 conformal, ix, x, 70-72, 115, 221 Euclidean, 72 of Grassmannian, 221 of hypersurface, 71 non-Euclidean, 127 projective differential, ix, 88, 97
pseudoconformal, 221 of submanifold, 77 of surface, 71 Grassmann coordinates, 222, 223, 232, 234, 235 d-web, 302-309, 320
curvature tensor of, 307, 309 torsion tensor of, 307-308 manifold, see Grassmannian mapping, 221, 223, 232, 238, 244, 250, 257
371
SUBJECT INDEX
PSL(n + 2), 151 structure, ix, x, xii, 221, 268, 290, of rotations of R,-, 127 320 Grasamannian, x, xii, 221-224, 237-244, 252, 260-265, 305-307, 319 basis forms of, 238, 253, 254, 257
G(m, m + 2), 242 G(m, n), 222, 244, 267, 268, 306
G(l,n-m+1), 242
G(1,3), 19-23, 108, 221, 242, 265 G(1,4), 232, 235
G(2,4), 235 G(2,5), 245 realization of, x rectilinear generator of, 224 of straight lines, xi submanifolds on, x, xii, 253 Grassmannizability condition, 314 Grassmannizability problem, 320 Grassmannizable webs, 302, 303, 309, 313, 314, 318, 319
Gravitational constant, 190 Gravitational radius, 206 Group, 2 of admissible transformations of second-order frames, 121, 124 of of Ine transformations , 147 of conformal transformations, 3, 6, 10-12, 14, 17 intransitive, 12 cyclic, 7
R', 269
SL(p), 151, 166, 271, 273, 300 SL(2, C), 171 SO(n), 120, 172 SO(p, q), 6, 7, 17, 27, 120, 127, 169, 172-174, 299 SU(2), 172
T(n) of parallel translations, 2, 34, 41,166,173,175,264,274,286, 299
of transformations of pseudoconformal space, 128 transitive, 2 Z2, 7, 17 G-structure, 120, 125, 218, 301 of finite type, 125, 160, 284
Harmonic function, 218 intersection, 24
normalization, xii, 255, 258, 261 Hermitian symmetric space, 320 Hodge operator, 176, 178, 181, 218 Hodge tensor, xii, 178, 218 Holomorphic geometry, 217 Holonomic distribution, 94 Holonomic net of curvature lines, 55, 56, 94-98
fundamental, ix, 7
Homogeneous
general linear, 34-36, 120, 254 G0, 166, 172
coordinates of hypersphere, 16, 25, 127 point, 6, 19, 25, 223 straight line, 20 domain, 262 geometric object, 128, 130, 152 normalization, 260, 262, 264 space, 2, 71, 77, 115, 260, 264 Homothety, 2, 3, 173 Horizontal distribution, 147 form(s), 11, 238 invariant distribution, 147 Hyperbolic congruence, 113 linear congruence, 22
Gp, 166, 172 GL(n), 120, 254 GL(q), 304 H of homotheties, 34, 37, 41, 44, 120, 126-128, 166, 169,
171-173,254,269,286,299,300 isotropy, 2, 12
of motions of compactified T:(M), 299, 300
of motions and homotheties of R", 33 Rq , 127
of motions of R", 2 of motions of Ry", 299 of motions of SAP, 286
O(n), 2, 172, 175 O(p, q), 7, 17, 120, 194 PO(n + 2,9 + 1), 17, 19, 127 PO(n + 2,1), 7, 10, 12, 75 invariants of, 75 of projective transformations, 115,
254, 273 pseudoorthogonal, 7, 27, 169, 299
point, Ill T11 led submanifold, 111
space, 7, 66 Hyperboloid of one sheet, 16 Hyperboloid of two sheets, 16 Hypercomplex of subspaces, 244 Hypercubic, 250 Hypergeometric function, xi, 221 Hyperplanar element(s), 26, 27
372
SUBJECT INDEX
Hyperplane(s), 1, 3, 4, 226, 253, 264 improper, 3 at infinity, 3, 264 polar, 21 proper, 3 tangent, see Tangent hyperplane Hyperquadric, 6-8, 10, 16, 17, 70, 87-90, 97, 153, 169, 204, 217, 260, 261, 265
equation of, 24, 261 imaginary, 260, 265 nondegenerate, 155, 158 f)(1,3), 21, 22, 109-114 submanifold of, 109 oval, 6 Plucker, 20, 223 of revolution, 159 tangent hyperplane to, 6 Hyperephere, 2, 3, 14-16, 25, 26, 42, 127, 160
center of, 2, 4 contact, 54, 56 imaginary, 4, 15, 66 improper, 4, 6 orthogonal to hypersurface, 31 orthogonal to submanifold, 143
proper, 4 real, 4, 7, 15, 66 tangent to hypersurface, 32 of zero radius, 127
Hypersurface(s), 45, 98 asymptotic cone of, 209
basis forms of, 34, 35, 46, 47, 49, 151
canal, xi, 57, 60, 61 of conformal space, xi, xii, 31-71 connected, 45 cubic, 250
invariant normalization of, xi, 71 moving frame of, 104, 105, 109, 151 normalizing object of, 40, 43 in Ps, 251, 252
plane generator of, 214-216 of projective space, xii, 72, 151, 153 real, 31 of revolution, 159
in Rn, 66 ruled, 251
of second order, 6, 20 second fundamental form of, 46, 150, 152, 210, 214 second fundamental tensor of, 38, 46, 52, 66, 152, 209, 215
simply connected, 45 smooth, 45 tangentially degenerate, 68
