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DX e T X(M) is anti-invariant, i.e., JDX E T t(M' for each x n M. In the sequel, :
put dim M = 2m, dim M = n, dim D = h, dim D' = q
and codim M = 2m-n = p. If q = 0, then a CR submanifold M is a Kaehlerian submanifold of M, and if h = 0, then M is an anti-invariant submanifold of M. If p = q, then a CR submanifold M is called a generic submanifold of M. If h > 0 and q > 0, then a CR submanifold
M is said to be n^n-trivial (or proper). Remark. Sometimes, the definitions of CR sutmanifolds, generic subnanifolds and anti-invariant subaznifolds (totally real sutmanifolds) are respectively given as follows (cf. Wells [1], [2]). Let M be a real n-dimensional submanifold of a complex m-dimensional complex manifold M. We define
217
n J' X(M),
H {(td) =
as holomorphic tangent space to M at x. If dimCHx(M) is constant on M, then M is called a CR subnanifold. It is well known that
max(n-m,O) < dirncH(M) < in. If dimeX(M) = max(n-m,0) at each point of M, then M is called a generic subinanifold. Moreover, if dim X(M) = 0, then M is called a totally real sutmanifold.
Remark. We now state the result which justifies the name of CP. sutrnanifold. Let M be a differentiable manifold and T(M)C its ccmplexi-
fied tangent bundle. A CR structure on M is a complex subbundle H of T(M)C such that Hx A Hx = {0}
and H is involutive, i.e., for con lex
-vector fields X and Y in H, [X,Y] is in H. It is well known that on
a CR manifold there exists a (real) distribution D and a field of endomorphisn p : D
> D such that p2 = -ID. D is just Re(H®H) and
Hx = {X - VrlpX : X E Dx}. Blair-Chen [1] prove the following:
Let M be a CR sutmanifold of a Haermitian manifold M. If M is non-trivial, then M is a CR manifold.
We next give some characterizations of CR submanifolds of Kaehlerian manifolds. First of all, we prove the following (Yano-Kon [12]) THEOREM 3,1. In order for a sub manifold M of a Kaehlerian manifold M to be a CR sulmanifold, it is necessary and sufficient that FP = 0.
Proof. Suppose that M is a CR sutmanifold of M. We denote by Z and l
the projection operators on Dx and DX respectively. Then
Z+Z` =I,
Zz=Z,
Z`2=l,
Z11=Z1Z=0.
Fran (3.1) we have r X = 0, FZ = 0 and PZ = P, from which and the second equation of (3.4), we find
FP=0.
if = 0.
From (3.3) and (3.14) we obtain
(3.15)
tf = 0,
and hence, from the first equation of (3.5)
(3.16)
Pt = 0.
Thus, the first equation of (3.4) implies
p3+P=0.
(3.17)
From the second equation of (3.5) we also have
f 3 + f = 0.
(3.18)
Conversely, for a sukmanifold M of a Kaehlerian manifold M, assume that we have (3.13), that is, FP = 0. Then we have (3.14), (3.15), (3.16), (3.17) and (3.18). We put
11=l-Z.
Z=-P2, Then we see
Z+l`=I,
Z2=Z,
lye=l
,
111=1:11 =0,
which show that Z and Z' are complementary projection operators and
219 consequently define complementary orthogonal distributions D and Db respectively. Fran equation Z = -P2 we have P1 = P. This equation can
be written as PV = 0. But g(PX,Y) is skew-symmetric and g(l X,Y) is symmetric and consequently t P = 0. Thus we have Zl P1 = 0. Moreover,
by Z = P2 we have FZ = 0. These equations show that the distribution D is invariant and the distribution Dl is anti-invariant.
QED.
Fran (3.17) and (3.18) we have (Yano-Kon [12] ) THEOREM 3.2. Let M be a CR submanifold of a Kaehlerian manifold M. Then P is an f-structure in M and f is an f-structure in the normal bundle of M.
Let R be the curvature tensor of a canplex space form M(c). Then we have (Blair-Chen [1]) THEOREM 3.3. Let M be a suhmanifold of a complex space form M(c) with c # 0. Then M is a CR submanifold of M if and only if the maximal holamorphic subspace Dx = Tc(M) A JTx(M) defines a non-trivial differentiable distribution D on M such that
(3.19)
g(R(X,Y)Z,W) = 0
for all X, Y c D and Z, W E D`, where D` denoting the orthogonal complementary distribution of D in M.
Proof. If M is a CR submanifold of M(c), then we have R(X,Y)Z = zcg(X,JY)JZ. From this equation we have (3.19). Conversely, fran (3.19) we have g(R(JX,X)Z,W) = -Icg(X,X)g(JZ,W) _
0 for all X c D and Z, W c D. Frcm this we see that JDD is perpendicular to D . Since D is holcmorphic, JL is also perpendicular to
D. Therefore, JDx a X(Mt. This shows that M is a CR sub nn fold. QED. Let us now suppose that M is a Hermitian manifold and let St be the fundamental 2-form of M, i.e., S2(X,Y) = g(X,JY). M is a Kaehlerian
manifold if and only if cM = 0. However we consider a class of Hermitian manifolds slightly larger than that of Kaehlerian manifolds for
which cI
= 52 A w, w being a 1-form called the Lee form. When w is
closed we call these manifolds locally conformal symplectic manifolds. They include the-well known Bopf manifolds. We prove the following theorem (Blair-Chen [1]).
THEOREM 3,4, Let fd be a Hermitian manifold with dS2 = S2 A w. Then in order for M to be a (R sutmanifold of U it is necessary that Dl- be integrable.
Proof. Let X be vector field in D and Z, W vector fields in D . Then S2(X,Z) = 0 and S2(Z,W) = 0. Therefore S2 A w(X,Z,W) = 0 and hence 0 = 3dc2(X,Z,W) = -g([Z,W],JX),
but X and hence JX is arbitrary in D and [Z,W] is tangent to M, thereQED.
fore [Z,W] is in D'.
From Theorem 3.4 we see that the distribution DL of a CR submanifold of a Kaehlerian manifold is integrable. We next consider the condition that the distribution D is integrable. If the distribution is integrable and moreover if the almost cc Alex structure P induced on each integral submanifold of D is integrable, then we say that the f-structure P is partially integrable.
THEOREM 3.5, Let M be a CR suYmanifold of a Kaehlerian manifold M. Then the f-structure P is partially integrable if and only if
B(PX,Y) = B(X,PY)
for any vector fields X and Y in D.
Proof. Let X and Y be vector fields in D. Then (3.7) implies
F[X,Y] = FOXY - F yX = -(OXF)Y + (VyF)X = B(X,PY) - B(PX,Y).
Thus D is integrable if and only if B(X,PY) = B(PX,Y). In this case
0 the integral submanifold of D is invariant in M and hence it is also a Kaehlerian manifold. Thus the almost complex structure induced from
P on the integral submanifold of D is integrable.
QED.
In the next place we give some examples of generic sutmanifolds and CR submanifolds of complex space forms.
Ercvnple 3.1. Let Cm be the corrplex number space of complex dimension m. Let M be a product Rienannian manifold of the form Cp x Mq, where Mq is a real q-dimensional anti-invariant submanifold of Cq
Cam. Then m is a generic submanifold of Coq, and M is moreover a CR submanifold of Cm with m > p+q.
Example 3.2. Let e(r) be an m-dimensional sphere of radius r. We consider an immersion:
x ... x S"(rk)
>
C(n+k)/2,
where m1,...,mk are odd numbers. Then n+k is evev. We now consider (m.+l)/2
M.
S 1(ri) C C
(i = 1,...,k)
1
and
C(n+k)/2 = C(mj,.+1)/2
x .., x C
(rr+1)/2
(m.+1)/2 M. 1 Then each S '(ri) is a real hypersurface of C . We denote by (m.+l)/2 M. . Then Jv i is tangent to vi the unit normal of S '(ri) in C 1
m S 1(ri). Therefore S (r1) x ... x S of
CP(n+k)/2,
(rk) is a generic submanifold
and hence CR submanifold of Cm (2m > n+k) with parallel
mean curvature vector and flat normal connection. Similarly, we can consider an immersion:
S"l(r1) x ... x S
(rk) x RP
> C(n+k) /2
x Cp C G!,
where (n+k)+2p < 2m. Then the su nifold is a generic submanifold of C(n+k+2p)/2 and CR submanifold of do with parallel an curvature vector and flat normal connection. In the following we need some results of Riemannian fibre bundles,
which will be proved in Chapter IX.
Let S1 be a (2m+1)-dimensional unit sphere and &n be a complex m-dimensional projective space with constant holcmorphic sectional
curvature 4. Then there is a fibering
Let N be an (n+l)-dimensional submanifold immersed in S
1 and M be
an n-dimensional sulinanifold immersed in &n. We assume that N is tangent to the vertical vector field fibration n : N
of 52n+1 and there exists a
> M such that the diagram
N
i
t
> S2nt+1
n
n
M
1
> cr
commutes and the immersion i' is a diffecmorphism on the fibres. We denote by a the second fundamental form of the immersion i'. Then we have the following lemmas. LFNhIA 3,1. The second fundamental form a of N is parallel if and
only if the second fundamental form B of M satisfies
(3.20)
(VXB)(Y,Z) = g(X,PY)FZ + g(X,PZ)FY
and
(3.21)
fB(X,Y) = B(X,PY) + B(Y,PX).
LE1 A 3.2. The normal connection of N is flat if and only if M satisfies the conditions
Rl(X,Y)V = 2g(X,PY)fV
(3.22)
and
(3.23)
(Vxf )V = 0.
ExwTle 3.3. Let e(r) be an m-dimensional sphere with radius r. We consider the following immersion:
k
Sn+k
N
n+1 =
ml,...,mk
where m1,...,mk are odd numbers and ri +
Nmj,...,mk
=
S'1(r1)
x ... x
mi, i=1
+ rk = 1 and we have put
Smi(rk).
Then we have the following commutative diagram:
Nml,...,mk IT
i
CP(n+k-l)/2
ml,...,mk
where we have put M ml,...,mk
M
ml,...1,
= 7r (N
mt,...,mk
is a generic suhmanifold of
). Then we see that CP(n+k-1)/2
and is a CR
suhmanifold of CPFn (2m+1 > n+k). Since the normal connection of
Nml,...,mg
is flat, we see that the normal connection of M,, mll...+mk
satisfies R`(X,Y) = 2g(X,PY)f and Vf = 0 by Lemur. 3.2. Moreover, the
second fundamental form B of Mm
... mk satisfies (3.20) and (3.21)
because of Lemma 3.1. (See Example 4.1 of Chapter VI.) Excvrrple 3.4. In Example 3.3, if ri = (mi/(n+l))l/2 (i = 1,...,k),
then N is minimal and hence M mi,...,mk
ml......
k
is also minimal.
Let M be an n-dimensional CR sutmanifold of a complex projective space ?P(c). If the curvature tensor Rl of the normal bundle of M satisfies
R`(X,Y) = icg(X,PY)fV
(3.24)
for any vector fields X, Y tangent to M and any vector field V normal to M, then the normal connection of M is said to be semi-flat. The
justification of this definition is given by Lei 3.2. We notice that, if M is a generic submanifold of ho(c), then f vanishes identically and hence RL = 0. LEMMA 3,3, Let M be an n-dimensional CR sutma.nifold of Cpm with
semi-flat normal connection. Suppose that dim D = h > 4 and the f-structure in the normal bundle of M is parallel. If B satisfies (3.20), then the second fundamental form a of N of S The proof of this lemma. will be given in §
l is parallel.
of Chapter IX.
LE44A 3.4. Let M be an n-dimensional CR submanifold of CPm. Then we have
g(VB,VB) > 2hq,
where h = dim D and q = dim Dl, and the equality holds if and only if (3.20) holds.
Proof. We put
T(X,Y,Z) = (VXB)(Y,Z) + g(Y,PX)FZ + g(Z,PX)FY.
Then T = 0 if and only if (3.20) holds. Let {ei) be an orthonormal basis of Tx(M). Then
IT12 = IVBI2 + 2hq + 4
g((Ve B)(ei,Pei),Fej i,j
i
On the other hand, equation of Codazzi implies
B)(ei,Pei),Fej) - hq.
Since B is symmetric and P is skew-symmetric, the first term in the right hand side of the equation above vanishes. Consequently, we have
ITI2 = IVB12 - 2hq,
which proves our assertion.
QED.
LEMhA 3,5, Let M be a CR sutmanifold of a Kaehlerian manifold M. Then the f-structure in the normal bundle of M is parallel if and
only if AVtU = AUtV
(3.25)
for any vector fields U and V normal to M. Proof. From (3.9) we have
g((OXf)V,U) _ -g(FAVX,U) - g(B(X,tV),U) = g(AVtU,X) - g(AUtV,X),
W.
which proves our lemma. LEI4IA 3.6. Let M be a CR suthanifold of a complex space form
1F(c) with semi-flat normal connection. If the f-structure in the
normal bundle of M is parallel, then AfV = 0 V normal to M.
for any vector field
Proof. From equation of Ricci we find
g(EAV,AU]X,Y) = tcfg(FY,U)g(FX,V) - g(FX,U)g(FY,V)).
Thus we see that AfVAU = AUAfV for any vector fields U and V normal to M. On the other hand, (3.7) implies
o = g((VXf)fV,FY) = -g(f2V,(V F)Y)
= g(Af2 X,PY) - g(AfVX,Y), from which
A2
= A PAf2V.
Therefore we have
TrA
= -TrA PAf2V = TrAf2VPAfV = TrAfVAf2VP = TrAf2VAfVP = TIAfVPAf2V = -TrA2f.V.
Consequently, we have TrA
= 0 and hence Afv = 0.
QJED.
THEOREM 3.6. Let M be an n-dimensional canplete CR sutma.nifold of
CPm with semi-flat normal connection and h-> 4. If the f-structure f in the normal bundle of M is parallel and if IVB1 2 = 2hq, then M is totally geodesic invariant suhmanifold C1?nl2 of CPFn or M is a generic
sukmanifold of CP(n+q)/2 in CEP and is
tr(Sml(rl) x ... X Smk(rk)),
Emi = n+l,
Eri = 1,
where 2 < k < n-3 and ml,...,mk are odd numbers and q = k-1.
Proof. We assume that q = 0. Then M is a Kaehlerian sutmanifold of Cam. Fran equation of Ricci we have AvAu - A
= 0 for any vector
fields U and V normal to M. Thus Lenma. 1.1 inplies 0 = AjUA[j - ALTAN
2J 2 , and hence AU = 0. Therefore, M is totally geodesic in e n and
is C/2 Let us next assume that q > 1. From Lemtes 3.3 and 3.4 we see S2m+1 that the second fundamental form a of N in is parallel. From Lemma 3.6 we also have AfV = 0. If AFx = 0 for some vector field X tangent to M, (3.12) implies that c = 0 when q.1 2. This is a contradiction. When q = 1, if AFX = 0, then M is totally geodesic in CP° and
hence M is invariant or anti-invariant in C1. This is also a contradiction. Thus the first normal space of M is of dimension q. Moreover, the first normal space of M is parallel. Indeed, from (3.7) and (3.21) we have (VXF')Y = B(Y,PX), and hence g(DXFY,fV) = g((VXF)Y,f\) =
g(B(Y,PX),fV) = 0. Then we can see that the first normal space of N S2m+1 in is also parallel. Therefore, there is a totally geodesic S2m+1, (n+l+q)-dimensional suhmanifold Sn+l+q of and hence Cp(n+q)/2
of C' containing N and M respectively. Then, from the definition, M CP(n+q)/2. is a generic submanifold of Now, using Lemma 3.2, Example 3.3, Theorem 5.7 of Chapter II and Example 4.3 of Chapter II, we have our assertion.
QED.
THEOREM 3.7. Let M be an n-dimensional complete generic subnanifold of (PM with flat normal connection. If h > 4 and
I
2hp
(p = codim M), then M is
n(S1(r
X ... x
Smi(rk)
Fmi = n+1,
Zr? = 1,
where 2 < k 1, Epi+p=n, 1 Cph+p+hp
by
(Zoo ...,Zh;wO,...,Wp)
> (ZOWO,...zlwj,...,ZhWp),
where (zO,...,zh) and (w O,..., P) are the homogeneous coordinates of CPh and CPP respectively. It is easy to see that fhp is a Kaehlerian
imbedding. Let My be a p-dimensional anti-invariant submanifold of CPh+p+hp. Cph x M' induces a natural CR product in
Cep. Then
A CR product submanifold M = MT x ML in CI is called a standard CR product if
(1) m = h+p+hp, and (2) MT is a totally geodesic Kaehlerian submanifold of &n, where h = dimCDx and p = dins"`. x
We shall prove that m = h+p+hp is in fact the smallest dimension
of C" for admitting a CR product. LEMMA 3,12, Let M be a CR product of CP. Then {B(XiX)}, i = 1,...,2h; x = 1,...,p, are orthonormal vectors in Nx, where {Xi} and {Zx} are orthonormal basis for Dx and DX respectively.
241
Proof. Since tB(X,PY) = 0, we obtain AFZPY = 0 for any vector fields X and Y tangent to M. From this, (3.7) and (3.11) we have
g(B(PX,Z),fB(X,Z)) = g(X,X)g(Z,Z)
for any X E Dx and Z E D X- On the other hand, (3.7) implies
0 = (VZF')X = -B(Z,PX) + fB(X,Z).
Thus we have
IB(X,Z)12 = g(X,X)g(Z,Z).
We suppose that JXJ = IZI = 1. Then IB(X,Z)l = 1. Therefore we obtain by linearlity that
g(B(Xi,Z),B(XX,Z)) = 0,
i # j.
Moreover, we see that B(X,Z) E N by AFZPY = 0. Hence, if dim Lx' = 1, the lesna is proved. If dim DX = p > 2, then
g(B(Xi, X),B( Xj,Zy)) + g(B(Xi,Zy),B(j, X)) = 0
for i # j, x # y. Since M is a CR product, we see that ) X,Zy) = 0 and hence, by (3.10) we obtain
g(R(Xi,
Xi
g(B(Xi,Zx)B(Xi, y)) - g(B(XiZy),B(j, X)) = 0.
Therefore we obtain g(B(Xi, X),B( Xiy)) = 0 and hence we have our
assertion.-
QED.
As an immediate consequence of Ira= 3.12 we have THEOREM 3.20. Let M be a CFi product submanifold of CPR. Then
m > h+p+hp.
242 THEOREM 3.21. Every CR product M of CL with m = h+p+hp is a standard CR product.
Proof. For any X, Y, Z in D and W cD; the Gauss equation (3.10) implies
g(B(X,W),B(Y,Z)) = g(B(X,X),B(Y,W)).
In particular, if Y = PX, then
g(B(X,X),B(PX,W)) = g(B(PX,Z),B(X,W)) = g(fB(X,Z),B(X,W)) _ -g(B(X,Z),BBPX,W)),
from which
g(B(X,Z),B(PX,W)) = g(B(X,PZ),B(X,W)) = 0.
Therefore we obtain
g(B(X,Z),B(X,W)) = 0
for any X, Z in D and W in Dl. Then by linearlity we have
g(B(X,Z),B(Y,W)) + g(B(Y,Z),B(X,W)) = 0.
Thus we have
g(B(X,Z),B(Y,W)) = 0.
On the other hand, Lemma. 3.12 implies that B(X,Z) lies in FD' for any
X, Z in D, because of m = h+p+hp. But we have AFXPY = 0, and hence we must have B(X,Z) a N. Consequently, we have B(X,Z) = 0 for any X, Z in D. Therefore, MT must be totally geodesic in CPm.
QED.
THEOREM 3,22, Let M be a CR product of CPS. Then we have
JA12 > 4hp.
If the equality holds, then MT and ML are both totally geodesic in
UP. Proof. First of all, we have JB(X,Z)I = 1 for any unit vectors X in D and Z in D`. Thus we have
2h
JAZZ = 4hp +
E
JB(X1.,X.)I2 +
i,j=1
I
X, y=1
IB(ZX Z
)12,
Y
where {Xi} and {ZX) are orthonornal basis of D and D'' respectively.
From this equation we have our assertion.
QED.
ExompZe 3.6. Let RPP be a real p-dimensional projective space. Then RPp is a totally geodesic anti-invariant submanifold of CPp. Then the composition of the immersions
CPh x RPP
>
> C,h+p+hp
Cpb x Cpp
gives a CR product in CPS with I A12 = 4hp. i
Theorems 3.20 and 3.21 are proved by Chen [2].
> CPm
244 EXERCISES
A. COMPLEX SPACE FORMS IMMERSED IN COMPLEX SPACE FORMS: Let Mn(k)
be a complex n-dimensional space form of constant holamrphic sectional curvature k immersed in a complex (n+p)-dimensional space form
Mn+p(c). Ogiue [4] proved the following THEOREM 1. If p = n(n+l)/2, then either c = k or c = 2k, the latter case arising only when c > 0. Furthermore, Nakagawa-0giue [1] proved the following theorems. THEOREM 2. If c > 0 and the immersion is full, then c = vk and n+p = (n+v) - 1 for some positive integer v. THEOREM 3. If c < 0, then c = k, that is, Mn(k) is totally geodesic in Mn+p(c).
These theorems are the local version of a classification theorem of Kaehlerian imbeddings of complete and simply connected complex space forms into complete and simply connected complex space forms (Calabi [1]).
In Theorem 1, the second fundamental form of M is parallel.
Nakagawa-Takagi [1] showed the classification theorem of complete Kaehlerian submanifolds imbedded in CPm with parallel second fundamental form.
B. KAEHLERIAN SUBMANIFOLDS WITH R(X,Y)S = 0 : Let M be a complex n-dimensional Kaehlerian submanifold of a complex space form bin+p(c). We denote by S the Ricci tensor of M and by R the Riemmannian curvature
tensor of M. We consider the condition
(*)
R(X,Y)S = 0
for any vector fields X and Y tangnet to M.
Then we have (Kon [5])
245
THEOREM 1. Iet M be a complex n-dimensional Kaehlerian subxrani-
fold of a complex space form MP(c) satisfying the condition (*). If c < 0, then M is totally geodesic.
When c > 0, Nakagawa-Takagi [2] proved the following THEOREM 2. Let M be a complex n-dimensional Kaehlerian submanifold of a complex space form Mn+p(c) (c > 0). If M satisfies the condition (*) and the codimension p is less than n-1, then M is Einstein.
When M is a Kaehlerian hypersurface, Ryan [2] proved THEOREM 3. The complete Kaehlerian manifolds with (*) which occur as hypersurfaces in a ccuplex space form Mn+1(c) (c 0 0) are (1) the complex projective space CPn and the complex quadric Q°, (2) the disk Dn of holomorphic sectional curvature c < 0.
When c = 0, Tsunero Takahashi [3] proved the following THEOREM 4. A complete complex hypersurface in Cn+1 satisfying the condition (*) is cylindrical.
C. POSITIVELY CURVED KAEHLERIAN SUB ANIFOLDS: Let M be a complex n-dimensional complete Kaehlerian submanifold in a complex projective
space C +p"with constant holomorphic sectional curvature 1. Then we have (Ogiue [2])
THEOREM 1, If every Ricci curvature of M is greater than n/2, then M is totally geodesic.
We denote by H the holomrphic sectional curvature of M. Then we have (Ogiue [1])
THEOREM 2. If H > 1 - (n+2)/2(n+2p), then M is totally geodesic. Moreover, we obtain the following theorem (Itoh [1]). THEOREM 3. If H > 3n/(4n+2), then M is totally geodesic. Thr positively curved Kaeblerian suhmanifolds, see Ogiue [5].
248
D. NORMAL CURVATURE: Let M be a ccaplex n-dimensional Kaehlerian
submanifold of a complex space form Mm(c) .. We denote by R. the curvature tensor in the normal bundle of M. We consider the following
condition on R' : (*)
Ri(%,Y) = fg(%,JY)J,
where X and Y are arbitrary vector fields tangent to M and f is a function on M. I. Ishihara [1] proved the following THEOREM. Let M be a complex n-dimensional (n > 2) Kaehlerian submanifold of a complex space form &P(c) and assume that M satisfies the condition (*). Then either M is totally geodesic or M is an Einstein Kaehlerian hypersurface of M1(c) with scalar curvature n2c.
The latter case occurs only when c > 0. Combining this theorem with Theorem 1.10, we can determine complete Kaehlerian sukmanifolds with (*) of simply connected complete
complex space fortes. E. KAEHLERIAN IMMERSIONS WITH VANISHING BOCHNER CURVATURE TENSOR: Kon [9] proved the following theorems:
THEOREM 1, Let M be a Kaehlerian manifold of complex dimension n+p with vanishing Bochner curvature tensor, and let M be a Kaehlerian suh:nanifold of M of complex dimension n with vanishing Bochner curva-
ture tensor. If p < (n+l)(n+2)/(4n+2), then M is totally geodesic in M.
THEOREM 2. Under the same assumption as in Theorem 1, if p = 1 and n > 2, then M is totally geodesic in M.
F. ANTI-INVARIANT SUBMANIFOLDS WITH FLAT NORMAL CONNECTION: Let M be an n-dimensional anti-invariant submanifold of a complex m-dimensional Kaehlerian manifold M. If n = in, by Lemma 2.3, we see that M is flat if and only if the normal crnnection of M is flat. When n < m, Yano-Kon-Ishihara [1] proved the following theorem.
247
THEOREM, Let M be an n-dimensional (n > 3) anti-invariant submanifold of a complex space form TP+p(c) (c 0 0) with parallel mean curvature vector. If the normal connection of M is flat, then M is a flat anti-invariant submanifold of some Mn(c) in Mn+p(c), where Mn(c) is a totally geodesic Kaehlerian submanifold of Mn+p(c) of complex dimension n. G. PARALLEL SECOND FUNDAMENTAL FORM: For an anti-invariant
minimal subrmnifold of a complex space form with parallel second fundamental form we have (Kon [6])
THEOREM. Let M be an n-dimensional anti-invariant minimal submanifold with parallel second fundamental form of a complex space form Mn(c). Then either M is totally geodesic or M has non-negative scalar curvature r > 0. Moreover, if r = 0, then M is flat. H. TOTALLY UMBILICAL ANTI-INVARIANT SUBMANIFOLDS: Yano [7] proved the following theorems. THEOREM 1. Let M be an n-dimensional (n > 3) totally umbilical,
anti-invariant submanifold of a complex m-dimensional Kaehlerian manifold with vanishing Bochner curvature tensor. Then M is conformally flat.
THEOREM 2. Let M be an n-dimensional (n > 4) anti-invariant
submanifold of a complex n-dimensional Kaehlerian manifold M with vanishing Bochner curvature tensor. If the second fundamental forms of M are commutative, then M is conformally flat. 1. CONFORMALLY FLAT ANTI-INVARIANT SUBMANIFOLDS: Jet M be an
n-dimensional anti-invariant submanifold of a complex projective space CPn with constant holomorphic sectional curvature c > 0. We denote by K and r the sectional curvature and the scalar curvature of M respectively. Verheyen-Verstraelen [1] proved the following theorems.
THEOREM 1. Let M be compact, conformally flat and of dimension n > 4. Then r > ((n-1)3(n+2)/4(n2+n-4))c implies that M is totally geodesic.
218 THEOREM 2. Let M be complete, conformally flat and of dimension
n > 4. Then K > ((n i)2/4n(n2fa-4))c
implies that M is totally geodesic.
J. TOTALLY UMBILICAL CR SUBMANIFOLDS: Bejancu 13] proved the
following theorem. THEOREM. Let M be a totally umbilical non-trivial CR submanifold
of a
M. If dim D+ > 1, then M is totally geodesic
in M.
K. REAL HYPERSURFACES: Let M be a real (2n-1)-dimensional real hypersurface of a Kaehlerian manifold M of complex dimension n (real
dimension 2n). Then M is obviously a generic submanifold of M. We give examples of real hypersurfaces in a complex projective space CPn with constant holomorphic sectional curvature 4 (see R. Takagi [2]). Let Cn+l be the space of (n+l)-tuples of complex numbers (z1,....zn+1). Put
-
= ((z1,...,z) n+1 E Cn+1
S2n+1
:
n+l + Iz.12 = 1}. j=1 J
For a positive number r we denote by MO(2n,r) a hypersurface of S2n+1 defined by n Iz.I2 = rizn+112,
I
j=l
n+1 I Iz.I2 = 1. j=1 J
For an integer m (2 < m 3) and M(2n-l,t) = 7(M'(2n,t)) (n > 2) are connected compact real hypersurfares of CPn with three constant principal curvatures. R. Takagi [1] [2] proved the following theorems.
THEOREM 1. If M is a connected complete real hypersurface in CPn (n > 2) with two constant principal curvatures, then M is a geodesic hypersphere.
THEOREM 2, If M is a connected complete real hypersurface in CPn (n > 3) with three constant principal curvatures, then M is congruent to some M(2n-l,m,s) or M(2n-l,t).
We denote by C a unit normal of a real hypersurface of a Kaehlerian manifold M. We put JC = -U. Then U is a unit vector field on M. We define a 1-form u by u(X) = g(X,U). If the Ricci tensor S of M is of the form
S(X,Y) = ag(X,Y) + bu(X)u(Y)
for some constants a and b, then M is called a pseudo-Einstein real hypersurface of M. We have (Kon [13])
THEOREM 3. If M is a connected complete pseudo-Einstein real hypersurface of (
(n > 3), then M is congruent to some geodesic
hypersphere MO(2n-l,r) or M(2n-1,m,(m-1)/(n-m)) or M(2n-1,1/(n-1)). When a and b are functions, see Cecil-Ryan [1]. L. GENERIC MINIMAL SUBMANIFOLDS: Let M be a compact n-dimensional generic minimal submanifold of a real (n+p)-dimensional complex projective space CP(n+p)/2 with constant holomorphic sectional curvature 4.
Then we have (Kon [15])
250 THEOREM. If the Ricci tensor S of M satisfies S(X,X) > (n-1)g(X,X) + 2g(PX,PX),
then M is a real projective space RPn (p = n), or M is the pseudoEinstein real hypersurface tr(Sm(r) x Sm(r)) (m = (n+l)/2, r = (1/2)1)
of CP(n+l)/2 (p = 1). M. SUBMANIFOLDS OF A QUATERNION KAEHLERIAN MANIFOLD: Let 0 be a 4n-dimensional quaternion Kaehlerian manifold with structure (F,G,H). Let M be a Riemannian manifold of dimension m (m < n) immersed in R by an isometric immersion f. We call M a totally real submanifold of R if Tx(M) J FTx(M), 7x(M) J GTx(M), TA(M) J H X(M) for any point x of M. Then Funabashi [1] proved the following THEOREM. Let HPn be a quaternion projective space of dimension 4n and M a connected and complete submanifold of dimension n immersed by f : M
> HPn. Assume M is a compact, totally real and minimal
submanifold satisfying the inequality !A12 HPn or, M is the
unit sphere Sn, f being congruent to the standard immersion > HPn, where n : Sn
> RPn is the natural projection.
Sn
251
CHAPTER V
CONTACT MANIFOLDS In this chapter, we study the various almost contact manifolds and contact manifolds. In §1, we give the definitions almost contact
manifolds and almost contact metric manifolds. §2 is devoted to the study of contact manifolds. We give some fundamental properties and examples of contact manifolds and contact metric manifolds. In §3,
we define the torsion tensor field of a almost contact manifold and study the normality of the manifolds. Moreover, we define a K-contact
Riemannian manifold and give some conditions for the manifold to be K-contact. In §4, we consider the contact distribution on a contact
manifold. §5 is devoted to the study of Sasakian manifolds. We define
a Sasakian manifold and give sane examples of Sasakian manifolds. Moreover, we define the q-sectional curvature of a Sasakian manifold and give the typical examples of Sasakian space forms, that is, Sasakian manifolds of constant 4-sectional curvature. In §6, we
discuss regular contact manifolds. We prove theorems of Boothby-Wang [1], and consider a principal fibre bundle, which is called the
Boothby-Wang fibration. Furthermore, we consider the relation of the Boothby-Wang fibration and Sasakian structures and prove a theorem of Hatakeyama [1]. In the last §7, we consider the Brieskorn manifold.
We give a contact structure on a Brieskorn manifold. Moreover, we show that there exists a Sasakian structure on a Brieskorn manifold
by using the deformation theorey of the standard Sasakian structure on a unit sphere.
For contact manifolds, we refered to Blair [1] and Sasaki [2].
252
1. ALMOST OONTACT MANIFOLDS
Let M be a (2n+1)-dmensional manifold and 0, g, n be a tensor field of type (1,1), a vector field, a 1-form on M respectively. If
E and n satisfy the conditions
n(E) = 1,
(1.1)
(1.2)
c 2X = X + n(X)E
for any vector field X on M, then M is said to have an almost contact structure (4,E,n) and is called an almost contact manifold.
By the definition every almost contact manifold must have a nonsingular vector field E over M. However, the Euler characteristic of any compact manifold is equal to zero and then there exists at least one non-singular vector field over the manifold. Therefore, the condition for an almost contact structure and that for an almost complex structure may be considered to impose almost the same degree of restrictions for odd and even dimensional manifolds respectively.
PROPOSITION 1.1. For almost contact structure (c,E,n) we have
(1.3)
$E = 0,
(1.4)
nW) = 0,
(1.5)
ranks = 2n.
Proof. From (1.1) and (1.2) we have 0ZE = 0, and hence OE = 0 or 4E is a non-trivial eigenvector of ¢ corresponding to the eigenvalue 0. Suppose that OE # 0. Then we have 0 = 02($E) _ -g + 71(0E)E, from which OE =
and hence
0. But we have 0 = 0ZE _
n(cE)c. This contradicts to the fact that 4E # 0 and n(oE) 0 0.
0 Therefore, we have F = 0. From this and (1.2) we easily see that 0.
In the next place, since cF = 0, rank¢ < 2n. If X is another vector of M such that cX = 0, then (1.2) implies that X = n(X)F, that is, X is proportional, to F. Therefore, we have ranlj = 2n.
QED.
We now prove that every almost contact manifold admits a Riemannian metric tensor field which plays an anologous role to an almost Hermitian metric tensor field. We first prove the following lemma.
LEMA 1.1, Every almost contact manifold M admits a Riemannian metric tensor field h such that
h(X,F) = n(X)
for any vector field X on if.
Proof. Since M admits a Riemannian metric tensor field f (which exists provided M is pa.racorrpact), we obtain h by settibg
h(X,Y) = f(X - n(X)F,Y - n(Y)F) + n(X)n(Y).
QED.
PROPOSITION 1,2. Every almost contact manifold ,lf admits a Riema-
nnian metric tensor field g such that
n(X) = g(X,E),
(1.6)
g(4X,cY) = g(X,Y) - n(X)n(Y).
(1.7)
Proof. We put
g(X,Y) = j(h(X,Y) +
n(X)n(Y)).
Then we can easily verify that this satisfies (1.6) and (1.7).
From (1.2), (1.6) and (1.7) we have
QED.
254
g(9,Y) + g(X,OY) = 0.
(1.8)
This means that 0 is a skew-symmetric tensor field with respect to g. We call the metric tensor field g, appearing in proposition 1.2, an associated Riemannian metric tensor field to the given almost contact structure
If M admits tensor field
g being an asso-
ciated Riemannian metric tensor field of an almost contact structure then M is said to have an almost contact metric structure and is called an almost contact metric manifold.
PROPOSITION 1.3. Let M be a (an+1)-dimensional manifold with Then the structure group of its
almost contact structure
tangent bundle reduces to U(n) x 1. The converse is also true.
Proof. First of all, we can choose 2n+1 mutually orthogonal unit
vectors e1,...,en0ej,...,0en,C, which form an orthononnal frame of M, and is called an adapted frame. Then with respect to this frame, we see
I In (1.9)
g =
0
01
In
0
-In
0
0
0
0
0J
0 1
0
In
0
0
0
1J
1
ng with
Now take another adapted frame
respect to which g and 0 have the same canponents as (1.9) an3 put
ei = rei,
=
0ei = r4ei,
then we can easily see that the orthogonal matrix
r = I Cn 0 must have the form
Dn
0
0
1
2'5
r=
Thus the structure group of the tangent bundle of M can be reduced to U(n) x 1.
Conversely, if the structure group of the tangent bundle of M can be reduced to U(n) x 1, then we can define g and
as tensors having
(1.9) as canponents with respect to the adapted frames. We can also give a 1-form n and a vector field & by (0,...,0,1) and t(0'...,O,1) respectively. They satisfy the desired properties.
QED.
Since the structure group of the tangent bundle of an almost contact manifold M reduces to U(n) x 1 and the determinant of every element of U(n) x 1 is positive,
have the following
PROPOSITION 1.4. Every almost contact manifold is orientable.
2. CONTACT MANIFOLDS
A (2n+1)-dimensional manifold M is said to have a contact structure and is called a contact manifold if it carries a global 1-form rj such that
(2.1)
n A (dn )n 0 0
everywhere on M, where the exponent denotes the nth exterior power. We call n a contact form of M.
A quadratic form 6 of the Grassuan algebra AV*, V* being dual to a vector space V, is said to have rank 2r if the exterior product 6r 0 0 and 6r+' = 0. Equivalently, ranks = dimV - dimV0, where V 0 {X : X e V, 6(X,V) = 01. It follows that on a contact manifold M the condition (2.1) implies that the quadratic form dry in the Grassman
algebra
has rank 2n. We then have that VO = {X : X E X(M)
256
do (X,T x(M)) = 0} is a subspace of dimension 1 on which n # 0, and which is thus complementary to the 2n-dimensional subspace on which n = 0. Let Ex be the element of VO on which n has value 1, then E is a vector field, which we call an associated vector field to n, defined over M by n, and which is never zero since n(g) = I.
THEOREM 2.1, Let M be a (2n+1)-dimensional manifold with contact structure n. Then there exists an almost contact metric structure
such that
g(X,4Y) = dn(X,Y).
Proof. For the contact form n there exists a vector field
such
1 and dn(C,Tc(M)) = 0 at every point x of M. We define a
that
skew-symmetric tensor field ¢ of type (1,1) as follows. First of all we can prove that there exists a Riemnannian metric
tensor field h such that n(X) = h(X,E). On the other hand, do is a syirplectic form on the orthogonal complement of E and hence that there
exists a metric g' and an endrxorphism 0 on the orthogonal complement of C such that g'(X,OY) = dn(X,Y) and 02 = -I. Extending g' to a metric g agreeing with h in the direction E and extending 0 so that 4E = 0, we have an almost contact structure (O,C,n,g).
For an almost contact metric structure
(2.2)
QED.
on M we put
7(X,Y) = g(X,cY).
We call
the fundamental 2-form of the almost contact metric structure.
Since 0 has rank 2n, we have n A @n # 0.
An almost contact metric structure constructed from a contact form n, appearing in Theorem 2.1, is called a contact metric structure
associated to n and a manifold with such a structure is called a contact metric manifold.
An almost contact metric structure with @ = do is a contact metric structure.
In the next place, we give a definition of contact structure due
257 to Spencer [1], which is called a contact structure in the wider sense. First of all, we notice that the following theorem of Darboux was obtained (see Cartan [1], Sternberg [1]).
THEOREM 2.2. Let w be a 1-form on an n-dimensional manifold M and
suppose that w A (du)p # 0 and (cb)P+1 = 0 on M. Then about every point there exists a coordinate system (x 1,...,xp,y1,...,yn-p) such that
w=
dy1 -
i=1
yidxi .
From this we see that for every point of a (2n+1)-dimensional contact manifold M, there exists coordinates (xl,yl,z), i = 1,...,n, such that
n yldxl.
n = dz i=1
n,z) be cartesian coordinates in (2n+1)-
Let (xl,...,x dimensional Euclidean space
R2n+1,
and let no be the 1-form on R2n+1
defined by
n0 = dz -
(2.3)
n I Yldxl. i
Then we can easily verify that
(2.4)
no A (dn0)n # 0.
R2n+1.
no is called a contact form on
A diffearorphisn f : U
> U' , where U and U' are open subsets
of R2n+1 is called a contact transformation if and only if f*n0 = Tno,
where T is a non-zero, real valued function on U. We denote by r the set of all contact transformations. r is a pseudo-group in the following sense:
(i) if f : U
> U' and g : V
mations and U' n V # 46, then
a contact transformation,
> V' are contact transfor-
f 1(U' n V)
g(U'rl V) is also
259 (ii) by the carposition in (i), r is associative and (iii) each element of r admits its inverse in r.
A contact transformation f e r such that
f*nO = n0
(2.5)
is called a strict contact transformation. The set r0 of all such transformations is a sub-pseudo-group of r.