tangentially nondegenerate, 150, 152, 154, 208 tangent hyperplane to, 35, 66, 151, 152
of third order, 236 ultrahyperbolic, 208-210 Identity of Lie group, 10 matrix, 230, 270 operator, 115, 179, 231 Imaginary asymptotic cone, 208 cone, 6
developable surface, 111 hyperquadric, 260, 265 hypersphere, 4, 15, 66 isotropic cone, 120 quadric, 7 radius, 15
Improper hyperplane, 3 hypersphere, 4, 6 Incidence condition, 237 Incident subspaces, 224 Indeterminate net, 92 Index-free notations, 131, 291 Index notations, xiii Infinitesimal displacement of frame of conformal space, 10, 14 CO(p,q)-structure, 126 null-pair, 193 projective space, 11, 23, 151, 271,
306
pseudoconformal space, 18 Cs , 24 7Z2(Vn-1), 44, 49
IZ3(Vn-1) 44 space p3, 23 submanifold, 74, 86, 90, 106, 142 Infinitesimal displacement of adapted frame, 274 invariant frame, 83 point frame, 237 tangential frame, 237 Inflectional center, 112 Integrability condition(s), 116, 315 Integrable distribution, 198, 200-202 Integrable almost Grassmann structure, 270
Integral curve(s), 188, 189 element, general, 100 geometry, xi, 221, 250 manifold(s), 100, 255, 270 submanifold, 95, 197, 243, 251
SUBJECT INDEX surface, 186, 200 Internal domain, 102 Intrinsic geometry of normalized V'", 161 Intrinsic normalization of submanifold, 115 Invariance of isotropic geodesics, 140 Invariant(s), 70, 146 bundle of normal hyperspheres, 43 bundle of second fundamental forms, 79, 88-91
373
subgroup, 2 subspace, 230 tangent m-sphere, 78 Weyl connection, 135 Inverse
matrix, 11, 241, 274 tensor, 32, 39, 60, 77, 106, 123 Inversion, 2, 15, 65, 66 center of, 3
bundle of tangent hyperspheres, 78
Involutive
circle, 115
distribution, 95, 186, 188 operator, 180 principal distribution, 187 transformation, 115 Irreducible net, 94, 98 Isoclinic
conformal connection, 128 conformal frame, 67 conformally symmetric connection, 135
connection, 134, 135 derivative, 70 differential form, 255
a-plane, 196 d-web, 318, 319
distribution, 197, 198
(p + 1)-web, 309-311, 313, 320
family of central m-spheres, 79 family of frames, 86
subspace, 305, 313
forms of
almost Graasmann structure, 285 conformal structure on hypersurface, 153 conformal structure on submanifold, 142 general linear group, 36, 121, 254 CO(p,q)-structure, 126 G-structure, 285
isotropy group, 127 group of motions of R.,-, 127 stationary subgroup H.' (Vm), 75 stationary subgroup Hs (V n- , ), 36, 41
stationary subgroup H=(Vm), 80 stationary subgroup H=(V^-I ), 41, 44
stationary subgroup H= (V n- 1), 44
stationary subgroup of m-plane, 238
structure group of CO(2, 2)-structure, 166
frame(s), 51, 63, 67, 83
of group PO(n + 2,1), 75 horizontal distribution, 147 infinitesimal operator, 70 local parameters, 116 normalization, of hypersurface, xi, 71 of submanifold, xi, 116 of surface, 115 point(s), 75, 80, 82, 229 quadratic form, 70, 112, 134, 136 relative, 50, 81-83, 86
stationary subgroup, 81
three-web, 197, 199, 201 Isothermic curvature lines, ix, 70 hypersurface, 63 surface, 63
Isotropic a-submanifold, 195, 215 Q-submanifold, 195, 215 bivector(s), 184 bundle, 171, 301 co-frame(s), 164 cone, 6, 7, 13-16, 102, 104, 112, 113,
120, 127, 139, 141, 164, 169, 170, 171, 194, 209
of CO(2, 2)-structure, 199-201, 279 plane generator of, 164, 170, 171, 270
of pseudoconformal space, 18 curve, 24, 102, 109
distribution, 186, 208 fiber bundle(s), xii, 164, 168-172, 174176, 182-197, 200, 202,
205-208
curvature tensor of, 168, 183 fibration(s), 218 foliation(s), 186-188, 202 four-web, 187
frame(s), 164, 180, 204 frame bundle, 188 geodesic congruence, 189 geodesic(s), 138-140, 161, 189 hypersurface, 102, 104, 108, 109 net, 113 submanifold, 101, 110 tangent elements, 109 Isotropy group, 2, 12, 127
374
SUBJECT INDEX of affine space, 264 invariant forms of, 127 of space SAP, 286
Kerr metric, 206, 219 Klein interpretation, 7 Kronecker symbol, 11, 12, 54, 91
Laguerre space, ix, 70 Law of transformation of basis forms, 34, 35, 254 connection forms, 161 curvature tensor, 160 invariant, 70 quadratic form, 70 Riemannian connection, 137 tensor, 35, 148 vector, 35
Left-invariant forms, 151, 271, 301 Levi-Civita connection, 136, 259, 262 Lie