A (2n+1)-dimensional manifold M will be called a contact manifold in the wider sense if there exists an open covering {ti} of M with homeonorphisn fi : Ui
> Vi c-
R2n+1 such that fij = fi-fi
for all pairs (i,j) such that fij is defined. Two such coordinate
systems {U ,fi} and {UJ,fl} will be called equivalent if
e r
whenever defined. An equivalence class will be called a contact structure in the wider sense on M.
From the definition we see that there exists a non-zero function
Pij on fj(Uitl i) such that
(fi-f3-)* n0 = PijnO.
Therefore we have
fIn0 =
If we define the 1-form ni on every Ui by setting
ni = fin0,
then we have
ni = on non-empty Ui n U3. Since no satisfies (2.4), we obtain
ni A (drai)n 0 0. Let D be the subbundle of the tangent bundle T(M) whose fibre D is given by
D = {X a T{(M): ni(X) = 0)
for x e U1. Recall that a vector bundle over a manifold with standard fibre lip is said to be orientable if the structure group of its asso-
ciated principal fibre bundle with group GL(p,R) can be reduced to GL+(p,R), which is a subgroup of GL(p,R) consisting of matrices with
We put ni = Tijnj on Ui n U3. Then we have
positive determinants.
ni A (drai)n = Tn+1 (nj A (dnj)n
and
is just the Jacobian of the coordinate transformation. Thus,
if M is orientable and n is even, Tij must be positive and hence vector bundle D is orientable.
We now prove the following theorem (G. W. Gray [1]). THEOREM 2.3. Let M be a (2n+1)-dimensional orientable contact manifold in the wider sense. If n is even, then M is a contact manifold.
Proof. From the assumption we see that T(M) and D are orientable, and hence the quotient bundle T(M)/D admits a global cross section S without zeros. On the other hand, ni defines a local cross section Si
over U1 by the equation ni(Si) = 1, and hence Si = hiS, where the Ws are non-vanishing functions of the same sign. We define n by n = h n i i on U1. Then we obtain a global 1-form n such that n A (dn)n # 0. QED. We now give some examples of contact manifolds. Excanple 2.1. Let
Then
R2a+1
be a (2n+1)-dimensional Euclidean space.
260
n dz - E yldxi i=1 is a contact form on R2n+1,
(xi,yi,z)
being cartesian coordinates.
Then the vector field & is given by a/az. Excmpte 2.2. Let M be a (2n+1)-dimensional regular hypersurface of R2n+2 (i.e., C with a unique tangent plane at every point). In R2n+2 (x1,...,)2n+2), we consider a 1-form with cartesian coordinates
defined by
a = xldx2 - x2dxl + ... +
x2n+1dx2n+2 - x2n+2&2n+1.
Then we have
dx2n+2),
da = 2(dxl A dx2 + ,,, + dx2n+1 A
from which
2n+1 a A (da)n =
2n-1n![
(-1)1-lxidxl
A dx2
A ...
i=1 ... A
dxi-1
A dxi+l A ... A dx2n+2].
We denote by v1,...,v2n+l 2n+1 linearly independent vectors which span the tangent space of M at x0 = (x.,...,x0 +2). We put
wi = *dxd(vl,...,v.+1),
where * denotes the Hodge star operator of the Euclidean metric on R2n+2.
Then a vector w with components w3 is normal to the hypersurface
spanned by v1,...,v2n+1. We also have
(a A (da)n)(vl,...,v2n+l) = xO.w,
where (
2n+2
) denotes the ordinary scalar product in R.
269
On the other hand, the equation of the tangent space of M at
is
x0 given by w (x-x0) = 0. Therefore, the tangent space of M at x passes
through the origin if and only if
0, that is, a A (da)
= 0 at
x0. Moreover, we see that n = i*a, i being an immersion of M into
R2n+2,
satisfies
n n (dn)n = i*(a A (da)m).
Therefore, we see that n A (dr1)n vanishes at x0 on M if and only if
the tangent space of M at x0 passes through the origin of R2n+2. Consequently, we have the following theorem (G. W. Gray [1]). THEOREM 2.4. Let M be a smooth hypersurafce immersed in R2n+2 If the tangent space of M does not pass through the origin of R2n+2 then M has a contact structure.
As a special case of Theorem 2.4, we see that an odd-dimensional R2n+2 carries a conatact structure. Furthermore, since (xl,...,x2n+2) > (-x1,..,-x2n+2), a is invariant under substitution sphere Stn+1
a also induces a contact structure on the real projective space RP2n+1 Exa.npZe 2.3. In On we put
S=
n
x6 n+i .
i=1 Let Ri be the subspace of R2n defined by xl = 0, i = 1,...,n and Rn the subspace of R2n defined by xJ = 0, j = n+1,...,2n. Then (3 induces
a contact form on a hypersurafce M of dimension 2n-1 immersed in R2n if and only if M n Ri = 5, dim(M (1 R2) = n-1 and no tangent space to
M q in R2 containes the origin in E.
Excmrple 2.4. Let M be an (n+l)-dimensional Riemrannian manifold and T(M)* its cotangent bundle. We denote by (x1,.. .,)cp+l) local coordinates on U and (p1,...,p +1) fibre coordinates over U defined with
respect to dxIs. If n : T(M)* then (pl,gl =
> M is the natural projection,
are local coordinates on T(M)*. We put
262
n+1
i pd4
i=1
on a coordinate neighborhood. We denote by T1(M)* the bundle of unit cotangent vectors. Then T1(M)* has empty intersection with the zero section of T(M)*. Moreover, the intersection with any fibre of T(M)* is an n-dimensional sphere and no tangent space to this intersection contains the origin of the fibre. Therefore, from Example 2.3, we see that S induces a contact structure on the hypersurface T1(M)*. Similarly, we obtain a contact structure a on the bundle T1(M) of
unit tangent vectors. We denote by g.. the carponents of the metric with respect to the J1 coordinates (xl,...,xn+1) and by (v1...... vn+1) the fibre coordinates
on T(M). We define a 1-form S locally by
S =
g1.vdgl,
where we have put ql = xi.7r. From this we have our assertion. Ezcvrrple 2.5. We now give an example of a contact manifold in the
wider sense.
Let M = 10+1 x RPn, where RP denotes n-dimensional real projective space. Let ()0,...,xn) be coordinates in Rn+1 and (t0,...,tn)
homogeneous coordinates in PPP. The subsets
{Ui},
i = 1,...,n defined
by ti # 0, form an open covering of M. In Ui we define a 1-form ni by
ni =
n
ti j=0E t.dxJ. J
Then we have ni ^ (drai)n # 0 and ni = (tj/ti)r1j. Thus, M has a contact
structure in the wider sense, but for n even, M is non-orientable and hence cannot carry a global contact form.
0 3. TORSION TENSOR OF ALMOST CONTACT MANIFOLDS
let M be a (2n+l)-dimensional almost contact manifold with almost contact structure (4),C,n). We consider a product manifold M x R, where
R denotes a real line. Then a vector field on M x R is given by (X,f(d/dt)), where X is a vector field tangent to M, t the coordinate of R and f a function on M x R. We define a linear map J on the tangent space of M x R by
(3.1)
J(X,fat) =
Then we have J2 = -I and hence J is an almost ccrrplex structure on
M x R. The almost carplex structure J is said to be integrable if its Nijenhuis torsion NJ vanishes, where
NJ(X,Y) = J2[X,Y] + [JX,JY] - J[JX,Y] - J[X,JY].
If the almost complex structure J on M x R is integrable, we say that the almost contact structure (4),E,n) is normal.
In the following we seek to express the condition of normality in term of the Nienhuis torsion N0 of ¢, which is defined by
N0(X,Y) = $2[X,Y] + [4X,4)Y] - $[4)X,Y] - [X,oY].
Since NJ is a tensor field of type (1,2), it sufficies to compute NJ((X,0),(Y,0)) and NJ((X,0),(O,d/dt))'for any vector fields X and Y on M. From (3.1) we have
NJ((X.0),(Y,0)) = -([X,Y],O) +
-(+[X,4)Y]-(Xn(Y))f,n([X,4)Y]}dt)
234
=
e)Y-(L,yn)R
NJ((R,0),(O,d/dt)) = (-[$X,E],
),
d )
=
dt).
Here we define four tensors N(1), N(2), N(3) and N(4) respectively by
N(1)(X,Y) = NN(X,Y) + N(2)(X,Y)
N(3)(X)
= (LLXn)Y - (LLYn)X,
= (L Ox,
N(4)(X) = (LLn)X.
It is clear that the almost contact structure (O,C,n) is nprmal if and only if these four tensors vanish. Larva 3.1, If N(1) = 0, then N(2) = N(3) = N(4) = 0.
Proof. If N(l) = 0, then we have
(3.2)
[C,X] + 4[E,0] - (Cn(X))C = 0.
Thus we have
n([C,X]) - sn(x) = 0,
which shows that N(4) = 0. From this equation we also have n([E,4Xl) = 0. On the other hand, applying ¢ to (3.2), we see that
0 = OLCX - LE¢X +
from which (LEO)X = 0 and hence N(3) = 0. Finally, from N(l) = 0 we
have
2S 0 = N1,(4X,Y) + 2dn(4X,Y)
= -[OX,Y] - [X,4Y] - (cYn(X))E - n(X)[OY,E] c[OX,$Y] + (cXr(Y))E.
Applying n to this equation and using n([E,4X]) = 0, we get
$y-X(Y) - n([4X,Y]) - 4Yn(X) + n([cY,X]) = 0.
Thus we have N(2) = 0.
QED.
In view of Team 3.1 we have PROPOSITION 3.1. The almost contact structure (c
,n) of M is
nonn3l if and only if
N+2dn®E=0. LEMMA 3.2. let M be a contact metric manifold with contact metric structure
Then N(2) and N(4) vanish. Moreover, N(3) vanishes
if and only if E is a Killing vector field with respect to g. Proof. We have
dn(cX,,Y) = 0(4X,4Y) =
-g(X,43Y) =
dn(X,Y),
from which
dn($X,Y) + dn(X,4Y) = 0.
This is equivalent to N(2) = 0. On the other hand, we have 0 = g(X,OE) = dn(X,E) = #(Xn(E) - En(X) - n([X,E])).
288
Thus we obtain
En(x) - n([E,X]) = o.
Therefore, we have LEn = 0 and hence N(4) = 0. Moreover, we see that
(LEg)(X,E) = Ei(X) - n([E,X]) = (I n)X = 0.
On the other hand, we easily see that LEdn = 0 and consequently
(LLdn)(X,Y) = (LE(D)(X,Y) = 0,
from which
0 = Eg(X,4Y) - g([E,X],$Y) - g(X,4[E,Y])
.
= (LEg)(X,,Y) + g(X,(LLc)Y) = (LEg)(X,cY) + g(X,N(3)(Y)).
Thus E is a Killing vector field if and only if N(3) = 0.
QED.
L DM 3,3, For an almost contact metric structure (4,E,n,g) of M
we have
2g((V4)Y,Z) = 3dO(X,bY,4Z) - 3d4(X,Y,Z) + g(N(1)(Y,Z),4X)
+ N(2)(Y,Z)n(X) + 2dn(cY,X)n(Z) - 2dn(4Z,X)n(Y).
Proof. The Riemannian connection V with respect to g is given by
2g(VXY,Z) = Xg(Y,Z) + Yg(X,Z) - Zg(X,Y) + g([X,Y],Z) + g([Z,X],Y) - g([Y,Z],X).
On the other hand, d$ is given by
297
3dt(X,Y,Z) = 74(Y,Z) +
74 (X,Y)
- 4'([X,Y],Z) - 0([Z,X],Y) - 4)([Y,Z],X)
Fran these equations and (2.2) we have our equation.
QED.
LEMMA 3.4. For a contact metric structure ($,E,r,g) of M with 0 = do and N(2) = 0, we have
2dn(4Y,X)n(Z) - 2dn(OZ,X)n(Y).
2g((V3&)Y,Z) =
Especially we have V0 = 0. Proof. The first equation is trivial by the assumption. We prove
that V = 0. Fran N(2) = 0 we have dn(X,&) = 0. Thus the first equation implies that V 0 = 0.
QED.
E
In the case of Lemma 3.4 it is also easy to see that the integral curves of F are geodesics, that is, V&& = 0.
Let M be a (2n+1)-dimensional contact metric manifold with contact metric structure
If the structure vector field E is a Kill-
ing vector field with respect to g, then the contact structure on M is called a K-contact structure and M is called a K-contact manifold.
Fan Lemma 3.2 we have the following PROPOSITION 3,2, Let M be a contact metric manifold. Then M is a K-contact manifold if and only if N(3) vanishes. Since we have
dr)(X,Y) = #(g(VXE,Y) - g(V,X)) = g(V,Y), for a K-contact structure, we obtain VX _ -4X. Conversely, if y _ as $ is skew-synnEtric, l; is a Killing vector field. Thus we have
268
PROPOSITION 3.3. Let M be a contact metric manifold. Then M is
a K-contact manifold if and only if VXE _ -4X.
We now give a geometric characterization of K-contact manifolds.
THEOREM 3.1, In order that a (2n+1)-dimensional Riemannian manifold M is K-contact, it is necessary and sufficient that the following two conditions are satisfied: (1) M admits a unit Killing vector field E;
(2) The sectional curvature for plane sections containing E are equal to 1 at every point of M. Proof. Let M be a K-contact manifold. Then
g(R(X,E)E,X) = g(-Q2X,X) = g(X,X) = 1,
where X is a unit vector field orthogonal to E.
Conversely, we suppose that M satisfies the conditions (1) and (2). Since E is a Killing vector field, we have
(3.3)
R(X,E)Y = VXVYE - VVXYE.
We put n(X) = g(X,E) and OX = -V. Then we easily see that 4E = 0. Fran (3.3) we also have
1 = g(R(X,E)E,X) = -g(c2X,X),
where X is a unit vector field on M orthogonal to E. Therefore, we obtain ¢2X = -X for every vector field X of M orthogonal to E and hence
02Y = -Y + n(Y)E
2" for any vector field Y of M. Moreover, we see that
drl(X,Y) = j(g(V
Consequently,
,Y) - g(VYE,X)) = -g(V 1 ,X) = g(X,cY).
is a K-contact structure on M.
QED.
From (3.3) we also have
R(X,E) _ZX = X - n(X)C.
(3.4)
We easily see that a (2n+l)-dimensional Riemannian manifold M admitting a unit Killing vector field C which satisfies (3.4) is a K-contact manifold.
4. CONTACT DISTRIBUTION
Let M be a (2n+1)-dimensional contact manifold with contact form n. Then n = 0 defines a 2n-dimensional distribution D of the tangent bundle. The distribution D is called the contact distribution which is as far from being integrable as possible from the fact that n A (dn)n
00. In the following we consider an integral submanifold of the distribution D (Sasaki [1]).
THEOREM 4.1. Let M be a (2n+1)-dimensional contact manifold with contact form n. Then there exist integral sub nnifolds of the contact distribution D of dimension n but no higher dimension. Proof. We can choose local coordinates (xl,yl,z) such that n = dz - Eyldxl on the coordinate neighborhood. Then for a point x with coordinates 4,y0',z(,) in the coordinate neighborhood, xl = xp, z = z0 defines an n-dimensional integral sukmanifold and a
maximal
integral sutmanifold containing this coordinate slice is an integral suhmanifold of D in M.
Let N be an r-dimensional integral sulmanifold of D. We suppose that r > n. We denote by e1, ...,er r linearly independent local vector
270
fields tangent to N and extend these to a basis er+1,...,eW2n+1 of M. Then we have
n(ei) = 0,
dn(ei.e? = 0,
i,j = 1,...,r.
Thus, since r > n, we see that (n A (dn)n)(e1,...,e2n+1) = 0, which is a contradiction.
QED.
PROPOSITION 4.1. Let N be an r-dimensional submanifold immersed in a (2n+1)-dimensional contact manifold M. Then N is an integral submanifold of D if and only if n and do vanish on N. Let (0,E,n,g) be an associated almost contact metric structure. Then N is an integral sukmanifold of D if and only if every tangent vector X of N belongs to D and X is normal to N in M. Procf. For any vector fields X and Y tangent to N we see that n(X) = n(Y) = 0 and hence dn(X,Y) = 0. Conversely, if n and do vanish
on N, we have
0 = dn(X,Y) = - ln([X,YI)
for any vector fields X and Y tangent to N. Thus N is an integral
suYmanifold of D. The second statement follows immediately fran the fact that dn(X,Y) = g(X,cY).
QED.
LB$'A 4.1. Let (xl,yl,z) be local coordinates for x = (xx,Yo, ZO)
such that n = dz - Eyldxl on the coordinate neighborhood. In order that
r linearly independent vectors Xt, t = 1,...,r < n, at x with canponents (a1,b1,ct) be tangent to an r-dimensional integral submanifold it is necessary and sufficient that n(Xt) = 0 and dn(Xt,Xs) = 0, that is,
ii
ct = YOat,
lait bi s
=
lasbit.
Proof. Since the necessity is clear, we prove the sufficiency.
0 We put c
= Ealbl and choose a sufficiently small neighborhood V of
the origin of R with coordinates (u 1 ...,u ) such that
xl
1
_
z = zo +
Call t t
ICtut
+
t
O y1 + tbtut,
Y1
+ tt
ctsutuS
t,s
define a mapping i of V into M. Then
axl/au = at,
ayl/aut = bt,
az/aut = Ct + 1ctsus = i.yI(axl/au ) + I (axu/aut)(ayu/auS)uS is s i
= 51(axl/au ) i
and hence the napping i defines an integral submanifold tangent to QED.
X1,...,X, at X. THEOREM 4.2.
be a vector at a point x of M belonging to D.
Then there exists an r-dimensional integral sub manifold N (1 < r < n) of D through x such that X is tangent to N.
Proof. Let'(bcl) be the conponents of X with respect to the local coordinates (xl,yl,z). Since X E D we have cl = Eiyoal. If not all *the ai's vanish, choose a2,...,al such that rank(al) = r and define
C2,...,cr by ct = Eiyoat, t = 2,...,r. We now define b2,...,b. inductively as follows. We suppose that b1,s are given. We take bs+1 (1 < s < r-1) as a set of solutions of
lfl = jal bl I
S+111
.... '
jalfl = jal bl 8+1 s s
which exists as rank (at) = r. Then {(t,bt,ct)} satisfy the conditions of the previous Lema 4.1 and hence we have an integral subnanifold N with X tangent as desired. If on the other hand, all of the si's vanish, then c1 also vanishes, so choosing b2, ...,bb such that rank(bt) = r we again have an
0 r-dimensional integral submanifold with X tangent to by Lemma 4.1. QED.
A diffeomorphism f: M
> M is called a contact transformation
if f*D = Tn for some non-vanishing function r on M. If moreover t = 1,
then f is called a strict contact transformation. The following lemma is trivial. LEMMA 4.2, A diffeomorphisn f on a contact manifold M is a contact transformation if and only if f4X belongs to D for every X in D. THEOREM 4.3. A diffeomnrphism f on a contact manifold M is a contact transformation if and only if f maps n-dimensional integral submanifolds of D onto n-dimensional integral sutmanifolds of D. Proof. Let f be a contact transformation and 11 be an n-dimensional
integral submanifold. Then we have f*(T{(N)) in D. Thus f(N) is an integral submanifold. Conversely, for any vector X at x in D, we have seen that there exists an integral sutmanifold N through x with X as a tangent vector. Since f(N) is also an integral sulmanifold, f *X is in
D and f is a contact transformation by Lemma 4.2.
QED.
5. SASAKIAN MANIFOLDS
Let M be a (2n+1)-dimensional contact metric manifold with contact
metric structure ($,C,n,g) If the contact metric structure of M is normal, then M is said to have a Sasakian structure (or normal contact metric structure) and M is called a Sasakian manifold (or normal contact metric manifold). We denote by V the operator of covariant differentiation with
respect to g. Then we have THEOREM 5.1, An almost contact metric structure is a Sasakian structure if and only if
(5.1)
(Y)Y = g(X,Y)l; - n(Y)X.
on M
Proof. If the structure is normal, we have (P = do and N(1) = N(2)
= 0. Thus lama 3.3 implies (5.1). Conversely, we suppose that the structure satisfies (5.1). Putting
Y =
in (5.1), we have --OV
this, we obtain V
= n(X)E - X, and hence, applying 0 to
= -X. Since 0 is skew-symmetric, we see that C is
a Killing vector field. Moreover, we obtain
dr,(X,Y) = #((VXrj)Y - (VYn)X) = g(X,4)Y) = 4)(X,Y).
Thus the structure is a contact metric structure. Furthermore, by a straightforward computation we have N4) + 2dn 0
= 0. Therefore, the
structure is Sasakian.
QED.
If M is a Sasakian manifold, from (5.1) we have
R(X,Y) = n(Y)X - n(X)Y,
(5.2)
where R denotes the Rie annian curvature tensor of M. From (5.2) we also have
(5.3)
R(X,E)Y = -g(X,Y)C + n(Y)X.
'THEOREM 5,2. let M be a (2n+1)-dimensional Riemannian manifold
admitting a unit Killing vector field C. Then M is a Sasakian manifold if and only if (5.3) holds.
Proof. From Theorem 3.1 we see that M has a K-contact structure Then we have
R(X,t)Y = VXVYE - VVgy, _ -(VX4))Y.
Thus Theorem 5.1 proves our assertion. We now give some examples of Sasakian manifolds.
QED.
274
Example 5.1. Let M be a real (2n+1)-dimensional hypersurface of a Kaehlerian manifold M of ccixplex dimension n+1. We denote by J the
almost complex structure tensor field of M and p the operator of covariant differentiation in M. The operator of covariant differentiation with respect to the induced connection on M will be denoted by V. We denote by C a unit normal of M in
and by A the second fundamental
tensor of M. We put JC = -C. Then
is a unit vector field on M. For
any vector field X tangent to M we put
JX = Ox + n(X)C,
where n is a 1-form dual to E. Then we have
42X = -x + n(X)C,
4
= 0,
n(4X) = 0,
n(X) = g(E,X)
Thus (O,C,n,g) defines an almost contact metric structure on M.
bbreover, the Gauss and Weingarten formulas for M are respectively given by
OXY = VXY + g(AX,Y)C
and
OXC = -AX.
Fran these equations we have
(Vx4)Y = n(Y)AX -
V
= 4,AX.
We suppose that AX = -X + Sn(X),, where a is a function on M. Then
we have
g(V,Y) + g(V,X) =
g(¢Y,X) = 0.
Therefore, E is a Killing vector field on M. Furthermore, we have
(Vxo)Y = g(X,Y)E - n(Y)X.
275 Thus Theorem 5.1 shows that M is a Sasakian manifold. Conversely, if M is a Sasakian manifold, we obtain
AX = -x + n(AX+x)E,
from which
AX = -X + On(x)E,
where we have put S = n(4) + 1. Consequently, a real hypersurface M of a Kaehlerian manifold M is a Sasakian manifold if and only if its second fundamental form A satisfies A = -I + an ® E. We notice that such a real hypersurface is an n-umbilical hypersurface. For a real hypersurface M of a Kaehlerian manifold 22 we see that
N0(X,Y) + 2dn(X,Y) = n(X)(bA-A$)Y - n(Y)(8A-Ac)X.
From this we easily see that 4A = A¢ if and only if N0(X,Y)+2dn(X,Y)F = 0. Therefore, the almost contact metric structure (0,&,n,g) on M is normal if and only if i; is a Killing vector field on M.
Excnnpie 5.2. Let Cn+1 be a caiplex (n+l)-dimensional number space. Stn+1 be a (2n+1)-
Then Cn+l admits a Kaehlerian structure J. Let dimensional unit sphere, i.e.,
Stn+1 = {Z E Cn+1
:
IZI = 1}.
Then 52n+1 is a real hypersurface of Cn+l and the second fundamental Cn+1 is given by A = -I. Therefore, from Example form A of g2n+1 in 1.1, Stn+1 admits a Sasakian structure (0,E,n,g) which is called a Stn+1 natural Sasakian structure on Excmrpte 5.3. It R2a+2 be a (2n+1)-dimensional number space. We
put
R2 n n = i(dz -
yidxi), i=1
(xl,yl,z) being cartesian coordinates. Then the structure vector field
E is given by E = -2a/az and the Riemannian metric tensor field g is given by
n ((dxl)2 + (dyl)z).
g = 4(n ®n +
i=1 This gives a contact metric structure on R2n+1 as follows. First of all we have
16iJ+ylyJ
0
0
6ij
0
-yl
0
1
g = 4
j
We give a tensor field c of type (1,1) by a matrix form
0
-6.. 0
6ij
0
0
0
yj
0
The vector fields Xi = 2a/ayl, Xn+i = 2(a/axl + yla/az),
form a
-basis for the contact metric structure. On the other hand, we can
see that N0 + 2dn ® = 0 and hence the contact metric structure is normal.
Let M be a (2n+1)-dimensional Sasakian manifold with Sasakian structure
From (5.1) we easily see
277 (5.4)
R(X,Y)4>Z = 4>R(X,Y)Z + g(4>X,Z)Y - g(Y,Z)4>X
+ g(X,Z)4>Y - g(tY,Z)X.
Erom (5.4) we also have the following equations:
(5.5)
g(Y,Z)X - g(X,Z)Y
R(X,Y)Z =
- g(QY,Z)4>X + g($X,Z)4>Y,
(5.6)
g(R($X,cY)4>Z,4>W) = g(R(X,Y)Z,W) - n(Y)n(Z)g(X,W)
- n(X)n(W)g(Y,Z) + n(Y)n(W)g(X,Z) + n(X)n(Z)g(Y,W).
A plane section in X(M) is called a 4>-section if there exists a
unit vector X in X(M) orthogonal to E such that {X,¢X) is an orthonormal basis of the plane section. Then the sectional curvature K(X,4>X) = g(R(X,4>X)4>X,X) is called a 4>-sectional curvature, which will
be denoted by H(X). We shall show that on a Sasakian manifold the 4>-sectional curvatures determine the curvature completely.
In the following we prepare some lemmas. We put
P(X,Y;Z,W) = g(Y,Z)g($X,W) - g(4>X,Z)g(Y,W)
+ g(4>Y,Z)g(X,W) - g(X,Z)g(4>Y,W).
Then we have
P(X,Y;Z,W) = -P(Z,W;X,Y).
If {X,Y} is an orthonormal pair orthogonal to E and if we put g(X,4>Y) = cos9, 0 < e < n, then
P(X,Y;X, Y) = -sin2O.
278
We now put B(X,Y) = g(R(X,Y)Y,X)
for any vectors X and Y and D(X) =
for any vector X orthogonal to E. Then we have LEMMA 5,1, For any vectors X and Y orthogonal to E we obtain
B(X,Y) = 32(D(X+4Y) + 3D(X-cY) - D(X+Y) - D(X-Y) - 4D(X) - 4D(Y) -
Proof. First of all we have
D(X+Y) + D(X-Y) = 2[D(X) + D(Y) + 2B(X, Y) + 2g(R(X,OX)OY,Y) + 2g(R(X,OY)$X,Y)].
Putting OY instead of Y in this equation we obtain
D(X+qY) + D(X-4Y) = 2[D(X) + D(cY) + 2B(X,Y) + 2g(R(X,¢X)cY,Y) + 2g(R(X,Y)4Y,4X)].
Since D(cY) = D(Y), we find
D(X+Y) - D(X-Y) - 4D(X) - 4D(Y)
= 12B(X,Y) - 4B(X,¢Y) + 8g(R(X,4X)OY,Y) +
On the other hand, from (5.4) and the Bianchi's identity, we have
0 8g(R(X,$X)$Y,Y) = 8[B(X,Y) + B(X,$Y) + 2P(X,Y;X,$Y)].
We also have the following equations:
12g(R(X,Y)$Y,$X) = 12[B(X,Y) + P(X,Y;X,$Y)],
4g(R(X,$Y)Y,$X) = 4[-B(X,tY) + P(X,$Y;X,Y)].
From these equations we have our assertion.
QED.
We notice that D(X) = H(X) if and only if X is a unit vector and B(X,Y) = K(X,Y) if and only if {X,Y} is an orthonormal pair.
LEMMA 5.2. Let {X,Y} be an orthonoru l pair of the tangent space
of a Sasakian manifold M orthogonal to . If we put g(X,$Y) = cose (0 < 6 < n), then
3(1-cos6)aH( X_$Y )
K(X,Y) = 8[3(l+cose)2H(
- H( X
YX+Y
) - H( X Y) - H(X) - H(Y) + 6sin2e ] .
Proof. Since D(Z) = IZ14H(Z/IZI) for any Z, we see that
D(X+$Y) = JX+$Y14H( X+$Y
4(1+cose)2H( X+$Y ).
Similarly, we obtain
D(X-$Y) = 4(1-cose)2H(
IX-9yj
D(X+Y) = 4E( X+Y ),
D(X-Y) = 4H( X Y ).
),
From these equations and Lemma. 5.1 we have our equation.
QED.
THEOREM 5.3. The 0-sectional curvatures determine the curvature of
a Sasakian manifold.
280
Proof. Since the sectional curvatures of a Rienannian manifold determine the curvature, it sufficies to show that the sectional curvatures are determined by the 0--sectional curvatures uniquely.
Let {X,Y} be an orthonormal pair. We put
X = n(X)g + aZ,
Y = n(Y)g + bW,
where a = (1 - 3(X)2)1/2, b = (1 -
n(Y)2)1/2,
Z, W being orthogonal
to E. Then Z and W are unit vectors. By a simple computation we find
K(X,Y) = a2n(Y)2 - 2abn(X)n(Y)g(Z,W) + b2n(X)2 + a2b2g(R(Z,W)W,Z).
Noticing that
g(Z,W)
isnWn(y),
g(R(Z,W)W,Z) = (1-g(Z,W)2]K(Z,W) = [1 - b2 a (X)2n(Y)2]K(Z,W),
we obtain
K(X,Y) = 3(X)2 + n(Y)2 + [1-3(X)2-n(Y)2]K(Z,W).
On the other hand, by Lemma 5.2, K(Z,W) is determined by 0-sectional curvatures. This ccmpletes the proof.
QED.
THEOREM 5.4, If the Q-sectional curvature at any point of a Sasakian manifold of dimension > 5 is independent of the choice of 0-section at the point, then it is constant on the manifold and the curvature tensor is given by
R(X,Y)Z = }(c+3)[g(Y,Z)-g(X,Z)Y] + *(c-1)[n(X)n(Z)Y-n(Y)n(Z)X +g(X,Z)n(Y)C-g(Y,Z)n(X)C+g(oy,Z)o% g(ox,Z)OY+2g(X,OY)OZ],
289
where c is the constant ¢-sectional curvature.
Proof. We notice that R(X,E)X = - and R(U,X)E _ -X for any vector X orthogonal to E. Thus we have actually prove that any vector field of type (5.3) on a Sasakian manifold satisfying the symmetries of the curvature tensor, the Bianchi's identity, (5.4) and which coincides with the values of the 4>-sectional curvatures must be the curvature tensor. From this we see that R(X,Y)Z is of the given form.
Conversely, we easily see that c is constant when dimension > 5. QED.
A Sasakian manifold M is called a Sasakian space form if M has constant ¢-sectional curvature c, and will be denoted by M(c).
Example 5.4. Let 52n+1 be a unit sphere with natural Sasakian structure (4>,F,n,g). We consider the deformed structure:
n* = an,
F* = a 1E,
g* = Cg + (a2 - a)n 0 n,
where a is a positive constant. We call this deformation D-homothetic deformation. Then
is a Sasakian structure on Stn+1 Stn+l(c)
with constant 4-sectional curvature c = 4/a - 3. We denote by the Sasakian manifold with this structure.
Example 5.5. Let R2n+1 be a (2n+1)-dimensional number space with Sasakian structure as in Example 5.3. It is checked that R2n+1 is of constant q-sectional curvature -3 and we denote it by R2n+1(-3). Example 5.6. let CDn be a simply connected bounded complex danai.n
in Cn with constant holomorphic sectional curvature c < 0. We denote by (J,G) a Kaehlerian structure on CEP. Since the fundamental 2-form S2 of the Saehlerian structure is closed, S2 = dw for some real analytic 1-
form w. Let t denote the coordinate on R and put Ti = 2w + dt on a product space R x CUP. If we consider R as an additive group, then n is an infinitesimal connection form on the trivial line bundle (R,CD"). We have F = 3/at and g = ,m*G + n 0 Ti, where w : (R,&')
> CEP is
282
the projection. T1 is also written as r = 2mr*w+ dt, and we have drt = 2n'. Therefore, these tensors define a Sasaan structure on (R,CD") with constant 0-sectional curvature k = c-3 < -3. We denote. this by
(R,&)(k). We show that three types of model spaces in these examples 5.4, 5.5 and 5.6 are unique up to isomorphism's, where an isomorphism means
a diffecmorphism which maps the structure tensors into the corresponding structure tensors (Tanno [41). THEOREM 5.5. Let M be a (2n+1) -dimensional complete simply connec-
ted Sasakian manifold with constant 0-sectional curvature c. (1) If c > -3, then M is isamorphic to S
+1(c) or M is D-harothe-
tic to S2n+1
(2) If c = -3, then M is isomorphic to
R2n+1(-3);
(3) If c < -3, then M is isomorphic to (R,C')(c). Prcof. From the assumption M admits local 0-holomorphic free mobility and hence M admits global 0-holarorphic free mobility because of M is complete and simply connected. Thus M admits an autarorphism group Aut(M) such that, for any point x and y, any 0-section at x carried to any other 0-section at y by some element of Aut(M). Aut(M) is of (n+1)2-dimension and M is diffeomorphic to a homogeneous space Aut(M)/(isotoropy group). Thus we can assume that M is real analytic, and also that g is real analytic. We denote by M* one of the model spaces corresponding to c > -3, = 3 or < -3 and by
the
structure tensors of M*. For any point x of M and x* of M*, let
(el,...en0el,...,0en,E) and
i,...,en,#
be orthonormal
0-basis at x :uld x* respectively. We define a linear isomorphism F: TY(M)
> TY*(M*) by Fei = e1 4l, Foei = 0*et (i = 1,...,n) and
FE = F*. Then we have F0 = 0*F and F is isanetric at x, that is, F is isomorphic at x. Since both 0- and 0*-sectional curvatures are equal to c, F maps R into R*, F being considered as a map of tensor algebra. The covariant derivatives of 0 and E are written in terms of and hence the covariant derivative of R is expressed by
$, E, g
0, E and g,
that is, F maps (VR) x into (V*R*)x*. Likewise, we see that F maps
283
(vkR)x into (v*kR*)x* for every positive integer k. Then we have an
isanetry f of M onto M* such that f(x) = x* and the differential of f at x is F (cf. Kobayashi-Nomizu [1; p.259-261]). We then have that (vE)x is mapped to (V*E*)x*. Thus we have
(V*(fl;))x* =
f is also a Killing vector field. By (f&)x*
f
_ CX* and (V*(fE))x* = (V*C*)x*, we get f _
and n*) are determined by g and
*. Because 0 and n (4*
(g* and *), f is an isomorphism
between M and M*.
QED.
In the following we study the properties of the Ricci tensor of a Sasakian manifold. First, let M be a (2n+1)-dimensional K-contact mani-
We denote by S and Q the Ricci
fold with stricture tensors
tensor and the Ricci operator of M respectively. We prove PROPOSITION 5.1. If the Ricci tensor S of a K-contact manifold M is parallel, then M is an Einstein manifold. Proof. Since
X - n(X)E, we obtain S(C,C) = 2n. Thus we
have (VXS)(C, ) = 2S(cX,E) = 0, and hence S(X,F) = 2nn(X). From this we see that S(X,gY) = 2ng(X,OY). Therefore, we obtain S(X,Y) = 2ng(X,Y) for any vector fields X and Y, which means that M is an Einstein mani-
fold.
QED.
PROPOSITION 5.2, Let M be a K-contact manifold. If M is locally symmetric, then M is a Sasakian manifold with constant curvature 1. Proof. By the assumption we have
-R(X,4Y) -
g(cY,X)E + n(X)¢Y.
Replacing Y by cY in this equation, we see that
R(X,Y)l; + R(X,E)Y = 2n(Y)X - n(X)F -
From this we find
284 R(X,Y)Z + R(X,Z)Y = 2g(Z,Y)X - g(X,Y)Z - g(Z,X)Y.
Thus, for any orthonormal pair {X,Y}, we obtain
K(X,Y) = g(R(X,Y)Y,X) = 1,
which shows that the sectional curvature of M is 1 and hence M is a Sasakian manifold.
QED.
From Propositions 5.1 and 5.2 we see that-the notion of parallel Ricci tensor and locally symmetric are not so essential on a K-contact manifold and consequently on a Sasakian manifold.
From (5.6) and the Bianchi's identity we have LEW 53, The Ricci tensor S of a (2n+1) -dimensional Sasakian manifold M is given by
2n+1 g(gR(X,AY)ei,ei) + (2n-1)g(X,Y) + n(X)n(Y).
S(X,Y)
i=1
Moreover, by (5.6), we easily see that LE('MA 5,4, The Ricci tensor S of a (2n+1)-dimensional Sasakian
manifold M satisfies the following equations:
S(X,E) = 2nn(X),
S(¢X,cY) = S(X,Y) - 2nn(X)n(Y).
From Lemma 5.3 and the Bianchi's identity we also have LE74MA 5.5. The Ricci tensor S of a (2n+1)-dimensional Sasakian
manifold M satisfies
(VzS)(X,Y) = (VxS)(Y,Z) +
n(X)S(4Y,Z)
- 24(Y)S(4X,Z) + 2nn(X)g(4Y,Z) + 4nn(Y)g(gX,Z).
0 If the Ricci tensor S of a K-contact manifold M is of the form
S(X,Y) = ag(X,Y) + bn(X)n(Y),
a and b being constant, then M is called an n-Einstein manifold. Obviously, we have PROPOSITION 5.3. If a Sasakian manifold M is of constant +-sectional curvature, then M is an n-Einstein manifold.
PROPOSITION 5.4. Let M be a (2n+1)-dimensional Sasakian manifold. If the Ricci tensor S of M satisfies S(X,Y) = ag(X,Y) + bn(X)n(Y), then
a and b are constant. Proof. From the assunuption on the Ricci tensor S we have a+b = 2n
and r = (2n+l)a + b, where r denotes the scalar curvature of M. Then we have Za = -Zb'and Zr = (2n+l)Za + Zb = -2nZb. On the other hand,
Team 5.5 implies
Zr = 2Za + 2(N71(Z) = -2Zb +
Therefore, we obtain
(n-1)(Zb) = -(Cb)n(Z).
Putting Z =
in this equation, we find b = 0 and hence Zb = 0, which
shows that b is constant. Then a is also constant.
QED.
298 6. REGULAR CONTACT MANIFOLDS
Let M be a (2n+1)-dimensional contact manifold with a contact form n. We denote by
the associated vector field to n. We say that
a contact structure n is regular if each point has a regular coordinate neighborhood , i.e., a cubical coordinate neighborhood U such that
the integral curvaes of
passing through U pass through the neighbor-
hood only once. Then the vector field E; is- said to be regular.
Hereafter we will assume that the manifold M to be compact. Then, if E is regular, & is a closed vector field, that is, the orbit of through an arbitrary point is a closed curve. Let B be the set of orbits of C. Then we have the natural projection
IT
: M
> B,
making correspond every point p of M to the orbit through p. Then we see that B with the quotient topology is a 2n-dimensional differentiable manifold and n is a differentiable map (see Palais [1]). We put
11E(p) = inf( t : t > 0,
0t(p) = p).
Clearly, VC is constant on every orbit. We call }r(p) a period functions of E. If & is regular, we see that µ,(p) is constant on M.
We now give the theorems of Boothby-Wang [1]. THEOREM 6.1, Let n be a regular contact form on a compact manifold M of dimension 2n+1. Then there exists a gauge transformation n' = on such that the vector field E' associated to n' has the following
properties: (1) The group of diffeomorphisms of M which is generated by E'
is a 1-dimensional ccepact Lie group; (2) Each element of this group except the identity does not leave
any point of M invariant.
0 Proof. We denote the period by y and put n' = (1/11)n. Then we
have &' = p&. Thus, F' has the same set of orbits as E. We may easily verify that p'(p) = pC,(p) = 1 for any p c M. Hence, the group of diffeomorphisms of M generated by &' depends only equivalence classes of the real variable t mod 1. Consequently, it is a 1-dimensional compact Lie group. We call this group the circle group and denote it by S1. The assertion (2) is evident.
QED.
If a 2n-dimensional manifold admits a closed exterior 2-form with maximal rank c, i.e., a 2-form 52 such that d0 = 0, (S2n) 0 0,
then we call the manifold a symplectic manifold with fundamental form 9 and the structure given by n the symplectic structure.
THEOREM 6.2. Let M be a (2n+1)-dimensional compact contact manifold with a regular contact form n. Then we have the following: (1) M is a principal circle bundle over B; (2) n defines a connection in this bundle;
(3) The base manifold B is a 2n-dimensional symplectic manifold whose fundamental form 0 is the curvature form of n, i.e., dQ = 7010 is the equation of structure of the connection; (4) S2 determines an integral cocycle on B.
Proof.-By Theorem 6.1 we can modify n so that E generates the circle group S1. (1) Since
is regular we can choose an open covering {U1) of (x1,...,x2n+1)
M such that on Ui we have coordinates
with the integral
curves of C being given by xl = const......x2n = const. Then {n(Ui)) is an open covering of B. We define local cross sections si : n(Ui) si(xl,...,x2n) _ (xl,...,x2n,c) for some l > M withasi = id. by constant c, x ,...,x
being regarded as the coordinates on rr(Ui).
We define the maps fi -: n(Ui) X S1
> M by fi(p,t) = ¢tsi(p)
p c M, 0t being elements of the I -parameter group of diffeomorphisms
generated by E. Then fi are coordinate functions for the bundle. (2) Since we have LEn = O and Ldn = 0, rj and do are invariant
under the action of Si. Now, the Lie algebra 6 of the group S1 is 1-dimensional vector space. So, we identifyrel with the line R, and
283 take A = d/dt as a basis of Cg
Then n = nA is an invariant form on
M which has its values in d1. For the sake of simplicity we write n instead of n again. To prove that in defines a connection we must show that the following two conditions: (a)
n(A*) = A,
where At is the fundamental vector field corresponds to A; (b)
Rtn = (ad t -')n,
where Rt is the right translation by t e S1, Rt is the dual map of Rt on the space of 1-forms on M.
First we have n(A*) = n(A*)A = nQ)A = A. This proves (a). Next, as Rt(p) = 4t(p) and n is invariant under the group S1, we have Rtn = n. Since S1 is abelian, we see that ad(t 1) = 1. Thus we have (b). (3) We denote by S2 the curvature form of the connection n. Then
dn(X,Y) _ - j[n(X),n(Y)] + Q(X,Y).
However, as the group S1 is abelian [n(X),n(Y)] = 0 and hence
do = S2. On the other hand, do is invariant and
0. Therefore there exists a 2-form Q on M such that do = n*St. Now, n*dit = dn*52 = den= 0, so that dSt = 0 and n*(Stn) _ (n*S2)n = (dn)n 0 O'giving Stn 0 0. Therefore M is symplectic. (4) Finally, as the transition functions fib : Ui
Uj
S1
are real (mod 1) valued, one can check that [S2] a H2(M,Z) (see
Kobayashi [1] for details).
QED.
Conversely we now prove the following THEOREM 6.3, bet B be a 2n-dimensional symplectic manifold such that its fundamental form Q determines an integral cohcmology class. Then there is a principal circle bundle it : M
> B and a connec-
tion n on M such that (1) n is a contact form on a (2n+1)--dimensional manifold M, and
(2) its associated vector field E generates the right translations of the structure group S' of the bundle.
Proof. We take A = d/dt as the basis of the Lie algebra of S1.
0 According to a theorem of Kobayashi [1], there exists a principal circle bundle M over B and a connection n on M such that n defined by n = nA satisfies do =Tr*r. Since Tr* is an isomorphism, we have (dn)n = (,1*Q)n = ,ff*O # 0. Therefore, if we denote the fundamental
vector field corresponding to A by A* and 2n linearly independent horizontal vectors by X1,.... X2n, then we have (n n (dn)n)(A*,X1,...,
X2n) 0 0. Thus, n is a contact form of M. Moreover, since n(A*) = A, we have n(A*) = 1. On the other hand, for arbitrary vector X of M we have dn(A*,X) = Tw*Q(A*,X) = Q(TiA*,TrX). Hence, we get d (A*,X) = 0.
Therefore, A* is the vector field associated to n, i.e., A* = C. Hence,
generates the group S1 of right translations of the bundle
W.
M. Let Tr : M
> B be a Boothby-Wang fibration (Theorem 6.2).
Since B carries a global symplectic form Q, there exists a Riemannian metric h and a Tensor field J of type (1,1) such that (J,h) is an almost Kaehlerian structure on M with Q as its fundamental 2-form. Let n be the contact form on M with do = Tr*Q and
vector field. We define a tensor field
OX = (JTr*X)*,
its associated
of type (1,1) on M by
X C Tx(M),
where * denotes the horizontal lift with respect to n. Then
02X = (JTr*(JT*X)*)* _ (J2Tr,'X)* = -(,T*X)* = -X + n(XX.
that is,
2 =-I+n0 C. Therefore,
is an almost contact structure on M. We define a
Riemannian metric g of X by g(X,Y) = h(n*X,TT*Y) + n(X)n(Y),
X,Y E x(M),
290 that is,
g=n*h+n0n Clearly E is a unit Killing vector field with respect to g. Moreover
we have
g(X,cY) = h(n*X,n*(JTr*Y)*) = h(Tr*X,JTr*Y) = S2(n, X,n*Y) = n*S2(X,Y) = dn(X,Y) and similarly
g(gX,OY) = h(n*X,n*Y) = g(X,Y) - n(X)n(Y).
Thus we have (Hatakeyama [1]) THEOREM 6.4, Compact regular contact manifold carries a K-contact structure.
Let N0 and NJ be the Nijenhuis torsions of 0 and J respectively. Since we have LEA = 0, we obtain
N(E,X) + 2dn(E,X)E = 42[E,X] - O[E,OX] = 0.
On the other hand, for projectable horizontal vector fields X and Y, we have
N0(X,Y) + 2dn(X,Y)E _ (J27r*[X,Y])* + [(Jn,X)*,(Jn*Y)*] - (Jn*[(J1T*X)*,Y])*
- (Jn*[X,(Jn*Y)*])* + 2dn(X,Y)E
_ (JZ[n*X,n*Y])* + [Jn+X,JTr*Y]* + n([(Jn*X)*,(Jn*Y)*l) - (J[Jn*X,n*Y])* - (J[Tr,,X,Jn*Y])* + 2dn(X,Y)E
291
= (NJ(n, C,n*Y))* - 2(0(Jn
,Jn*Y))* - (SZ(w*X,n*Y))*)
= (NJ(n*X,Tf*Y))*.
Thus we see that K-contact structure (c,C,n,g) is Sasakian if and only if M is Kaehlerian.
A Hodge manifold is by definition a compact Kaehlerian manifold such that the fundamental 2-form 0 determines an integral cocycle over the manifold. Thus, by the previous considerations, we have (Hatakeyama [1])
THEOREM 6.5, In order that a (2n+1)-dimensional manifold M with a regular contact structure admit a Sasakian structure is that the base manifold B of the Boothby-Wang fibration of M is a Hodge manifold.
7. BRIESKORN MANIFOLDS
Let Cn+1 be the complex vector space of (n+l)-tuples of complex numbers (z 0,zl,...,zn). A Brieskorn manifold is by definition a (2n-1)-
dimensional submanifold
BBn-1
(aO,al,...,an) in Cn+1 defined by equa-
tions
z0 +za,+...+zan=0
(7.1)
and
zOzO + zlzl + ... + znzn = 1,
(7.2)
where a0,al,...,an are positive integers. We denote by B2n
B2n-l(a0,al'".,an)
for simplicity. We also denote by B2n the complex hypersurface
in Cn+l - {0) defined by (7.1). The Brieskorn manifold 52n+1.
intersection of B2n with the unit sphere Let us consider the C-action on Cn+1 defined by
B2n-1
is the
292
mw/atz
j,
z =e
(7.3)
where m is the least carmon multiple of integers a0,al,...pan and w
is a complex variable. We can easily see that the Faction fixes the origin 0 and transforms 92n onto itself. Therefore, restricting w to at s = 0, we see that
its real part s and differentiating
ze Oa
ul = (a z3),
is a tangent vector of B2n at z. Similarly, restricting w to its purely
imaginary part it (t E R), we see that
U2 = iu1 = (a izj),
z E
B2n
J
is a tangent vector of On at z orthogonal to ul. When we restrict w to it, (7.3) gives a S1-action on Cn+1 and the S1-action leaves T3 2n, S2n+1 and so B2n-1. Therefore, if z E B2n-1, the orbit of z under this B2n1 and so u2 is a tangent vector of B2n-1 action lies on
We denote by dz the differential at a point z on On. Then we have
1Z-dz=0 j , J where f(z0,zl,...,zn) is the polynomial on the left hand side of (7.1). This is equivalent to = 0, where the bracket means the inner product of two vectors 3f/3z (the canplex conjugate of of/az) and dz
in C. Thus we have Re = 0
and
Re = 0.
Therefore we see that
a -l v1 = (a-) _ (ajZjJ
a.-1 ),
v2 = (izj) _ (iajzjJ
) = ivl
29 3
are normal vectors of B2n at the point z. We easily see that u1, u2, v1 and v2 are mutually orthogonal.
We now restricted the point z to the one on
B2n-1.
Then the unit
normal vector n of S2n+1 has z. as its caxponents. We see that v1, v2 -1 and n are normals to B2n in Cn+1 and they are linearly independent. We put
v=n+avl+uv2, where a. A = -(Re(Eai zj3)/),
a.
u = Im(Eajzj3)/.
Then v1, v2 and v are normal vectors of B2n-1 in Cn+1 orthogonal with B2n-1 which each other, which shows that v is a normal vector of tangent to B2n at each point z c
B2n-1.
B2n inherits the ccinplex structure from that of Cn+l. If we denote
by > the Kaehlerian inner product, then
= Re.
Then we have
n
= jiJI (zjdzj - zjdzi-). We define a real 1-form tt on Bin-1 by
Ti = jij(zi dzj - zi dzj).
We notice that n = = .
Sasaki-Hsu [1] proved that the form n is a contact form on B2n-1. THEOREM 7.1. Every Brieskorn manifold is a contact manifold.
Proof. We shall show that the 1-form n on 4-1 is a contact form.
294
Since
n
do = i j! dz4 A dz j we have
(7.4)
n A
(dn)n-1
= din{I J (zjdzj - -Zjd )} A
dZk A dzk) n-1
n _ i(n-1)!in[{J) (zJ.d =:O
J
- dz.)}
A {E (dz0 A dz0) A ... A (dzj Aid-zj -)
j) ' Tp(B2n-1) = (Dp it Tp(B2n-1))+< fp where <EP> is the 1-dimensional space generated by C at p. Since
¢ = 0, we see that
maps Tp(B2n-1) into itself. Therefore,
B2n-1
is
MG an invariant suhmanifold of Stn+1 with respect to
Any
invariant suLmanifold of a Sasakian manifold is also a Sasakian manifold (see §1 of Chapter VI). Consequently, B2n-1 admits a Sasakian
structure.
QED.
306
EXERCISES
A. K-CONTACT RIEMANNIAN MANIFOLDS: Let M be a K-contact Riemannian
manifold with structure tensors
We denote by R and S the
Riemannian curvature tensor and the Ricci tensor of M respectively. Then we have (Tanno [1]) THEOREM. Let M be a K-contact Riemannian manifold. (1) If
0 for any X, then M is an Einstein manifold;
(2) If R(X,F)R = 0 for any X, then M is of constant curvature. B. IMMERSED K-CONTACT RIEMANNIAN MANIFOLDS: Takahashi-Tanno [1] studied K-contact Riemannian manifolds immersed in a space form as hypersurfaces and proved the following theorems. THEOREM 1. If an n-dimensional K-contact Riemannian manifold M is isometrically immersed in an (n+1)-dimensional space form M, then M is a Sasakian manifold.
This theorem gives a sufficient condition for a K-contact Riema-
nnian manifold to be Sasakian. THEOREM 2. Let M be an n-dimensional K-contact Riemannian manifold isometrically immersed in an (n+l)-dimensional space form M of constant curvature 1. Then (1) the type number t < 2, and (2) M is of constant curvature 1 if and only if the scalar curvature r = n(n-1).
THEOREM 3. Let M be an n-dimensional K-contact Riemannian manifold isometrically immersed in an (n+l)-dimensional space form M of constant curvature c # 1. Then c < 1 and M is of constant curvature 1. C. PRODUCT OF ALMOST CONTACT MANIFOLDS: bet M and M be manifolds of dimension 2n+1 and 2m+1 and let E =
and E =
be
almost contact structures on M and M, respectively. We define J by
J(X,X) _ (% Then, it is easily seen that Jz = -I, which shows that J is an almost complex structure on M x M. We call J the induced almost complex struc-
ture onMxMbyEandE. If the induced almost complex structure J on M x M by E is integ-
rable (i.e., complex structure), we call E is integrable. Morimoto [1] proved the following theorems.
THEOREM 1. The induced almost complex structure on M x M is integrable if and only if z and E are both integrable.
THEOREM 2. An almost contact structure E is integrable if and only if E is normal.
Since an odd dimensional sphere is normal, we obtain (CalabiEckmann [1]) THEOREM 3. The direct product of two spheres of odd dimension
has a complex structure. D. TRIPLE OF KILLING VECTORS: Assume that a Riemannian manifold M admits three unit Killing vectors F, n and S which are mutually orthogonal and satisfy
F;
=
[n10 ,
n =
[;,F],
; = #R,n]
We call such a set {F,ri,l} a triple of Killing vectors. A triple {F,n,c} of Killing vectors is called a K-contact 3-structure, if F,
n and r are K-contact structures. A K-contact 3-structure is called a Sasakian 3-structure if all of F, n and c are Sasakian structures.
If, for a K-contact 3-structure
any two of C. n and L are
Sasakian structures, then {F,r,r} is necessarily a Sasakian 3-structure (see Kuo [1], Kuo-Tachibana [1], Tachibana-Yu [1]). Kashiwada [1]
proved
0 THEOREM. A Riemannian manifold with Sasakian 3-structure is an Einstein manifold. E. SASAKIAN +-SVhMETRIC MANIFOLDS: In view Proposition 5.2 a symmetric manifold condition is too strong for a Sasakian manifold.
Toshio Takahashi [1] introduced the notion of Sasakian 0-symmetric manifolds. Let M be a Sasakian manifold with structure tensors (0,g,7j,g).
We denote by R the Riemannian curvature tensor of M. A Sasakian manifold M is said to be a locally 4-symmetric manifold if
2[(OVR)(X,Y)Z] = 0
for-any horizontal vectors X, Y, Z and V, where a horizontal vector means that it is horizontal with respect to the connection form n of the local fibering; namely a horizontal vector is nothing but a vector which is orthogonal to C. Takahashi [1] proved the following
THEOREM 1, A Sasakian manifold is a locally -symmetric manifold if and only if each Kaehlerian manifold, which is a base space of local fibering, is a Hermitian locally symmetric manifold.
A complete and simply connected Sasakian locally 0-symmetric manifold is called a globally 4-symmetric manifold.
THEOREM 2. A Sasakian globally 4-symmetric manifold is a principal G1-bundle over a Hermitian globally symmetric space, where G1 is a
1-dimensional Lie group which is isomorphic to the 1-parameter group of global transformations generated by E.
F. CONTACT BOCHNER CURVATURE TENSOR: Let M be a (2n+1)-dimensional Sasakian manifold with structure tensors
We denote by R, Q
and r the Riemannian curvature tensor, the Ricci operator and the scalr curvature of M respectively. Then the contact Bochner curvature
tensor B of M is defined by (Matsumoto-C ann [1])
309 B(X,Y) = R(X,Y) + t1 {QY A X - QX A Y + QTY A OX - Q¢X A ¢Y
+ 2g(QQX,Y)4 + 2g(4X,Y)% + n(Y)gX A F + n(X)& A 4Y)
AY A X
- m+4 Y A OX + m+4 n(Y)FA X + n(X)YA F),
where we have put m = 2n, k = (r+m)/(m+2) and (X A Y)Z = g(Y,Z)X g(X,Z)Y. Then we have
[1])
THEOREM 1. Let M be a Sasakian manifold with vanishing contact Bochner curvature tensor. If M is an n-Einstein manifold, then M is of constant 4-sectional curvature. We define the contact Ricci tensor L of M by setting
L(X,Y) = g(QX,Y) + 2g(X,Y) - 2(n+1)n(X)n(Y).
Then we have (Ikawa-Kon [1]) THEOREM 2. Let M be a (2n+1)-dimensional Sasakian manifold with constant scalar curvature and vanishing Bochner curvature tensor. If the contact Ricci tensor of M is positive semi-definite or negative semi-definite, then M is of constant $-sectional curvature. G. FIBRATION OF COMPACT NORMAL. ALMOST CONTACT MANIFOLDS: Similar
to the Boothby Wang fibration of compact regular contact manifolds, we can construct a fibration of compact normal almost contact manifolds with F regular (see Hatakeyama [1], Morimoto [21). Conversely we have the following (Morimoto [2]) THEOREM. Let M be a (2n+1)-dimensional compact normal almost contact manifold with structure tensors
and suppose that r; is
a regular vector field. Then M is the bundle space of a principal
circle bundle n : M
> B over a real 2n-dimensional almost complex
manifold B. Moreover n is a connection form and 2-form '' on M such that do = n*'Y is of bidegree (1,1).
310 H. BRIESKORN MANIFOLDS: Let B2n-1 be the Brieskorn manifold with contact form n = jiE(z.dz.-z.dz.). We denote by g the metric tensor J J J J B2n-1 field of induced from the natural metric of Then n is Cn+l.
given by n(X) = g(Jv,X), X e X(B2-1), where v = n + av1 + uv2, and Cn+1.
J is an almost complex structure of
We put N = v/Ivi and
n' = n/IvI. Kashiwada [2] proved the following THEOREM. For a Brieskorn manifold B2n-1(a0,a1,...,an) the following conditions are equivalent:
(1) The induced structure (2)
B2n-1
of B2n-1 is Sasakian;
is n-umbilical in the Kaehlerian manifold
On;
(3)a0=a1=...=an; (4) B2n-1 is umbilical in the Kaehlerian manifold B2n.
311
CHAPTER VI
SUBMANIFOLDS OF SASAKIAN MANIFOLDS
In this chapter, we study submanifolds of Sasakian manifolds, especially those of Sasakian space forms. In §1, we study invariant submanifolds of Sasakian manifolds. We compute the Laplacian for the second fundamental form and the Ricci tensor of an invariant submanifold of a Sasakian space form, and give some integral formulas. As applications of these integral formulas we prove the classification theorem of invariant submanifolds of codimension 2 of Sasakian space forms with n-parallel Ricci tensor. We also study invariant submanifolds of codimension 2 with constant scalar curvature. In §2, we discuss anti-invariant submanifolds, tangent to the structure vector field, of Sasakian manifolds. We give sane examples of anti-invariant suhnanifolds tangent to the structure vector field of Sasakian space forms and prove some theorems which characterize these examples. In §3, we study submanifolds of Sasakian manifolds which are normal to the structure vector field. Then the submanifolds are anti-invariant submanifolds. §4 is devoted to the study of contact CR submanifolds of Sasakian manifolds. We construct some examples of contact CR submanifolds of a unit sphere, which have parallel mean curvature vector and flat normal connection. We also prove some theorems concerning these examples. In the last §5, we discuss the structure induced on
sutmanifolds.
312
1. INVARIANT SUBMANIFOLDS OF SASAKIAN MANIFOLDS
Let M be a (2m+1)-dimensional Sasakian manifold with structure tensors ($,E,n,g). A (2n+1)-dimensional silhnanifold M of I is said to
be invariant if the structure vector field E is tangent to M everywhere
on M and g is tangent to M for any vector X tangent to M a every point of M, that is, gTx(M) c Tx(M) for all x e M. We easily see that any invariant submanifold M with induced structure tensors, which will
as M, is also a Sasakian mani-
be denoted by the same letters fold.
Let V (resp. V) be the operator of covariant differentiation with respect to the Levi-Civita connection of rd (resp. M). Since
is tan-
gent to M, for any vector field X tangent to M, we have -$X = OX _ 0
+ B(X,E). Thus we obtain B(X,E) = 0. For any vector fields X and Y
tangent to M we obtain OX,Y = V Y + B(X,W = (OXO)Y + JOXY + B(X,OY) = g(X,Y)E - n(Y)X + OVXY + B(X,$Y),
vx¢Y = (OX4)Y + ¢OXY = g(X,Y) - n(Y)X + BOXY + OB(X,Y).
From these equations we have ¢B(X,Y) = B(X,4Y). Since B is symmetric,
we also have B(X,Y) = B(OX,Y). Consequently, we have the following lemma for the second fundamental form B of M, or equivalently, for the associated second fundamental form A of M.
LEJT A 1,1, Let M be an invariant submanifold of a Sasakian manifold M. Then A
(1.1)
B(X,C) = 0,
(1.2)
B(X,cY) = B(cX,Y) = OB(X,Y),
(1.3)
cAOX = -AOX = AO X.
= 0,
313
PROPOSITION 1.1. Any invariant sutmanifold M of a Sasakian manifold M is a minimal sutmanifold. Proof.
Let el,...en4el,...,0en,E be a -basis of Tt(M). By (1.1)
and (1.2) we see that
I[B(ei,ei) + B(gei,¢ei)] +
0
because of B(Oei, ei) = 02B(ei,ei) _ -B(ei,ei). This means that M is a minimal sutmanifold of M.
QED.
PROPOSITION 1.2. If the second fundamental form of an invariant
suimanifold M of a Sasakian manifold M is parallel, then M is totally geodesic.
Proof. By the assumption on B we have
0 = (OXB)(Y,E) = B(Y,cX) _ OB(X,Y),
from which
B(X,Y) = -¢2B(X,Y) - n(B(X,Y))C = 0.
Therefore, M is totally geodesic.
QED.
Let R (resp. R) be the Riemannian curvature tensor of M (resp. M). From equation of Gauss and (1.2) we have
(1.4)
g(R(X,¢X)QX,X) =
2g(B(X,X),B(X,X)).
Thus we have the following PROPOSITION 1.3, bet M be an invariant submanifold of a Sasakian space form M(c). Then M is totally geodesic if and only if M is of
constant -sectional curvature c. We now suppose that the ambient manifold M is a Sasakian space form with constant 4-sectional curvature c. Then we have
314 (1.5)
R(X,Y)Z = *(c+3)[g(Y,Z)X-g(X,Z)Y] + j(c-1)[n(X)n(Z)Y
-n(Y)n(Z)X+g(X,Z)n(Y)g-g(Y,Z)n(X)t+g(oY,Z)+X
AB(Y,Z)X - AB(X,Z)Y. (1.6)
(VXB)(Y,Z) - (V B)(X,Z) = 0.
We denote by S and r the Rioci tensor and the scalar curvature of M respectively. Then, from (1.5) we have
(1.7)
S(X,Y) = i(n(c+3)+(c-1))g(X,Y) - l(n+l)(c-l)n(X)n(Y) - lg(B(X,ei),B(Y,ei)), i
(1.8)
r = n2(c+3)+n(c+1) -
E g(B(eiej ),B(ei,ei)),
i,J where {ei} is an orthonormal basis of M. In view of Proposition 1.2, the notion of parallel second funda-
mental form of an invariant submanifold of a Sasakian manifold is not essential. So, we need the following. Definition. Let M be an invariant sutmanifold of a Sasakian manifold M. The second fundamental form B of M is said to be n-pcrallel if
0 for all vector fields X, Y and Z tangent to M. It will be proved that the notion of n-parallel second fundamental
form corresponds to that of parallel second fundamental form for invari-
ant submanifolds of Kaehlerian manifolds. We easily see that the second fundamental form B is n-parallel if and only if
(1.9)
(OXB)(Y,Z) = n(X)gB(Y,Z) + n(Y)¢B(X,Z) + n(Z)¢B(X,Y).
315
mw 1,2, Let M be an invariant sulmanifold of a Sasakian manifold M. Then the second fundamental form of M is n-parallel if and only
if IVAI2 = 31A12. Proof. For the proof of this we take a cu-basis e1,...,e2n+1
(en+t = Oet, e2n+1 = E) We also take a basis vl,...,v2p of the normal space of M, where 2p = codim M. Then we have
A)aC) = g(Aaei,Aaei),
g((De i
i
g((oe1A)a4ej,(VelA)aOe.)
= g(ei,Aaej)2 + g(O(oelA)aejO(VelA)aej),
where A. = Av
(a = 1,...,2p). These equations imply
a
2n+1 E
g((o A)e.,(D A)ej)
a=1 i,j=1 2n
I a=1 I a=1
I
i
g((Ve A)a$ej(Ve
i,j=1 2n I
i
21A12 i
i
g(O(De A)aejO(Ve A)aej
i,j=1
i
+ 31A12.
i
Therefore we have 2n (1.10)
IVA12 - 31A12
=
I
a=1 i,3=1
g(O(0 A)a eO(o A) ae) > 0, ei ei
which proves our asertion.
QED.
LE41A 1.3. If the second fundamental form of an invariant sutrmnifold M of a Sasakian manifold M is n-parallel, then the square of the length of the second fundamental form of M is constant, that is, IA 12
is constant. Proof. Let {ei} be an orthonoimal basis of M. Then we have the following equation.
316 VXJAl 2 = vX[iIJg(B(ei,ee),B(ei,ee))] = 2i1 g((v33)(0i,eJ),B(ei,eJ))
=2
[n(X)g( B(ei,ei 1, +n(e.)g(4B(X,ei),13(t.i,t.J)) )
= 2Z[g(gB(X,ei),B(E,ei)) + g(pB(X,ei),B(,ei))] = 0. 1
Therefore, JA12 is a constant.
QED.
In the sequel we give an integral formula, so-called the Simons' type formula, for an invariant sulznanifold of a Sasakian space form
by computing the Laplacian of the square of the length of the second fundamental form.
Let M be a (2n+1)-dimensional invariant sukmanifold of a Sasakian space form M¢1(c). Hereafter we put p = m-n. We take air orthonormal basis el,...,e2n+1 of M such that en+t = cet (t = 1,...,n), e2n+1 = ' We take also an orthonormal basis v1,...,v2p of the normal space of M such that vp+s = Ovs (s = 1,...,p). In view of Proposition 3.1 of Chapter II, by a straightforward
computation, we have
(1.11)
-g(V2A,A) - 31A12 =
(TrAaAb)2 +
I[a,A,]12 a,b=l
a,b=1
- J(n+2)(c+3)JA12.
LE" 1.4, Let M be a (2n+1)-dimensional invariant submanifold of a (2m+1)-dimensional Sasakian manifold M. Then we have
(1.12)
1JAI4
I
n(c+3). If an invariant submanifold M is of constant 4-sectional curvature k, by using the Gauss equation, we have
(1.15)
n(n+1
1 (TrAaAb)2 = (c-k)IAI2 = a,b
l)
IAI4.
319
Therefore we have THEOREM 1.5. Let M be a (2n+1)-dimensional invariant submanifold 22m+1 (c). If M is of constant 4>-sectional of a Sasakian space form curvature k, then
0 < IVA12 - 31A12 = n(n+l)(n+2)(c-k)(>(c+3)-(k+3)).
THEOREM 1.6. Let M be a (2n+1)-dimensional invariant submanifold of a Sasakian space form M
1(c). If M is of constant 4>-sectional
curvature k, and if c > -3, then either M is totally geodesic, that is, c = k, or (c+3) > 2(k+3).
THEOREM 1.7. Let M be a (2n+1)-dimensional invariant submanifold 22m+1(c) with constant 4>-sectional curvature
of a Sasakian space form
k. If the second fundamental form of M is n-parallel, then M is totally geodesic, or (c+3) = 2(k+3), the latter case arising only when c > -3. From (1.12) and (1.15) we have
THEOREM 1.8. Let M be a (2n+1)-dimensional invariant submanifold
of a Sasakian space form M1(c) with constant 4>-sectional curvature k. If p < n(n.+l)/2, then M is totally geodesic, where p = m--n.
In the next place, we consider a (2n+1)-dimensional invariant
subn=ifold M of a Sasakian space form i(c) with respect to the Ricci tensor S of M. First of all we ccinpute the Laplacian for the Ricci operator Q of M (Kon [8]). We put
T=Q-al-bn0E, a, b being constant and a+b = 2n, (2n+l)a+b = r. We notice that T = 0 if and only if M is n-Einstein.
LEMMA 1.5. Let M be a (2n+1)-dimensional invariant submanifold of a Sasakian space form M2m+1(c). If the scalar curvature r of M is
constant, then
320
-g(V2Q,Q) - 21QJ2 - 16n3 - 8n2 + 8nr
_ -n(c+3)ITI2 + I I[Q,Aa]I2. a=1 Proof. Fran Lemur. 5.5 of Chapter V we obtain
2n+1 (V2S)(X,Y) _
(VeV S)(X,Y) i=1
i
i
= 1[(R(ei,X)S)(ei,Y) + (R(ei,$Y)S)(ei,$X)] - 4S(X,Y) + 8ng(X,Y) + (3r-12n2-6n)n(X)n(Y).
On the other hand, by equation of Gauss, we have
l(R(ei,X)S)(ei,Y) = -F[S(R(ei,X)ei,Y) + S(ei,R(ei,X)Y)
+ S(AB(X,ei)ei,Y) - S(AB(ei,Y)X,ei) +
S(AB(X,Y)ei,ei)].
Moreover, we have 2n+1
-IS(AB(X,Y)ei,ei) _
[g(AaAB(X,Y)Aaei,ei) a=1 i=1
+
i,ei)] = 0,
[S(AB(ei,Y)X,ei) - S(AB(X,eei,Y)]
_ l[g(AQAa,Y) - g(QAaAaX,Y)] a
From these equations and Theorem5.4 of Chapter V we obtain
(R(ei,X)S)(ei,Y) = 1(n(c+3)+2)S(X,Y) - *(c+3)rg(X,Y) + jn(c-i)g(X,Y) - Jn(2n+1)(c-1)n(X)n(Y) + *(c-1)ri(X)n(Y)
+ )[g(AaQAaX,Y) - g(QAAaX,Y)]. a
321
Similarly we have
i(n(c+3)+2)S(X,Y) - *(c+3)rg(X,Y) + Jn(c-l)g(X,Y) - n(n(c+3)+2)n(X)n(Y) + *(c+3)rn(X)n(Y) - jn(c-l)n(X)n(Y) + J[g(AaQAaX,Y) - g(QAaAa ,Y)]. a Fran these equations we have our assertion.
QED.
If the Ricci tensor S of a Sasakian manifold M satisfies (VXS)(gY,gZ) = 0 for all vector fields X, Y and Z on M, then we say that the"Ricci tensor S of M is n-parallel.
We now prepare the following lemras.
LEMMA 1,6. Let M be a Sasakian manifold with n-parallel Ricci tensor. Then we have the following: (1) The scalar curvature r of -M is constant;
(2) The square of the length of the Ricci operator Q of M is
constant, that is, IQ12 = constant. Proof. The Ricci tensor S of M is n-parallel if and only if
(VXS)(Y,Z) = 2n[g(X,Y)n(Z) + g(¢X,Z)n(Y)]
+ n(Y)S(X4Z) +
Thus we have VXr = E(VX.S)(ei,ei) = 0, which shows that r is constant.
Moreover, since g(QX,Y) = S(X,Y), we obtain VXIQ12 = 2Eg((VXQ)ei'Qei)
= 0 and hence IQ12 is constant.
QED.
LEMMA 1.7, The Ricci tensor S of a (2n+1)-dimensional Sasakian manifold M is n-parallel if and only if the following equation is satisfied:
IVQ12 - 21Q12 - 16n3 - 8n2 + 8nr = 0.
Proof. By using a 4-basis el,...e2,+1
(en+t = 4et, e2n+1
we have
2n+1 IVQI2 =
g((V Q)e.,(V Q)e.)
E ei J i,j=1 J ei 2n+1 2n 2n+1 £ g((Ve Q)eJ.,(V Q)e) + I E g((Ve Q)F,(Ve Q)F) J i=1 j=1 i i i=1 ei i 2n+1 2n+1 2n JI g((Ve Q)F,(e1Q)E) =
2n+1 2n = 2IQI2 + 16n3 + 8n2 - Bar +
I g($(0 Q)e,¢(V Q)e.).
E
j=1
i=1
ei
J
ei
J
On the other hand, we can see easily that the Ricci tensor S of M is n-parallel if and only if ui
Q)ej,4(De Q)ej) = 0. Thus we have 1
our assertion.
1
QED.
Here we notice that
ITI2 = IQI2 - r2/2n + 2r - 4n2 - 2n.
Fran this we also have
fI[Aa,Ab]I2 = 2Tr(A*)2 =
nIAI4
+ 2ITI2.
PROPOSITION 1.4. Let M be a (2n+1)-dimensional invariant su mani22m+1(c) (c < -3). If the Ricci tensor
fold of a Sasakian space form
S of M is n-parallel, then M is ri-Einstein.
Proof. From Lemma 1.6, IQI2 is constant. Thus we have -g(V2Q,Q)
IVQI2 Therefore Lemmas 1.5 and 1.7 imply
-n (c+3) ITI2 + 11 [Q,Aa] 12 = 0. Since c < -3, we have ITI2 = 0 and hence T = 0, which shows that M is n-Einstein.
QED.
PROPOSITION 1.5. Let M be a (2n+1)-dimensional compact invariant submanifold of a Sasakian space form
Vm+1
(c > -3) with constant
scalar curvature. If QAa = AaQ (a = 1,...,p), then M is n-Einstein. Proof. By the assumption and lenma. 1.5 we easily see that 1T12= 0.
Therefore M is n-Einstein.
hp
QED.
an invariant submanifold of codimension 2 of be a Sasakian space form Vn+3(c). Then the Ricci tensor S of M is 1,8. let M
n-parallel if and only if the second fundamental form of M is n-parallel.
Proof. Suppose that the second fundamental form of M is n-parallel. Then it is easy to see that S is n-parallel. Conversely, we assurne that S is n-parallel. We shall prove that
B is n-parallel. To prove this it sufficies to show that, at each point x of M,
O(VoXA)OvoY = 0
for any X, Y e TT(M), where {v,cv} is an orthonormal basis of the
normal space of M. There exists a 1-form s such that DXv = s(X)O.
From (1.7) we have ¢(VXAv)Z$Y = c(VXA)vAv4Y + s(X)02AA0 + 4Av(VXA)v4Y +
0.
Since 4Av = -Avd, the above equation reduces to
(1.16)
¢(VXA)vAvlY + (Av(VXA)voY = 0.
We take any two A, p of characteristic roots of Av at a point x of M.
We naw define spaces by setting
TX = {X c TT(M): AAX = AX},
T- = {X a TA(M): AvX = lpX}.
334
Then TA A 71 = {0} when A # u. If Y E 71, then $Y e T u. let Y e Tu.
Then (1.16) implies
v$(VXA)v$Y = -U$(VXA)vvY,
which shows that c(VXA)vcY e T T. let X E TX, Y c TV and A # tit. Then,
from equation of Codazzi, we obtain
¢(VOXA)v$Y E T_X (% T_4.
Thus we have (VOXA)vOY = 0. On the other hand, if X E T, then
anda=u#0. Then we find
02XETX. Let XcTX , Y
C)2(VOXA)v$Y c TX.
From this and equation of Codazzi we have
c TX,, T_X,
0. From this we have
which means that 0.
Let A = µ = 0. We take X, Y e TO at x E M, and extend these to local vector fields on M which are covariant constant with respect to V at x. Here we notice that if X e T0, then OX c T0. From (1.16), we obtain, at x e M,
gWVOXA)v(VOXA)v$Y,Y) = 0,
and hence
0. Consequently we have
cases. On the other hand, if
(V
0, then
-g(cX,AviY)F,
0 for any
325
0. This completes the proof of our assertion.
from which
QED.
LENS 1.9. Let M be an invariant submanifold of codimension 2 of a Sasakian space form on+3(c). If the Ricci tensor S of M is rI-parallel, then M is an 71-Einstein manifold.
Proof. Fran Lemmas 1.5, 1.6 and 1.7 we have 2
n(c+3)ITI2 =
I I[Q,Aa]12. a=1
Since M is of codimension 2, we can take a basis {v,qv) for T(M)L. Then (1.7) implies
S(AVX,Y) = i(n(c+3)+(c-1))g(AvX,Y) - 2g(AV3X,Y) = S(X,AvY),
which shows that QAv = AvQ and hence n(c+3)IT12 = 0. Thus if c # -3, then M is n-Einstein. Let c = -3. Then (1.11) and Lemma 1.8 show that
M is totally geodesic and hence M is n-Einstein.
QED.
From these lemmas we have (Kon [83) THEOREM 1.9. Let M be an invariant sutmanifold of codimension 2 of a Sasakian space form
jP+3(c).
Then the following conditions are
equivalent: (1) The Ricci tensor of M is n-parallel; (2) The second fundamental form of M is n-parallel; (3) M is n-Einstein.
THEOREM 1.10, Let M be an invariant subnanifold of codimension 2 K?n+3 (C). If
with n-parallel Ricci tensor of a Sasakian space form
c < -3, then M is totally geodesic. If c > -3, then either M is totally geodesic, or an n-Einstein manifold with JAl2 = n(c+3) and hence r = n(n(c+3)-2).
Proof. From Theorem 1.9, M is n-Einstein. Then (1.11) and
Lemma, 1.4 imply
326
j(n+2)11IAI2 - (c+3)]IAIZ = 0.
This proves our theorem.
QED.
THEOREM L.U. bet M be a compact invariant suhrmnifold of codimension 2 of a Sasakian space form I1+3(c) (c > -3). If the scalar curvature r of M is constant, then either M is totally geodesic, or an n-Einstein manifold with the scalar curvature r = n(n(c+3)-2). Proof. In this case we have already seen that QAv = AAQ. Then Proposition 1.5 states that m is an n-Einstein manifold. Thus our theorem follows from Theorem 1.10.
QED.
Here we consider the case that the second fundamental form of M satisfies IAI2 = (n+2)(c+3)/3. We prove
THEOREM 1.12. Let M be a (2n+1)-dimensional invariant submanifold 0r'+1 of a Sasakian space form (c) (c > -3). If IAI2 = (n+a)(c+3)/3, then M is an n-Einstein manifold of dimension 3 and has the scalar curvature r = (c+l).
Proof. Since IAI2 is constant, we have -g(V2A,A) = IVAI2. Then (1.11) implies
0 < IDAI2 - 31A12 = I(TrAaAb)2 + 1l(Aa,Ab]I2 - J(n+2)(c+3)IAI2 = ',-[3IAI2-(n+2)(c+3)]IAI2
n - 4 1 a a TrA2TrA2. b a i s t#s aft
From the assumption we have
n I A X = 0 i s t#s
and
TrA2T-rAb = 0.
aft
a
Thus we may assume that at = 0 for t = 2,...,n and 'Na = 0 for a = 2,..
.,p. Therefore we have
327
QX = i(n(c+3)+(c-1))X - 2(n+1)(c-l)n(X)E - 2A1X.
From this we see that QAa = AaQ for all a. From this and Lemma 1.5, M is n-Einstein because of the second fundamental form of M is n-parallel and hence Q is n-parallel. Then Theorem 1.4 implies
3(n+2)(c+3) = IA12 > n(c+3),
from which n = 1 and hence M is of dimension 3. Thus we have r = (c+l) by (1.8).
QED.
We now assume that M is a regular Sasakian manifold. Let M/C denote the set of orbits of E. Then M/t is a real 2n-dimensional Kaehlerian manifold. Then there is a fibering 7r : M
> M/F.
Henceforth X*, Y* and Z* on M will be horizontal lifts of X, Y and Z over M/C respectively with respect to the connection n. Then we have
(S'(Y,Z))* = S(Y*,Z*) + 2g(Y*,Z*),
where S' denotes the Ricci tensor of M/&. From this we have
((VXS')(Y,Z))* = (VX*S)(Y*,Z*),
where V' denotes the operator of covariant differentiation in M/E. therefore we see that the Ricci tensor S' of M/E is parallel if and only if (VX*S)(Y*,Z*) = 0 which is equivalent to
0
for any U, V, W c T{(M) because of the horizontal space is spanned by {4U: U c TX(M)). On the other hand, (V,US)(cV,¢W) = 0 implies that (V02US)(¢V,¢W) = -(VUS)(gV,¢W) = 0 and the converse is also true. Therefore the Ricci tensor S' is parallel if and only if (VTS)(4V,4W) = 0, which states the meaning of the definition of n-parallel Ricci tensor.