algebra, 10, 320 group, 10, 71, 72, 151 hyperquadric, 25-27 hypersphere, xi mapping, 25, 26 sphere geometry, 24-26, 29 Light cone(s), 15, 102
Light impulse, 140 Lighting surface, 108 Lightlike hypersurface, 102 Lightlike submanifold, 101 Light tetrad(s), 218 Line(s) geometry, 19 of propagation of light, 108 submanifolds, xii, 108, 111, 117 Linear complex, 21 special, 22, 23 congruence,22 -fractional function, 3 homogeneous object, 286 mapping, 46, 179, 229 operator, 178, 229 scalar, 231
pencil of subspaces, 224 span, 231, 233, 237 subspace, 28 transformation(a), 6, 316 Liouville theorem, 2, 3, 28 Lobachevsky space, 7, 66 Local conformal space, 126-128, 132 diffeomorphism, 302 projective space, 127
pseudoconformal space, 128 space, 301 twistor theory, 320 Locally flat almost Grassmann structure, xii, 290, 301
conformal structure, 144 Grassmann manifold, 319 Lorentz group, 169, 173, 174 Lorentzian signature, xi, 102, 140 Lorentzian structure, 171 Mainardi-Codazzi equations, 52 Manifold(s) algebraic, 240 with conformal structure, 119 integral, 100, 255 of null-pairs, 193 of oriented hyperplanar elements, 27
real, xi Mapping bijective, 245, 267, 270 differentiable, 20 rank of, 253, 255 Matrix coordinate of subspace, 226, 227, 255,
256, 265
Grassmann, see Grassmann mapping invariant, 260 inverse, 11, 241, 274 1-forms, 131, 291, 292 PlOcker, 20, 28, 109 symmetric, 261
Maurer-Cartan equations, 151 Maximum rank d-web, 321 Maximum rank four-web, 219 Maxwell-Einstein equations, 219 m-canal hypersurface, 60, 61 m-conjugate system, Method of exterior differential forms, x of moving frames, x, xiii tensor, x, xiii Metric, xii, 119 form, 15 Riemannian, ix, 63 of Riemannian manifold, 119 tensor, 136, 139, 140, 218, 262 Middle curvature, 66 Minimal hypersurface, 66 Minkowski space, 15, 16, 28, 102, 116 Mdbius geometry, 3 M6bius space, 3 Motions of Euclidean space, 1-3 homogeneous space, 77
SUBJECT INDEX pseudoelliptic apace, 262 Moving frame of conformal space, 8, 28 hypersurface, 104, 105, 109, 151
of null-pair, 193 projective space, 271 pseudoconformal space, 18 space p3, 23 submanifold, 73, 86, 90, 106
375
form, 152 matrix, 59, 74, 268 Nonsymmetric Ricci tensor, 134 Non-umbilical point, 81 Normal
Moving frames method, x, xi, xiii
bundle of hyperspheres, 75 circle, 43 conformal connection, 128, 132, 146 congruence of circles, 64 first, 67
m-pair(s), 115, 227, 260
focal family, 69
degenerate, 227
in general position, 228, 229 nondegenerate, 227-229, 260 m-plane, 115, 237, 265 m-quasiaffine space, 263, 266 m-sphere, 81, 82, 98, 116, 144, 145
Multidimensional web(s), x, xi, 221, 270, 301, 304
Multiple eigenvalue, 58 Multiple root, 194, 198 Multiplicative group of reals, 269, 273
hypersphere, 43, 46, 80, 87, 149 m-sphere, 82
(n - m)-sphere, 149 of submanifold, 88 Normalization(s), complete invariant, 82 of Grassmannian, 252, 257, 265 harmonic, xii, 255, , 258, 261 homogeneous, 260, 262, 264 intrinsic, 115 invariant, see Invariant,
normalization Net(s)
nondegenerate, 253
conjugate, 213, 249
of conjugate lines, 67, 90, 98, 116 holonomic, 98 of curvature lines, 55, 56, 67, 89-99, 116
holonomic, 55, 56, 94-98 indeterminate, 92 irreducible, 94, 98 totally holonomic, 94, 95, 97 of developable surfaces, 113 holonomic, 55, 56, 94-98 of isotropic lines, 102 Newman-Penrose tetrad(s), 170, 218
Nondegenerate hyperquadric, 155, 158 m-pair, 227-229, 260 normalization, 253 null-pair, 193, 194, 219 projective transformation, 229 quadratic form, 119, 176, 255 tensor, 53 Non-Euclidean geometry, 127 Nonholonomic submanifold, 161 Nonhomogeneous coordinates of point, 12
coordinates of subspace, 223 projective coordinates, 164
Nonisotropic complex, 112, 113 hypersurface, 72, 109 submanifold, 141, 150 Nonsingular
polar, 260, 262, 265 of submanifold, 146 Normalized domain of Grassmannian, 253-258, 260-264 Normalized submanifold, 72, 115, 149, 150 Normalizing condition, 74 mapping, 253 of zero rank, 263 (n - m)-sphere, 149 object 40, 43, 78, 82 subspace(s), 253-256, 263, 264 (n + 3)-spherical coordinates, 26 Null-pair, 193, 194 Object C,.,,, 125, 126, 128-131, 135, 141,
145, 146, 147 complete, 71, 77, 87 fundamental, see Fundamental geometric object normalizing, 40, 43, 78, 82 1-canal hypersurface, 61 1-canal submanifolds, 145 1-form, differential, 63, 120, 123, 125, 126, 128, 131, 132