0 THEOREM 1,13, Let M be a regular Sasakian manifold. Then the Ricci tensor S of M is ri-parallel if and only if the Ricci tensor S' of M/g is parallel. Excurrple 1.1. Let R2n+3 be a (2n+3)-dimensional Euclidean space
with standard Sasakian structure of constant q-sectional curvature c = -3. A (2n+3)-dimensional unit sphere S2n+3 has the standard Sasakian structure of constant 4-sectional curvature c > -3. By Can+l, R and (R,CDn+1) we denote the open uni ball in a complex (n+l)-dimensioCan+1 nal Euclidean space Cn+l, a real line and the product bundle R x
Then (R,CDn+1) also has a Sasakian structure with constant -sectional curvature_c < -3. Let (P be an n-dimensional complex quadric in a complex projectivae space C
+1. We denote by (S,Qn) a circle bundle
over Qn. Then (S,Qn) define a Sasakian structure which is n-Einstein S2n+3. R2n+1, (R,CDn) and (S,Qn) is an invariant submanifold of and Stn+1 are totally geodesic invariant submanifolds of (R,CDn+1) R2n+3,
and S2n+3 respectively. We prove the following (Kon 18])
THEOREM 1.14. (1)
S2n+l
and (S,(p) are the only connected complete
invariant submanifolds of codin ension 2 in S2n+3 which have n-parallel Ricci tensor;
(2) (R,C) (resp. R2n+1) is the only connected complete invariant R2n+3) sukmanifold of codimension 2 in (R,CDn+l) (resp. which has n-parallel Ricci tensor. S2n+3
Proof. Let M be one of the
and (R,CDn+l) and B be
Cn+l (if M = R2n+3), CPn+1 (if i = S2n+3) and (fin+1 (if M =
(R,a)n+1)).
Then M is a principal G1-bundle over B, where G1 is a circle or a line. Then an invariant sub manifold M of codimension 2 of M is also regular. We can consider the following commutative diagram:
M
>M
7Tl
mg
lTr
>
M By Theorem 1.13, the Ricci tensor S' of M/l; is parallel. Since M/F is
an invariant submanifold of B of codimension 2, by Theorem 1.6 of
Chapter IV, WE is Einstein. From Theorem 1.10 of Chapter IV our theorem reduces to the following lemma.
LEMMA 1.10, Let M be a non-totally geodesic connected complete n-Einstein invariant submanifold of codimension 2 of S2n+3. Then there is an automorphisn 6 of S2n+3 such that 6M = (S,QW). Proof. By Theorem 1.10 of Chapter IV, B = nM is holomorphically isemetric to Qn. Frcm Theorem 1.8 of Chapter IV, there is a holomorphic iscmetry 6 of CPn+l such that eB = Qn. Let x c B, Ox = y and T(t), 0 < t < 1, be a curve joining x and y. Then we have a continuous family of J-basis (T(t),ei(t),Jei(t)), i = 1,...,n+1, on T(t) such that ei(1) = 6*(ei(0)). Thus a is contained in the connected component e+1.
of the automorphism group of
Bence there are finite numbers of
infinitesimal autonorphismis X1,X2,...,Xs of CPn+1 such that 6 =
expts s...expt1X1. By Lemma 5.1 of Tanno [4] there are infinitesimal automorphisms
Y1..... Ys of S2n+3 such that nYk = Xk, n(exptkYk)(u)
= exptkXk(mi), u c M, k = 1,...,s. Putting 8 = expts s...exptlY1, we have
9M = Qn. Since SM and (S,(;") have the same fibre, we have QED.
9M = (S,(P) By Theorem 1.11 and Theorem 1.14 we have (Kon [8]) THEOREM 1.15.
S2n+1 and (S,(P) are the only compact invariant
sutmanifolds of codimension 2 in S2n+3 which have constant scalar curvature.
2. ANTI-INVARIANT SIIRMANIFOLDS TANGENT TO THE STRUCTURE
VECTOR FIELD OF SASAKIAN MANIFOLDS
let rd be a (2m+1)-dimensional almost contact metric manifold with structure tensors (4),E,n,g). An n-dimensional sutmanifold M immersed in ti is said to be anti-invariant in 2 if OTx(M) c T{(M)' for each x c M.
Then we see that, 0 being of rank 2m, n < m+1. When n = m+l we have
330
PROPOSITION 2.1. let M be an almost contact metric manifold of
dimension 2n+1 and let M be an anti-invariant si
nifold of M of
dimension n+l. Then the structure vector field E is tangent to M. Proof. By the assumption we have ¢T{(M) = T{(M)- at each point x of M. For any vector field X tangent to M we have gQ,$X) = -g(c
,X)
= 0, which shows that the structure vector field E is tangent to M. QED.
PROPOSITION 2.2. Let M be a submanifold tangent to the structure vector field E of a Sasakian manifold M. Then E is parallel with respect
to the induced connection on M, i.e., OE = 0, if and only if M is an anti-invariant submanifold of M.
Proof. Since g is tangent to M, we have X = VV - B(X,g). Thus 9 is parallel if and only if X =
and hence X is normal to M. QED.
In this section we sail study anti-invariant su
n folds tangent
to the structure vector field of a Sasakian manifold. Therefore, in this section, we mean by an anti-invariant submanifold U of a Sasakian manifold M an anti-invariant submanifold M tangent to the structure vector field
of M.
PROPOSITION 2.3. let M be an (n+l)-dimensional anti-invariant subnanifold of a (2n+1)-dimensional Sasakian manifold M. If n
1,
then M is not totally umbilical.
Proof. If M is totally umbilical, then B(X,Y) = g(X,Y)u, u being the mean curvature vector of M. Then B(E,C) = 0 implies u = 0 and hence M is totally geodesic. This contradicts to the fact that OX = -B(X,E)
# 0.
QED.
Here we choose a local field of orthonormal frames e0 E,el,...,en; en+1,...,em;el*= ell...,en* en;e(n+l)* en+l,.... em* em in M such
that, restricted to M, e0,el,...,en are tangent to M. With respect to r ,wl,...own;wn+l,...,(d,3 ...,ten* this frame field of, let
W
(n+l)*
,...,
u
be the dual frames. Unless otherwise stated, we use
the conventions that the ranges of indices are respectively:
331
A,B,C,D = i,j,k,l,s,t = 0,1,...,n,
x,y,z,v,w = 1,...,n,
a,b,c,d = X,u,v = n+l.... m,(n+l)*,...,m*, a,a y = n+l,.... m.
Then we have the following equations:
* Wy =
Wy*,
Wa = Wu s
* y* Wy = Wx
,
a = WS
,
U)
s*'
x _ x*
Wa - Wa*,
s
a
x* _
a*
W x= -W0 ,
W x
= WO,
Wa = -WO o
Wa
= Wa 0,
Wa - Wx
We restrict these forms to M and then we have Wa = 0. Thus we have
WO = WO* = wo = 0. Therefore (2.13) of Chapter II implies
x_ y
z
hyz - hxz = hxy,
x
a
h00 = 0,
h0i - 0,
x h i = -'xi'
where we use h-ij in place of hid to simplify the notation. For each a,
the second fundamental form Aa is represented by a symmetric (n+l,n+l)matrix Aa = (ha.). Then we have
x
0 0
-1
0
10 hx
for all x,
yz
for all A.
AA =
0
332 Hereafter we put
a =
n7), iihich is a symmetric (n,n)-matrix. We
notice that
IAi 2 = IH12 + 2n, where I H12 = ETrH2a. We also have a
Tra -Tra for all a, and hence M is minimal if and only if Tr a = 0 for all a. Since 07{(M) c T{(M)- at each point x of M, we have the following decanposition:
x(M),
T X(Mt =
where NA(M) is the orthogonal conplement of $X(M) in X(M)y. If V e a(M), then 4V a NS(M). For any vector field V normal to M we put
¢V = tV + fV,
where tV is the tangential part of cV and fV the normal part of 4V. Then t is a tangent bundle valued 1-form on the normal bundle T(M)'
and f is an endomorphisn on T(M). Then we have
tfV = 0,
f2V = -V - 4tV,
tqX = -X + n(X)
foX = 0.
From these we also have
f3+f=0. Since f is of constant rank (see Stong [1]), if f does not vanish,
it defines an f-structure in the normal bundle T(M)r' (see Chapter VII).
By the Gauss and Weingarten formulas we have
(Vxf)V = -B(X,tV) - lAVX.
If (OXf)V = 0 for all X and V, then the f-structure f in the normal bundle of M is said to be paraZZel. LE14 A 2.1. It M be an (n+l)-dimensional anti-invariant sutrnani-
fold of a (2m+l)-dimensional Sasakian manifold M. If the f-structure f in the normal bundle of M is parallel, then AV = 0 for V E NX(M), or equivalently, AA = 0.
Proof. If V E X(M), then tV = 0. Thus we have (AVX = 0 and hence ¢2AVX = -AVX + n(AVX)E = 0. On the other hand, we see that n(AVX) = g(B(X,E),V) = -g(gX,V) = 0. Thus we have AV = 0.
QED.
We can easily see that if Ha = 0 for all a, then f is parallel. We denote by R and R the Riemannian curvature tensors of M and M respectively. Then we have
(2.1)
y*ij = Ryij R0
iXjy - iyjX(2.2)
i
= ROjk = 0.
Suppose that the ambient manifold M is of constant ¢-sectional curvature c. Then the Gauss equation is given by
(2.3)
Rjkl = *(c+3)(6 ik6jl - 6il6 jk) +
+ nirlldjk - ninkdjl) +
(c-1)(njnkdil - njnldik
a
fran which we find the Ricci tensor Rij of M is given by-
(2.4)
Rij = *[n(c+3)-(c-1)]dij - j(n-1)(c-1)ninj
+
I (hkkhij a,k
334
On the other hand, we have
a,k(
h x
hil -
fix)
- lij .+ nine.
a,x
Thus (2.4) reduces to
(2.5)
R.. = i(n-1)(c+3)6.. - a[(n-1)c+(n+3)]ninj
+
a,x
(ha ha -
Since we have
(h7 )2 =
(hXy)2 + n,
a,x,y
a,i,x
the scalar curvature r of M is given by
r = $n(n-1)(c+3) +
(2.6)
F
a,x,y
(ha hay
-
ha hay
From (2.5) and (2.6) we have PROPOSITION 2.4. Let M be an (n+l)-dimensional anti-invariant submanifold of a Sasakian space form M
1(c). If M is minimal, then
the Ricci tensor S and the scalar curvature r of M satisfiy (1) S - J(n-1)(c+3)g +
;[(n-1)c-(n+3)]n O n is negative semi-
definite;
(2) r < jn(n-1)(c+3).
PROPOSITION 2,5. Let M be an (n+l)-dimensional anti-invariant suirnanifold of a (2n+1)-dimensional Sasakian manifold M. Then M is
flat if and only if the normal connection of M is flat.
Proof. First of all we have
g Y#i]
L(hixhjy - hix hiy)
Ryij
Yi.] + z
because of hix = h' and (2.1). This combined with (2.2) proves our
assertion.
QED.
In the following we compute the Laplacian of the square of the length of the second fundamental form of M. Since hljk - hikj = Of (3.1) of Chapter II reduces to
(v2B)(X,Y) = j(R(ei,X)B)(ei,Y) + DXDy(TrB), i
from which
ha.6ha. _ i3
CC
a,l,j
1J
a,i,j,k
a
+ ha
a
(hai3 kki_i
ij
+ hhtiRkjk ij
t
ijk
- h
ijNiRbjk).
Therefore, (2.16) and (2.18) of Chapter II imply
Rijkhijhkt + Rkjkhijhti
£ a,i, j hi.jnhij a,i,,7,k - jkhljhid) -
[(h hkj -
£
t
a,b,i,j,k,t -
hijhbjhtkhbk - hijhithtihkk3.
Consequently we obtain
(2.7)
jha
hij = lhijh
j + Q(n+1)(c+3)lTrAa - 4(c-1)jTrA2
- 12(c+1)1(Tr X)2 - 14(c+3)1(TrA )2 - jn(n+l)(c-1)
+ J[Tr(AaAb - AbAa)2 - (TrAaAb)2 + TrAbTrAa2Ab].
PROPOSITION 2,6. Let M be an (n+l)-dimensional anti-invariant suboanifold of a Sasakian space form M
iAIA12 - J(ba
)2 = jha h
1(c). Then
j + a(c+3)1[nTrHa - (TrHa)2]
+ *(c+3)1[TrHX - (TrHX)2] + J[Tr(H Hb - HbHa)2 - (Tr aHb)2 + TrHbTrHaHb].
338 Proof. First of all, we have
TrAX = TrH + 2,
Trq =
TrAa = WE.-
We also have the following equations:
JTr(AaAb - AbAa)2 = JTr(HaHb - HbHa)2 - 4E(Tr X)2 - 41Tr a + 81TrHX - 2n(n-1),
J(TrAaAb)2 = I(TrAa)2 = J(TrHa)2 + 41TrH2 + 4n,
ITrAbTrAa2Ab = JTrHbTrHaHb + 21(TrHX)2 + 1(mra)2.
Moreover, we obtain
le-Ah'J = JAIA12 - Z(haJk)2
(2.8)
= JAIAI2 - E(hayZ)2 - 31TrH2,
(2.9)
- I(TrH,)2.
j = IhX]h
jhijh
J
Substituting these equations into (2.7), we have our equation.
QED.
FYcm (2.8) and (2.9) we also have PROPOSITION 2.7. Let M be an (n+l)-dimensional anti-invariant suhnanifold of a (2m+1)-dimensional Sasakian manifold M. (1) If the second fundamental form of M is parallel, then HA = 0 for all A;
(2) If the mean curvature vector of M is parallel, then TrHx = 0 for all A.
We now put
Tab =
hayhb xi y
,
Ta =T an,
T = ITa = IH12.
a
THEOREM 2.1, Let M be an (n+1) -dimensional compact anti-invariant
minimal sutmanifold of a Sasakian space form
FP+1(c).
Then
0 < fMIVAI2*1 < fM[(2 - n)T - y
Ty)2.
QED.
Therefore, we have our assertion.
THEOREM 2.2. Let M be an (n+l)-dimensional compact anti-invariant minimal sulmanifold of a Sasakian space form M2n+1(c). Then, either T = 0, or T = n(n+l)(c+3)/4(2n-1), or at some point x of M, T(x) > n(n+l)(c+3)/4(2n-1). THEOREM 2.3. Let M be an (n+l)-dimensional anti-invariant minimal
suhmanifold of a Sasakian space form Mn+1(1). If 1A12 = (5n2-n)/(2n-1), then n = 2 and M is flat. With respect to an adapted dual orthonormal
frame field wO,wlw2,wl*,w2*, the connection form (wB) of M5(1), restricted to M, is given by
0
0
0
0
0
0
0
0
0
w1
--OO+aw
awl
0
0
w2
w0+Aw1
Aw2
0
0
-wl up-Aw2
-awl
-w2
-Aw1 _w0-Aw2
, A=
338
goof. Since 1 ALI 2 is constant, we have 0 < IVA12 < [(2 - n)T - (n+1)]T.
Thus, by the assumption and T = IAl2 - 2n, we see that the second fundamental form of M is parallel. Moreover, we have, from Lemma. 5.1 of Chapter II,
T")2 = 0,
E (T X>y
- Tr(HXHy - HyHX) 2 = 2TrHXTrH2,
and hence Tx = Ty for all x, y and we may assume that x = 0 for x = 3,...,n. Therefore, we mist have n = 2 and we obtain
0
1
1
0
1
0
0
-1
H1=X by putting h12 = h1
X. Then we have
=
0
-1
0
Al = -1
0
X
0
X
0
A2
,
0
0
0
X
-1
0
-1
0. -X
On the other hand, we find
x
x
d
xd
yx
dhab = hadwb + hdbwa - habwy.
Putting x = 1, a = 1 and b = 0, we see that dX = wo = 0, which shows that X is a constant. Since T = 2, we get 4X2 = 2. Thus we may assume that X = 1/v!E. Moreover, we have
w0 = -wl* = 0 1 w2* = Xw l
1
w0 - -w2* = 2 w1* _ -w0 + Xw2, 1
0,'
0
1
01*
= w ,
0 2 w2*=w, 2
W2 2
_ -w0 - Xw
w1 2 - 0.
KK On the other hand, from equation of Gauss,
easily see that M is
flat and hence the normal connection of M also flat. These prove our theorem.
QED.
Example 2.1. Let J = (ats) (t,s = 1,...,6) be the almost complex structure of.C3 such that a2i,2i-1
= li a2i-1,2i = -1 (i = 1,2,3) and the other components being zero. Let S (1//) _ {z E C: I ZI2 = 1/3), a plane circle of radius
M3 =
Sl(1//)
We consider
X
Sl(1//)
x
S1(1/15)
in 55 in C3,which is obviously flat. The position vector X of M3 in S5 in C3 has components given by
X = (1//)(cosul,sinul,cosu2,sinu2,cosu3,sinu3),
ul, u2, u3 being parameters on each S1(l//). Putting Xi = aix/aui,
we have
X1 = (1//)(-sinul,cosu1,0,0,0,0), X2 = (1/,/3)(0,0,-sinu2,cosu2,0,0),
X3 = (1//)(0,0,0,0,-sinu3,cosu3).
The structure vector field & on S5 is given by
t = JX = (1/v)9)(sinul,-cosu1,sinu2'-cosu ,sinu3,-cosu3).
Since C = -(X1 + X2 + X3), C is tangent to M3. On the other hand, the structure tensors (c,E,n) of S5 satisfy
4Xi = JXi - n(Xi)X,
i = 1,2,3,
which shows that 4%j is normal to M3 for all i. Therefore, M3 is an
anti-invariant suhinifold of S5. Moreover, M3 is a minimal suhmanifold
340
of S5 with I Al2 = 6 and the normal connection of M3 is flat. Since the connection form (4AA) of S5, restricted to M3, coincides
with that in Theorem 3.2, we have THEOREM 2.4. Let M be an (n+1)-dimensional compact anti-invariant
minimal submanifold of S +1. If 1A12 = (5n2-n)/(2n-1), then M is
S1(1/,/)
x
S1(1/,/) x S1(1/T)
in S5.
THEOREM 2.5. Let M be an (n+l)-dimensional compact anti-invariant
minimal such nifold of S S. If the second fundamental form of M is parallel and if JA12 = (5n2-n)/(2n-1), then M is
S1(1/T) x S1(1/f)
x
S1(1/T)
in an S5 in S
1.
Proof. Since the second fundamental form of M is parallel, by Proposition 2.7, AA = 0 for all X. Thus we have the same inequality as in the proof of Theorem 2.1 and then we have n = 2. We also have T1 = T2 0 0. Thus the first normal space of M is spanned by el*, e2*, that is, the first normal space of M is 4Tx(M). For any vector fields
X and Y tangent to M and any vector field V in N(M) we have g(DXpY,V) =
g((OX4)Y,V) + g(fv XY,V)
= g(OVXY,V) + g(OB(X,Y),V) = -g(AoX,Y) = 0,
which shows that the first normal space of M is parallel. Thus M is S2m+l in an S5 in and M is anti-invariant in S5. Therefore our theorem follows from Theorem 2.4.
QED.
Example 2.2. Let S1(ri) _ (zi a C: Izil2 = ri}, i = 1,...,n+1. We consider
Mn+1 = S1(rl) x ... x S1(rn+l)
'S1
in Cn+1 such that ri + ... + rn+1 = 1. Then
0+1 is a flat subm nifold
of S2n+1 with parallel mean curvature vector and with flat normal connection. The position vector X of Mn+1 in Cn+1 has components given
by
+1,rn+1sinun+1 ).
X = (r1cosu1,rlsinu ,...,rn+lcosu
Then X is an outward unit normal vector of
Stn+1
in C. Putting
Xi = aiX = ax/au l, we have
X1 = r1(-sinul,cosu ,O,...,O), .......................... Xn+1 = rn+1(0,.... 0,-siniin+1 ,,,,n+l).
The structure vector field C on S2n+1 is given by its carponents
C = -JX =
(rlsinu1,-rlcosul,...,rn+lsinun+l,-rn+lcosu +1 ).
Therefore we see that
= -(X1 + ... +
Xn+1),
which means that the
vector field E is tangent to M. So the structure tensors (4,E,n,g) Stn+1 of satisfy
cXi = JXi - n(X1)X,
i = 1,...,n+1.
Thus M is normal to M for all i. Therefore Mn+1 is anti-invariant in S2n+}.
THEOREM 2.6. It M be an (n+l)-dimensional compact anti-invariant sulmanifold of S2n+1 with parallel mean curvature vector. If the normal connection of M is flat, then M is
S1(r1) X ... X S1(rn+l)'
Erl = 1.
Proof. Since the normal connection of M is flat, Proposition 2.5 implies that M is flat. On the other hand, as the mean curvature vector
342
of M is parallel, IAl2 is constant. From these and Proposition 4.4 of Chapter II we see that the second fundamental form of M is parallel. Therefore our theorem follows from Theorem 4.4 of Chapter II.
QED.
THEOREM 2.7. Let M be an (n+l)-dimensional ccapact anti-invariant
sutmanifold with flat normal connection of 521. If the second fundamental form of M is parallel, then M is
S1(rl) x ... x Sl(rn+l)
in an S2,,+1 in
S3n+1 .
where Zr? = 1.
Proof. From the assumption we have AA = 0 for all A and we see that M is flat. We shall show that the first normal space of M is of dimension n and parallel with respect to the connection induced in the normal bundle. Since M is flat, by equation of Gauss, we have
0 = (dikdjl - 8il8jk) + a
If the dimension of the first normal space N1(M) of M is less than n,
then for sane x, Ax = 0. Thus we have
J(hixhjl -
a
y _ I(hiyhS Y
Therefore we have 0 = (6 ix6jl
-
- hilhx
= 0.
6il6jx), which is a contradiction.
We also see that N1(M) is parallel by the similar method used in the proof of Theorem 2.5.
QED.
THEOREM 2.8. Let M be an (n+l)-dimensional compact anti-invariant
minimal submanifold of a Sasakian space form M2
1(c) (c > -3). If
T < }nq(c+3)/(2q-1) (q = 2m-n), then T = 0. Proof. From Proposition 2.6 and Lemma. 5.1 of Chapter II we have
343
J(hXyz)2 - JAIA12 = -ITT(HaHb
1T2
HbHa)2 +
EL
- }n(c+3)T - }(c+3)ITrHX [(2 - q)T - }n(c+3)]T -
1 (Ta - Tb)2 - *(c+3)
Therefore, by the assertion, we obtain
1(Ta - Tb)2 = 0,
ITx = 0.
Thus we see that x = 0 for all x and Ta = Tb for all a and b. Hence we have Ta = 0 for all a. Consequently, we obtain T = 0.
QED.
We give examples of anti-invariant submanifolds with T = 0.
Example 2.3. Let Stn+1 be a unit sphere of dimension 2n+1 with standard Sasakian structure and let CPn be a complex projective space of real dimension 2n with constant holarorphic sectional curvature 4.
A real projective space RPn of dimension n with constant curvature 1 is imbedded in CPn as an anti-invariant and totally geodesic submanifold (see Abe [1]). We consider the following cormutative diagram:
(S,RP1)
>
S2n+1
V
where (S,RPn) denotes a circle bundle over RPn. Then (S,RPn) is an
anti-invariant subnanifold with T = 0 of
52n+1.
Example 2.4. Let R2n+1 be an Euclidean space with cartesian coordinates ()l,...,xn,yl,.... yn,z). As in Example 5.5 of Chapter V
we derive the standard Sasakian structure in R2n+1 with constant -sectional curvature -3. We consider the following natural imbedding of Rn+1 into R2n+1:
344
Then we easily see that Rn+1 is an anti-invariant submanifold of R2n+1 which has T = 0.
3. ANTI-INVARIANT SUF1ANIFOLDS NORMAL TO THE STRUCTURE VECTOR FIELD OF SASAKIAN MANIFOLDS
From the consideration of §4 of Chapter V it is interesting to study the following suhmanifolds of contact manifolds, especially
those of Sasakian manifolds. First of all, we prove THEOREM 3,1. Let M be an n-dimensional sutrnanifold of a (2n+l)-
dimensional K-contact manifold M. If the structure vector field E is normal to M, then M is an anti-invariant submanifold of M, and n < m. Proof. From the Weingarten formula we have
g(¢X,Y) = -g(V,Y) =
for any vector fields X and Y tangent to M. Since AC is symmetric and
is skew-symmetric, we have AE = 0 and X is normal to M. Thus M is an anti-invariant sub manifold of M. We also easily see that n < m. QED.
Throughout in this section we mean by an anti-invariant submanifold M of an almost contact metric manifold M, a sutmanifold M of M normal to the structure vector field E of M. Especially we consider a submanifold M of a Sasakian manifold M normal to E. We state some fundamental properties of the second fundamental
form of an anti-invariant submanifold Mof a Sasakian manifold M. We already have the following
(3.1)
AE = 0.
345
Moreover, we easily see
(3.2)
for any vector fields X and Y tangent to M.
LEhMA 3.1, Let M be an n-dimensional anti-invariant sutmanifold of a (2m+l)-dimensional Sasakian manifold M. If the second fundamental form of M is parallel, then
AqX = 0
for any vector field X tangent to M. If moreover n = m, then M is totally geodesic in M.
Proof. Prom the assumption on the second fundamental form of M we obtain
0 = g((OXB)(Y,Z),0 = g(B(Y,Z),OX) = g(AOXY,Z),
from which
0. If n = m, the normal space Tx(M)L is equal to
cT{(M) 9 {F}..Therefore M is totally geodesic in M.
QED.
Since 4Tx(M) C TA(M)'' at each point x of M, we have the decompo-
sition of TX(M)` into the direct sum
Tx(M)` = $TT(M) ® N(M),
N(M) being the orthogonal complement of X(M) in Tx(M)L. We have
N(M) = ON(M) 0 W. For any vector field V normal to M we put
4V=tV+fV, where tV is the tangential part of V and fV the normal part of 4V. Then t is a tangent bundle valued 1-form on the normal bundle of M and f is an endomorphisn of the normal bundle of M. We find
346 f2V = V - $tV + n(V)g,
tfV = 0,
t9 = X,
f$X = 0.
Moreover, we have
f3 + f = 0, which means that f defines an f-structure in the normal bundle of M.
LE" 3,2. If the f-structure f in the normal bundle of M is parallel, then
AV = 0
for V E NA(M).
Proof. First of all, we have (VXf)V = -B(X,tV) - cAVX. Since
Vf = 0 and tV = 0, we obtain AVX = 0. Thus we have
o2AVX = -AVX + n(AVX)C = -AVX = 0,
which proves our equation.
QED.
Let u be the mean curvature vector of M. Then, from (3.1), we have
g(DRu,E) = g(u,cX).
From this we have the following
LEMM 3,3, Let M be an n-dimensional anti-invariant sutmanifold of a (2m+l)-dimensional Sasakian manifold A. If the mean curvature
vector u of M is parallel, then u E Nx(M) at each point x of M. Moreover, if n = m, then M is minimal.
From Lamas 3.1, 3.2 and 3.3 we have
347
PROPOSITION 3.1. let M be an n-dimensional anti-invariant su)manifold of a (2m+1)-dimensional Sasakian manifold M with parallel f-structure f in the normal bundle of M.
(1) If the second fundamental form of M is parallel, then M is totally geodesic;
(2) If the mean curvature vector of M is parallel, then M is minimal.
We choose a local field of orthonormal frames el,...,en;en+1,"
.,
em*4em in M in such a
n;e(n+i)* n+l1...1 l,...,en* em;eO* E,el* way that, restricted to M, el,...,en are tangent to M. Unless otherwise stated, we use the conventions that the ranges of indices are respectively:
i,j,k,l,t,s = 1..... n,
a,b,c,d = p,q,r = n+l,...,m,l*,.... m*.
In the following we put h0j =
h0
= g(Aei,ej) and hlj = hid _
g(e ei,ej) to simplify the notation. t* We now assume that the ambient manifold M is a Sasakian space
form M
(3.3)
l(c). Then the Gauss equation of M is given by
Rjkl = }(c+3)(dikdjl
- dilajk) +
J(hikhjl -
hik hjk)
Rij = *(n-1)(c+3)bij +
a,k r = in(n-1)(c+3) +
E
a,i,j
ij ij
(ha ha- - ha-ha.). ii ii
If M is minimal, we have Z1hii = 0 for all a, and hence we have
the following
348
PROPOSITION 3.2. Let M be an n-dimensional anti-invariant submanifold of a Sasakian space form M
l(c). Then M is totally geodesic
if and only if M satisfies one of the following conditions: (1) M is of constant curvature 4(c+3); (2) S = 4(n-1)(c+3)g;
(3) r = 4n(n-1)(c+3),
where S and r be the Ricci tensor and scalar curvature of M respectively.
Let M be an n-dimensional anti-invariant sutmanifold of a Sasakian space form M
l(c). Then we have (see §2)
haAha ij ij a.i.j
=
+ 4(c+3)F[nTrA2 - (TrA )2] ha. a i,7hkkij a a a a.i.j.k
+ 4(c-l)t[TrAt2 - (TrAt)2] + aF[Tr(AaAb - AbAa)2 b
- (TrAaAb)2 + TrAbTrAaA.].
On the other hand, we have
E
hij&i
= JAIA12 -
a,i,j
(hljk)2 F a,i,j,k
= iAIA12 -
p.i.j.k(
i.jk)
2 _ TEA2 . t
t
From these equations we obtain
PROPOSITION 3.3. Let M be an n-dimensional anti-invariant submanifold of a Sasakian space form 1faD+1(c). Then
#AIA12 - F(hpjk)2 = Fhijhkkij + 4n(c+3)IA12 - F(TrAa)2
- 4(c+3)1(TrAa)2 + 4(c+3)1TrA2 - 4(c-l)F(TrAt)2
+ F[T-(AaA. - AbAa)2 + TrAbTrAa2Ab].
349
THEOREM 3.2. Let M be an n-dimensional caiipact anti-invariant sutmaniiffold with parallel mean curvature vector of a Sasakian space 22n+1 form (c). Then
0 < JM (hp.k)2*1 < 1M[(2 - i)IA12 - *(n+1)(c+3)]IA12*l. I n p,i,j,k Proof. From Lemma 3.3 we see that M is minimal. Thus Proposition 3.3 implies
oIAI2 - 1(hpjk)z = *(n+1)(c+3)IAI2 - F(TrAt)2+ 1Tr(AtAS AsAt)2.
On the other hand, we obtain
-ITr(AtAs - ASAt)2 + I(TrA2)2 - *(n+l)(c+3)IAI2
.S2
_
I
TrA2A2 + 1(TrA2)2 - }(n+l)(c+3)IAI2
n)IAI2 - *(n+l)(c+3)]JAI2 - n (TrA2 - TrA2)2. t>s
Thus we have our inequality.
QED.
THEOREM 3.3. Let M be an n-dimensional compact anti-invariant submanifold with parallel mean curvature vector of a Sasakian space form M2n+1(c). Then either M is totally geodesic, or IAI2 = n(n+l)(c+ 3)/(4(2n-1), or at some point x of M, IAI2(x) > n(n+l)(c+3)/4(2n-1). We now take a unit sphere S2n+1 as an ambient manifold. Then we
have THEOREM 3.4. Let M be an n-dimensional anti-invariant submanifold with parallel mean curvature vector of 52n+1 (n > 1). If IAI2 = n(n+l)/(2n-1), then n = 2 and M is a flat surface of S5. Proof. Fran the assumption we have
j(TrAA - TrAs) = 0,
350
Tr(AtAS - ASAt) = 2TrAATrAs.
Therefore, TrAt = TrAs for all t, s. E Qn Iemma 5.1 of Chapter II we m a y a s s um e
that At = 0 for t = 3,...,n. Then we must have n = 2 and we
can put
A0=0,
0
1
1
0
1
0
0
-1
A2=A
Al = X
Since I AI2 = 2, we obtain 2A2 = 1. Thus we may assus
that k = 1/a.
Moreover, the Gauss equation (3.3) shows that M is flat. Ermr:e 3.1. Let C:n+1 be a complex (n+l)-dimensional number space with almost complex structure J and let S2n+1 be a (2n+1)-d n ensional unit sphere in G'n+1 with standard Sasakian structure (4,,E,-n,g). Let S1
be a circle of radius 1. Let us consider
Si.
T n = S1 x ... x
Then we can constract an isometric minimal immersion of Tn into
S2n+1
which is anti-invariant in the following way. Let 1: Tn
X=
1
>
n+l( cosu
1
S2n+1 be a minimal immersion represented by
sinu
1 ,
n+1 n+1 ...,cosun,sinun,cosu sinu
where we have put un+1 = -(ul + ... + un). We may regard X as a posiCn+1. tion vector of S2n+1 in The structure vector field E of S2n+1' restricted to Tn, is then given by
=
-JX
n1(sinus,-cosul,...,sinun+l'-'cosun+1).
Putting X1 = aX/aul, we have
0,-sinus,cosui,0,...,0,sinun+l,-cosun+l).
Xi =
n+l(0,....
51
i = 1,...,n. Thus X1,...,)cn are linearly independent and rI(Xi) = 0
for i = 1,...,n. Therefore the immersion X is anti-invariant. Moreover, the immersion X is a minimal immersion with 1A12 = n(n-1). From these considerations we have (Yano-Kon [4]) THEOREM 3.5, Let M be an n-dimensional compact anti-invariant
sutmanifold with parallel men curvature vector of S2n+1 (n > 1). If IA12 = n(n+1)/(2n-1), then M is S1 x Si.
4. CONTACT CR SU3IMIFOLDS
Let M be a (2m+l)-dimensional Sasakian manifold with structure tensors (p,E,n,g). We consider a Riemannian manifold M isometrically immersed in M with induced metric tensor field g.
Throughout in this section, we assume that the sutmanifold M is tangent to the structure vector field C of a Sasakian manifold M. Fbr any vector field X tangent to M, we put
qX = PX + FX,
(4.1)
where PX is the tangential part of OX and FX the normal part of OX. Similarly, for any vector field V normal to M, we put
V = tV + fV,
(4.2)
where tV is the tangential part of OV and fV the normal part of 4V. We easily see that P and f are skew-symmetr;.c. Fran (4.1) and (4.2)
we have
g(FXtV) + g(X,tV) = 0.
Moreover, we obtain
P2=-I-tF+n0 C,
FP+fF=O,
352
Pt + tf = 0,
f2 = -I - Ft.
Since we have 4F = PP + FE = 0, we find PP = 0 and FE = 0. For any vector field X tangent to M, we have 0X4 = -4X = 0
+ B(X,&). Thus,
we have the following
0
(4.3)
= -PX,
B(X,C) = -FX,
Especially, we have
= tV.
A.
0. Let X and Y be vector fields tangent
to M. Then we obtain
(4.4)_:-
(VxP)Y = AF X + tB(X,Y) + g(X,Y)E - n(Y)C,
(7XF)Y = -B(X,PY) + fB(X,Y).
(4.5)
For any vector field X tangent to M and any vector field V normal to
M, we also have
(4.6)
(TXt)V = AfVX - PA.X,
(4.7)
(7xf)V = -FAUX - B(X,tV).
Leett+M be an (n+l)-dimensional sub manifold of a Sasakian space
form M' 1(c). Then we have the following equations of Gauss and Codazzi respectively:
(4.8)
R(X,Y)Z =i,(c+3)[g(Y,Z)X-g(X,Z)Y] + 4'(c-1)1n(X)n(Z)Y
+2g(X,PY)PZ] +
(4.9)
AB(Y,Z)X
AB(X,Z)
Y,
(VXB)(Y,Z)-(VYB)(X,Z) = }(c-1)[g(PY,Z)FX-g(PX,Z)PY+2g(X,PY)FZ].
Moreover, we have equation of Ricci
(4.10)
g(Rl(X,Y)U,V) + g([AV,A)X,Y) = l(c-1)[g(FY,U)g(FX,V)-g(FX,U)g(FY,V)+2g(X,PY)g(fU,V)).
Definition. Let M be a submanifold tangent to the structure vector field ; isanetrically inmersed in a Sasakian manifold M. Then M is called a contact CR submanifold of M if there exists a differentiable distribution D : x
> Dx c Tx(M) on M satisfying the following
conditions:
(1) D is invariant with respect to 0, i.e., 4Dx c Dx for each
xcM, and (2) the canplementary orthogonal distribution D`: x
DX C_
X(M) is anti-invariant with respect to 0, i.e., 4DX c X(M)` for each x c M.
In the sequel, we put
dim M = 2n+1,
dim .M = 2nt1,
dimD`=p,
dim D = h,
codimM=2n-n=q.
If p = 0, then a contact CR suhmanifold is an invariant sut=nanifold
of M, and if h = 0, then M is an anti-invariant submanifold of M tangent to the structure vector field E. If q = p, then a contact CR submanifold M is called a generic submanifoZd of M. In this case we
have OX(M)l c T{(M) for every point x of M. If h > 0 and p > 0, then a contact CR sukmanifold M is said to be non-trivial (proper).
Let M be a contact CR snh nifold of a Sasakian manifold M. We denote by Z and l
the projection operators on D and DL- respectively.
Then we have
Z+l =I,
Z2=Z,
Z`2=Z`,
Since we have 4ZX = PZX + FIX, we obtain
ZZI =l l=0.
354
l'pl =O,
FL =0.
From $ZtX = WX + Fl-X we have PV- = 0, and hence PZ = P. Moreover we have
(4.11)
FP=O,
fF=0,
(4.12)
tf = 0,
Pt = 0.
Thus we find
(4.13)
P3 +P=01
(4.14)
f3 + f = 0.
These equations show that P is an f-structure in M and f is an fstructure in the normal bundle. Conversely, for a sutra nifold M, tangent to the structure vector
field F, of a Sasakian manifold M, we assume that we have FP = 0. Then we have IF = 0, (4.12), (4.13) and (4.14). We put
Z=-Pz+n0E,
l =I - Z.
Then we can easily verify that
Z+Z`=I,
Z2=Z,
Z`2=Z1,
ll`=ll=0,
which mean that Z and Z- are complementary projection operators and consequently define orthogonal distributions D and D` respectively. From Z = -P2 + n 0 E, we have PI = P because of p3 = -P and PE = 0.
This equation can also be written as Pt' = 0. But g(PX,Y) is skewsymmetric and g(V-X;Y) is symmetric and consequently Z''P = 0. Thus
we have Z'Pl = 0. On the other hand, we obtain FZ = 0 because of FP = 0
and
= 0. Consequently, the distribution D is invariant and DLis
anti-invariant with respect to 0. Moreover, we have Z&
1-
= 0
and consequently the distribution D contains E.
On the other hand, putting
V. = I+P2, we still see that Z and Zt define canplementary orthogonal distributions D and D1 respectively. We also have P1 = P, l P = 0, FZ = 0
and PZ'' = 0. Thus we see that b is invariant and D1 is anti-invariant with respect to 0 and also that ZE = 0, r = E, which mean that Dcontains F.
From these considerations we have (Yano-Kon [16])
THEOREM 41. In order for a sutmanifold M, tangent to the structure vector field F, of a Sasakian manifold M to be a contact (fit sub-
manifold, it is necessary and sufficient that FP = 0.
THEOREM 4.2. Let M be a contact B sutmanifold of a Sasakian manifold M. Then P is an f-structure in M and f is an f-structure in the normal bundle. LEM'1A 4.1. Let M be a contact CR suhnanifold of M. Then
AMY - A X = n(Y)X - n(X)Y
for X, Y E Dl.
Proof. Let X, Y be in D. Then PX = PY = 0, and hence
g((OZP)X,Y) = g(OZPX,Y) - g(PVZX,Y) = 0
for any vector field Z tangent to M. Frcm this and (4.4) we find
g(AZ,Y) + g(tB(Z,X),Y) = n(Y)g(Z,X) - n(X)g(Z,Y)
Thus we have our equation.
QED.
356
THEOREM 4.3. Let M be an (n+l)-dimensional contact CR sutmanifold of a (2m+1)-dimensional Sasakian manifold 2. The distribution D` is completely integrable and its maximal integral submanifold is a qdimensional anti-invariant submanifold of M normal to F or a (q+l)dimensional anti-invariant submanifold of M tangent to E. Proof. For any vector fields X and Y in D` we have
[X,Y] = P[X,Y] + F[X,Y] = -(VXP)Y + (VYP)X + F[X,Y]
= AMY - AX - n(Y)X + n(X)Y + F[X,Y] = F[X,Y].
Thus we have ¢[X,Y] E T(M), and consequently [X,Y] E D.
QED.
THEOREM 4.4, Let M be an (n+l)-dimensional contact CR submanifold of a (2m+1)-dimensional Sasakian manifold M. Then the distribution D is completely integrable if and only if
B(X,PY) = B(Y,PX)
for any vector fields X, Y e D, and then F E D. bbreover, the maximal integral suhnanifold of D is an (n+l-q)-dimensional invariant submanifold of M.
Proof. Let X, Y E D. Then (4.5) implies
q[X,YJ = P[X,Y] + F[X,Y] = P[X,Y] + (VYF)X - (VXF)Y = P[X.Y] + B(X,PY) - B(Y,PX).
Thus we see that [X,Y] E D if and only if B(X,PY) = B(Y,PX) for every X, Y e D. If D is normal to F, then g([X,Y],F) = 2g(PX,Y) for X,Y E D.
Thus, if D is completely integrable, we have g(PX,Y) = 0, which shows that dim D = 0. This proves our assertion.
QED.
We now give examples of contact CR sutmanifolds and generic submanifolds of a Sasakian manifold.
357 Exanple 4.1. Let S
l be a (2m+1)-dimensional unit sphere with
standard Sasakian structure (0,C,rl,g). We denote by Sm(r) an m-dimen-
sional sphere with radius r. We consider the following immersion:
S'(rl) x ... x
k
Smi(rk)
>
n+1 =
Sn+k,
m. i=1 1
where ml,...,mk are odd numbers and ri + ... + rk = 1. Here n+k is m. m.+1 (m +1)/2 m. 1 1 also odd. Let v. be a point of S 1(r.) in R = C . S 1(r 1 1 (m1+1)/2 -1 1 is a real hypersurface of C with unit normal ri vi. Thus v
= C(n+k+l)/2. We restrict (mi+l)/2 the almost caoplex structure of C (n+k+l)/2 to C . Then each _ (v1,...,vk) is a unit vector in
Jv
1
Rn+k+l
is tangent to S 1(r). Thus Jv is tangent to M 1 -n1,...,mk
x ... x Smk(rk). We then consider the normal space of
%
= S(r ) "1 1
,...,mk
in
Sn+k which is the orthogonal complement of the 1-dimensional space spanned by v in the space spanned by v1,...,vk. That is,
)`' $ T x(M ml P... 1Mk Let w1,...,wk
1
= ). Then wi
be an orthonormal frame for Tx(M
is given by a linear combination od v1,...,vk. Thus Jwi 1s tangent to
M
, and hence we see that
¢wi=Jwi-rl(wi)v=Jwi. Therefore
i is tangent to Mml,...,mk for T (
x ml,.,mk )A.
Consequently,
1,...,mk
c T (M
all i = 1,...,k-1. Thus
x ml,...,mk
).
is a generic submanifold of Sn+k. M ml,...,mk
has parallel second fundamental form and flat normal connection. Flu thermorer
r...r
is a contact CR submranifold of S1
K
(2m+1 > n+k) with parallel second fundamental form and flat normal connection. Example 4.2. In Example 4.1, if ri = (m./(n+l))1/2 (i = 1,...,k),
then M
n]., ... ,mlc
is a generic minimal submanifold of Sn+k and hence
minimal contact CR submanifold of S2"+1 (2m+1 > n+k). Then the square of the length of the second fundamental form of M is given by I Al 2 = (n+l)(k-1).
We notice that in Examples 4.1 and 4.2, if we put mi = 1 for all
i = 1,...,k (k = n+l), then M
is an anti-invariant submanifold
m1,...mk
of S2n+1
In the following we consider a contact CR submanifold with flat
normal connection. Let M be an (n+l)-dimensional contact (R submanifold of a (2m+1)-dimensional Sasakian manifold M. Then we have the following decomposition of the tangent space Tx(M) at each point x of M:
Tx(M) = H{(M) ®
Nx(M),
where Hx(M) = OHt(M) and Nx(M) is the orthogonal complement of
Hx(M) ®
in Tx(M). Then 4Nx(M) = F x(M) C Tx(M)1. Similarly, we
have
Tx(Mf = %(M) ® X(M)l, where Nx(M)- is the orthogonal complement of FN (M) in Tx(Mf . Then X
ONx(Mf = f x(M)1 = N (M)''. We now take an orthonormal basis el,...,e2m+1 of M such that,
restricted to M, el,...en+1 are tangent to M. Then el,...en+1 form an orthonormal basis of M. We can take el,...'en+l such that el,...ep form an orthonormal basis of N{(M) and ep+1,...,en form an orthonormal
basis of H{(M) and en+1 = F, where p = dim X(M). Moreover, we can of an orthonormal basis of Tx(MY such that 2m+1 form an orthonormal basis of FN (M) and en+2+p,..., en+2"** en+l+p x form an orthonormal basis of NX(MY. In case of need, we can e2m+1 take en+2' ...,e
take en+2"* ., en+l+p such that en+2 Fe V .. " en+l+p Fep .
35
Unless otherwise stated, we use the conventions that the ranges of indices are respectively:
i,j,k = 1,...,n+l;
x,y,z = 1,...,p;
a,b,c = p+1,...,n;
a,s,Y = n+2,...,n+l+p.
Here, we take S2n+1 as an ambient manifold M. Then we have
LDM 4.2. If the normal connection of M is flat, then
AfV=0
for any vector field V normal to M.
Proof. Since R` = 0, (4.10) implies that AVAU = AUAV. Thus, from (4.3), we obtain AVtU = AUtV. From this and tf = 0, we see that AfVtU = 0 and AfVA = 0. Moreover, from (4.7), we obtain (VXf)fV = 0. Thus, from (4.5) and (4.14), we have
g((VXf)fV,FY) = -g(f2V,(VXF)Y) _ -g(AfVX,Y) + g(Af2VX,PY) = 0.
From this and the fact that AfVAf2V = Af2VAfV, we have
TrAzfV = TrAfVPAfV = -TrAtVPAf2V = -TrAf2VAfVP = -TrAfVAf2VP = -TrAf2VPAfV = -TrA}.V:
Consequently, we have TrAf2.V = 0 and hence AfV = 0.
QED.
LEM V1 4.3. Let M be an (n+l)-dimensional contact CR submanifold of
S
1 with flat normal connection. If PAV = AVP for any vector field V
normal to M, then
g(AOX,AVY) = g(X,Y)g(tU,tV) -
Jg(AUtV,ei)g(A.Fe
i
X,Y).
i
Proof. From the assumption we have g(AUPX,tV) = 0, which implies
EE g((V A)UPX,tV) + g(AU(V P)X,tV) + g(AUPX,(VYt)V) = 0.
Thus, from (4.4) and (4.6), wee have
g((V A) PX,tV) - g(X,Y)g(AUg,tV) + n(X)g(AUY,tV)
+ g(AUAY,tV) + g(AUtB(Y,X),tV) + g(AUPX,AY) - g(AUPX,PAVY) = 0,
from which and Lemma 4.2, we find
g((4.,A)UPX,tV) + g(X,PY)g(tU,tV) + g(AUtV,tB(PY,X)) - g(AUPX,PAVPY) = 0.
On the other hand, we have
g(AUtV,tB(PY,X) = -lg(AUtV,ei)g(AFe X,PY),
-g(AUPX,PAVPY) = g(A PX,AVY).
From these equations we obtain
g((V
A)UPX,tV) + g(X,PY)g(tU,tV) - lg(AUtV,ei)g(AFe1X,PY) + g(AUPX,AVY) = 0.
Therefore, the Codazzi equation implies
g(X,PY)g(tU,tV) - lg(AUtV,ei)g(AF,e X,PY) + g(AUPX,AVY) = 0,
i
from which
i
369
(4.15)
g(PX,PY)g(tU,tV) - g(AUtV,ei)g(AFe PX,PY) + g(AUP2X,AVY) = 0.
On the other hand, we have
g(PX,PY)g(tU,tV) = g(X,Y)g(tU,tV) - n(X)n(Y)g(tU,tV) - g(FX,FY)g(tU,tV),
-lg(AUtV,ei)g(AFe.PX,PY) = -lg(AUtV,ei)g(AFe.X,Y) + n(Y)g(AUtV,X) + n(X)n(Y)g(tU,tV) - Jg(AUtV,ei)g(AFe X,tFY),
i
i
g(AUP2X,AVY) = -g(AUx,AVY) - n(Y)g(AUtv,x) - g(A.X A tFY).
Substituting these equations into (4.15), we find
g(X,Y)g(tU,tV) - lg(AUtV,ei)g(AFeX,Y) - g(AUX,AVY)
i
i
- g(FX,FY)g(tU,tV) - lg(AUtV,ei)g(AFe X,tFY) - g(AUX,AVtFY) = 0.
i
i
Moreover, we obtain
-lg(AUtV,ei)g(Ape X,tFY) = g(AUtV,AFYX) + g(FX,FY)g(tU,tV) - g(AUX,AVtFY)
g(AUtV,AFYX).
Fran these, we obtain our equation.
QED.
LEA 4,4, Let M be an (n+l)-dimensional contact CR sutmanifold of S
l with flat normal connection. If the mean curvature vector
of M is parallel, and if PAV = AVP for any vector field V normal to M, then the square of the length of -the second fundamental form of M is constant. Proof. Since AfV = 0, we have IA 12 = aTrAa2. On the other hand,
3'82
Le= 4.3 gives JA12 = (n+l)p +
E g(A(% te(I ,te0)TrA8.
a,$
Since the normal connection of M is flat, we can take {ea} such that Dea = 0 for each a, because, for any V F- FN(M) we have DXV a FN(M)
by (4.7) and AVtU = AUtV. Then we have
VXIAI2 =
E g((VXA)atea,tea)TrAa = I g((Vte A)atea,X)TrA3 a,s a,a a
by using VX(tea) _ (VXT)ea = Afe X - PAaX and Pt = 0. On the other
hand, using PAV = AVP, we have a
g((VPe A)aPei,X) = )[g((VPe P)A(X ei,X) + g(P(VPe A)aei,X) 1 1 1
i
- g( a(VPe P)ei,X)]. 1
Since Aa is symmetric and P is skew-symmetric, using (4.4), we see that
jg(Aa(VPe P)ei,X) = 0. 1
Jg((VPe P)Aaei,X) = 0, 1
i
i
Therefore, we have
lg((Vpe A)aPei,X) _ Ig(P(VPe A)oei,X) 1 1
i
Jg((Vpe A)aei,PX)
Ig((VpXA)aPei,ei) = 0,
i
i
1
where we have used the Codazzi equation and the fact that (VPXA)a is symmetric and P is skew-symmetric. Since we have E(VeaA)aea = E(Vpe A)aPei, the equation above implies 1
a (VeA)aea = 0.
Moreover, we see that
(VA)
= 0.
Since the mean curvature vector of M is parallel, we have
0 = E(Ve A)aei = j(Ve A)aea + (V&A) + J(Ve A)aex i x x i a a A)ate8. = }(Ve A)aex = Y(Vte
x
x
B
B
Therefore, the square of the length of the second fundamental form of M is constant.
QED.
Frcm Lemmas 4.2, 4.4 and Proposition 3.1 of Chapter II we have LF1l4A 4.5. Let M be an (n+1)-dimensional contact CR submanifold
of S1 with flat normal connection. If the mean curvature vector of M is parallel, and if PAV = AvP for any vector field V normal to M, then
IVA12 = -(n+1)ITTA22 + J(Tr a)2 a a
+
TrABTrAaAB. F (TrAaAB)2 a,B a,B
LEIT1A 4.6. Under the same assumptions as those of Lemma 4.5, the second fundamental form of M is parallel.
Proof. From Lemma 4.3 we obtain TrA2AB = TrAag(ea,eB) + jTr(AYAa)g(AYteateB),
Y TrAaAB = (n+l)g(ea,e8) + :TrAyg(AyteateB).
Y Fran these equations we have
(TrAaAB)2 = (n+1)jTrAa +
a,B
a
E TrAaABTrA a,B,Y
TrASTrA2A0 = -I(TYA a)2 a a,0
F
TrA A0TrA
Yg(Y
a,Y
a,te8)
Substituting these equations into the equation of Iemm 4.5, we find IVAI2 = 0, which shows that the second fundamental form of M is paraQED.
llel.
We prove the following theorems (Yano-Kon [16]). THEOREM 4.5. Let M be an (n+l)-dimensional ccnplete contact CR
submanifold of S1 with flat normal connection. If the mean curvature vector of M is parallel, and if PAV
AVP for any vector field V normal
to M, then M is Sn+l or
S"(r1) x ... X Smk(rk),
n+l = Imi,
1,
2 < k < n+l,
i
in some
Sn+1+P,
where m1,...,mk are odd numbers.
Proof. We first assume that F = 0, thst is, M is an invariant
submanifold of S
1. Then we have PAV + AVP = 0. Thus we have P.AV = 0
and hence AV = 0. Consequently, M is totally geodesic in S1 and M is Sn+1.
We next assume that F # 0. Since the second fundamental form of M is parallel and the normal connection of M is falt, by Theorem 5.5 of Chapter II, the sectional curvature of M is non-negative. On the other hand, from Lemma. 4.3, we see that AV # 0 for any V E EN (M). Thus
X Lemma 4.2 shows that the first normal space of M is of dimension p. Therefore, by Theorem 4.3 of Chapter II and Example 4.1, we have our QED.
assertion.
THEOREM 4.6. Let M be an (n+l)-dimensional complete generic
submanifold of S
1 with flat normal connection. If the mean curvature
vector of M is parallel, and if PAV = AvP for any vector field V normal
to M, then M is
Sml(rl) X ... X Smi(rk),
n+l = Imi, i
ri = 1, 1
2 < k < n+l,
0 where m1,...,mk are odd numbers. Let M be an (n+l)-dimensional contact CR suhmanifold of
S2m+1
with flat normal connection. Let V be a parallel vector field normal
to M. Then we have VXtV = PAvX. Hence we have div tV = -TrPAV = 0. Therefore, from Theorem 4.3 of Chapter I, we find
div(Vt,7tV) = S(tV,tV) + JIL(tv)gl2 - IvtVl2.
In the following we suppose that M is minimal. Then the Ricci tensor S of M is given by
S(X,Y) = ng(X,Y) - lg(A22X,Y). a
On the other hand, we have
IvtV12 = TrAA - g(tV,tV) - jg(AZatV,tV). a
From these equations we obtain
div(vtvtV) = (n+l)g(tV,tV) - TrA + JIL(tV)gl2.
Since the normal connection of M is flat, we can take {e such that De
a
of FN(M)
= 0 for each a. Thus we have
div(Pte tea) = (n+l)p - IA12 + JEIL(tea)gl2. a a a THEOREM 4.7. Let M be an (n+1)-dimensional contact minimal contact S2m+l -CR submanifold of with flat normal connection. Then
0 < J fMf IL(te(x )g12*1= IM[ IAI2 - (n+l)p]*1. a As an application of Theorem 4.7, we have
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THEOREM 4.8. Let M be an (n+l)-dimensional ccmpabt minimal contact S2m+1 CR submanifold of with flat normal connection. If I Al 2 = (n+1)p, then M is
S71(rl) x ... X Smi(rk),
ri = (mi/(n+1))1/2
Sn+l+p,
n+l = Flni, Err = 1, 2 < i < n+1, in some
(i = 1,...,k),
where ml,...,nk are
odd numbers.
Proof. From the asswption ae have JL(tea)gj = 0 for each a. Thus we have
0 = (L(tea)g)(X,Y) = g(OXtea,Y) + g(VYtea,X) = g((AaP - PAa)X,Y)
for each a. Consequently, PAV = AvP for any vector field V normal to M. Therefore, our assertion follows from Theorem 4.5.
QED.
5. INDUCED STRUCTURES ON SUWANIFOLDS
Let M be an n-dimensional manifold. We assume that there exist on M a tensor field f of type (1,1), vector fields U and V, 1-forms u and v, and a function A satisfying the conditions:
(5.1)
f2X = -X + u(X)U + v(X)V,
(5.2)
u(fX) = Av(X),
(5.3)
u(U) = v(V) = 1 - A2,
v(fX) = -Au(X),
fU = -AV,
fV = AU,
u(V) = v(U) = 0,
for any vector field X on If. In this case, we say that the manifold M has an (f,U,V,u,v,A)-structure. Then we have (Yano-Okiunura (1])
367
THEOREM 5.1. A manifold M with (f,U,V,u,v,X)-structure is of even dimensional. Proof. Let x be a point of x at which A2 # 1. Then we see that U # 0 and V # 0 at x. Two vectors U and V are linearly independent.
Fbr, if there are two numbers a and b such that aU + bV = 0, then u(aU + bV) = au(U) = a(l - A2) = 0, v(aU + bV) = bv(V) = b(1 - A2) = 0.
Thus we have a = b = 0. Thus U and V being linearly independent at x, we can choose n linearly independent vectors X1 = U, X2 = V, X3,...,
a which span the tangent space Tx(M) and such that u(Xa) = 0, v(Xa) _ 0, for a = 3,...,n. Consequently, we have f2Xa = -Xa for all a, which shows that f is an almost complex structure in the subspace Ex of TT(M)
at x spanned by X3,...,Xn and that Ex is even dimensional. Thus Tx(M) is also even dimensional.
Next, let x be a point of M at which A2 = 1. In this case, we see that u(U) = u(V) = v(U) = v(V) = 0. We also see that if u # 0,
then v # 0, and if u = 0, then v = 0. We first consider the case in which u # 0, v # 0. Then u and v are linearly independent. Because, if there are two numbers a and b such that au + by = 0, then (au + bv) (fX) = A(bu - av)(X) = 0 and hence bu - av = 0, A being different from zero. Thus we have (a2 + b2)u = 0, from which a = 0, b = 0. Thus, u and v being linearly independent at x, we can choose n linearly independent covectors wl = u, w2 = v, w3,...,w1 which span the cotangent space X(M)* of M at x. We denote the dual basis by X11X21 ..., a. If U and V are linearly independent at x, we can assume that
n-1 = U,
Xn = V. Then we have
f2Xa = -Xa + u(Xa)U + v(a)V = -Xa,
a = 3,...,n
which shows that f is an almost complex structure in the subspace Ex of TA(M) at x spanned by X3,...,Xn and that Ex is even dimensional and consiquently T{(M) is also even dimensional.
If U and V are linearly dependent, there exist two numbers a and b such that aU + bV = 0 and a2 + b2 # 0. Applying f to this equation, we find A(-aV + bU) = 0, and hence bU - aV = 0. Thus, we must have
360
U = V = 0. Consequently, we have f2X = X for any vector X in X(M) and Tx(M) is even dimensional.
If u = 0, v = 0, we also have fzX = X for any vector X in TX(M) and consequently TA(M) is even dimensional. Thus we have our assertion.
W. The structure (f,U,V,u,v,A) is said to be normal if
S(X,Y) = N(X,Y) + du(X,Y)U + dv(X,Y)V = 0
for any vector fields X and Y on M, N being the Nijenhuis torsion of f.
We consider a product manifold M x R2, where R2 is a 2-dimensional Euclidean space. Then, (f,U,V,u,v,A)-structure gives rise to an almost carplex structure J on M x Rz:
J =
as we can easily check using (5.1), (5.2) and (5.3).
Computing the Nijenhuis torsion of J, we can easily prove PROPOSITION 5.1. If J is integrable, then (f,U,V,u,v,A)-structure is normal.
We assume that, in M with (f,U,V,u,v,A)-structure, there exists a positive definite Riemannian metric g such that
v(X) = g(V,X),
(5.4)
u(X) = g(U,X),
(5.5)
g(fX,fY) = g(X,Y) - u(X)u(Y) - v(X)v(Y).
We call such a structure a metric (f,U,V,u,v,X)-structure and denote it by (f,g,u,v,A).
Example 5.1. Let M be a (2n+1)-dimensional almost contact metric manifold with structure tensors (q,C,n,g). Then the structure tensors
0 satisfy
¢2=-I+n®&, n(X) = gQ,X),
OE=0,
n(4X) = 0,
n(;) = 1,
g(gX,¢Y) = g(X,Y) - n(X)n(Y)
for any vector field X on M.
Let M be a 2n-dimensional hypersurface of M. We denote by the same g the induced metric tensor field on M. The unit normal of M in M will be denoted by C.
Fbr any vector field X tangent to M we put
0 = fX + u(X)C,
v(X) = n(X),
C = V + AC,
4C = -U,
A = n(C) = g(c,C),
where f is a tensor field of type (1,1), u, v 1-forms, U, V vector fields and A a scalar function on M. Then they satisfy
f2X = -X + u(X)U + v(X)V,
fU = -AV,
fV = AU,
u(U) = 1 - A2,
u(fX) = Av(X),
u(V) = 0,
v(fX) = -Xu(X),
v(U) = 0,
v(V) = 1 - A2.
Moreover, we have
g(U,X) = u(X),
g(V,X) = v(X),
g(fX,Y) = -g(X,fY),
g(fX,fY) = g(X,Y) - u(X)u(Y) - v(X)v(Y).
Therefore, the hypersurface M admits (f,g,u,v,A)-structure. Example 5.2. Let M be a real (2n+2)-dimensional almost Hermitian manifold with structure tensors (J,g). Then J2 = -I and g(JX,JY) =
370
g(X,Y) for any vector fields X and Y on R. Let M be a si+hnifold of cod.imension 2 of M with orthonormal frame C, D for X(M -. We put
JX = fX + u(X)C + v(X)D,
JC = -U + AD,
JD = V - AC.
Then the induced structure (f,g,u,v,A) on M satisfies (5.1) - (5.5). Therefore, the sutmanifold M admits an (f,g,u,v,A)-structure. ExcvnpZe 5.3. Let M be a (2n+1)-dimensional Sasakian manifold and
let M be a hypersurface of M. We have the Gauss and Weingarten formulas
OXY = VXY + g(AX,Y)C,
VXC = -AX.
Then we have
(VXf)Y = -v(Y)X + u(Y)AX + g(X,Y)V - g(AX,Y)U,
VXU = AX + fAX,
VXV = -fX + AAX.
On the other hand, we obtain
S(X,Y) = (VfXF)Y - (Vfy f)X + f(VYf)X - f(VXf)Y
+ g(VXU,Y)U - g(VYU,X)U + g(VXV,Y)V - g(VYV,X)V = u(Y)(AfX - fAX) - u(X)(AfY - fAY).
Thus, if Af = fA, then the hypersurface M is normal. We shall prove the converse. Let S(X,Y) = 0 for all X and Y and put PX = (Af-fA)X. Then
u(U)PX = u(X)PU.
Also, it can be shown that g(PX,Y) = g(X,PY) so that
31 u(X)g(PU,Y) = u(Y)g(PU,X),
that is to say,
g(PU,Y) = au(Y)
for some a. Thus we have
u(U)g(PX,Y) = u(X)g(PU,Y) = au(X)u(Y),
but since the trace of P is zero, we have a = 0, i.e., P = 0, which means that Af = fA. Consequently, M is normal if and only if Af = fA.
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EXERCISES
A. K-CONTACT SUBMANIFOLDS: Let Si be a (2n+1)-dimensional K-contact
Riemannian manifold with structure tensors
A sutmanifold M
of M is said to be invariant in Si if OX is tangent to M for any tangent
vector field X to M, and
is always tangent to H. Any invariant sub-
manifold M of a K-contact Riemannian manifold M is also a K-contact Riemannian manifold with respect to the induced structure on M. Then we have (Endo [1]) THEOREM 1, Any invariant submanifold M of a K-contact Riemannian manifold Si is a minimal submanifold.
On the other hand, we obtain (cf. Kon [1]) THEOREM 2. Let M be an invariant submanifold of a K-contact
Riemannian manifold M. Then M is totally geodesic if and only if the second fundamental form of M is parallel.
B. COSYMPLECTIC MANIFOLDS: Let M be a normal almost contact metric manifold such that the fundamental 2-form 0 is closed and do = 0. Then M is called a cosymplectic manifold. The cosymplectic structure is characterized by a
= 0 and OXn = 0. Let M be an
invariant submanifold of a cosyrrplectic manifold M. Then M is also a cosymplectic manifold with respect to the induced structure on M.
Ludden [1] proved the following THEOREM. If M is a cosymplectic manifold of constant 0-sectional curvature and M is an invariant submanifold of codimension 2 of M which is n-Einstein, then M is locally symmetric. C. FLAT NORMAL CONNECTION OF INVARIANT SU3 ANIFOLDS: Let M be
a Sasakian space form of constant -sectional curvature k and M be an invariant submanifold of Si. Then we have
THEOREM. The following conditions are equivalent: (a) The normal connection of M is flat, i.e., R` = 0;
373
(b) k = 1 and M is totally geodesic in M.
In the case of codimension 2, Kenmotsu [2] proved the above theorem. The theorem above was proved by Kon [3] when the codimension of M is greater than 2.
D. AXIOM OF 4-HOLOMORPHIC PLANES: Let M be a (2n+1)-dimensional Sasakian manifold with structure tensors
We say that m
admits the axiom of 4-holcooorphic (2r+1)-planes if, for each point x
of M and any (2r+1)-dimensional ¢-holamrphic subspace S of Tx(M), 1 < r < n, there exists a (2r+1)-dimensional totally geodesic suhnanifold N passing through x and satisfying T{(N) = s, where we mean a 0-holomorphic subspace S by a subspace of Tx(M) satisfying OS C S. I. Ishihara [2] proved the following
THEOREM. A Sasakian manifold is of constant -sectional curvature if and only if the manifold satisfies the axiom of ct-holomorphic (2r+1)-planes.
E. REDUCTION THEOREMS OF ANTI-INVARIANT SUBMANIFOLDS: I. Ishihara [2] studied the reduction theorems of codimension of anti-invariant sukmanifolds of Sasakian space forms. Let M be an (n+l)-dimensional
anti-invariant sutmanifold, tangent to the structure vector field of a (2m+1)-dimensional Sasakian manifold M. If Eiha
,
= 0 for all
indices a and x, then the mean curvature vector of M is said to be pseudo-parallel. (For the ranges of indices, see §2.) The normal
connection of M is said to be pseudo-flat if R&y = 0 for all indices. Then THEOREM 1. Let M be an (n+l)-dimensional (n > 3) anti-invariant sutmanifold, tangent to the structure vector field F, of a Sasakian
space form M1(c) (c # -3) with pseudo-parallel mean curvature vector. If the normal connection of M is pseudo-flat, then there is
in Ml(c) a totally geodesic and invariant sutmanifold M2n+1(c) of dimension 2n+1 in such a way that M is immersed in M2n+1(c) as a flat
anti-invariant sutmanifold. Let M be an n-dimensional anti-invariant sutrranifold, normal to
the structure vector field F, of a (2m+1)-dimensional Sasakian
374 manifold M. If Ekh
d
= 0 for all indices i and p, then the mean
curvature vector of M is said to be n paraZZeZ. (Fbr the ranges of indices, see §3.) For the normal curvature tensor It we consider the condition
(*)
g(E' (X,Y)U,V) = g(4Y,V)g(4X,U) - g(0X,V)g(¢Y,U)
for any vector fields X, Y tangent to M and any vector fields normal
to M. Then we have (I. Ishihara [3]) THEOREM 2. Let M be an n-dimensional (n > 3) anti-invariant sukmanifold, normal to the structure vector field E, of a Sasakian space.-. form
M2n+1(c) (c # -3) with n -parallel mean curvature vector.
If it satisfies (*), then there is in M2n+l(c) a totally geodesic and invariant submanifold M2n+1(c) of dimension 2n+1 in such a way M2n+l(c) that M is immersed in as a flat anti-invariant minimal suhmanifold.
F. CONFORMALLY FLAT ANTI-INVARIANT SUBMANIFOLDS: Let M be an (n+l)-dimensional anti-invariant suhmanifold, tangent to the structure vector field E, of a (2m+l)-di.nensional Sasakian manifold M. If the
second fundamental form B of M satisfies
B(X,Y) = [g(X,Y)-n(X)n(Y)]a + n(X)B(X,E) + n(Y)B(X,E)
for any vector fields X, Y tangent to M, where a denotes a normal vector field to M, then M is said to be a contact totaZZy wnbilical.
We have (Kon [10]) THEOREM 1, Let M be an (n+l)-dimensional (n > 3) contact totally
umbilical, anti-invariant submmifold, tangent to the structure vector field E, of a (2m+l)-dimensional Sasakian manifold M with vanishing contact Bochner curvature tensor. Then M is locally a product of a confonnally flat Riemannian manifold Mn and a 1-dimensional space M1.
Yano [8] proved the following
375
THEOREM 2, Let M be an n-dimensional (n > 3) totally umbilical anti-invariant sutxmanifold, normal to the structure vector field E,
of a (2m+1)-dimensional Sasakian manifold M with vanishing contact Bochner curvature tensor. Then M is conformally flat.
G. GENERIC SUBMANIFOLDS: Yano-Kon [la] proved the following THEOREM, Let M be an (n+l)-dimensional complete generic minimal
submanifold of
t+1
with parallel second fundamental form. If M is
Einstein, then M is
Sq(r) x ... x Sq(r) (N-times),
r = (q/(n+1))1/2,
where q is an odd number and 2m-n = N-1, Nq = n+1. H. PSEUDO-UMBILICAL HYPERSURFACE: Let M be a hypersurface of a Sasakian manifold M, tangent to the structure vector field. If the second fundamental form A of M is of the form
AX = a[X-n(X)E] + bu(X)U + n(X)U + u(X)C
for any vector field X tangent to M, a and b being functions, then M is called a pseudo-umbilical hypersurface of M, where a vector field U and a 1-form u are defined to be U = -4C, u(X) = g(U,X) respectively for a unit normal C of M. The notion of pseudo-umbilical hypersurfaces of Sasakian manifolds corresponds to that of n-umbilical real hypersurfaces of Kaehlerian manifolds. If M is a pseudo-umbilical hypersurface of
52n+1 (n
> 2), then
M has two constant principal curvatures with multiplicities 2n-1 and 1 respectively. Then we have (Yano-Kon [101) THEOREM 1. Let M be a compact pseudo.-umbilical hypersurface of
52n+1 (n > 2). Then M is
S2n-t(rl) x S1(r2),
r1 + r2 = 1.
376
A Sasakian manifold M of dimension 2n+1 is said to satisfy the P-axiom if for each x e M and each 2n-dimensional subspace S of Tx(M),
a S, there exists a pseudo-umbilical hypersurface N such that Tc(N) = S, x e N and g(AU,U) = a+b = constant. Yano-Kon [10] proved the following
THEOREM 2. If a (2n+1)-dimensional Sasakian manifold M (n > 2) satisfies the P-axiom,
is a Sasakian space form.
To prove the theorem above, we need the following theorem of Tanno [7].
THEOREM 3. A (2n+1)-dimensional (n > 2) Sasakian manifold M is a Sasakian space form if and only if R(X,gX)X is proportional to OX for any vector field X of M such that n(X) = 0. 1. PSEUDO-EINSTEIN HYPERSURFACES: Let M be a 2n-dimensional
hypersurface of 52+1 tangent to the structure vector field. If the Ricci tensor S of M is of the form
S(X,Y) = a[g(X,Y)-n(X)n(Y)] + bu(X)u(Y) + n(X)S(l;,Y) + n(Y)S(i;,X) - n(X)n(Y)SQ,E),
a and b being constant, then M is called a pseudo-Einstein hypersurface S2n+1.
of
Yano-Kon [10] proved the following theorems (see also Kon
[12]).
THEOREM 1. Let M be a pseudo-Einstein hypersurface of
S2n+l
(n > 3). Then M has two constant principal curvatures or four constant principal curvatures. 5"n+1.
We give examples of pseudo-Einstein hypersurfaces of
Let
Cn+l be the space of (n+l)-tuples of complex numbers
Put S2n+1 = {(zl,...,zn+l) E 0+1 : EIz.I2 = 1}. For a positive number r we denote by MO(?rn,r) a hypersurface of 52n+1 defined by n+1
n F.
j=1
Iz.I2 = 1.
rizn+ll2,
j=1
J
377
For an integer m (2 < m < n-1) and a positive number s a hypersurface M(2n,m,s) of S2n+1 is defined by
m
n+1 IZjl2 = S
n+1 j1llzjl2 = 1.
Izj12,
jI
For a number t (0 < t < 1) we denote by M(2n,t) a hypersurface of S2n+1 defined by
n+1 I
n(+'1
ZJIZ = t,
L
j=1
IZjl2 = 1.
j=1
MO(2n,r) and M(2n,m,s) have two constant principal curvatures and M(2n,t) has four constant principal curvatures (Nomizu [ ], R. Takagi [ ]). MO(2n,r) is always pseudo-Einstein. M(2n,m,s) is pseudo-
Einstein if s = (m-l)/(n-m) and M(2n,t) is pseudo-Einstein if t = 1/(n-1).
THEOREM 2. If M is a canplete pseudo-Einstein hypersurface in S2n+1 (n > 3), then M is congruent to some MO(2n,r) or to some M(2n,m,(m-1)/(n-m)) or to M(2n,l/(n-1)).
J. HYPERSURFACES WITH (f,g,u,v,A)-STRUCTURE: Let M be a 2n-dimensional hypersurface of
S2n+1.
Then M admits an (f,g,u,v,A)-structure.
Nakagawa-Yokote [1] proved the following THEOREM 1. Let M be a compact hypersurface of S2n+1 satisfying Af = fA = 2af, a being function. If n > 3, then one of the following two assertions (a) and (b) is true: (a) M is isometric to one of the following spaces: (1) the great sphere S2n; (2) the small sphere S2n(c), where c = 1+a2;
(3) the product manifold S2n-1(c1) x S1(c2), where c1 = 1+a2, c2 = 1+1/a2;
(4) the product manifold Sn(c1) x Sn(c ), where c1 = 2(1+x2 +a(1+a2)1/2), c2 = 2(1+x2-a(1+a.2)1/2);
(b) M has exactly four distinct constant principal curvatures a±(1+a2)1/2, (-1±(1+a2)1/2 )/a with maltiplicities n-1, n-1, 1 and 1, respectively.
378
In the case that (b) holds, the hypersurface M is congruent to
some M(2n,t) in Exercise (I) (see R. Takagi. (3]). When the second fundamental tensor A com sites to f, Bon [12]
proved THEOREM 2, Let M be a coaplete hypersurface of Stn+1 (n > 2). If Af = fA, then M is congruent to S2n(a2+1), a = v(AV)/(1-X2), or to S2p+1(r1) X S2q+1(r2). p+q = n-1.
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CHAPTER VII
f-STRUCTURES
In this chapter, we study a manifold which admits an f-structure. In §1, we define an f-structure on a manifold and give a necessary and sufficient condition for a manifold to admit an f-structure (Yano [41). We also give integrability conditions of an f-structure (Ishihara-Yano [1].
§2 is devoted to the study of the normality of an
f-structure (S. Ishihara [1]). In §3, we consider a manifold with a globally framed f-structure (Goldberg-Yano [2]). In the last §4, we discuss hypersurfaces of framed manifolds and give some theorems (Goldberg [2], Goldberg-Yano [1]).
1. f-STRUCTURE ON MANIFOLDS
A structure on an n-dimensional manifold M given by a non-null tensor field f satisfying
f3 + f = 0 is called an f-structure (Yano [4]). Then the rank of f is a constant, say r (Stong [1]). If n = r, then an f-structure gives an almost complex structure of the manifold M and n = r is necessary even. If M is orientable and n-1 = r, then an f-structure gives an almost contact structure of the manifold M and n is necessary odd.
We put
Z=-f2,
m=f2+I,
380 I denoting the identity operator, then we have
Z+m=I,
12=Z,
ft = If = f,
m2=m,
mf = mf = 0.
These equations show that the operators Z and m applied to the tangent space at each point of the manifold M are complementary projection operators. Then there exist in M two distributions L and N corresponding to the projection operators Z and m respectively. When the rank of f is r, L is r-dimensional and N (n-r)-dimensional. We now introduce in M a local coordinate system and denote by fi, 1i, Ti the local components of the tensors f, Z, m respectively. We also introduce a positive definite Riemannian metric in M and take r mutually orthogonal unit vectors ua (a,b,c,... = 1,2,...,r) in L and n-r rutually orthogonal unit vectors uA (A,B,C,... = r+l,...,n)
in M. We then have Ziub
ub h i h miUB = UB.
miub = 0,
From fm = 0, that is, fim = 0, we find, contracting with uB and taking account of the last equation of (1.1),
fiuB = 0.
If we denote by (vi,vi) the matrix inverse to (ub,uB), then vi and
vi are both components of linearly independent covariant vectors and satisfy
ai
a
viuB = 0,
viub = 0,
viub = 6B
v iu = b (1.2)
301
and (1.3)
iu+viA=bi.
Now from (1.1) and (1.3), we find
(Zhva)
h a
i = day
x hub hvA
i
(Zivh)uB = 0
b
(mivh)uB = dB,
= 0,
which show that
Zhlvha
mivh = vi,
= vi,
from which
(1.4)
mivh = vA.
mivh = 0,
From mf = 0, that is, fimh = 0, we find, contracting with vA and taking account of the last equation of (1.4),
ua, we find
On the other hand, from
Zjiua
via'
Zj(di
that is,
(1.5)
Zi
i a
by (1.1) and (1.3). Similarly, we get
- viA)
Vial
382
h= viuA. Ah
(1.6)
mi
If we change ub and uB into ub and uB respectively by orthogonal transformations
ub
cb a'
uB
A'
where
c
= d cb, cc*b
cCCB = 'CB,
then v a and vi are change into va and vA respectively by the rules
-a V.
bb
= cavi,
-A BB vi = cAvi,
and we have vivi = vivi,
vivi = vivi.
Consequently, if we put
v
(1.7)
v i +
a globally defined positive definite Riemannian metric
with respect to which (ub,uB) form an orthogonal frame such that
(1.8)
i a v = aj iua,
A
as we can easily verify it. If we put
Zji =
we find, from (1.5) and (1.6),
i
vj = aj iuA,
mji =
383
mji =
Zji =
because (1.8). Consequently
Zji + mji = aji,
that is, Zji and mji are both symmetric and their scan is equal to aji.
We can easily verify the following relations:
Zji,
0,
mji.
If we put
gji =
(aji +
then gji is again a globally defined positive definite Riemannian metric satisfying
j =
9J-ii
mji = mjgti-
Thus the distributions L and N which were orthogonal with respect to aji are still orthogonal with respect to gji and uA which were mutually orthogonal unit vectors with respect to aji are still mutually orthogonal unit vectors with respect to gji. We can easily verify that gji satisfies
If we put figst = fit-
we have
384
mji = gji
and
mji = -gji.
Hence we have
ft(fit + fti) = 0. The rank of f being r and n-r linearly independent solutions of 0 being given by vt, these equation give
fit + fti = vAW
for certain WA, from which
= 0, or
fit+fti=0 by fituB = fivi = 0 and ftiuB = (ftuB)gji = 0. Thus fji is skewsymmetric tensor of rank r, hence r must be even. Gathering the results,
we have (Yano [4]) THEOREM 1,1, Let M be an n-dimensional manifold with f-structure f of rank r. Then there exist complementary distributions L of dimen-
sion r and N of dimension n-r and a positive definite Riemannian metric g with respect to which L and N are orthogonal and such that
fjfigts + mjgti = gji' fji = -fij,
fji = fjgti
Thus the rank r of f must be even. Take a vector uh in the distribution L, then the vector fhui is also in L and orthogonal to uh, and moreover has the same length as uh with respect to gji. Consequently, we can choose in L r = 2m mutually orthogonal unit vectors ub such that
395
+l -
iul ' um +2 -
i2....... u2m = f um.
Then with respect to the orthogonal frame (ub,uB), the tensors gji and fii have components:
(1.9)
f=
,
g =
IM denoting (m,m)-identity matrix. We call such a frame an adapted frame of the structure f.
Now take another adapted frame (ub,uB) with respect to which 9ji
and fii have the same components as (1.9) and put
h =
nb
-h- Ah
ah
rbua
uB = rBuA,
then vie-can easily see that the orthogonal matrix
m0
A I
r
mm 0
0
1
0
n-2m
must have the form
An
r=
Bm
0
m Am
0
0
0n-2rn
0
396
Thus the group of the tangent bundle of M can be reduced to U(m) x O(n-2n). Conversely, if the group of the tangent bundle of M can be reduced to U(m) x 0(n-2n), then we can define a positive definite Riemannian metric g and a tensor f of type (1,1) and of rank 2m as tensors having (1.9) as corponents with respect to the adapted frames. Tben we have
f3 + f = 0. Thus we have (Yano [4] ) THEOREM 1.2, A necessary and sufficient condition for an n-dimensional manifold M admit an f-structure f of rank r is that r is even (r = 2m) and the group of tangent bundle of M be reduced to the group U(m) x 0(n-2n). In the next place we consider the integrability conditions of the distributions L and N.
The Nijenhuis tensor Nf of f is given by
Nf(X,Y) = [fX,fY] - f[fX,Y] - f[X,fY] - Z[X,Y].