One-parameter foliation, 94 group, 166, 173 subgroup, 34 One-to-one correspondence, 45, 46, 223 One-to-one mapping, 20
SUBJECT INDEX
376
Open domain of Grassmannian, 253, 260 Open neighborhood, 224 Operation of complex conjugacy, 189 Operator of covariant differentiation, 260
of differentiation with respect to fiber parameters, 274 V, 76 e, 178, 218 w, 229 Optical directions, 219 Orientation, 25 Oriented hyperplanar element, 26 hypersphere, 25, 26 hypersurface, 45 manifold, 176 Orthonormal frame(s), 12, 14, 75 Orthogonal frame(s), 12 group, 2 hypersphere, 73 m-hedron, 79 trajectories, 69 transformation, 2 Osculating circle, 43 hypersphere, 84, 93
sphere, 85, 87, 89, 93, 96, 97 subspace, of developable surface, 213 of isotropic a-submanifold, 211-214 of submanifold, 89, 98 of surface, 211-214 of tangentially nondegenerate
submanifold, 213 of variety fl(m,n), 238, 239 Oval hyperquadric, 6 Pair of congruences, 117 Parabolic congruence, 113 linear congruence, 22 pencil of hyperspheres, 26, 27 point, 111
submanifold, 111 Paraconformal structure(s), 320 Parallel d-web, 302
translation(s), 2, 148, 274 transport, 132, 264 vector field, 148
Parallelizability condition for three-webs, 304 Parallelizable webs, 302, 304
Parameters, fiber, 77, 143, 176, 204 principal, 135, 277 p-dimensional direction, 54 Pencil of characteristic straight lines, 251, 252 hyperplanes, 236 hyperquadrics, 157 hyperspheres, 40, 65, 95, 98 normal hyperspheres, 48 oriented hyperspheres, 26, 27 second fundamental forms, 37, 67 straight lines, 21, 22 tangent hyperspheres, 33 tangent linear complexes, 112 tangents, 117 tensors, 37
Pentaspherical coordinates, ix, 4, 28, 70 Petrov classification, 189-193, 206, 218,
219
Petrov's type(s), 191-193 Pfaffian derivative(s), 287-289 equation(s), 14, 55, 69, 186 equations, system of, 71, 74, 186 completely integrable, ii, 56, 60,
64, 71, 148, 186 in involution, 100 Planar hypersurface, 245 Plane(s), 215 field of straight lines, 22, 232 generator of asymptotic cone, 208, 241-243 cone C=, 113, 168, 171, 195, 216
hypercubic, 250 hyperquadric, 17 hyperquadric fl(1,3), 22 hypersurface, 214-216 isotropic fiber bundle, 164 Segre cone, 232, 240, 245, 267-269, 300, 305, 317 Segre variety, 225, 230, 231, 264
variety f1(m, n), 224, 225 generators, family of, 168, 224, 242
at infinity, 7 Plucker coordinates, 20 hyperquadric, 20, 223 manifold, 19 mapping, 20, 28, 109 Poincar4 space, 28 Point(s), 1, 115, 116 conjugate, 69 elliptic, 111 focal, 246 hyperbolic, 111
SUBJECT INDEX at infinity, 3, 16, 66, 126, 127, 286,
377 space, 3, 10, 117, 127, 150, 154, 155,
300
158-161,216,221,223,225,237, 238, 244, 253, 262, 263, 267, 269, 271, 272, 291, 299, 300,
invariant, 82, 229 non-umbilical, 81 parabolic, 111 singular, see Singular points of tangency, 112
302
dual, 224 infinitesimal displacement of frame of, 11, 151
umbilical, 42, 46, 81
Polar
structure, 320
bilinear form, 5
structure equations of, 151, 237 P3, 232, 234, 244 P4, 232, 233, 235, 236, 251 P5, 169, 216, 223, 232, 244, 245,
-conjugate subspaces, 67, 261 hyperplane, 21
normalization, 260, 262, 265 -normalized Grassmannian, 262 Pole of hyperplane, 7, 26 Polynomials CQ(a) and Ca(p), 184-190, 194, 214
Polyspherical coordinates, xi, 3, 4, 6, 15, 28
Principal
250
Ps, 250 P9, 232-236
transformation(s), 6, 159, 229, 260 Projectivization of asymptotic cone, 241, 243 cone, 225, 251
a-plane(s), 185 13-plane(s), 185
bivector, 259
direction(s), 53, 67, 89, 188, 191, 192, 219
of affinor, 113 domain of, 53 double, 219 of hypersurface, 53, 113
orthogonal, 53 subspace of, 58
distribution, 186, 191-193 double, 189, 191-192, 206 of general type, 191-193 triple, 192, 193 forms, 135
isotropic distribution, 186-193, 209 parameters, 135, 277 subbundle, 128 two-dimensional direction, 188 Product direct, 225, 264 exterior, 19, 177 Projection, 237 center, 216, 217 of Segre variety, 216, 217 Projective coordinates of point, 227 coordinate system, ix differential geometry, ix, 88, 97 frame, 66, 222, 260, 306 line(s), 171, 230
matrix coordinates, 226, 265 plane, 216 point frame(s), 236 realization, xiii