Then we have the following identities:
Nf(mX,mY) = ZNf(mX,mY) = -Z[mX,mY],
mNf(X,Y) = m[fX,fY],
mNf(ZX,ZY) = m[fX,fY],
f(P) = m[ZX,ZY]. Since Z = -f2, If = f, if we have Nf(ZX,ZY) = 0 for all vector fields X and Y, then Nf(fX,fY) = 0 and conversely. We also see that the following three conditions are equivalent:
(i) mNf(X,Y) = 0, (ii) mNf(ZX,ZY) = 0, (iii) mNf(fX,fY) = 0
for any vector fields X and Y. The Lie derivative Lyf is, by definition, given by
387
(LYf)X = f[X,Y] - [fX,Y].
We also have
Nf(ZX,mY) = f(LYf)ZX = f[Z(Lyf)ZX],
from which
fNf(ZX,mY) _ -Z(Lmyf)ZX.
The distribution N is integrable if and only if Z[mX,mY] = 0 for any vector fields X and Y. Thus we have
PROPOSITION 1,1. A necessary and sufficient condition for N to be integrable is that
Nf(mX,mY) = 0
or equivalently
ZNf(mX,mY) = 0
for any vector fields X and Y.
The distribution L is integrable if and only if m[ZX,ZY] = 0 for any vector fields X and Y. Thus we have
PROPOSITION 1.2. A necessary and sufficient condition for the distribution L to be integrable is that one of the conditions (i), (ii) or (iii) be satisfied.
Because of Z + m = I, Nf(X,Y) can be written in the form
Nf(X,Y) = ZNf(ZX,ZY) + mNf(ZX,ZY) + Nf(ZX,mY) + Nf(mX,ZY) + Nf(mX,mY).
Thus we have
388
PROPOSITION 1,3. A necessary and sufficient condition for both of two distributions L and N to be integrable is that
Nf(X,Y) = INf(ZX,ZY) + Nf(ZX,mY) + Nf(nX,ZY)
for any vector fields X and Y. Suppose that the distribution L is integrable and take an arbitrary vector field X' in an integral manifold of L. We define an operator f' by f'X' = fX'. Then f' leaves invariant tangent spaces of every
integral manifolds of L, and f' is an almost complex structure on each integral manifold of L. For any vector fields X' and Y' tangent to integral manifold of L, we denote by N'(X',Y') the vector-valued two form corresponding to the Nijenhuis tensor of the almost complex structure induced on each integral manifold of L from the structure f. Then
we have
Nf(ZX,ZY) = N'(ZX,ZY)
for any vector fields X and Y on M.
If the distribution L is integrable and moreover if the almost
complex structure f' induced from f on each integral manifold of L is integrable, then we say that the f-structure is partially integrable. We have the following theorem.
THEOREM 1.3. A necessary and sufficient condition for an f-structure to be partially integrable is that one of the following equivalent conditions be satisfied:
Nf(ZX,ZY) = 0,
or
Nf(fX,fY) = 0
for any vector fields X and Y on M.
Next, from Propositions 1.1, 1.3 and Theorem 1.3, we have
389
PROPOSITION 1.4. In order that the distribution N is integrable and the f-structure be partially integrable, it is necessary and sufficient that
Nf(X,Y) = Nf(ZX,mY) + Nf(mX,ZY)
for any vector fields X and Y.
PROPOSITION 1.5. The tensor field Z(Lnyf)Z vanishes identically for any vector field Y if and only if
Nf(ZX,mY) = 0
for any vector fields X and Y.
When two distributions L and N are both integrable, we can choose a local coordinate system in such a way that L's are represented by putting n-r local coordinates constnat and N's by putting the other r coordinates constant. Then the projection operators Z and m can be supposed to have the ccagonents
0
0
respectively. Since f satisfies fl = if = Z and fm = mf = 0, f has the components
f =
0
0
where fr is an (r,r)-n atria. Thus, for a vector field mY in M, the
Lie derivative LWf has components
390
LmYf =
Hence, if we assume that the tensor field Z(LmYf)Z vanishes identically for any vector field Y, then we have L Yf = 0, which means that the
m
components of f are independent of the coordinates which are constant along the integral manifold of L in an adapted coordinate system. Conversely, if the components of f are independent of these coordinates,
then we easily see that Z(LmYf)Z vanishes. Thus we have PROPOSITION 1.6. Suppose that the two distributions L and N are both integrable and that an adapted coordinate system has been chosen. A necessary and sufficient condition for the local components of the f-structure to be functions independent of the coordinates which are constant along the integral manifold of L is that
Nf(ZX,mY) = 0
for any vector fields X and Y.
From Propositions 1.3 and 1.6 we have PROPOSITION 1.7. Suppose that L and N are both integrable and that an adapted coordinate system has been choosen. The carponents of f are independent of the coordinates which are constant along the
integral manifold of L if and only if
Nf(X,Y) = INf(ZX,ZY)
for any vector fields X and Y. We now assume the following: (a) f is partially integrable, that is, Nf(X,Y) = Nf(ZX,mY) + Nf(mX,ZY);
391
(b) The distribution N is integrable, that is,
Nf(mX,mY) = 0; (c) The carponents of f are independent of the coordinates which
are constant along the integral manifolds of L in an adapted coordinate system.
In this case we say that the structure f is integrable. Combining Propositions 1.1, 1.7 and Theorem 1.3 we have (Ishihara-Yano [1]) THEOREM 1,4, A necessary and sufficient condition for the structure f to be integrable is that
Nf(X,Y) = 0
for any vector fields X and Y on M.
When the structure f is integrable, the components of f have the
f
foam
01
Or 0
in an adapted coordinate system and fr is an (r,r)-
matrix whose elements are functions independent of the coordinates which are constant along the integral manifolds of L. Since fr defines a complex structure on an integral manifold of L, we can effect a change of adapted coordinate system in such a way that fr becomes 0
fr =
-Im I
M
0
where r = an. The converse being evident, we have (Ishihara-Yano [1]) THEOREM 1.5. A necessary and sufficient condition for an f-structure to be integrable is that there exist a coordinate system in which f has the constant components
0
f=
Im 0
r = an being rank of f.
m 0
0
0
0
-I
0
,
392
2, NORMAL f-STRUCi1RE
Let M be an n-dimensional manifold with an f-structure f. We denote by L and N the distributions corresponding to the projection operators Z and m respectively. We put r = rank f. Then L is r-dimensional and N (n-r)-dimensional. Let U be an arbitrary coordinate neighborhood of M. If we take in M arbitrary an adapted set {fx} of n-r vector fields fx spanning the distribution N at each point, then there exists uniquely in U an ordered set {fy} of n-r 1-forms fy such that 2n-r +l Efx
0 fx
= m,
fx(fy)
= Y
where the indices x,y,z,... run over the range {n+1,n+2,...,2n-r}.
We then have fy(fX) = 0,
ffx = 0
for any vector X at each point of M. We call such an ordered set {fx} an (n-r)-frame and the ordered set {fy} an (n-r)-coframe being dual to {fx}.
If a 1-form 0, global or local, satisfies OX = 0 for any X E L, 4>
is said to be transversal to L. It is expressed uniquely by 0 = 0yfy.
Similarly, any vector field v in N is expressed uniquely by v = v fx. Denoting by fb, fy, fb respectively the components of f, fy, fx with respect to local coordinates (na) in U, where the indices a,b,c,... run over the range {1,2,...,n}. Then we have
a c a_= fbfy y a - db,
fbfc
fafx = O,
ca
fyfc = 0,
fyfx = dy.
393
The set of all tangent vectors belonging to the distribution N > M over M, which is a subbundle
forms a vector bundle p : N(M)
of the tangent bundle T(M) of M. Let N(M)* be the vector bundle dual belonging to the fibre N* of N(M)*
to N(M). If we take an element
at x e M, then there exists at x uniquely a 1-form
of M, which is
transversal to L, such that qv = Ov for any element v belonging to the fibre x of N(M) at x. Conversely, for any 4> transversal to L at x, there exists uniquely
of N* such that cv = Ov for any v c N.
Thus N(M)* can be identified naturally with the set of all 1-forms transversal to L, and hence the bundle N*(M) can be regarded as a subbundle of the cotangent bundle T(M)* of M. In a coordinate neighborhood U, {fx} is a basis of N{ and {fy} is a basis of N*. Let v be a vector field of N and 4, be a transversal 1-form to L. Then we have v = vxfx, 4> = 0f3' in U with functions vx and 0
defined in U. We call
(vx) and (0y) the components of v and p respectively with respect to f
x Let U and U' be twv coordinate neighborhoods of M such that
U a U' # 6. If Ifx) and {fx,} be (n-r)-frames in U and U' respectively, then we have
fy, - AY, y fy
in U A U', where the matrix (Ay,) is a function in U A U'. Taking v mr N and a 1-form 0 transversal to L, we have Oy'fyo
V=vfx,
0 _ 4yfY,
v = vx'fx "
0 =
respectively in U and in U', and
V = AX,v in U A U', where (AX') =
, ,
4y = Ayo0y,
X,)-1
let there be given a connection w* in N(M). Then w* has n(n-r)2 components
y with respect to local coordinates (na) of M and an
c
394
(n-r)-frame { X} in U. Denotong by ray and ra.x0y, the carponents of the given connection w* respectively with respect to {na,fx} and
to
we find in U A U' rc,xvy,
i AX'(r
_
X AY, + 3 Ay.),
an
where ac = mane. Taking a vector X and a 1-form p at a point x of M, we consider an element Tx(X,p) of the fibre FR of the vector bundle
N(M)* 0 N(M) at x and suppose for VX,p) to be bilinear with respect to its arguments X and p. The correspondence s : (X,p)
> x(X,p)
is called an Fr_valued tensor of type (1,1) at x. If there is given
a correspondence T : x
> T{, it is called an N(M)* 0 N(M)-valued
tensor field of type (1,1) and its differentiability is naturally defined. Let v and ¢ be respectively a vector field and a 1-form and T an N(M)* 0 N(M)-valued tensor field of type (1,1). Denote by T(V,c) a cross-section of N(M)* 0 N(M) such that its value at x is given by
Tx(vxOx), where vx and ¢x are respectively the values of v and 0 at x E M. Then T(av,T¢) = aTT(v,q) for any functions a and T. T(X,p) is locally expressed by
2nr T(X,p) _
n
7Ccp Tb,
b,c=1
b
c
Tb =
c
bxfy y 0 f x
x,y n+lTc
with functions Tcbyx defined in U and Xa, ph are respectively canponents
of X and p with respect to (rya). Fbr any vector field v = vxfx E N any transversal 1-form
yfy,
if we put Vcvx = acvx + rcxyvy,
V 0 = achy - rcxyoxI c y
then it is easily verified that (Vcvx)fx and
are globally
defined covariant vector fields in M which take their values respectively in N(M) and,in N(M)*. In this sense, we call
Vcv = Vcvxfx
and
Vc¢ = Vc¢yf
395
or simply Vcvx and Vcy, respectively the covariant derivatives of v and 0 with respect to the connection w*.
Let there be given a linear connection w in M and denote by rc
a b
its components with respect'to local coordinates (r1a) in U. If we
consider now an N(M) 0 N(M)*-valued vector field Ta, then we can put
Ta = Tayxfx 0 fy
in U with components Tayx with respect to {na,fx}. On putting in U
VcTayx
z 7A x ya = acr x + rcabT"y + rc z1 yz - rc y z .
then the tensor field
VcTa = (VcTayx)fx 0 fy
defined in each neighborhood U determines globally in M a tensor field of type (1,1) which takes values in N(M) 0 N(M)*. In this sense, we call VcTa, or VCTayx, the covariant derivative of Ta with respect to connections w and w*.
In the same way, we can define the covariant derivatives of tensor field of any mixed type. Sunning up, if there are given connections w and w* respectively in M and N(M), we can introduce the covariant differentiation Vc operating on tensor fields of any mixed type. In general case, two connections w and w* may be given independently. However, if there is given a linear connection w in M, then there exists in N(M) a connection w* defined by components
rcxy = (2cfy + re dfy)
e,
where rc b are the components of the given linear connection w. By identifying each tangent space of the fibre of N(M) with the fibre itself, the tangent space TQ(N(M)) of N(M) at a e N(M) is
expressible as a direct sum by
396
To(N(M)) = TT(M) ® x = Lk ® Nx ®
x,
x being the point p(a) of M. There exists naturally an identification
Nx
> Fx.
Let there be given a connection w* in the vector bundle N(M). Taking a tangent vector X of the base space M at x, we denote by X*
the horizontal lift of X at each point a of the fibre p1(x) with respect to the connection w*. We define a linear operator a applied to the tangent space a(N(M)) of the manifold N(M) at a point x by
JQ(X*) = (fx)*,
JQ(Y*) = j(Y),
Ja(Z) = -0 -1(Z))*,
where X, Y and Z belong respectively to Lx, N and
x being the
point p(a). It is easily verified that the operator Ja defined in each tangent space Ta(N(M)) determine an almost coaplex structure J in the manifold N(M), i.e., J2 = -I. We shall now give the tensor representation Ji of the almost corrplex structure J. The fibre of N(M) being (n-r)-dimensional vector
space Rn-r, the collection P-1 (U) = U x Rn-r of local product of N(M) over U's forms an open covering of N(M). In p-1 (U), any element v of N(M) such that p(v) c U is expressed by (na,vx), where (na) are coordinates of p(v) and v = vxfx. Any tangent vector of the bundle space N(M) is expressed by
if the tangent space of Rn-r are identified with Rn-r itself. That
is to say, (navx) are local coordinates defined in each p-1 (U) of N(M).
Let there be given a linear connection w* iq N(M) and rcxy its components with respect to local coordinates (na) and (n-r)-frame {fx} in U of M. Then, in the tangent space of N(M) at any point (na,vx) of p 1(U) x Rn-r , the horizontal plane is defined by a linear
39!
equation
Vx + ra`va
(ra' = ra yvy),
and the vertical plane is defined by a linear equation
Va = 0.
If, in each tangent space of N(M), we consider a frame consisting of 2n-r vectors V(i) with carponents V(i) such that
V(b)
cv(b))
(V(y)
V(y)
1V(b)
then V(b) are horizontal and V(y) are vertical, where the indices h,i,j,... run over the range {l,2,...,n,n+l,...,2n-r}. We now define in each tangent space of N(M) a linear operator J by 2n-r
n
J(V(b))
+1
b
(x)'
n faV(a).
J(V(Y)) aE=1
We see that J has the components
(db -rb
0
fb
-fYl
fb
0
a
0 l-1
f db
x b
-r
y
,
-faY fZ'ex x
x yre
398 We can easily verify that
- a 2 0
ab
0
0
ax y
Consequently, we obtain J2 = -I. Hence J is an almost canplex structure in the bundle space N(M). Stunning up, we have
THEOREM 2,1, If a manifold M admits an f-structure f of rank r, then there exist almost complex structures in the bundle space of the (n-r)-dimensional vector bundle N(M) over M. Given a connection with
components Icxy in N(M), then an almost canplex structure J = (J) is determined by (*).
In the next place, we define a tensor field Scba of type (1,2) by
Scba = Ncba + (2cfb -
-
-
abfc)fa
(fcrb u
fur
zc
u)fa,
where Ncba is the Nijenhuis tensor of f, i.e.,
eba = fc2efb - fb2efe - (2cfb - abfc)fe.
We put
Scbx = fe(a e fb
- fb(2ef - 2c e)
2b e)
b
Scat' = fy2efc - fc2efy
+
Scyx = fy(2efC
- 2c e)
Saxy = fx2efy - fy2efx -
fe2cfy
+
- (fefb -
fcfzl e Zy,
+ fcfyrexz - Tcxy'
(fxI'ezy - fylezx)fz
Then we have
Scba + Sbca = 0,
Scbx + Sbcx = 0,
Saxy + Sayx = 0.
exz
In term of covariant differentiation Oc these are expressed respectively as follows:
Scba = Ncba + (V fb - Vbf,)fa,
Scbx = fe
b - Ob
e) - fb(o efc - O c fe
Sa c y = fy e fea - fc e fya + fe cfy , Scyx = fe(oefx - Oce), a
Saxy = fxe Oefy -
fy efx
We denote by Hjih the Nijenhuis tensor of the almost ccrrplex
structure J of N(M), which is given by
Hjih =
J181J - (aji -
with respect to local coordinates (rla,vx) in P 1(U) = U X Rn-r, where nx
we have put
= vx and 2i = Vans. We find
Hcba = Scba - (Tcz3baz - TbzS. z) + - (fe R c
ebz - fb eC z)fa
z
+ (Tczfb ebu
cbx = Scb - (Tczczx - TbzScz ) -
a
Tczrbus
- Tb feR
Scberex
cu)fu,
+ (TczSbez
- TbzScez)Tex - TczTb SzyeTex + (Rcbx - fCb ed )
zfeF UfdR x + (Tczfefd z c edx - T z b - T bzfefd)R
c zb ued
+
(feR
- fbbRecz
z
d-
ze a a a cy = Sc Y + TczSzy + ce fy z
(Tczfz eb -
e
)f;rd
400
Hcy
a
= Sc
a
y
+ T czSzya + R
ce
zfyfa ,
Hcyx = Scyx - Scey"ex + rczSyzerex +
+
fcfy
edx
zfdrd
x + feeR
ed
c z
y eC
,
Hzya=Sazy, Hx zy =
zy e Serx
-
efd - fz yRcd
cbyxVY and Rebyx is the curvature tensor of the linear byx being defined by
where Rcbx
connection rcy, R Rcby
x
x
x
= 8crb y - abrc y + rc
x
z
x
HlihJ
= 0,
z
r - rb zrc y. z b y
Noticing that
Hj.lhJi + HiikJk = 0,
Hi.lhJi -
we obtain
Scb
x
ae
xfey
= Sce fbfa - ey
c b'
Shay = Sdeafbfy - Se zyfbfe, x
= Sdef cy f
Sa
= -S
zy
e
x d e
Scy
f + S zycfxe,
afcfb.
cb z Y
These identities show that
Scba = 0 implies Scbx = 0, Scay = 0, Scyx = 0, Sazy = 0
(cf. Nakagawa [1], Ishihara-Yano [1]).
401
When the almost complex structure J is conplex in the bundle space N(M), we say that the given f-structure f is normal with respect to a connection w* given in the vector bundle N(M). f is normal if and only if Hjih vanishes identically. If Hjih = 0, then
(2.1)
Scba = 0, Scb = 0, Scya = 0, Scyx = 0, S
= 0.
Therefore we have Rcbx = 0, and hence Rcbyx
(2.2)
= 0.
Conversely, if we assume that (2.1) and (2.2), then Hjih = 0.
Therefore, we obtain PROPOSITION 2.1. A necessary and sufficient condition for an f-structure f to be normal in M with respect to a connection w* given in the vector bundle N(M) is that the tensor fields Scba' Scbx' Scay' Scyx' SaZY
vanish identically and the connection w* is of zero curvature. If Scba vanishes, then Scbx, Scay, Scyx, SaZ, are equal identically to zero: Thus we have THEOREM 2.2. A necessary and sufficient condition for an f-structure f to be normal in M with respect to a connection w* given in the vector bundle N(M) is that the tensor field Scba vanishes identically and the connection w* is of zero curvature.
If an f-structure in M is normal with respect to a connection given in N(M), then the connection has zero curvature by means of
Theorem 2.2. Thus, M assumed to be simply connected, the vector bundle N(M) is trivial, that is, it is a product bundle. Therefore, we have PROPOSITION 2.2. If an f-structure in M is normal with respect to a connection given in the vector bundle N(M), then the vector
bundle N(M) induced from N(M) by the covering projection n : M -> M is trivial,- where M is the universal covering space of M.
402
If the vector bundle N(M) is trivial, then it is naturally identi-
fied with the product space M x Rn-r. Then there exists naturally a
connection m of zero curvature in N(M). In this case, a necessary and sufficient condition for f to be normal with respect to WS is
Scba = cba + (2cfb - 8bfe)fa = 0 (cf. Nakagawa [1]).
3. FRAMED f-STRUCTURE
Let M be an n-dimensional manifold with an f-structure f of rank r. If there are n-r vector fields Ea spanning the distribution N at
each point of M, and if there exist n-r differential form na satisfying
a(Eb) = db,
where a,b,c,... = 1,...,n-r, and if
f2=-I+na0Ea, then M is said to have a globally framed f-structure or, simply, a framed structure. We call such a manifold M a framed manifold, and denote it by M(f,Ea,na). We see that
fEa = 0,
0,
a = 1,...,n-r.
A framed manifold M(f,Ea,,a) is called a framed metric manifold if a Riemannian metric g on M is distinguished such that (i) na = a = 1,...,n-r, and (ii) f is skew-symmetric with respect to g. It can be shown that a framed manifold carries a metric with these properties.
We put F(X,Y) = g(fX,Y) and call it fundamental 2-form of the framed
manifold. A framed metric manifold M(f,n,g) a is said to be covariant
constant if the covariant derivatives with respect to g of its structure tensors are zero. For a framed manifold M, if we put
f = f + n2i
n2i-1
® Eli-1 -
0 E
i = 1,...,[(n-r)/2],
,
an almost contact structure f is defined on M if dim M = n = 2m, and an almost contact structure
1) if n = 2m-l. If M is
a framed metric manifold, then an almost complex structure f is defined on M with n = 2m in terms of which the metric g is Hermitian. Setting
F(X,Y) = g(fX,Y), we obtain
F = F +
21n2i
n2i-1
A
i
If the fundamental 2-form F and the na are closed forms, the almost Hermitian structure on M is almost Kaehlerian. It is Kaehlerian if f has vanishing covariant derivative with respect to g, that is, if the structure tensors f and Ea are covariant constant with respect to the metric g.
THEOREM 3.1. A covariant constant even dimensional framed manifold M carries a Kaehlerian structure. M(f,na,g)
In the odd dimensional case the framed metric structure gives rise to the almost contact metric structure M(f,1 - 2m -r-1, g). Then
g(fX,fY) = g(X,Y) -
n2n-r-1(%)n2m-r-1(Y).
We put
_ =
f,
E = Z2m-r-1'
71 = n
2m_r--1
then 2i-1 = F + 21n2i A 'n
,
4(X,Y) = g(fX,Y),
404
If the fundamental 2-form @ and the 1-form n are closed, the almost contact structure on M is almost cosymplectic. It is cosymplectic if and only if the almost contact structure is normal.
THEOREM 3.2. A covariant constant odd dimensional framed manifold M carries a cosymplectic structure.
Proof. Since f and na have vanishing covariant derivatives with respect to the Riemannian metric g, so does 0. Thus we can easily verify that the torsion of 0 vanishes. Therefore, M(q,E,n) is normal. QED.
In the next place vie consider the relation of the normality of
the framed structure and the integrability (normality) of the induced almost complex (almost contact) structure. First of all, we have
LEMMA 3.1. Let M(f,Ea,na) be a framed manifold. Then
X(na(Y)) _ (LXna)(Y) + na([X.Y]).
dna(Eb,X) _ (LEbna)(X),
dna(fX,Y) _ (LfXna)(Y).
Since the f-structure is framed, there exists a connection of
zero curvature in N(M). Then a framed f-structure f is normal if
S=Nf+dna'0Ea vanishes. LEMMA 3.2. Let M(f,Ea,na) be a normal framed manifold. Then
LEnb=0, a LE'`f = 0,
[Ea,Eb] = 0,
dna(fX,Y) + dna(X,fY) = 0
a for any vector fields X and Y and a,b = 1,...,n-r.
405 Proof. Since the structure of M is normal, we have
Nf(X,Y) + dna(X,Y)Ea = 0,
from which
-f[fX,Eb] + f2[X,Eb] + da(XEb)Ea = 0,
that is,
f(
f)X + da(XEb)EzL = 0.
Taking the interior product of the both sides of the equation above nb = 0.
by roc, we obtain dnc(X,Eb) = 0 which is equivalent to I
a
Moreover, we obtain
f2[Ea,Eb] + dnc(Ea,Eb)Ec = f2[Ea,Eb] = 0,
and hence f[Ea,Eb] = 0. Thus, [Ea,Eb] =
for some function dab
on M. Then we get nc([Ea,Eb]) = acab = 0.
We also have fi.E f = 0, so (I M = ua(X)Eb for some function
a
a
)i a(X). Consequently, we get r1b((I
0 = (I,E a
n f))X = ((I,Ear1b)f)X + fb(,Eaf)X), f = 0 for all a.
the ya(X) vanish, that is, l
Next, we find
f)X) = ua(X). Thus, since
a
na([fX.fY]) + dna(X,Y) = 0,
so that
0 = na([fX,f2Y]) + dna(X,fy)
_ -r1a([fX,Y]) + fX(na(Y)) + drla(X, fY) .
0 On the other hand,
have
.
fX(na(Y)) - Y(na(fX)) - na([fX,YI) = dna(fX,Y).
This proves the last equation.
QED.
THEOREM 3.3. Let M(f,Ea,na) be a an-dimensional normal framed manifold of rank r, a = 1,...,2m-r. Then the induced almost complex n2i-1 structure f = f + n2i ® E2i-1 ® E2i on M is integrable. Proof. Let Nf be the torsion tensor of f. By a straightforward
computation, vve find dn21(X,fY))E21 Nf(X,Y) = Nf(X,Y) + dna(X,Y)Ea + (dn2i(fX,Y) + (dn2i-1(fX,Y)
+ dn2i-1(X,fY))E21 + n2i(X)(LE f)Y 2i-1 - n 2i(Y)(LE2i-1 n2i-1 f)X 2i -
2i
E
2i-1].
Our theorem is now a consequence of Lemma 3.2.
QED.
We state the following converse of Theorem 3.3.
THEOREM 3.4. Let M(f,Ea,na) be an even dimensional framed manifold of rank r, a = 1,...,n-r, where induced almost complex structure f = n2i n2i-1 ® Eli-1 ® E2i is integrable. Then, if (a) the dna are of bidegree (1,1) with respect to f, i.e., dna(fX,Y) + dna(X,fY) = 0, (b) the vector fields Ea are holomnrphic and (c) [E2i-1,E2i] = 0, the f-structure is normal. If M(f,Ea,na) is an even dimensional normal framed manifold, dna then the are of bidegree (1,1) with respect to the induced almost
complex structure f. We have
dna(fX,Y) = dna(fX,Y) - r
(X)dna(E2i-1,Y)
+ n2i-1(X)dna(E2i,Y).
407
But, by Lemma 3.2,
dna(Eb,Y) = Eb(na(Y)) - na([Eb,Y])
= %(rja(y))
- a(V )
= Eb(na(Y)) - LEb(na(Y)) = 0.
THEOREM 3,5, Let M(f,Ea,na) be a (2m-1)-dimensional normal framed manifold of rank r, a = 1,...,2m-r-1. Then, the induced almost contact - n2i-1 ® structure f = f + n2i 0 Eli on M is normal. EZi-1 Proof. Let S be the torsion tensor of f. Then
S(X,Y) = [fX,fY] - f[fX,Y] - f[X,fY] - [X,Y]
+ dn2m-r-1(X,Y)EZm-r-1 +
[fX,fY] - f[fX,Y] - f[X,fY] - [X,Y] + (na([X,Y]) + dna(XY))% - (,,r-1([X,Y]) + dn2m-r-1(X,Y))E2r-1 +
(n2m-r-1([X,Y]
+ dn2m-r-1(X,Y))E2m-r-1
= Nf(X,Y) + dna(X,Y)Ea = 0.
Thus,
(f,E2m-r-1'n2m-r-1)
is normal.
QED.
We also have the following converse.
THEOREM 3.6. Let M(f,E,na) be an odd dimensional framed manifold a n2i ® E2 of rank r where induced almost contact structure f = f + ;_1 nai dna 1 0 E2i is normal. Then, if (a) the are of bidegree (1,1) with respect to f, (b) LE f vanishes and (c) [E2i-i,E21] 0, the
=
f-structure is normal.
a
408 4. WYPERSURFACES of FRMIED MANIFOLDS
Let M(?,Ea,na) be a framed manifold of dimension n and rank r, a = 1,...,n-r. We consider an (n-1)-dimensional hypersurface M immersed in d such that: Fbr each x e M the vectors Ea, a = 1,...,n-r-1 belong to the tangent space TX(M) and :!En_r 4 Tx(M). The vector field E
En-r is then an affine normal to M, so we may write
Ix =fX+B(%)E,
TE =0,
where f and B are tensor fields on M of type (1,1) and (0,1), respectively. If B = 0, the submanifold is an invariant hypersurface of M. On the other hand, if 0 # 0, it provides a measure of the deviation of M from this property. Such a hypersurface will be called noninvariant or a normal variation of M. A hypersurface may, of course, be neither invariant or noninvariant. However, in the sequel, unless otherwise specified, M will be a noninvariant hypersurface of the framed manifold M.
First of all, we have -x + na'(X)Ea = f2X + B(fX)E.
Since there are vector fields E on M such that
Ex = Ex,
x = 1,...,n-r-1,
we obtain
f2 = -I + nx 0 Ex,
cB=n,
nx =
nx,
n=n-n-r
cB is the 1-form on M defined by c8(X) = B(fX).
x = 1,...,n-r-1,
409 Moreover, we obtain
x(Ey = nx(Ey) = 6y ,
T1
f X = 0,
e(Ex) = 0,
fEx = fEx + e(Ex)E,
x = 1,...,n-r-1.
THEOREM 4,1. A noninvariant hypersurf ace of a framed manifold
admits a framed structure of the same rank as the ambient manifold. Moreover, it admits a 1-form a determining an (n-r-1)-dimensional distribution complementary to the distribution determined by the Pfaffian system nx = 0, x = 1,...,n-r-1.
If M is integrable, we obtain THEOREM 4.2. A noninvariant hypersurface of a normal framed
manifold M(,Ea,na) is a normal framed manifold of the same rank r carring a 1-form whose differential has bidegree (1,1) with respect to the induced f-structure.
Proof. Given a symmetric affine connection 0 on M, an affine connection V is defined on M with respect to the affine normal E by the Gauss formula
pXY = VXY + h(X,Y)E,
where h is the second fundamental tensor of the immersion with respect to E. We express ST in the form
Sf(XX,Y) = (O
f)Y - (V T)X + 1(DY7f )X - 1(OXf)Y
+ [(OXna)(Y) - (DYna)(X)]Ea
for any vector fields X and Y on M. For any vector fields X and Y on M, by a straightforward computation, we obtain
Sf(X,Y) = Sf(X,Y) + [de(fX,Y) + de(X,fY)JE.
410 From the equation above we have our assertion.
QED.
Let M(f,ja,g) be an n-dimensional framed metric manifold with rank r. Let F be the fundamental 2-form of M defined by F(X,Y) = g(i!,Y). Let F be the fundamental 2-form of the induced f-structure
on M. We then have
F(X,Y) = g(fX,Y) = g(fX,Y) + e(X)j(Y) = F(X,Y) + (e A c6)(X,Y),
that is,
F=F+6Ac6. Since f is not of maximal rank, the tensor field
G=g-c6®c6 is not a Riemannian metric. However, if n = 2m+1 and r = 2m, f is of maximal rank, so G defines a positive definite metric. In this case, it is easily checked that G is Hermitian with respect to f. In fact,
if F is closed, G is an almost Kaehlerian metric and F + 0 A CO is the fundamental 2-form of the almost Kaehlerian manifold M(f,G). If the structure on M is integrable, then M is Kaehlerian.
THEOREM 4.3. In addition to the canonical framed metric structure
(f,rlx,g) the noninvariant hypersurface M(f,Exrix) admits the framed metric structure (f,nx,g*), where g* = g + 6 ® 6.
Proof. We have
g(fX,Y) + 6(X)n(Y) = -g(X,fY) - 6(Y)n(X).
Hence we have g(fX,Y) + 6(X)ce(Y) = -g(X,fY) - e(Y)c6(X),
411
that is,
(g + 6 0 e)(fX,Y) = -(g + 6 0 e)(X,fY).
Moreover, we obtain r1X(X) = g(X,EX) = g*(X,£X) - e(X)e(EX) = g*(X,EX)
W.
412 EXERCISES
A. AUTOMORPHISMS: Let M(f,Ea,rla) and M'(f',Ea,n'a) be framed
manifolds of the same rank. A diffeemorphism p of M onto M' is called an isomorphism of M onto M' if
f' u#
µ*Ea = Ea.
and
If M' = M and f' = f, Ea = .,
,a
=
a, a = 1,...,n-r, then p is
said to be an automorphism of M. Then we have (Goldberg-Yano [2]) THEOREM. The group of automorphisms of a compact framed structure is a Lie group.
B. PRODUCT FRAMED MANIFOLDS: Let M(f,Ea,na), a = 1,...,n-r, and M(f,Ex,nx), x = i.....n-r, be framed manifolds. An f-structure may be defined on the product manifold M x M canonically as follows: For any X E Tp(M) and X e Tp--(M), we put
f(X,%) = (fX,fX).
Then
f2 = -(I,I) + (na e Ea,O) + (O,nx ® Ex),
where 0 is the zero vector and I, I the identity tensors of M, M respectively. Clearly, f3 + f = 0 and f has rank r + I'. We put
Ex)'
Ea = (E a.10),
En-r+x = (0,
na
= (na'0)'
Then
fEA = 0,
A(EB) = dB'
n-r+x
= (0,nx).
413 where A,B =
Thus M x M carries a framed structure
(f,EA,nA) of rank r+r. Then we have (Goldberg [3)) THEOREM. The direct product of two normal framed manifolds is a normal framed manifold.
For the product framed manifolds see also Millman [1] and Nakagawa [1].
C. QUASI-SYMPLECTIC MANIFOLDS: An even dimensional framed metric manifold M(f,Ea,g) of rank r is called quasi-symplectic if the fundamental 2-form F is closed and parallel along the integral curves of
the vector fields Ea. It is symplectic if dim M = 2n and r = 2n. A quasi-symplectic manifold with zero torsion will be called an integral quasi-symplectic manifold. Goldberg [4] proved
THEOREM, The betti number b2q(M) of a conpact integral quasisymplectic manifold M is different from zero for q = 0,1,...,r/2. For the quasi-symplectic manifolds see also Goldberg-Yano [3].
D. HYPERSURFACES: Let M be a hypersurface of a framed manifold M(f,Ea;na). We suppose that M is covariant constant, that is, Of = 0 and ona = 0 for all a. If for any vector field X on M, OXE is propor-
tional to E, then M is said to be totally flat, where E is the affine normal of M. Then we have (Goldberg [2]) THEOREM, A noninvariant hypersurface of a covariant constant
framed manifold is a totally flat covariant constant framed manifold and the connection in the affine normal bundle is trivial. If the hypersurface is invariant, it is also totally geodesic.
414
CHAPTER VIII
PRODUCT MANIFOLDS
In this chapter we give the fundamental results concerning geometry of product manifolds. We also prove some theorems of submanifolds of product manifolds. In §1, we discuss locally product manifolds. §2 is devoted to the study of locally decomposable Riemannian manifolds. We give some properties of Riemannian curvature tensors of locally decomposable
Riemannian manifolds. Moreover, we define an almost product Riemannian manifold. In §3, we discuss st
n folds of almost product Riemannian
manifolds which are invariant with respect to an almost product structure. In the last §4, we consider Ka.ehlerian product manifolds
and its sulinanifolds. For the theory of product manifolds we refer to Tachibana [2]
and Yano [5]. 1. LOCALLY PRODUCT MANIFOLDS
Let us consider an n-dimensional manifold M which is covered by such a system of coordinate neighborhoods (Eh) that in any intersec.
tion of two coordinate neighborhoods
al = Eas(E;a), with
and (E
Cx, _
) we have
e, (E,),
415
IaxEx'I # 0,
Iaaa"I # 0,
where the indices a,b,c,... run over the range 1,2,...,p and the indices x,y,z,... run over p+1,...,p+q=n. Then we say that. the manifold
M admits a locally product structure defined by the existence of such a system of coordinate neighborhoods called separating coordinate system. We call locally product manifold a manifold which admits a locally product structure.
Let vh = (va,vx) are components of a contravariant vector, then
(Va,0) and (0,v) are also components of contravariant vectors. Similarly, if wi = (wb,wy) are components of covariant vectors and also if, for example,
b
Ty
IX
Y
are components of a tensor, then f74b
0
0
10
0
to
Ty
0 x)
are all components of tensors of the same type as T. For vh = (va,v we see that (v ,-vx) are also components of a contravariant vector. This process may be represented by vh
Then we have
F-OFi =
Ihj,
> Five, where
416 where I denotes the, identity tensor, I = 6
.
If we put 0
db
0
0
0
6x
}y,,
_
,
0
Qh =
0
in a separating coordinate system, then Ph is an operator which
projects any vector
vh = (va,vx)
into
Phvl = (va,0)
and Qh is an operator which projects
into
vh = 2), a necessary and sufficient condition that the two components are both Einstein is that the Ricci tensor of the manifold has the form
Rji=agji+bFjl, a and b being necessarily constant.
We next suppose that the two components are both of constant curvature. Then we have
Rdcba = A(gdagcb - geagdb)'
cyxw = u(gzwgyx - gywgzx)
for certain constants X and u. The equations above may also written in the form
Rkjih = *(X+u)[(gkhgji - gjhgki) + (FkhFji - FjhFki)1
+ *U-11)[(F
gji - Fjhgki) + (gkhFji - gjhFki)1.
Conversely suppose that the curvature tensor of a locally decomposable Riemannian manifold has the form
Rkjih = a[(gkhgji - gjhgki) +
(F
Fji
- FjhFki)]
+ b[(Fkhgji - Fjhgki) + (gkhFji - gjhFki)].
Then we have
Rdcba = 2 (a+b) (gd-.gcb - gcagdb) Raw = 2(a-b)(gzwgyx - gywgzx). Thus if p,q > 2, then gcb and gzy are both metrics of manifolds of constant curvature, and 2(a+b) = X. 2(a-b) = u are both constants. Thus we have THEOREM 2,5. In a locally decomposable Riemannian manifold
MP x 0 (p,q > 2), a necessary and sufficient condition that two components are both of constant curvature is that the curvature
tensor of the manifold has the form
Rkjih = a[(gkhgji - gjhgki) + (FkhFji - FjhFki)] + b[(Fkhgji - Fjhgki) + (gkhFji - gjhgki)]'
a and b being necessarily constant. In the following we define an almost product manifold. Let M be an n-dimensional manifold with a tensor field F of type (1,1) such
that
F2 = I. Then we say that M is an almost product manifold with almost product structure F. We put
Q = j(I - F).
p = }(I + F),
Then
P+Q= I,
P2 = P,
Q2 = Q,
PQ=QP=0,
F = P - Q. Thus P and Q define two complementary distributions P and Q globally. We easily see that the eigenvalues of F are +1 or -1. An eigenvector corresponding to the eigenvalue +1 is in P and an eigenvector corresponding to -1 is in Q. Thus, if F has eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q, then the dimension of P is p and that of Q is q.
Conversely, if there exist in M two globally caiplementary distributions P and Q of dimension p and q respectively, where p + q = n and p,q > 1. Then we can define an almost product structure
F on M by F = P - Q. If an almost product manifold M admits a Riemannian metric g such that
g(FX,FY) = g(X,Y)
for any vector fields X and Y on M, then M is called an almost product Riemannian manifold.
424
3. SUBMANIFOLDS OF PRODUCT MANIFOLDS
Let 2 be an m-dimensional almost product Riemannian manifold with structure tensors (F, g). Let M be an n-dimensional Riemannian
manifold isometrically immersed in M. For any vector field X tangent
to Mwe put F% = fX + hX,
where fX is the tangential part of FX and hX the normal part of FX.
For any vector field V normal to M we put
FV = tV + sV,
Where tV is the tangential part of FV and sV the normal part of FV.
We then have
f2X = X - thX,
hfX + shX = 0,
s2V = V - htV,
ftV + tsV = 0.
We easily see that
g(fX,Y) = g(X,fY),
g(fX,fY) = g(X,Y) - g(hX,hY).
If Frx(M) C X(M) for each x E M, then M is said to be Finvariant in M. Then h vanishes identically, and hence f2 = I and g(fX,fY) = g(X,Y). Therefore, (f,g) is an almost product Riemannian structure on M. Conversely, if (f,g) is an almost product Riemannian structure on M, then h = 0 and M is F-invariant in M. Consequently,
we obtain (Adati [1])
43 THEOREM 3,1, Let M be a sutmanifold of an almost product Riemannian manifold M. A necessary and sufficient condition for M to be Finvariant is that the induced structure (f,g) of M is an almost product Riemannian structure.
Let M be a locally decomposable Riemannian manifold, that is, VXF = 0, where V denotes the operator of covariant differentiation in M. We denote by V the operator of covariant differentiation in M with respect to the induced connection on M. If M is F-invariant in
M, then we easily see that F X(MY'c Tx(M)` for each x e M. Then we have
VXFY = F9XY = FDXY + FB(X,Y) = fVXY + sB(X,Y),
VXFY = VXfY = (VXf)Y + fVXY + B(X,fY).
Canparing the tangential and normal parts of these equations, we obtain (Vxf)Y = 0 and sB(X,Y) = B(X,fY). Thus M is locally decomposable.
THEOREM 3.2. Let M be an F-invariant submanifold of a locally decomposable Riemannian manifold M1 x T d-1. Then M is a locally decan-
posable Riemannian manifold M1 x M 1, where M1 is a submanifold of M1 and M-1 is a sutmanifold of R-1, M1 and M 1 being both totally geodesic in M.
Proof. We put
T1(x) = {X c Tx(M) : fX = X},
T 1(x) = {X a T;(M) : fX = -X}.
Then the correspondence of x c M to T1(x) and that to T 1(x) define two distributions T1 and T 1 in M respectively. Let Y e T1. Then for any vector field X tangent to M, we have fVXY = VXfY = VXY and hence VXY E T1. This shows that the distribution T1 is parallel. Similarly, we see that T 1 is also parallel. consequently, the integral manifolds of T1 and T-1 are both totally geodesic in M. We denote than by M1
426
and M 1 respectively. We now stew that M1 is in M1. Let X e T1. Then
QX = J(IX - FX) = J(X - fX) =0. Thus X belongs to the tangent space Tx(M1). Therefore M1 is a sr
M
1
nifold of M1. Similarly we see that
is a submanifold of M 1.
QED.
A sutmanifold M of an almost product Riemannian manifold M is
said to be F-anti-invariant if 7x(M)
X(Mf for each x e M.
THEOREM 3,3. Let M be a submanifold of a locally decomposable Riemannian manifold M. If M is anti-invariant with respect to F,'then AhXY = 0. Moreover, if 2dim M = dim M, then M is totally geodesic. Proof. Let X and Y be vector fields tangent to M. Then
VXFY = OXhY = -Ahy,X + DXhY, OXFY = F7XY = hVXY + tB(X,Y) + sB(X,Y).
Fran these equations we obtain
-g(AhyX,Z) = g(tB(X,Y),Z) = g(B(X,Y),hZ).
Since B is symmetric, we have A. Y = AhYX. Thus we obtain
-g(& ,X,Z) = g(AhZR,Y) = g(AhYX,Z).
Thus we have AhXY = 0. If dim M = 2dim M, the normal space Tx(M)` is spanned by {hX : X E Tx(M)}. Therefore AhgY = 0 means that M is totally geodesic. QED.
Let Sn be an n-dimensional sphere of radius 1, and consider Sn X Sn as an ambient manifold M. We denote by P and Q the projection operators of the tangent space of M to each components SP respectively.
The almost product structure F of M satisfies TrF = 0, where TrF is the trace of F. The Riemannian curvature tensor of M is given by
W (3.1)
R(X,Y)Z = i[g(Y,Z)X-g(X,Z)Y+g(FY,Z)FX-g(AX,Z)AY]
for any vector fields X, Y and Z on M. We can easily see that M is an Einstein manifold.
Let M be a hypersurface of M. We denote by N the unit normal of M in M. We can put
FN=U+XN.
FX=fX+u(X)N,
Then f, u, U and A define a symmetric linear transformation of the tangent bundle of M, a 1-form, a vector field and a function on M respectively. Moreover, we easily see that
g(U,X) = u(X).
The Gauss and Weingarten formulas of M are given by
VXY = OXY + g(AX,Y)N,
VXN = -AX,
where A is the second fundamental tensor of M with respect to N.
The Gauss and Codazzi equations of M are given respectively by
(3.2)
R(X,Y)Z = i[g(Y,Z)X-g(X,Z)Y+g(fY,Z)fX-g(fX,Z)fY] + g(AY,Z)AX - g(AX,Z)AY,
and
(3.3)
(VXA)Y - (VyA)X = J[u(X)fY - u(Y)fX].
Moreover, we obtain
f2X = X - u(X)U, fU = -AU,
u(fX) = -Xu(X),
Trf = -A,
XX = -2u(AX),
u(U) = 1 - A2, VXU = -fAX + AAX.
428
We also have
(VXf)Y = g(AX,Y)U + u(Y)AX,
(VXu)(Y) = Xg(AX,Y) - g(AfX,Y).
We now asstmm that the hypersurface M has constant mean curvature. Then, by a straightforward computation, we have
(3.4)
JAIA12 = -XTrfA2 + Tr(fA)2 + A(TrA)g(AU,U) - (TrAf)2 + 2a(TrA)(TrfA) + g(AU,AU) - 2(TrA)2 - IA12(IA12-(n-1)) + (TrA)(TrA3) + IVA12.
On the other hand, we obtain
(3.5)
div((TrfA)U - fAU) = g(AU,AU) - (TrfA)2 + X(TrA)(TrfA) - (TrA)g(AU,U) + Tr(fA)2 - XTrfA2 + (n-1)(i-X2),
(3.6)
div((TrA)U) = -(TrA)(TrfA) + X(TrA)2.
From these equations we obtain (Ludden-Okumura [1]) THEOREM 3.4. Let M be a hypersurface of SP x Sn. If the mean curvature vector of M is constant, then
2AIA12 - div((TrfA)U - fAU) - Idiv((TrA)U)
= 2(TrA)g(AU,U) - 2(X-1)(TrA)(TrfA) - 2(1+X)(TrA)2 - IA12(1AI2-(n-1)) + (TrA)(TrA3) - (n-1)(1-X2) + JVAI2.
40 In the following we shall give some applications of Theorem 3.4
(Ludden-0kumira [1]). THEOREM 3.5. A compact minimal hypersurface M of SP x SP (n > 1)
satisfying fM[`AI2 - (n-l)]IA12*1 > JMIVAI2*1
is an F-invariant hypersurface. Proof. From the assumption TrA = 0 and hence
fM[IA12(n-1) - 1AI4 - (n-1)(1-A2) + IVAJ2]*1 = 0.
Fran this we see that 12 = 1. Therefore, M is an F-invariant hyper-
surface of SP x e.
QED.
In view of Theorem 3.2 we can prove LE T1A 3.1, A complete F-invariant hypersurf ace M of SP x Sn is a
Riemannian product manifold U' x Sn, where M' is a hypersurf ace of Sn.
Since M' is totally geodesic in M' x SP, the second fundamental form of M' in.Sn has the quite similar properties to A. From this and Theorem 5.8 of Chapter II, we have THEOREM 3.6. The n((m/(n-1))1/2) x Snl(((n-m-1)/(n-1))1/2)xSn in Sn x SP are the only compact F-invariant minimal hypersurf aces of Sn x SP satisfying JAI2 = n-1.
4. SUBMANIFOLDS OF KAEHL.ERIAN PRODUCT MANIFOLDS
In this section we study sibmanifolds of Kaehlerian product manifolds (see Yano-Kon [7]). Let 1m be a Kaehlerian manifold of complex dimension m (of real dimension 2m) and if be a Kaehlerian manifold of caiplex dimension n (of real dimension 2n). We denote by Jm and Jn almost complex structures of SP and Mn respectively. We consider the Kaehlerian product
430
M=K°xMn and put
ix =
JnP7C + JJQX
for any vector field X on M, where P and Q denote the projection operators. Then vm see that
JnPP=PJ, J2 = -I,
Jn@=QJ,
FJ=JF,
g(JX,JY) = g(X,Y),
V$J = 0,
F being an almost product structure on M. Thus J is a Kaehlerian structure on M. If Mm is of constant holocmrphic sectional curvature c1 and Mn is of constant holarnrphic sectional curvature c2, then the Riemannian curvature tensor R of M is given by
(4.1)
R(X,Y)Z =
16(cl+c2)[g(Y,Z)X-g(X,Z)Y+g(JY,Z)JX-g(JX,Z)JY
+2g(X,JY)JZ+2g(FY,Z)FX-g(FX,Z)FY+g(FJY,Z)FJX -g(FJX,Z)FJY+2g(FX,JY)FJZ] + 16(c1 c2)[g(FY,Z)X-g(FX,Z)Y+g(Y,Z)FX-g(X,Z)FY
+ g(FJY,Z)JX-g(FJX,Z)JY+g(JY,Z)FJX-g(JX,Z)FJY +2g(FX,JY)JZ+2g(X,JY)JFZ]
for any vector fields X, Y and Z on M.
We now consider an F-invariant submanifold M of a Kaehlerian product manifold M = On x Mn. Suppose that M is a Kaehlerian suth nifold of M. Since M is F-invariant, M is a Riemannian product manifold
Mp x Mq and MP is a sub mnifold of ff° and Mq is a sutmanifold of Mn. We now show that MP is a Kaehlerian subrmnifold of Mm and Mq is that of Mn. Let X E TX(M#). Then
431
JX = JmfiX + JnQX = Jm C X(
) n T{(M) = TX(Mp).
Therefore MP is a Kaehlerian submanifold of SP. Similarly, Mq is a Kaehlerian submanifold of Mn. Thus we have
THEOREM 4,1, Let M be an F-invariant Kaehlerian sutrrsnifold of a Kaehlerian product manifold M = PP x SP, Then M is a Kaehlerian product manifold MP x Mq, where IF is a Kaehlerian submanifold of Mr0 and Mq is a Kaehlerian su}manifold of Mn.
We next assume that M is an F-invariant, anti-invariant submanifold of M. Then M is the product MP x Mq. Let X E Tx(Mp). Then we have
JX= JmPX+ JnQX= J 11 c TX(M)''.
Since QJX = JnX = 0, we see that QJm = 0. This means that JmX is in Tr(f'). Thus IF is anti-invariant in X . Similarly, Mq is antiinvariant in Mn. THEOREM 4.2. Let M be an F-invariant submanifold of a Kaehlerian product manifold M = Mm x Mn. If M is anti-invariant in M, then M is a Riemannian product manifold MP x Mq, where IF is an anti-invariant
submanifold of M 4JMIVA12*1
is an F-invariant hypersurface.
x CV3
438 THEOREM 2. CPn-1 x C P" and
Q°-1
x CP are the only Cixpact F_
invariant Kaehlerian hypersurfaces of CPn x (PP' with constant scalar
curvature, where Qn-1 is the complex quadric.
For anti-invariant such nifolds we have (Yano-Eon [7] ) THEOREM 3. Let M be a real (m+n)-dimensional (m > n) coapact
F-invariant, anti-invariant minimal suhmanifold of CPn x Cam. If JAI2 = n(n+1)/(2m-1), then m = n = 2 and M is of the form
S1
x
S1 x Rp2 or Rp2 x S1 x S1.
439
CHAPTER IX SUBMERSIONS
In this chapter we study the submersions and its applications to the theory of suh anifolds.
In §1, following O'Neill [2] we give the fundamental equations and some examples of submersions. §2 is devoted to the study of
almost Hermitian submersions (Watson [1]). In the last §3, we discuss the relation between Sasakian manifolds and Kaehlerian manifolds which is a special case of the submersion with totally geodesic fibres. We also consider the relation between submanifolds of Sasakian manifolds and submanifolds of Kaehlerian manifolds. Then the various notions in sulmanifolds of Sasakian manifolds which correspond to that of Kaehlerian manifolds will be clear.
1. FUNDAMENTAL EQUATIONS OF SUBMERSIONS
Let M and B be Riemannian manifolds. A surjective mapping 1r
: M
> B is called a Riemannian submersion (O'Neill [2]) if: (Si) n has maximal rank;
that is, each derivative map n* of n is onto; hence, for each x E B, n 1(x) is a suimanifold of M of dimension dim M - dim B. The sutmani-
folds 7T(x) are called fibres, and a vector field on M is vertical
if it is always tangent to the fibres, horizontal if always orthogonal to fibres; (S2) n* preserves lengths of horizontal vectors.
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For a submersion n : M
B, let H and V denote the projec-
tions of the tangent spaces of M onto the subspaces of horizontal and vertical vectors, respectively. We denote by V the operator of covariant differentiation of M. Following O'Neill [2] we define a tensor field T of type (1,2) for arbitrary vector fields E and F on M by
TEF = EVVE(VF) + WVE(HF).
We shall make frequent use of the following three properties of T: (a) TE is a skew-symmetric operator on the tangent space of M, and it reverses the horizontal and vertical subspaces; (b) T is vertical, that is, TE = TVE;
(c) For vertical vector fields, T has the symmetric property, that is, for vertical vector fields V and W, TVW = TWV.
This last fact, well known for second fundamental forms, follows immediately from the integrability of the vertical distribution.
Next we define a tensor field A of type (1,2) on M by
AEF
= WHE(HF) +
HVHE(VF).
A has the following properties: (a') AE is a skew-symmetric operator on the tangent space of M, and it reverses the horizontal and vertical subspaces; (b') A is horizontal, that is, AE
= AHE; (c') For horizontal vector fields X, Y, A has the alternation
property AXY = -.X. The last property (c') will be proved in the proof of Lemma 1.2. A basic vector field on M is a horizontal vector field X which is n-related to a vector field XX on B, that is, n, Cp = X*,m(p) for
all p E M. Every vector field X* on B has a unique horizontal lift X to M, and X is basic. Thus X
X* is a one-to-one correspondence
between basic vector fields on M and arbitrary vector fields on M.
In the following we prepare same lemmas which give the basic
formulas for submersions.
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LEWIA 1,1, Let X and Y be vector fields on M. Then (1) G(X,Y) = g(X*,Y*).r, where G is the metric tensor field on M and g the metric tensor field on B; (2) H[X,Y] is the basic vector field corresponding to [X*,Y*]; (3) HVXY is the basic vector field corresponding to V* Y*9 X* where v* is the operator of covariant differentiation of B.
Proof. The first assertion (1) follows from (S2), the second from the identity r*[X,Y] = [X*,Y*]. We shall prove (3). For any basic vector fields X, Y and Z on M we have
2G(VXY,Z) = XG(Y,Z) + YG(X,Z) - ZG(X,Y)
- G(X,[Y,Z]) - G(Y,[X,Z]) + G(Z,[X,Y])
Thus we
But, for example, XG(Y,Z) = have
G(VXY,Z).
g(V*X*
Therefore, VXY is r-related to V*X Y*. Thus we have (3).
QED.
LEMMA 1.2. If X and Y are horizontal vector fields on M, then
AXY = JV[X,Y].
Proof. First of all we have
V[X,Y] = VVXY - VVYX = AXY - AyX.
Thus it sufficies to prove the alternation property (c'), or equiva-
lently, to show that AX = 0. We may assume that X is basic, hence that 0 = VG(X,X) = 2G(VVX,X) for any vertical vector field V. Since V is r-related to the zero vector field, we see that [V,X] = VVX-VXV is vertical. Hence
442
G(Vvx,X) = G(Vxv,x) = -G(V,Vxx) = -G(v,AXx).
Since AxX is vertival, the result follows.
QED.
We denote by p the Riemannian connection along a fibre with respect to the induced metric. We see that pVW = vvVW for any vertical
vector fields V and W. We easily see the following LEhMA 1,3, Let X and Y be horizontal vector fields, and V and W
vertical vector fields on M. Then (1) VVW = TvW + VVW,
(2) VVX = HVVX + TVX,
(3) VXV = AxV + IVVXV,
(4) V Y = HVXY + AxY,
(5) if X is basic, HVVX = AxV.
L64 M 1. 4, Let X, Y be horizontal vector fields and V, W be
vertical vector fields on M. Then. (1) (VVA)W = -ATVW,
(VXA)W = -A W,
(2) (VxT )y = -TAXY,
(VVT)y = -TT
Y.
V
Proof. We will only prove (1). Let E be an arbitrary vector field on M. Then
(VVA)WE = VV(AWE) - AVVWE - AW(VVE).
Since A is horizontal, we see that AW = 0. On the other hand, we have
AVVWE = AHVVWE = ATVWE.
Thus we obtain (11.
QED.
443
Furthermore, we prepare the following lemmas. LEMMA 1.5. If X is a horizontal vector field and U, V, W are vertical vector fields, then
G((VUA)XV,W) = G(TUV,AXW) - G(TUW,AXV).
LEMMA 1.6. Let X and Y be horizontal vector fields and V and W be vertical vector fields. Then we have (1) G((VEA)XY,V) is alternate in X and Y; (2) G((VET)VW,X) is symmetric in V and W.
LEMMA 1.7. Let X, Y and Z be horizontal vector fields and V be a vertical vector field. Then we have
G G((VZA)XY,V) = GG(AXY,TVZ),
where G denotes the cyclic scan of over the horizontal vector fields
X, Y and Z. Proof: Since this is a tensor equation, we can assume that X, Y and Z are basix, and even that all three brackets [X,Y],.. are vertical. Thus, Lemma. 1.2 implies that j[X,YJ = AxY. Hence
#G([[X,Y],z],V) = G([AXY,Z],V) = G(VAXYZ,V) - G(VZ(AXY),V).
On the other hand, we have 7,V) = G(TAXYZ,V) = -G(Z,TAxYV) G(VISF
= -G(Z,TV(AXY)) = G(TVZ,AXY).
Thus, using the Jacobi's identity, we find G G(VZ(AXY),V) = SG(TVZ,AXY).
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Thus it remains only to show that SG(VZ(AXY),V) = GG((VZA)XY,V). But
G(VZ(AXY),V) - G((VZA)XY,V) = G(AVZXY,V) + G(AX(VZY),V).
The first term on the right hand side of this equation equals to
-G(Ay(HVg),V),
and since we assume that [X,Z] = 0, this becomes -G(Ay(HVXZ),V), from which the projection H may now be deleted. From this we have our assertion. For a submersion it : M
QED.
> B we now derive the equations
analogous to the Gauss and Codazzi equations of an immersion.
Since we can consider the fibres as sutmanifolds of M, we have the following theorem. Let R be the Riemannian curvature tensor of M and R the Riemannian curvature tensor of the fibre. Then THEOREM 1.1. If U, V, W, F are vertical vector fields and X is
horizontal, then G(R(U,V)W,F) = G(R(U,V)W,F) + G(TUW,TVF) - G(TVW,TUF), G(R(U,V)W,X) = G((VUT)VW,X) - G((VVT)UW,X).
Let R* be the Riemannian curvature tensor of B. Since there is no danger of ambiguity, we denote the horizontal lift of R* by R* as well. Explicitely, if X, Y, Z and W are horizontal vectors of M, we set
G(R*(X,Y)Z,W) = g(R*(n*X,n*Y)r*Z,n*W).
THEOREM 1.2. If X, Y, Z, H are horizontal vector fields and V is vertical, then-,
G(R(X,Y)Z,H) = G(R*(X,Y)Z,H) + 2G(AXY,AZH) - G(AyZ,AXH) + G(AXZ,AyH),
G(R(X,Y)Z,V) = -G((VZA)XY,V) - G(AXY,TVZ) + G(AyZ,TVX) - G(AXZ,TVY).
Proof. Since the two equations are tensor equation, we can assume that X, Y and Z are basic vector fields whose brakets are vertical. Then [X,Y] = 2AxY. We write the basic vector field AVyZ as V*yZ. Then VYZ = V*YZ + A.Z. From IR ma 1.3 we obtain
VXVyZ = V*XV*yZ + AXV*yZ + A
Z + VVX
Z,
V[X,y]Z = 2AZAXY + 2T zZ.
Therefore we have
R(X,Y)Z = VXVyZ - VYVXZ - V[X,Y]Z
= V*XV*YZ - V*yV*XZ + AXAYZ - AyAXZ - 2AZAXY - 2TpyZ + VVXAYZ - VVyAXZ + AXV*yZ - Ayv*XZ
= R*(X,Y)Z + AXAyZ - AyAXZ - 2AZAXY - Zr yZ + V(VXAyZ - VyAXZ) + AXV*yZ - AyV*XZ,
where we have used the fact that A[X,Y] = 0, i.e., n*[X,Y] = 0. Taking the inner product in the equation above with H, we have the first equation. Taking the inner product with V, we obtain
G(R(X,Y)Z,V) = -2G(TYZ,V) + G(VXAYZ,V) - G(V,yAXZ,V) + G(AXVYZ,V) - G(AYVXZ,V).
446
On the other hand, in the proof of Lema 1.7, we have
G(TA_YZ,V) = G(TVZ,AXY).
Moreover, we find
G((VXA)yZ,V) - G((V A)XZ,V) = G(VXAyZ,V) - G(AyVXZ,V) - G(VyAXZ,V) + G(AXVy,Z,V),
because of [X,Y] is vertical. Thus we have
G(R(X,Y)Z,V) = -2g(TVZ,AXY) + G((VXA)YZ,V) - G((VYA)XZ,V).
From this and Lemmas 1.6, 1.7 we have the second equation.
QED.
By the similar argument as that of the preceding theorem we have the following
THEOREM 1.3. If X and Y are horizontal vector fields, and V and W are vertical, then
G(R(X,V)Y,W) = G((VVA)XY,W) - G((VXT)VW,Y) + G(AXV,AYW) - G(TVX,TWY).
We denote by K, K* and k the sectional curvatures of M, B and the fibre, respectively. Then we have THEOREM 1.4. Let X and Y be orthononnal horizontal vectors and V and W be orthonormal vertical vectors. Then
K(V,W) = K(V,W) + G(TVW,TVW) - G(TVV,TWW), K(X,V) = G((VXT)VV,X) + G(AXV,AXV) - G(TVX,TVX), K(X,Y) = K*(n*X,n*Y) - 3G(AXY,AXY).
447
The first equation of Theorem 1.4 is one formulation of the Gauss equation of the fibre. Example 1.1. Let Cn+1 be a ccuplex (n+l)-dimensional number space
with the natural almost complex structure J. We consider a (2n+1)Let N be the outward unit normal of dimensional unit sphere Stn+l Then the integral curves of the tangent vector field in Stn+1.
Cn+l.
JN are great circles in S2n+1 that are the fibres of a bundle mapping : Stn+1 > cpn, & being the complex projective space (see n Example 2.6 of Chapter III). The usual Riemannian structure on CPn is characterized by the fact that n is a submersion. Since the fibres are totally geodesic in S2n+1, the tensor T vanishes. If X and Y are horizontal vector fields, then
AXY = G(X,JY)JN,
AXJN = JX.
Thus we have
G(AXY,JN) = G(X,JY).
Example 1.2. Let G be a Lie group furnished with two-sided invariant Riemannian structure. If K is a closed subgroup, then the usual Riemannian structure on G/K is characterized by the fact that the natural mapping n : G
> G/K is a submersion. The fibres,
left cosets of G mod K, are totally geodesic and hence T = 0. Let X and Y be left invariant horizontal vector fields on G, that is, X and Y be in the orthogonal complement of the Lie algebra of K in
the Lie algebra of G. By Lemna 1.1, AXY = JV[X,Y] is in the Lie algebra of K, and it is known that K(p) = *I[X,Y]I2 for plane section
p spanned by orthonormal vectors X and Y. Then for plane section p* = n*(p) tangent to G/K, Theorem 1.4 implies that
I(p*) = JI[X,Y]I2 +
4IV[X,Y]I2 = JIH[X,Y]I2 + IV[X,Y]I2:
448
Example 1.3. Let F(B) be the frame bundle over B with structure group 0(n). We identify the elements of the Lie algebra o(n) with skew-symmetric matrices, and use the inner product
= -Trace(ab) =
Then there exists a natural Riemannian structure on F(B) such that the projection a : F(B)
> B is a submersion. To define it, let
w be the Riemannian connection form on F(B) taking values in o(n), and let H = Kernelw be the Riemannian connection on F(B). If v is a vertical vector and w(h) = 0, define
IvI = Iw(v)I,
= 0,
IhI = In*(h)I.
Clearly, n is a submersion with H as its horizontal distribution. By a straightforward computation we see that the fibres are totally ,aodesic. Next, we compute A. If X and Y are horizontal vector fields on F(B), then w([X,Y]) = -S2(X,Y), where S2 is the curvature form of B
on F(B). Hence, by Lemma. 1.2, if x and y are horizontal vectors, Axy is the vertical vector such that w(Axy) = -jc2(x,y). If x, y and a vertical vector v are all tangent to F(B) at f = (fit ...,fn), then
_ - = -<w(v),w(Axy)> = i<w(v),SZ(x,y)>
_ i
i IJwij(v)
This implies that Axv is the horizontal vector at f.
2. ALMOST HERMITIAN SUBMERSIONS
Let M be a complex m-dimensional almost Hermitian manifold with Hermitian metric G and almost complex structure J, and B be a ccrrplex
n-dimensional almost Hermitian manifold with Hermitian metric g and almost complex structure J'. We suppose that there exists a submersion
449
IT
: M
> B such that n is an almost cornplex mapping, i.e.,
n*J = J'n*. Then we say that IT is an almost Hermitian submersion. We denote the vertical and horizontal distributions in the tangent bundle of M by V(M) and H(M), respectively. Then T(M) = V(M) 0 H(M). The orthogonal projection mappings are denoted by v : T(M) > H(M) respectively.
> V(M) and H : T(M)
PROPOSITION 2.1. Let n : M
> B be an almost Hermitian
submersion. Then the horizontal and vertical distributions are Jinvariant, i.e., JH(M) = H(M) and JV(M) = V(M). Proof. Since it is almost carplex, JW is vertical for W E V(M).
Let X be in H(M). Then G(JX,W) = -G(X,JW) = 0 and therefore JX is in H(M).
QED.
From Proposition 2.1 we see that any fibre of the submersion it
: M
> B is a complex subrnanifold of M. We notice here that
if X is a basic vector field, then JX is the basic vector field associated to J'X*.
We prove the following (Watson [1)) THEOREM 2.1. Let it : M
> B be an almost Hermitian submersion.
If M is quasi-Kaehlerian, nearly Kaehlerian, Kaehlerian, or Hermitian, then B has the same property. Proof. Let 4) and V be the fundamental 2-forms of M and B respec-
tively. We first claim
= n*(p' on basic vector fields. If X and Y are
basic vector fields on M, and X* and Y* are their associated vector fields on B, then
O(X,Y) = G(X,JY) =
n*4'(X,Y).
Since n* commutes with d on differential forms, we also see that ft = n*(d'@').
If M is almost Kaehlerian, then n*d'4' = 0. Since n* is a linear isometry, we obtain d'@' = 0 and therefore B is almost Kaehlerian. Suppose that M is nearly Kaehlerian. It is easy to see that the
450
basic vector field associated to p'X*J'X. for any vector field X* on B is JVXJX which vanishes on M. Thus B is nearly Kaehlerian. Similarly, we see that if M is quasi-Kaehlerian, then B is quasi-Kaehlerian. bbreover, the basic vector field on M associated to the Nijenhuis tensor N'(X*,Y*) on B is iN(X,Y). Therefore, if M is Hermitian, B is also Hermitian. Furthermore, when M is Kaehlerian, B is Kaehlerian. QED.
Now we can begin to examin haw the almost Hermitian structure on M places restriction on T and A.
LENIA 2.1. Let Tr : M
> B be a quasi-Kaehlerian submersion,
V and W vertical vectors, and X and Y horizontal vectors. Then
(a) TVJW = TJVW,
(b) TJVX = -TVX,
(c) AXJX = 0,
(d) AXJY = -A1JX.
Proof. (a) follows from the similar result on the second fundamental tensor of complex sutmanifolds. To see (b), note that
G(TJVX,W) = -G(TJVW,X) = -G(TVJW,X) = -G(JTVX,W).
Thus we have TJVX = -JTVX. Since M is quasi-Kaehlerian, we ot;.ain
vxJX - VJXX = JVXX + JVJXJX.
Taking the vertical part of this equation, we find
AXJX - AJXX = JAXX + JAJXTX = 0.
Therefore, we obtain AXJX = 0. Assertion (d). follows from (c) by the standard polarization trick. QED.
451
LQ11MA 2.2. Let n : M
> B be a nearly Kaehlerian submersion.
Then, for all vertical vectors V and W,
(a) TVJW = JTVW,
(c) TVJX = JTVX
(b) TJVW = JTVW,
for all horizontal X.
Proof. For (a) and (b) it will be sufficient to show
that
TVJV = JTVV. By a straightforward computation we have
TVJV = VV_' - OV" = (VVJ)V + JVVV - (VVJ)V - JVVV = J(VVV - VVV) = JTVV.
Assertion (c) is an easy calculation.
QED.
Clearly, we have the following
Lem 2.3. Let Tr : M
> B be a Kaehlerian submersion. If V
is a vertical vector field and E is any vector field, then
TVJE = JTVE.
We prove the following theorem (Watson [1]).
> B be an almost Hermitian submersion
THEOREM 2,2. Let n : M
with M, an almost semi-Kaehlerian manifold. Then B is almost semiKaehlerian if and only if the fibres of Ti are minimal subman;folds
of M. Proof. Let {E1,...'Fln-n`TE1,...,JEM-nF,,...,Fn,JF1,...,JFn} be a local J-basis on M whose horizontal vector fields are basic.
Then
n
60(X)
0 =
=
O(X) i=1
[VF O(Fi,X) + VJF 4V(JFi1X)1 , i i
462
where X is a basic vector field and 30(X) is given by
m-n 6O(X) _ -JI [VEJO(Ej-,X) + VJE s(JE.,X)].
On the other hand, we obtain
VF (D(Fi,X) = -G(VF JFi,X) + G(JVF Fi,X)
= -G(HVF JF.,X) + G(HJVF FL,X)
=
VI Fi* (D'(Fi*,X*).
Similarly, we have
VJ. (D(JFi,X) = V'J'F. (D '(J'Fi*,X*).
Therefore
0 = aVX) + a'V(X*).
Since we have d.D
(X) _
m-n I [G((VE J)EJ,X) + G((VJE J)JEj,X)] J j=1 j rn-n
E G(TE JE.-JTE E.-TJE E.-JTE E.,X) j=1 j J
JJ
JJ
JJ
m-n
_ -2 1 G(JT E
j=l
j
d@(X) = 0 if and only if a fibre of n is minimal. Therefore we have our assertion. THEOREM 2.3. Let M be semi-Kaehlerian, and it : M
QED. > B an
almost Hermitian submersion. Then B is sari-Kaehlerian if and only if the fibres of it are minimal.
463 We next consider the integrability of the horizontal distribution of a Kaehlerian submersion (see Johnson [1], Watson [1]). THEOREM 2.4. The horizontal distribution of a Kaehlerian submersion is carpletely integrable. Proof. Let Tr : M
> B be a Kaehlerian submersion, X a basic
horizontal vector field on M, Y horizontal and V vertical. Then
G(AJXY,V) = G(AXJY,V) _ -G(JY,HVVX)
= G(Y,HVVJX) = G(Y,AJXV) = -G(AjXY,V).
Therefore AJXY = 0, and hence AxY = 0. From this we see that the distribution is completely integrable. Moreover, the integral manifold is totally geodesic.
QED.
In the sequel, we consider the Riemannian curvature tensors of an almost Hermitian submersion.
The holomorphic bisectionaZ curvature of an almost Hermitian manifold M is defined for any pair of unit vectors E and F on M by (Goldberg-Kobayashi [1])
B(E,F) = G(R(E,JE)JF,F).
Then the holcmorphic sectional curvature of M is given by H(E) _ B(E,E). We have the following theorems.
THEOREM 2.5. Let v : M
> B be an almost Hermitian submersion.
Let X and Y be horizontal unit vectors, and V and W be unit vertical vectors. Then
B(V,W) = B(V,W) + G(TvJW,TJVW) - G(TVW,TJVJW), B(X,V) = G((VVA)XJX,JV) - G(AXJV,AJXV) + G(AXV,AJXJV) - G((V, A)XJX,V) + G(TjVX,TVJX) - G(TVX,TJVJX),
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B(X,Y) = B' (X*,Y*) - 2G(I JX,A.JY) + G(AJ Y,AXJY) + G(AX,AJJY).
> B is a quasi-Kaehlerian submersion,
THEOREM 2.6. If n : M then
B(V,W) = B(V,W) + ITVJW12 + JTVW12,
B(X,V) = G((VVA)XJX,JV) - G(AJV,AJXV) + G(AXV,AjXJV)
- G((V
1)XJX,V) - 2G(TVX,TjJX),
B(X,Y) = B'(X*,Y*) + IA__N1 2 + JAXYJ2.
From this we have THEOREM 2.7. Let n : M
B be a quasi-Kaehlerian submersion.
Then we have the following: (a) B(V,W) > B(V,W), and equality holds if and only if the fibres of it are totally geodesic;
(b) B(X,Y) > B'(X*,Y*), and equality holds if and only if the horizontal distribution is completely integrable.
We also have the following THEOREM 2,8. If it : M
> B is a nearly Kaehlerian submersion.
then
B(V,W) = B(V,W) + 2ITVWI2, B(X,V) = G((VVA)XJX,JV) - G(AXJV,AJXV) + G(AXV,AJ JV)
- G((VNA)X.JX,V) + 2 JTVXJ 2.
THEORBI 2.9. If n : M
> B is a Kaehlerian submersion, then
B(X,Y) = B'(X*,Y*).
B(X,V) = 2ITVXI2,
an this we obtain THEOREM 2.10. Let n : M
> B is an almost Hermitian submer-
sion. If M is of constant holomorphic bisectional curvature b, then B is of constant holorrorphic bisectional curvature b.
THEOREM 2.11. If n : M
> B is an almost Hermitian submersion,
then
H(V) = H(V) + ITVJVI2 - G(TVV,TJVJV), H(X) = H'.(X*) - 3IAXJXI2.
THEOREM 2.12. If n : M
> B is a quasi-Kaehlerian submersion,
then
H(V) = H(V) + ITVVI2 + ITVJVI2, H(X) = H'(X*).
THEOREM 2.13. If it : M
> B is a nearly Kaehlerian submersion,
then
H(V) = H(V) + 2ITVVI2.
3. SUBMERSIONS AND SUBMANIFOLDS
Let M be a (2m+1)-dimensional Sasakian manifold with structure tensors
such that there exists a fibering n :
Id
> M/!;
= N, where N denote the set of orbits of E and is a real 2m-dimensional Kaehlerian manifold (see §6 of Chapter V). This is a special case of a submersion -with totally geodesic fibres. We denote by (J,g) the
456
Kaehlerian structure of N. We denote by * the horizontal lift with respect to the connection n. Then we have
-G(X*,Y*) = g(X,Y)
(JX) * = q,X*,
g(X,Y)
for any vector fields X and Y on !, where we write
by
the operator of
covariant differentiation with respect to G (resp. g). Then we obtain (see Example 1.1)
(0'XY)* _ -¢2VX*Y* = V Y* - G(OX*Y*,E)E = OX*Y* - G(Y*,$X*)E.
Let R and R' be the Riemannian curvature tensors of R and N respectively. Then we have PROPOSITION 3.1. The Riemannian curvature tensors R and R' satisfy
(R'(X,Y)Z)* = R(X*,Y*)Z* + G(Z*,¢Y*)OX* -
2G(Y*,4,X*)OZ*
for any vector fields X, Y and Z on M. Proof. First of all we obtain
(0'XV'yZ)* = OX*(0'YZ)* - G((0'YZ)*,$X*)C = OX*DY*Z* -
G(Z*,OoX*Y*)C
+ G(Z*,4)Y*)4)X* - G(0Y*Z*,4)X*)E.
From this we have t4 similar expression of (D'YO'XZ)*. Moreover, we have
0'MY]Z)* = V[X*,Y*]Z* + 2G(Y*,$X*)4Z* - G(Z*,[X*,Y*))l;.
Fran these equations we have the equation of proposition.
QED.
Let S and S' be the Ricci tensors of M and N respectively. Then Proposition 3.1 implies
S'(X,Y) = S(X*,Y*) + 2G(X*,Y*).
(3.1)
Therefore, the scalar curvature r of M and the scalar curvature r' of
N satisfy (3.2)
r'
+ 2m.
Moreover, the sectional curvatures of M and N determined by orthonormal vectors X and Y on N satisfy
K'(X,Y) = K(X*,Y*) + 3G(X*,g*)2.
(3.3)
Especially, we have
R'(X,JX) = K(X*,cX*) + 3.
Thus we have THEOREM 3.1, M is of constant 4-sectional curvature c if and only if N is of constant holomorphic sectional curvature (c+3) Let M be an (n+l)-dimensional suYmanifold tangent to the structure vector field ( of M and N be an n-dimensional sutrmanifold of N.
Throughout in this section we assume that the diagram
M
1
>f
N
1
>N
canmutes. let V (resp. V') be the operator of covariant differentiation
458
in M (resp. N). We denote by the same G and g the induced metric tensor fields on M and N respectively. We denote by B (resp. B') the second fundamental form of the immersion i (resp. i') and the associated second fundamental tensors of B and B' will be denoted by A and A'
respectively. Let X and Y be vector fields tangent to N. Then we have the following Gauss formulas:
V'XY = V'XY + B'(X,Y)
and
pX*Y* = OX*Y* + B(X*,Y*).
Therefore we obtain
(V'XY)* + (B'X,Y))* = -02VX*Y* + B(X*,Y*).
Comparing the tangential and normal parts of this equation, we have respectively
(V'XY)* = -c2VX*Y* = VX*Y* -
(3.4)
(3.5)
(B'(X,Y))* = B(X*,Y*).
Let D and D' be the operators of covariant differentiation with respect to the linear connection induced in the normal bundles of M and N respectively. For any vector field X tangent to N and vector field V normal to N, we obtain
V'XV = -A'VX + D'XV
and
VX*V* = -AV*X* + DX*V*.
Thus we have
-(A'VX)* + (D'XV)* = 2AV,.X* + DX*V*,
from which
(3.6)
(A'VX)* = -$2AV*X* =
n(AV*X*)E,
4W (D'XV)* = DX*V*.
(3.7)
For any vector field X tangent to M we put
4)X = PX + FX,
where PX is the tangential part of OX and FX the normal part of qX. Similarly, for any vector field V normal to M, we put
OV=tV+fV, where tV is the tangential part of 4V and fV the normal part of V (see 93 of Chapter IV). We can define the operators P', F', t' and f' on N corresponding respectively to P, F, t and f (see §4 of Chapter VI). Then we have
(P'X)* = PX*,
(F'X)* = FX*
for any vector field X tangent to N. Moreover, we obtain
(t'V)* = tV*,
(f'V)* = fV*
for any vector field V normal to N. Thus we have PROPOSITION 3.2. (1) M is a contact CR submanifold of M if and only if N is a CR sulmanifold of N;
(2) M is a generic sutmanifold of M if and only if N is a generic sutmanifold of N;
(3) M is an anti-invariant submanifold of rd tangent to C if and only if N is an anti-invariant sub manifold of N; (4) M is an invariant submanifold of M if and only if N is an invariant submanifold (a complex sulmanifold) of N. We now study the relation between covariant differentiations of the second fundamental forms B and B'. From (3.4), (3.5) and (3.7)
460
we have
((VXB')(Y,Z))* - G(Y*,4X*)BQ,Z*) -G(Z*,4X*)B(Y*,E) = (VX*B) (Y*, Z*).
On the other hand, we see that
VXE = -PX,
B(X,O = -FX,
0.
Thus we obtain
(3.8)
(VX*B)(Y*,Z*) = [(VXB')(Y,Z)+g(Y,P'X)F'Z+g(Z,P'X)F'Y]*.
Moreover, from (3.7) of Chapter IV, we obtain
(VX*B)(Y*,E) = (VX*F)Y* - B(Y*,PX*) = fB(X*,Y*) - B(X*,PY*) - B(Y*,PX*),
from which
(3.9)
(VX*B)(Y*,E) = [f'B'(X,Y)-B'(X,P'Y)-B'(Y,P'X)]*.
From (3.8) and (3.9) we see that B is parallel if and only if B' satisfies
(VXB')(Y,Z) = g(X,P'Y)F'Z + g(X,P'Z)F'Y
and
f'B'(X,Y) = B'(X,P'Y) + B'(Y,P'X).
This is the proof of Lemma 3.1 of Chapter IV.
'E6I In the next place, we consider the normal connections of M and N. We denote by K` and R` the curvature tensors of the normal bundles of
M and N respectively. We give the relation of Kl and R. Let X and Y be vector fields tangent to N and U and V be vector fields normal to N. Then (3.7) implies
(D'XD'YV)*
DX*DY*V*,
(D'YD'XV)* =
DY*DX*V*.
Since [X,Y]* = [X*,Y*] - 2G(Y*,PX*)E, we find
(D'[X,Y]V)* = D[X* Y*]V* - 2G(Y*,PX*)DV*.
From these equations we have
(3.10)
G(KL(X*,Y*)V*,U*) = [g(R'(X,Y)V,U)+2g(Y,P'X)g(f'V,U)]*.