center of, 225 Grassmannian, 221 isotropic cone, 23 Segre cone, 232, 240 system of bivectors, 316 tangent space, 315 tangent subspace, 23 variety n(2,5), 247 Prolongation, 34, 71 Prolonged G-structure, 128, 166, 173
structure equations of 128 Prolonged structure group, 301
Proper
conformal geometry, x space, xi, 16, 19, 31, 74, 89, 100, 103, 141
structure, xi, 102, 120, 128, 153, 172, 175
domain of Grassmannian, 253 hyperplane, 3 hyperaphere, 3, 4 Riemannian metric, 136, 161 subspace, 1
Pseudoconformal geometry, x space, x, xi, 14, 16, 18, 19, 21, 27, 28, 31, 72, 100, 103, 116, 127, 128, 141, 221, 264, 265 C2', 108, 109, 111, 169, 221, 265
four-dimensional, xi structure, x-xii, 102-120, 127, 142, 217, 270, 299 Pseudocongruence of m-spheres, 116 Pseudoelliptic space, 262
SUBJECT INDEX
378
Pseudo-Euclidean space, 14, 15, 18, 28, 127, 128, 131, 264, 265, 299
four-dimensional, 28 group of motions of, 127, 128
group of motions and homotheties of, 128 R4, 166
Pseudogroup of contact transformations, 27
Pseudoorthogonal frame(s), 19, 163, 299 group, 7, 27, 169, 299 transformation, 6 Pseudoorthonormal frame(s), 169 Pseudo- Riemannian manifold, 102, 138-141, 160 Pseudo- Riemannian metric, x, 136, 194, 255, 258, 262 Pseudo-Riemannian structure 0(2, 2), 194 Pure imaginary function, 203 Pure imaginary isotropic cone, 13 (p + 1)-web, 271, 301, 305, 306, 320 of codimension one, 311 structure equations of, 271, 305 q-parameter family of submanifolds, 98 Quadratic congruence, 117
form(s), 1, 9, 13, 141, 163, 255, 257, 259, 299
relations, 223 Quadric(s), 17, 18, 22 double, 155, 157 four-dimensional, 20
imaginary, 7 real, 18 three-dimensional, 22 Quadrilateral web, 94 Quadruple of plane generators, 230
of points, 230, 231 principal distributions, 192, 193
root, 200 of subspaces, 230
Quasiaffine space, 263, 265, 266, 273
Quaternionic structure, 320 Quotient, 17, 254 Range of normalizing mapping, 253 Rank of mapping, 253, 255 quadratic form, 53, 110 system of forms, 93 system of hyperspheres, 93, 96 system of tensors, 80 tangentially degenerate submanifold, 68, 108
tensor(s), 45, 58, 80, 81, 258
Real
conformal space, 46 cross-ratio, 231 Curvature tensor, 176 eigensubspace(e), 180 eigenvalue(s), 180, 181 fiber bundle, 176 four-dimensional conformal structure, 217
generator of cone C, 188 isotropic cone, 18, 104, 120 isotropic directions, 100, 102, 103 isotropic fiber bundle, 165 plane generator, 111, 113
principal isotropic directions, 188, 191 quadric, 18 rectilinear generator(s), 6, 171 root(s), 107, 187 singular point(s), 107, 108
submanifold, 100 subspace, 230
tangent space, 171 tensor of conformal curvature, 218 transformation of coordinates, 100, 164, 169
Realization of Grassmannian, x Realization, projective, xiii Rectilinear a-generator, 250 fl-generator, 250 generator(s), 14
of asymptotic cone, 208 of cone C., 113, 141, 170 of developable surface, 213
of Grassmannian, 224 of hyperquadric, 17 of hyperquadric f1(1,3), 21, 22,
111, 117
of hypersurface UI_1, 68 Lie hyperquadric, 26 of Segre variety, 230, 231
of submanifold, 107, 108 of tangentially nondegenerate submanifold, 213 of variety 11(m, n), 224 Recurrent CO(2,2)-structure, 188 Reduced family of fourth-order frames, 284 second-order frames, 278, 279, 281 third-order frames, 283 Reduced group of admissible transformations, 124 Reduction of group of admissible transformations, 134 Reissner-Nordstr8m metric, 206, 207, 219
SUBJECT INDEX Relative conformal curvature, 184, 185, 205 invariant, 50, 81-83, 86, 318
tensor, 36-39, 130, 152, 177, 279, 280, 287
Relatively invariant form, 14, 37, 46, 119, 122, 132, 148, 150, 153 Relativity theory, 108 Representation, 71
complex, 173, 218
379 submanifold, 67, 79, 249 surface, 110 Second fundamental tensor of hypersurface, 38, 46, 153 Second order asymptotic direction(s), 250 envelope, 155, 157-159 frame(s), 41, 49, 80, 121, 123, 128, 131, 146, 204, 278, 279 tangency, 41, 42, 53, 54, 79-81, 84,
Restriction of Darboux form, 212, 214 Ribaucour congruence, 63
Second sheet of envelope, 45
Ricci
Second structure object, 281, 286, 287,
112, 156
identities, 129
tensor, 51, 133, 134, 190, 258, 262 symmetric, 258 Riemannian connection, 64, 134-140 geometry, x, 71, 129, 265 manifold, 116, 130, 137-141, 155, 161, 218, 252, 259
four-dimensional, xi, 217 metric, ix, 63, 119, 132, 134, 136, 137, 140, 161, 189, 202, 218, 255, 258, 262 structure, 134
tensor, 51, 161 Rigidity
conformal, 46, 72 problem, 72 theorem, xi, 46, 71 Rotation, 160 Ruled
hypersurface, 251 submanifold(s), xii, 108, 111, 117 surface, 24, 108, 109, 112, 113, 215 of second order, 23
Scalar
298 Sectional curvature of
normalized Grassmannian, 259 polar-normalized Grassmannian, 262, 263 Segre-afine space, 264-266, 286, 300 Segre cone
with complex generators, 270 Cp(m + 1, m), 240 Cp(2, 3), 232 dimension of, 240 directrix of, 232, 271
equations of, 268 plane generator(s) of, 232, 245, 267-269 projectivization of, 232, 240
SC=(p,q), 269, 271, 286, 291, 300,
301, 305, 316, 317 vertex of, 231, 264 Segre theorem, 97 Segre variety, 216, 217, 225, 230, 231, 240, 241, 264, 265, 271, 300, 316 degree of, 225, 245
dimension of, 225 equations of, 230 plane generator(s) of, 225, 230, 231,
curvature, 52, 133, 134, 190
264
linear operator, 231
projection of, 216, 217
1-form, 131, 291
rectilinear generator(s) of, 230, 231 S(k,l), 225, 230-232, 240, 241, 243, 245, 247, 256, 316
product of elements of conformal frame, 126 elements of projective frame, 237 hyperspheres, 5, 74 points, 8 vectors, 1, 262 web, 321
Schwarzschild metric, 206, 207, 219 Secant, 233 Second differential of point, 238 Second fundamental form(s) of completely isotropic submanifold, 212, 213
hypersurface, 46, 150, 152, 210, 214
Segrean, see Segre variety Self-conjugate curvature tensor, 176 subspace, 182 Self-dual classes of complexes, 252
CO(2,2)-structure, 194, 198, 215 part of Weyl tensor, 217 structure, 183, 185, 186 subspace, 182, 183
Semidirect product, 2, 41, 128, 264, 274
Semiflat
SUBJECT INDEX
380
asymptotic conformal structure, 162 CO(1, 3)-structure, 175 CO(2, 2)-structure, 169, 186, 209, 210, 215, 216, 218
four-dimensional structures, 183 Semifocal family, 251 Semiintegrable almost Grassmann, structure, xii, 270, 292, 300 conformal structure, 300 CO(2, 2)-structure, 198 four-dimensional structure, 183, 186 Semi-Riemannian manifold, 252, 265 Semi-Riemannian metric, 255, 265 Sequence of geometric objects, 77 Signature (p,q), 18, 100, 119 Simplex, coordinate, 224
Simply connected domain, 86, 87 hypersurface, 45 submanifold, 73 Singular point(s), 107, 108, 241, 243, 244, 246, 247, 251
vector, 19, 238 Spacelike direction, 103 Spacelike hypersurface, 102, 103 Space-time, 28, 140, 190, 202, 206, 217 Span, linear, 231, 233, 237 Special complex, 109-111 linear complex, 22, 23
linear group, 151, 166, 268, 271 orthogonal group, 120 pseudoorthogonal group, 120 relativity, 28, 102 three-web, 197
Specialization of fourth-order frames, 283 Specialization of second-order frames, 277 Spectrum of Hodge operator, 218 Sphere, 17, 71 n-dimensional, 3 Spherical coordinates, 206 Spherical curvature lines, ix, 70 Spherically symmetric body, 206, 207 Spherically symmetric solution, 219 Square of Hodge operator, 178 Stationary subgroup
straight line(s), 232
of element of Grassmannian, 274
subspace, 242
of element of normalized domain of Grassmannian, 264 Hl(Vm), 75, 81 H.(V"-1)333, 34, 41 HZ(Vm), 80 H=(V"-I), 41, 44 H=(V"-1), 44
Skew-symmetric bilinear form, 210 part of torsion tensor, 313 tensor, 129, 293, 294
Smooth curve, 92, 109, 138 family, 244 hypersurface, 45 submanifold, 73, 77, 109, 245 Space with affine connection, 115, 258, 289, 290
conformal, see Conformal space with conformal connection, 115, 160 dual, 224, 235, 236 elliptic, 66 Euclidean, see Euclidean space of exterior 2-forms, 178, 180 with group connection, 71 homogeneous, 2, 71, 77, 115, 260, 264
of m-pair, 254 of m-plane, 238 of point, 12, 33
Stationary value of sectional curvature, 259 Stereographic projection, xii of conformal space, 7, 17, 18, 264 of Grassmannian, 264-266
of variety fl(m, n), 286, 300 Structure conformal, see Conformal structure of Grassmannian, 224 pseudoconformal, x-xii, 102-120, 127, 142, 217
of space Cz, 221 Structure equations of
hyperbolic, 7, 66 Laguerre, ix, 70 projective, see Projective space pseudoconformal see Pseudoconformal apace R(m+l)(n-m), 224
almost Grassmann structure, xii,
R4, 198, 200, 201 tangent, see Tangent space
differentiable manifold, 121 d-web, 315
of twistors, 218
Grassmannian, 257, 271, 273, 276
274-276, 285, 300, 320 conformal connection, 128 conformal apace, 11, 14, 87
conformal structure, xii, 126, 131, 299
SUBJECT INDEX
Grassmann (p + 1)-web, 309 group G', 128 plane RP2, 194 projective space, 23, 151, 237, 271,
306
prolonged G-structure, 128 pseudoconformal space, 18, 128 pseudoconformal structure, xii, 126, 142
(p + 1)-web, 303, 304
Structure group of
almost Grassmann structure, 268, 286, 301, 305 CO(p,q)-structure, 301 CO(1, 3)-structure, 171 CO(2, 2)-structure, 166 d-web, 315
fiber bundles E. and E, 166 Grassmannian, 273 (p + 1)-web, 305 prolonged G-structure, 128
three-web, 271 Structure tensor of almost Grassmann
structure, xii Subfamily of
orthogonal frames, 12 orthonormal frames, 75 projective frames, 237 Subgroup, invariant, 2, 34 Submanifold(s), 71, 144, 149, 150 carrying conjugate net, 90 carrying net of curvature lines,
381
singular, 242 Subtensors C. and Cp, 168, 169, 173-176,
183, 184, 186, 187, 209, 210,
218, 300
Subtensors ao and ap, 293 Subweb,316
Summation convention, 9 Surface(s), 71 asymptotic line of, 110, 111 of light absorption, 108 second fundamental form of, 110 Symbol of covariant differentiation, 261 differentiation with respect to fiber parameters, 308 exterior multiplication, 132 Symmetric affinor, 52, 107 function, 82 linear operator, 108, 179 matrix, 261
part of curvature tensor, 304 part of torsion tensor, 312 tensor, 60, 62, 68, 137, 146, 152, 255 relative, 60, 152 Symmetrization, 76, 292, 293
Symmetry, xi, 170 Symmetry figure, 115, 116 System of circles, 70 hyperspheres, 93, 96
Pfaffian equations in involution, 100
89-99, 250
completely isotropic, xii, 183, 185, 186, 194, 211-215
of conformal space, x, xi, 115, 141 connected and simply connected, 73 on Grassmannian, x, xii, 253 integral, 95, 197, 243, 251 moving frame of, 73, 86, 90, 106 normalized, 72, 115, 149, 150
parabolic, 111 ruled, xii, 108, 111, 117
second fundamental form of, 67, 79, 249 smooth, 73, 77, 109, 245 of space with conformal connection, 115 tangent subspace to, 88, 100, 107, 141
V3 C Ps, 109 Subspace(s), 1, 221-226, 237, 253 characteristic, 251
invariant, 230
linear, 28 normalizing, 253-256, 263, 264
Tangency of second order, 41, 42, 53, 54, 79-81, 84, 112, 156 Tangent bundle of hypersurface, 46, 75 bundle of variety fl(m, n), 238 bundle of second order of variety ft(m, n), 238 cone,242
hyperplane to hyperquadric, 6 hypersphere, 18
hypersurface, 35, 66, 151, 152, 212 hyperquadric, 18 hypersphere, 13, 32, 46, 61, 73, 84,
87, 145
linear complex, 112 m-sphere, 145 space, xi, 1, 120, 177, 268, 274, 286, 305, 315 subspace to
leave of web, 315 submanifold, 88, 100, 107, 141 torse, 247, 251
SUBJECT INDEX
382
two-dimensional submanifold, 245 variety fl(m, n), 238, 267
of second order to variety fl(m, n), 239
2-plane to isotropic submanifold, 211, 212 Tangential coordinates, 236
frame, 233, 237 matrix coordinate of subspace, 226,
227, 229, 256
Tangentially degenerate hypersurface, 68 submanifold, 107, 108, 111 ruled submanifold, 111 Tensor(s), 35, 36 analysis, ix, 70 apolar, 38, 39, 58, 61, 66 C. and Cg, 168, 169, 171-176, 183, 184, 186, 199, 208, 209, 218 of conformal curvature, xii, 125, 128, 130,133-136,142,144-147,153, 154,161,162,166-168,184,187,
194, 218, 300
of CO(2,2)-structure, 209 curvature, see Curvature tensor Darboux, 152, 154, 157, 209 differential equations of, 39, 77, 125, 130, 168
field, xiii invariant, 320 inverse, 32, 36, 60, 77, 123 law of transformation, 148 nondegenerate, 53
Third fundamental form of isotropic submanifold, 212, 213 Third order asymptotic line, 244, 246 cone, 241 frame, 86, 121
hypersurface, 236 object(s), 45, 82, 86 Third structure object, 281, 287 Three-dimensional cubic submanifold, 250 projective space, 232, 234, 244 quadric, 22 submanifold, 246, 249 Three-parameter group, 166 Three-web, 195-202, 219, 245, 270, 271, 304, 311, 314
Time coordinate, 15 Timelike direction, 102 Timelike hypersurface, 102, 104 Torse, 108, 109, 247-251 Torsion form, 291, 300 Torsion-free, 301 affine connection, 115, 132, 136, 137, 147, 148, 258
CO(2, 2)-structure, 314 Torsion tensor of affine connection, 147, 303 almost Grassmann structure, 274, 279, 287, 293 subtensors of, 293 Grassmann (p+1)-web, 307-309, 313 (p + 1)-web, 303, 304
rank of, 45, 58, 80, 81, 258
three-web, 196, 199, 311 Torus, 17
relative, 36-39, 130, 152, 177, 279,
Total differential, 63, 66, 84, 128, 133,
280, 287
of relative conformal curvature, 206 Ricci, 51, 134, 190, 258, 262
second fundamental, see Second fundamental tensor skew-symmetric, 129, 293, 294 symmetric, 60, 62, 68, 137, 146, 152, 255 torsion, see Torsion tensor trace-free, 130, 131, 144, 294 (p, 4)-, 36 (0, p)-, 32, 35, 38, 39, 50, 152 (0,2)-, 30, 36, 37-39 (0,3)-, 39 (1,2)-, 47
(2,0)-,36 Theorem Frobenius, 12, 249, 318 Segre, 97 Third-class congruence, 236
134, 139, 166 Totally focal congruence, 247, 250 Totally geodesic submanifold, 311 Totally holonomic net, 94, 95, 97 Totally isotropic surface, 113 Transformation(s), of basis forms, 254 complex, 170
linear, 6 projective, 6, 159, 229, 260
Transitive group, 2
Transitive subfamily of frames, 12, 19 Transversal(s) #-plane, 196 bivector, 316, 317 subspace, 305, 316
of two m-pairs, 229-231 Transversally geodesic distribution, 198 d-web, 318, 319
SUBJECT INDEX four-web, 202 (p + 1)-web, 309-311, 320 submanifold(s), 311, 312 three-web, 198, 202
Triple principal distributions, 192, 193 Triply orthogonal system of surfaces, ix,
383
Vertical form(s), 12 Vlasov configuration, 233-236, 265 Vlasov congruence, 236
equation of, 236 Vlasov hypersurface, 236
Volume element, 177
70
Twistor, 217 fibration, 218 Two-dimensional developable surface, 213 isotropic direction, 184
Web(s) AGW (d, p, q), 318, 319
AGW(d,2,q), 321 AGW(6,3,2), 321 GW (p + 1, p, q), 307-309
of maximum rank, 219, 321 multidimensional, x, xi, 221, 270, 301,
plane generator, 22, 170, 171, 214,
215
submanifold, 245, 249
304
W(d,p,q), 301, 302, 314-321
tangent subspace of, 245
three-web, 245
W(p + 1,p,q), 271, 303-305, 309-314, 320
Two-fold hyperplane, 157 Two-web, 94
W (4, 2, q), 315, 320, 321
W(4,2,2), 187, 197, 219 W(3,2,q), 270, 271, 304, 311, 320 W(3, 2, 2), 195-202, 219, 314
Ultrahyperbolic hypersurface, 208-210 Umbilical point, 42, 46, 81
Unimodular matrix, 268 Unit tensor, 292
Weight of tensor, 36, 80, 82, 130, 177, 279, 280, 287
Weingarten formulas, 70, 161 Variety
Weyl connection, 51, 64, 132, 133, 136,
algebraic, see Algebraic variety
148-150, 161, 205
Cartan, 98, 213, 215 determinantal, 240, 242, 265
curvature tensor of, 133 geometry, 161
Vector(s)
law transformation of, 35
structure, 134
space, 19, 238
tensor, 125, 130, 161, 190, 191, 217
tangent, 35, 38 Vectorial frame, 120, 170 Vertex of cone, 54, 70, 250 Segre cone, 231, 264
Zero
matrix, 226, 256 -tensor, 92
Comprehensive coverage of the foundations, applications, recent developments, and future of conformal differential geometry Conformal Differential Geometry and its Generalizations is the first and only
text that systematically presents the foundations and manifestations of conformal differential geometry. It offers the first unified presentation of the subject, which was established more than a century ago. The text is divided into seven chapters, each containing figures, formulas, and historical and bibliographical notes, while numerous examples elucidate the necessary theory.
Clear, focused, and expertly synthesized, Conformal Differential Geometry and Its Generalizations
Develops the theory of hypersurfaces and submanifolds of any dimension of conformal and pseudoconformal spaces Investigates conformal and pseudoconformal structures on a manifold of arbitrary dimension, derives their structure equations, and explores their tensor of conformal curvature
Analyzes the real theory of four-dimensional conformal structures of all possible signatures Considers the analytic and differential geometry of Grassmann and almost Grassmann structures Draws connections between almost Grassmann structures and web theory
90000
WILEY-INTERSCIENCE John Wiley & Sons, Inc. Professional, Reference arid Trade New York, N.Y. 101u58.0012 60S ThIrd Aven New York Chichester Brisbane -Toronto SlNapme.
II I
9 780471 149583
I
I