On the other hand, the Ricci equation of M implies
G(KL(X*,E)V*,U*) = G([AU*,AV*]X*,E) = G(AV*X*,tU*) - G(A. X*,tV*) _ -[g(F'A'VX,U) + g(B'(X,t'V),U)]*.
From this and (3.9) of Chapter IV we obtain
(3.11)
G(KL(X*, )V*,U*) = g((VXf')V,U)*.
Consequently, the normal connection of M is flat if and only if
R1(X,Y)V = 2g(X,P'Y)f'V
and
(y, )V = 0.
This gives the proof of Leama 3.2 of Chapter IV.
462
PROPOSITION 3.3. Let M and N be invariant suhmanifolds of M and N respectively. Then the second fundamental form B of M is n-parallel if and only if the second fundamental form B' of N is parallel.
Proof. From the assumption we see that F' = 0. Thus (3.8) shows that (VX*B)(Y*,Z*) = ((VXB')(Y,Z))*. From this we have our assertion. QED.
When M is anti-invariant in M, the normal connection of M is
pseudo-flat if and only if K(X*.Y*)V* = 0 (see Exercise E of Chapter VI). Thus, fran (3.10), we have PROPOSITION 3.4, If M and N are anti-invariant submanifolds of R and N respectively, then the normal connection of M is pseudo-flat if and-only if the normal connection of N is flat.
When M is a generic submanifold of M, f vanishes identically, and hence f' on N vanishes identically. Therefore, we have PROPOSITION 3.5. If M and N are generic submanifolds of M and IN respectively, then the normal connection of M is flat if and only if the normal connection of N is flat.
3.1. Let u and u' be the mean curvature vectors of M and N respectively. Then
n+l
Proof. If we take an orthonormal basis {ei} for Tx(N). Then {et,E} forms an orthonormal basis for Ty(M) (a(y) = x). Thus (3.5)
implies (u')* = n(TrB')* = 1[EB(et,et) + B(E,E)] = 1TYB = nnl u.
In view of Learna 3.1 we obtain
PROPOSITION 3.6. M is a minimal sutmanifold of M if and only if N is a minimal submanifold of N.
MK From (3.7) and Lemma 3.1 we have DX*u
n+1
*(u')*
nnl(D'XU')*.
Therefore we have PROPOSITION 3.7. If the mean curvature vector u of M is parallel, then the mean curvature vector u' of N is also parallel. We consider the converse of this proposition. We prove the follow-
ing LE4 A 3,2, Let M be a sutmanifold tangent to E of a Sasakian
manifold M. Then DEu = -fu.
Proof. We first notice that the second fundamental form B satis-
fies (VXB)(E,Y) = (VB)(X,Y). Then we have
(n+1)DEu = J(VEB)(ei,ei) = J(ve B)(E.ei) i
= J[ D B(E,ei) - B(Ve E.ei)) = -j(Ve F)ei i
= -7.[-B(ei,Pei) + fB(ei,ei)]
In the above we have denoted by the same ei local, orthonormal vector fields on M which extend ei of the orthonormal basis {ei} and which
are covariant constant with respect to V at x of M.
QED.
From Lemma 3.2 we have PROPOSITION 3.8. Let M and N be generic submanifolds of &'S and N
respectively. Then the mean curvature vector U.of M is parallel if and only if the mean curvature vector p' of N is parallel.
464
When M is invariant in M, the mean curvature vector p of M is pseudo-parallel if and only if DX*v = 0 (see Exercise E of Chapter VI). Thus we have PROPOSITION 3.9. Let M and N be anti-invariant suYmanifolds of M and N respectively. Then the mean curvature vector p of M is pseudoparallel if and only if the mean curvature vector u' of N is parallel. Let R and R' be the Riemannian curvature tensors of M and N respectively. From Proposition 3.1 we obtain
(R'(X,Y)Z)* - (A'B,(Y Z)X)* + (A'B'(X Z)Y)* ((OXB')(Y,Z))* - ((VyB)(X,Z))*
= R(X*,Y*)Z* - AB(Y*,Z*)X* + %(X*,Z*)Y* + (oX*B)(Y*,Z*) - (VY*B)(X*,Z*) + G(Z*,PY*)¢X* - G(Z*,PX*)OY* - 2G(Y*,PX*)¢Z*.
Taking the tangential part of this equation and using (3.6), we obtain PROPOSITION 3.10. The Riemannian curvature tensors R and R' satisfy
(R'(X,Y)Z)* = R(X*,Y*)Z* + G(Y*,tB(X*,Z*))E - G(X*,tB(Y*,Z*))E + G(Z*,PY*)PX* - G(Z*,PX*)PY* - 2G(Y*,PX*)PZ*.
We denote by S and S' the Ricci tensors of M and N respectively. Then we have
(3.12)
S'(X,Y) = S(X*,Y*) + 2G(PX*,PY*).
Let {ei} be annorthonormal basis of N. Then, by (3.12), the scalar curvature r of M and the scalar curvature r' of N satisfy
40 (3.13)
r' = r + YG(Pet,Pet),
r' = r + n - IG(Fei,Fei).
Therefore we obtain the following PROPOSITION 3.11, (1) M and N are anti-invariant sub manifolds;
if and only if r = r';
(2) M and N are invariant sukmanifolds if and only if r = r' - n. We now ccapute the square of the length of the second fundamental forms. We obtain
G(A,A) _
2JG(B(et,E),B(et,E)) i
g(B'(ei,e),B'(ei,ej)) + 21G(Fet,Fet)
= g(A',A') + 21g(F'ei,F'ei), i
that is,
(3.14)
1A12 = JA1I2 + 21g(F'ei,F'ei) IA'12 + 2n - 21g(F'ei,F'ei).
i
From (3.14) we have PROPOSITION 3,12. (1) M and N are invariant sutmanifolds if and only if 1A12 = JA'12;
(2) M and N are anti-invariant sutmanifolds if and only if JA!2 = IA'12 + 2n. As an application of submersions with totally geodesic fibre, we consider sulmanifolds of a complex projective space (see Lawson [1]). THEOREM 3.2. Let N be an n-dimensional compact minimal sutmanifold
of &n. If the scalar curvature r' of N satisfies the inequality
p
r' > n(n+2) - 2-i
(p = 2m-n),
466
then M is a totally geodesic eamplex projective space
CPn/2.
S2m+1 Proof. We consider the fibering n : > CPm. By S2m}1 Proposition 3.6 the submanifold N of is minimal. Thus the scalr curvature r of N is given by
r = n(n+1) - G(A,A).
From this and (3.13) we obtain
G(A,A) = n(n+2) - r' - JG(Fet,Fei) < n(n+2) - r'. i
Therefore, by the assumption, we have
G(A,A) < n+l - 2-1/p
Hence Theorem 5.4 of Chapter II implies that A = 0 or JA{2= S2m+l (n+l)/(2-1/p). If A = 0, then M is totally geodesic in and hence N is a totally geodesic complex projective space
Cpn/2.
If
JA12 = (n+1)/(2-1/p), then F = 0, which shows that M is an invariant S2m+1. Moreover, from Theorem 5.8 of Chapter II, M is sub manifold of Sn+2.
a Veronese surface in S4 or a Clifford minimal hypersurface in S2m+1 But the ambient manifold is odd dimensional and any hypersurface S2m+1 of is not invariant. From these considerations we have our
assertion.
QED.
467 EXERCISES
A. POSITIVE RICCI CURVATURE: Nash [1] construct complete metrices of positive Ricci curvature on a large class of fibre bundles. THEOREM. Let it
: M -> B be a vector bundle over B, a compact
manifold admitting a metric of positive Ricci curvature. If the fibre dimension is greater than two, M admits a carrplete metric of positive Ricci curvature.
B. FIBRE BUNDLES AND SUBMERSIONS: A mapping f : M
> B of
Riemannian manifolds is said to be totally geodesic if for each geodesic xt in M the image f(xt) is a geodesic in B. Vilms [1] proved THEOREM 1. A Riemannian submersion is totally geodesic if and only if the fibres are totally geodesic sulznanifolds and the horizontal
subbundle is integrable. THEOREM 2. Let IT : M
> B be a fibre bundle with standard
fibre F and Lie structure group G. Assume the bundle admits a connection in the sense of Ehresman [1]. Endow B and F with Riemannian metrices, and assume F is G invariant. Then there exists a natural metric on M such that it is a Rienannian submersion with totally geodesic fibres.
We also have (Hermann [1], Muto [1], Nagano [1])
THEOREM 3. Let it : M
> B be a Riemannian submersion, and
assume M to be connected. If M is complete, so is B, and n is locally trivial fibre space. If, in addition, the fibres are totally geodesic, then it is a fibre bundle with structure group the Lie group of iso-
metries of the fibre. For a geodesic we have (Hermann [1], O'Neill [3]) THEOREM 4. Let it : M
> B be a Riemannian submersion. If x
is a geodesic of M which is borizontal at one point, then it is always
horizontal, and hence
is a geodesic of B.
40 C. EQUIVALENCE PROBLEM: Let Trl and
n2
be Riemannian submersions
from a complete M onto B. Assume the fibres of Trl and Tr2 are totally geodesic. Trl and Tr2 are said to be equivalent provided there exists
an isometry f of M which induces an isometry f' of B, so that the
following diagram commutes: M Trl
> M
f
I
1112 tl
.
B
>B
Escobales [1] proved THEOREM 1. Let Trl and r2 be Riemannian submersions from M onto B
satisfying the above hypotheses. Suppose f is an isometry of M satisfying the following two conditions: (1) f*p : ylp
> y2f(p) is an isometry from the horizontal
distribution yip of Trl at p onto the horizontal distribution 92f(p) of ?2 at f(p);
(2) For E, F E Tp(M), f*(AIEF) = A2f*Ef*F, where Ai are the integrability tensors of Tri.
Then f induces an isometry f' of B so that Trl and Tr2 are equiva-
lent. O'Neill [2] gives the following
THEOREM 2, Let Trl and 72 be submersions of a Riemannian manifold M and B. If Tr1 and Tr2 have the same tensors T and A, and if their derivation maps agree at one point of M, then Tr1 = 7r2. D. SUBMERSIONS WITH TOTALLY GEODESIC FIBRE: Escobales [1] gives the classification those B for which there is a Riemannian submersion TT
: SP
desic.
> B, where Sn is a square and the fibres are totally geo-
0 > B be a Riemannian submersion with
THEOREM 1. Let n : Sn
connected totally geodesic fibres, and assume 1 < dim fibre < n-i. Then, as a fibre bundle, n is one of the following types:
(a)
S1
S2m+1
(b)
>
S3
(c)
(e)
S1
S7
HPm
for n > 2
> S3
>
S4m+3
In
lIT
Cpm
>
(d)
S3
for n > 2
> S7
In
In
S2(i)
S4(1)
S15
In case (a) and (b), B is isometric to complex and quaternion projective space with sectional curvature 1 < K, < 4. In case (c), (d) and (e), B is isometric to a sphere of curvature 4.
Moreover, if
1 and
_
2 are say two submersions both in class
(a), (b) or Cc), then nl is equivalent to n2.
Escobales [2) proved the following
> B with connected complex
THEOREM 2, Any submersion it : CPn
totally geodesic fibres and with 2 < dim fibre < 2n-2 must fall into one of the following two classes:
(a)
S2
>
Cpl lIT BPFv
(b)
CP3
> CP7 lIT
S8()
In fact, 1 < K < 4, where K,4 denotes the curvature of BPFv, and S8( ) denotes the sphere of radius J. Moreover, class (a) is not empty. Finally, if n > 2, any two subversions in class (a) are equivalent.
41U
E. SUBMERSION WITH MINIMAL FIBRE: When a submersion has minimal fibres we obtain
THEOREM 1. Let n : M
> B be a Riemannian submersion with
minimal fibres. Then a closed hypersurface P of B has constant mean
curvature in B if and only if
a-1(P)
has constant mean curvature in
M.
We also have (see Escobales [21) THEOREM 2. Let it : M
> B be a Riemannian submersion with
totally geodesic fibres. Assume P is a totally geodesic submanifold of B. Then n-1(P) is totally geodesic provided AxY = 0 whenever Y is horizontal and tangent to n 1(P) and X is normal to n_1 (p).
For the submersion with totally umbilical fibres see Bishop [1]. F. REAL SUBMANIFOLDS AND SUBMERSIONS: Let 1r
:
> Nn be
Mn+1
a submersion with 1-dimensional fibre. Then the fundamental tensor
P of the submersion is a skew symmetric tensor of type (1,1) on N defined by
Vx*Y* = (0'XY)* + g(PY,X)*V,
where V is a unit vertical vector. Then Okumura [1] proved the following
THEOREM, Let Nn be a hypersurface of CP(n+l)/2 and
:
it
n+1
_
-> Nn the submersion which is compatible with the fibration > Sn+2 > CP (n+l)/2. In order that the second fundamental Si tensor A' of Nn commutes wit:, the fundamental tensor P of the submersion it, it is necessary and sufficient that the second fundamental Mn+1 tenor A of is parallel.
G. QUATERNION KAEHLERIAN MANIFOLDS AND SASAKIAN 3-STRUCTURE: Let it
: M
> B be a Riemannian submersion. Then we have the following
(see Ishihara-Konishi [1])
471
THEOREM. (1) If M admits a K-contact 3-structure, then B admits an almost quaternion structure;
(2) If M admits a Sasakian 3-structure, then B admits a quaternion Kaehlerian structure; (3) If M admits a Sasakian structure with constant curvature c,
then c = 1 and the induced quaternion Kaehlerian structure of B is of constant Q-sectional curvature 4.
473
BIBLIOGRAPHY The following bibliography includes all those references that are quoted in the text. The bibliography also contains books and
papers on the topics closely related to the content of our treatises. Adati, T.
[1] Sulxnanifolds of an almost product Riemannian manifold,
Kodai Math. J. 4 (19810t 327-343.
Ako, M. [1] Submanifolds of a Kaehlerian manifold, Kodai Math. Sem. Rep. 20 (1968), 1-11.
Alexander, S. and Maltz, R.
[1] Isometric immersions of Riemannian product in Euclidean space, J. Differential Geometry 11 (1976), 47-57. Ambrose, W. and Singer, I. M.
[1] A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428-443.
Auslander, L. [1] Four dimensional compact locally Hermitian manifolds, Trans. Amer. Math. Soc. 84 (1957), 379-391.
Bejancu, A. [1] CR submanifolds of a Kaehler manifold I, Proc. Amer. Math. Soc. 69 (1978), 135-142. [2] CR submani.folds of a Kaehler manifold II, Trans. Amer. Math.
Soc. 250 (1979), 333-345.
[3] Umbilical CR-submanifolds of a Kaehler manifold, Rendiconti di Mat. (3) 15 serie VI (1980), 431-446. Bejancu, A., Kon, M. and Yano, K.
[1] CR-submanifolds of a complex space form, J. Differential Geometry 16 (1981), 137-145.
474
Berger, M.
[1] Sur les variet'es d'Einstein compactes, C. R. IIIe Reunion
Math. Expression Latine, Namur (1965), 35-55. Bishop, R. L.
[1] Clairaut submersion, Differential Geometry, in Honor of K. Yano, Kinokumiya, Tokyo, 1972, 21-31. Bishop, R. L. and Crittenden, R. J. [1] Geometry of Manifolds, Academic Press, New York, 1964. Bishop, R. L. and Goldberg, S. I. [1) On the topology of positively curved Kaehler manifolds II, TBhoku Math. J. 17 (1965), 310-318. Blair, D. E.
[1] Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer Verlag, 1976. Blair, D. E. and Chen, B. Y. [1] On CR-submanifolds of Hermitian manifolds, Israel J. Math. 34 (1979), 353-363. Bochner, S.
[1] Vector field and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776-797.
Boothby, W. M. and Wang, H. C. [1] On contact manifolds, Ann. of Math. 68 (1958), 721-734. Calabi, E.
[1] Isometric imbedding of complex manifolds, Ann. of Math. 58 (1953), 1-23.
Calabi, E. and Eckmann, B. [1] A class of compact, complex manifolds which are not algebraic, Ann. of Math. 58 (1953), 494-500.
Cartan, E. [1] Lecons sue les Invariants Integraux, Hermann Paris, 1922.
Cecil, T. E. and Ryan, P. J. [1] Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), 481-499.
475
Chen, B. Y.
[1] Geometry of Sulmanifolds, Marcel Dekker, INC. New York,1973.
[2] CR-sukmanifolds of a Kaehlerian manifold. I, J. Differential Geometry 16 (1981), 305-322.
[3] CR-sutmanifolds of a Kaehler manifolds. II, J. Differential Geometry 16 (1981), 493-509. Chen, B. Y. and Ogiue, K. [1] Sane characterizations of complex space forms, Duke Math. J. 44 (1973), 797-799.
[2] On totally real submanifolds, Trans. Amer. Math. Soc. 193 (1974), 257-266.
[3] Some characterizations of complex space forms in terms of Chern classes, Quart. J. Math. ?6 (1975), 459-464. Chern, S. S.
[1] Einstein hypersurfaces in a Kaehlerian manifold of constant holanorphic curvature, J. Differential Geometry 1 (1967), 21-31.
Chern, S. S., Do Carme, M. and Kobayashi, S.
[1] Minimal sutmanifolds of a sphere with second fundamental form of constant length, Functional analysis and related fields, Pros. Conf. in Honor of Marshall Stone, Springer, Berlin, 1970.
Chevalley, C.
[1] Theory of Lie Groups, Princeton Univ. Press, 1946.
Ehresmann, C. [1] Les connexions infinitesimales duns un espace fibre differentiable, Coll. de Tbpologie, Bruxelles, 1950, 29-55.
Ejiri, N. [1] Tbtally real minimal
of n-dimensional real space
forms into n-dimensional complex space forms, Proc. Amer. ,Math. Soc. 84 (1982), 243-246.
476
Endo, H.
[1] Invariant siftnanifolds in a K-contact Riemannian manifold, Tensor N. S. 28 (1974), 154-156. Erbacher, J.
[1] Reduction of the co imension of an isometric immersion, J. Differential Geometry 5 (1971), 333-340. [2] Isometric im,nersions of constant mean curvature and triviality
of the normal connection, Nagoya Math. J. 45 (1971), 139-165. Escobales, Jr., R. H. [1] Riemannian submersions with totally geodesic fibers, J.
Differential Geometry 10 (1975), 253-276. [2] Riemannian submersions from complex projective space, J. Differential Geometry 13 (1978), 93-107. Fialkow, A.
(1] Hypersurfaces of a space of constant curvature, Ann. of Math.
39 (1938), 762-785. Fukanu, T.
[1] Affine connections in almost product manifolds with some structure, Tohoku Math. J. 11 (1959), 430-446.
Fukami, T. and Ishihara, S. [1] Almost Hermitian structure on S6, T6hoku Math. J. 7 (1955), 151-156. Funabashi, S.
[1] Totally real submanifolds of a quaternionic Kaehlerian manifold, K6dai Math. Sem. Rep. 29 (1978), 261-270. [2] Totally complex sutrnanifolds of a quaternionic Kaehlerian
manifold, Kodai Math. J. 2 (1979), 314-336. Goldberg, S. I.
[1] Curvature and Homology, Academic Press, New York, 1962.
[2] Normal variations of invariant hypersurfaces of framed manifolds, Canad. Math. Bull. 15 (1972), 513-521. [3] Framed manifolds, Differential Geometry, in honor of K. Yano, Kinokuniya, Tokyo, 1972, 121-132.
477
Goldberg, S. I. [4] A generaralizatiion of Kaehler geanetry, J. Differential Geometry 6 (1972), 343-355.
[5] On conformally flat space with definite Ricci curvature II, Kodai Math. Sam. Rep. 27 ( 1976), 445-448.
Goldberg, S. I. and Kobayashi, S. [1] Holomorphic bisectional curvature, J. Differential Geometry 1 (1967), 225-233.
Goldberg, S. I. and Oku ura, M. [1] Conformally flat manifolds and a pinching problem on the Ricci tensor, Proc. Amer. Math. Soc. 58 (1976), 234-236.
Goldberg, S. I. and Yano, K. [1] Non invariant hypersurfaces of almost contact manifolds, J.
Math. Soc. Japan 22 (1970), 25-34.
[2] On normal globally framed f-manifolds, Tbhoku Math. J. 22 (1970), 362-370.
[3] Globally framed f-manifolds, Illinois J. Math. 15 (1971), 456-474.
Gray, A.
[1] Vector cross products on manifolds, Trans. Amer. Math. Soc. 141 (1969), 465-504.
[2] Nearly Kaehler manifolds, J. Differential Geometry 4 (1970), 283-309.
[3] The structure of nearly Kaehlerian manifolds, Math. Ann. 223 (1976), 233-248.
[4] Compact Kaehler manifolds with nonnegative sectional curvature, Inventions Math. 41 (1977), 33-43.
Gray, J. W. [1] S ne global properties of contact structures, Ann. of Math. 69 (1959), 421-450.
Harada, M.
[1] On Sasakian suhTmifolds, Tbboku Math. J. 25 (1973), 103-109.
478
Harada, M.
[2] On Kaehler manifolds satisfying the axiom of antiholomcrphic 2-spheres, Proc. Amer. Math. Soc. 43 (1974), 186-189. Hatakeyama, Y.
[1] Some notes on differentiable manifolds with almost contact structure, l6boku Math. J. 15 (1963), 176-181. Hawley, N. S.
[1] Constant holomorphic curvature, Canad. J. Math. 5 (1953), 53-56.
Helgason, S.
[1] Differential Geometry and Symmetric Spaces, Academic Press,
New York, 1962. Hermann, R.
[1] A sufficient condition that a mapping of Riemannian manifolds be a fiber bundle, Proc. Amer. Math. Soc. 11 (1960), 236-242. [2] On the differential geometry of foliations, Ann. of Math. 72 (1960), 445-457. Hodge, W. V. D.
[1] Theory and Applications of Harmonic Integrals, Cambridge Univ. Press, London, 1941; 2 nd ed. 1951. Hong, S. L.
[1] Isometric immersions of manifolds with plane geodesics into
Euclidean space, J. Differential Geometry 8 (1973), 259-278. Hopf, H.
[1] Zur Topologie der kcaplexen Mannigfeltigkeiten, Studies and Essays presented to R. Courant, Interscience, New York, 1948, 167-185.
Hopf, H. and Rinow, W. [1]
Uber den Begriff der vollstandigen differentialgeometrischen 11
Flachen, Comm. Math. Helv. 3 (1931), 209-225. Igusa, J.
[1] On the structure of a certain class of Kaehler manifolds, Amer. J. Math. 76 (1954), 669-678.
47
Ikawa, T. and Kop, M.
[1] Sasakian manifolds with vanishing contact Bochner curvature tensor and constant scalar curvature, Oo11. Math. 37 (1977), 113-122.
Ishihara, I.
[1] Kaehler su1manifolds satisfying a certain condition on normal bundle, Atti della Accad nia Nazionale dei Lincei LXII (1977), 30-35.
[2] Anti-invariant sutzmanifolds of a Sasakian space form, Kodai Math. J. 2 (1979), 171-186. [2] Anti-invariant sub manifolds satisfying a certain condition
on normal connection, Kodai Math. J. 2 (1979), 371-383. Ishihara, S.
[1] Normal structure f satisfying f3 + f = 0, Kodai Math. Sem. Rep. 18 (1966), 36-47.
[2] Quaternion Kaehlerian manifolds and fibred Riemannian spaces with Sasakian 3-structure, Kodai Math. Sem. Rep. 25 (1973), 321-329.
[3] Quaternion Kaehlerian manifolds, J. Differential Geometry 9 (1974), 483-500.
Ishihara, S. and Konishi, M.
[1] Differential Geometry of Fibred Spaces, Publications of the study group of geometry, Tokyo, Vol. 8, 1973.
Ishihara, S. and Yano, K. [1]
On integrability conditions of a structure f satisfying f3 + f = 0, Quart. J. Math. 15 (1964), 217-222.
Itoh, T.
[1] Positively curved complex suhmanifolds inmersed in a ccn lex space form, Proc. Amer. Math. Soc. 72 (1978), 341-345. Itch, T. and Ogiue, K.
[1] Isotropic immersions, J. Differential Geometry 8 (1973), 305-316.
Johnson, D. L.
[1] Kaehler submersions and holomorphic connections, J. Differential Geometry 15 (1980), 71-79.
480
Kashiwada, T.
[1] A note on a Riemannian space with Sasakian 3-structure, Nat. Sci. Rep. Ochancmizu Univ. 22 (1971), 1-2. [2] On the induced almost contact structure of a Brieskorn manifold, Gecmetriae Dedicata 6 (1977), 415-418. Kenmotsu, K.
[1] Invariant sulmanifolds in a Sasakian manifold, T3boku Math.
J. 21 (1969), 495-500. [2] On Sasakian immersions, Sem. on Contact Manifolds, Pub. of the study group of geometry 4 (1970), 42-59. [3] Local classification of invariant n-Einstein submanifolds of codimension 2 in a Sasakian manifold with constant ct-sectional curvature, Tohoku Math. J. 22 (1970), 270-272.
[4] Some remarks on minimal sutmanifolds, Tohoku Math. J. 22 (1970), 240-248.
Ki, U. H. and Pak, J. S. [1] Generic submanifolds of an even-dimensional Euclidean space, J. Differential Geometry 16 (1981), 293-303. Ki, U. H., Pak, J. S. and Kim, Y. H. [1] Generic sutmanifolds of complex projective spaces with parallel mean curvature vector, Kodai Math. J. 4 (1981), 137-155. Kobayashi, S.
(1] Principal fibre bundles with 1-dimensional toroidal group, Tohoku Math. J. 8 (1956), 29-45. [2] Hypersurfaces of complex projective space with constant scalar curvature, J. Differential Geometry 1 (1967), 369-370.
Kobayashi, S. and Nanizu, K. [1] Foundations of Differential Geometry, Vol. I, Interscience
Publishers, 1963. [2] Foundations of Differential Geometry, Vol. II, Interscience
Publishers, 1969.
a9 Kon, M.
[1] Invariant subnanifolds of normal contact metric manifolds, Kodai Math. Sem. Rep. 25 (1973), 330-336. [2] Kaehler immersions with trivial normal connection, TRU Math. 9 (1973), 29-33. [3] On some invariant submanifolds of normal contact metric manifolds, Tensor N. S. 28 (1974), 133-138. [4] On invariant sub=manifolds in a Sasakian manifold of constant
4-sectional curvature, TRU Math. 10 (1974), 1-9. [5] On sane complex submanifolds in Kaehler manifolds, Canad. J. Math. 26 (1974), 1442-1449.
[6] Totally real minimal submanifolds with parallel second fundamental form, Atti della Accademia Nazionale dei Lincei 57 (1974), 187-189.
[7] Caxplex sutmanifolds with constant scalar curvature in a Kaehler manifold, J. Math. Soc. Japan 27 (1975), 76-81. [8] Invariant sutmanifolds in Sasakian manifolds, Math. Ann. 219 (1975), 277-290.
[9] Kaehler immersions with vanishing Bochner curvature tensors, Kodai Math. Sen. Rep. 27 (1976), 329-333. [10] Remarks on anti-invariant sutmanifolds of a Sasakian manifold, Tensor N. S. 30 (1976), 239-246.
[11] Totally real sutmanifolds in a Kaehler manifold, J. Differential (Banetry 11 1(1976), 251-257.
[12] On hypersurfaces immersed in S2n}1, Ann. Fac. Sci. de Kinshasa 4 (1978), 1-24. [13] Pseudo-Einstein real hypersurfaces in complex space forms, J. Differential Geometry 14 (1979), 339-354. [14] Real minimal hypersurfaces in a ccuplex projective space, Proc. Amer. Math. Soc. 79 (1980), 285-288. [15] Generic minimal sutmanifolds of a ccmplex projective space, Bull. London Math. Soc. 12 (1980), 355-360.
482 Kubo, Y.
[1] Kaehlerian manifolds with vanishing Bochner curvature tensor,
Kodai Math. Sem. Rep. 28 (1976), 85-89. Kuo, Y. Y.
[1] On almost contact 3-structure, 'It oku Math. J. 22 (1970),
325-332.
Kuo, Y. Y. and Tachibana, S. [1] On the distribution appeared in contact 3-structure, Taita J. of Math. 2 (1970), 17-24. Lawson, Jr., A. B. [1] Rigidity theorems in rank-1 symmetric spaces, J. Differential
Geometry 4 (1970), 349-357. Lichnerowicz, A. [1] Theoremes de reductiblite des varietes kahleriennes et
applications, C. R. Acad. Sci. Paris 231 (1950), 1280-1282. [2] Sur les forms harmoniques des varibtes riemanniennes localement rlductibles, C. R. Acad. Sci. Paris 232 (1951), 1634-1636. [3] Sur la reductibilite des espaces hanog6nes riemanniens,
C. R. Acad. Sci. Paris 243 (1956), 646-672. [4] Gean6trie des Groupes de Transformations, Dunod, Paris, 1958. Ludden, G. D.
[1] Sutmanifolds of cosymplectic manifolds, J. Differential
Geometry 4 (1970), 237-244. Ludden, G. D. and Okumura, M. [1] Some integral formulas and their applications for hypersurfaces of Sn x sn, J. Differential Geometry 9 (1974), 617-631.
Ludden, G. D., Okumura, M. and Yano, K. [1] A totally real surface in CP2 that is not totally geodesic,
Proc. Amer. Math. Soc. 53 (1975), 186-190. [2] Totally real sutznanifolds of complex manifolds, Atti dells Accademia' Nazionale dei Lincei LVIII (1975), 346-353.
M Matsumoto, M.
[1] On 6-dimensional almost Tachibana spaces, Tensor N. S. 23 (1972), 250-252.
[2] A remark on 10-dimensional almost Tachibana spaces, Bull. of St. Marianna Univ. 9 (1980), 1-11. [3] On 10-dimensional almost Tachibana spaces, Bull. of St. Marianna Univ. 10 (1981), 91-100.
Matsumoto, M. and Chuman, G. [1] On the C-Bochner curvature tensor, TRU Math. 5 (1969), 21-30. Matsumoto, M. and Tanno, S. [1] Kaehlerian spaces with parallel or vanishing Bochner curvature tensor, Tensor N. S. 27 (1973), 291-294.
Matsushima, Y. [1] Differentiable Manifolds, Marcel Dekker, Inc. New York, 1972. Matsuyama, Y.
[1] Invariant hypersurfaces of SP X Sn with constant mean curvature, Hokkaido Math. J. 5 (1976), 210-217. [2] Couplex hypersurfaces of Pn(C) x Pn(C), fl hoku Math. J. 28 (1976), 541-552. [3] Minimal sutinanifolds in SN and RN, Math. Z. 175 (1980),
275-282. Millman, R. S. [1] f-structures with parallelizable kernel on manifolds, J. Differential Geometry 9 (1974), 531-535. Moore, J. D.
[1] Isometric i versions of Riemannian products, J. Differential Geometry 5 (1971), 159-168.
Morimoto, A. [1] On normal almost contact structures, J. Math. Soc. Japan 15 (1963), 420-436.
[2] On normal almost contact structure with a regularity, 16boku Math. J. 16 (1964), 90-104.
Morrow, J. and Kodaira, K. [1] Complex Manifolds, Bolt, Rinehart and Winston, Inc. New York, 1971.
484 Muto, Y.
[1] On some properties of a fibred Riemannian manifold, Sci. Rep. Yokohama Nat. Univ. Sect. I, 1 (1952), 1-14. Myers, S. B. [1] Riemannian manifolds with positive mean curvature, Duke Math.
J. 8 (1941), 401-404.
Nagano, T. [1] On fibered Riemannian manifolds, Sci. Papers College Gen. Ed. Univ. Tokyo 10 (1960), 17-27. Nakagawa, H.
{1] On framed f-manifolds, Kodai Math. Sem. Rep. 18 (1966), 293-306.
[2] Einstein-Kaehler manifolds imnersed in a complex projective space, Canad. J. Math. 28 (1976), 1-8. Nakagawa, H. and Ogiue, K. [1] Complex space forms unversed in complex space forms, Trans. Amer. Math. Soc. 219 (1976), 289-297.
Nakagawa, H. and Takagi, R. [1] On locally symmetric Kaehler submanifolds in a complex projective space, J. Math. Soc. Japan 28 (1976), 638-667. [2] Kaehler submanifolds with RS = 0 in a complex projective space, Hokkaido Math. J. 5 (1976), 67-70.
Nakagawa, H. and Yokote, I. [1] Corpact hypersurfaces in an odd dimensional sphere, K4dai Math. Sem. Rep. 25 (1973), 225-245.
N:Lsh, J. C. [1] Positive Ricci curvature on fibre bundles, J. Differential Geometry 14 (1979), 241-254. Newlander, A. and Nirenberg, L. 1] Complex analytic coordinates in almost complex manifolds, Ann. of Math. 65 (1957), 391-404.
[1] Lie Groups and Differential Geometry, Publ. Math. Soc.
Japan, 1956.
es Nomizu, K.
[2] On hypersurfaces satisfying a certain condition of the curvature tensor, 76boku Math. J. 20 (1968), 46-59. [3] On the rank and curvature of non-singular complex hypersur-
faces in a complex projective space, J. Math. Soc. Japan 21 (1969), 266-269.
[4] Some results in E. Cartan's theory of isoparametric families of hypersurfaces, Bull. Amer. Math. Soc. 79 (1973), 1184-1188. [5] Conditions for constancy of the holomorphic sectional
curvature, J. Differential Geometry 8 (1973), 335-339. Nomizu, K. and Smyth, B. [1] Differential geometry of complex hypersurfaces II, J. Math. Soc. Japan 20 (1968), 499-521 [2] A formula of Simons' type and hypersurfaces with constant
man curvature, J. Differential Geometry 3 (1969), 367-377. Ogawa, Y.
[1] A condition for a compact Kaehlerian space to be locally
symmetric, Nat. Sci. Rep. Ochanonizu Univ. 28 (1977), 21-23. Ogiue, K.
[1] Positively curved complex suYmanifolds immersed in a complex projective space, J. Differential Geometry 7 (1972), 603-606. [2] Positively curved complex submanifolds immersed in a ccuplex projective space II, Hokkaisdo Math. J. 1 (1972),16-20. [3] On Kaehler immersions, Canad. J. Math. 24 (1972), 1178-1182.
[4] n-dimensional complex space forms immersed in {n+n(n+1)/2}dimensional complex space forms, J. Math. Soc. Japan 24 (1972), 518-526.
[5] Differential geometry of Kaehler submanifolds, Advances in Math. 13 (1974), 73-114. [6] Positively curved totally real minimal subnranifolds immersed
in a complex projective space, Proc. Amer. Math. Soc. 56 (1976), 264-266.
485
Oku u ra, M. [1] On sane real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355-364. O'Neill, B. [1] Isotropic and Kaehler immersions, Canad. J. Math. 17 (1965), 907-915. [2] The fundamental equations of a submersion, Michigan Math. J.
13 (1966), 459-469. [3] Submersions and geodesics, Duke Math. J. 34 (1967), 459-469. Palais, R. S. [1] A Global Formulation of the Lie Theory of Transformation Groups, Mem. Amer. Math. Soc. 22, 1957. Reinhart, B. ...[l] Foliated manifolds with bundle-like metrics, Ann. of Math.
69 (1959), 119-131. de Rham, G.
[1] Sur la reductibilit6 d'un espace de Riemann, Comm. Math.
Helv. 26 (1952), 328-344. [2] Variete's Differentiables, Hermann, Paris, 1955.
Ryan, P. J. [1] Homogeneity and some curvature conditions for hypersurfaces,
Tohoku Math. J. 21 (1969), 363-388. [2] A class of complex hypersurfaces, Coll. Math. 26 (1972), 175-182.
Sakamoto, K. [1] Complex sutmanifolds with certain condition, Kodai Math.
Sem. Rep. 27 (1976), 334-344. [2] Planar geodesic immersions, Tohoku Math. J. 29 (1977), 25-56. Sasaki, S.
[1] A characterization of contact transformations, T6hoku Math.
J. 16 (1964), 285-290. [2] Almost contact manifolds, Lecture Notes, T&ioku Univ. Vol. 1, 1965, vol. 2, 1967, vol. 3, 1968.
487
Sasaki, S. and Hsu, C. J.
[1] On a property of Brieskorn manifolds, T6boku Math. J. 28 (1976), 67-78.
Sawaki, S., Watanabe, Y. and Sato, T.
[1] Notes on a K-space of constant holomorphic sectional curvature, K&lai Math. Sen. Rep. 26 (1975), 438-445. Schouten, J. A. and Struik, D. J.
[1] On some properties of general manifolds relating to Einstein's theory of gravitation, Amer. J. Math. 43 (1921), 213-216. Schur, F. iI
t,
II
[1] Uber den Zu samnenhang der Raune konstanter Kruarmungsnasses mit den projectiven Raumen, Math. Ann. 27 (1886), 537-567. Sekigawa, K.
[1] Almost Hermitian manifolds satisfying some curvature conditions, Kodai Math. J. 2 (1979), 384-405. Simons, J.
[1] Minimal varieties in riemannian manifolds. Ann. of Math. 88 (1968), 62-105.
S,nyth, B. [1] Differential geometry of complex hypersurfaces, Ann. of Math: 85 (1967), 246-266. [2] Homogeneous complex hypersurfaces, J. Math. Soc. Japan 20 (1968), 643-647.
[3] Submanifolds of constant mean curvature, Math. Ann. 205 (1973), 265-280. Spencer, D. C.
[1] Some remarks on pertubation of structure, Analytic Functions, Princeton University Press, 1960, 67-87. Steenrod, N.
[1] Topology of Fibre Bundles, Princeton Univ. Press, 1951. Sternberg, S.
[1] Lectures on Differential Geometry, Printice-Hall, Englewood Cliffs, New Jersey, 1964.
me Stong, R. E.
(1] The rank of an f-structure, Sodai Math. Sm. Rep. 29 (1977), 207-209.
Tachibana, S.
(1] On almost-analytic vectors in certain almost-Hermitian manifolds, Tohoku Math. J. 11 (1959), 351-363.
[2] Some theorems on locally product Riemannian spaces, TBhoku Math. J. 12 (1960), 281-292. [3] On infinitesimal conformal and projective transformations of compact K-space, Tohoku Math. J. 13 (1961), 386-392.
Tachibana, S. and Yu, W. N. (1] On a Riemannian space admitting more than one Sasakian Structure, T8hoku Math. J. 22 (1970), 536-540. Takagi, H.
[1] An example of Riemannian manifolds satisfying R(X,Y)R = 0 but not VR = 0, T8hoku Math. J. 24 (1972), 105-108. Takagi, R.
[1] Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975), 43-53.
[2] Real hypersurfaces in a complex projective space with constant principal curvatures II, J. Math. Soc. Japan 27 (1975), 507-516.
[3] A class of hypersurfaces with constant principal curvatures in a sphere, J. DIfferential Geometry 11 (1976), 225-233.
Takahashi Takao [1] Deformations of Sasakian structures and its application to the Brieskorn manifolds, T8hoku Math. J. 30 (1978), 37-43. Takahashi Toshio [1] Sasakian 0-symmetric spaces, Tohoku Math. J. 29 (1977), 91-113 Takahashi Toshio and Tanno, S. [1] K-contact Riemannian manifolds isanetrically immersed in a space of constant curvature, T8hoku Math. J. 23 (1971), 535-539.
0 Takahashi Tsunero
[1] Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385. [2] Hypersurface with parallel Ricci tensor in a space of
constant holomrphic sectional curvature, J. Math. Soc. Japan 19 (1967), 199-204. [3] Complex hypersurfaces with RS = 0 in Cn*1, Tohoku Math. J. 25 (1973), 527-533. Takamatsu, K.
[1] Same properties of 6-dimensional K-spaces, Kodai Math. Sem. Rep. 23 (1971), 215-232. Takamatsu, K. and Sato, T.
[1] A K-space of constant holomorphic sectional curvature, Kodai Math. Sem. Rep. 27 (1976), 116-127. Tanno, S.
[1] Locally symmetric K-contact Riemannian manifolds, Proc. Japan Acad. 43 (1967), 581-583. [2] Isometric immersions of Sasakian manifolds in spheres, Kodai Math. Sem. Rep. 21 (1969), 448-458. [3] Hypersurfaces satisfying a certain condition on the Ricci tensor, 7Uhoku Math. J. 21 (1969), 297-303. [4] Sasakian manifolds with constant c-holomorphic sectional curvature, Tohoku Math. J. 21 (1969), 501-507. [5] 2-dimensional complex submanifolds immersed in complex projective spaces, T8hoku Math. J. 24 (1972), 71-78. [6] Compact complex submanifolds immersed in complex projective spaces, J. Differential Geometry 8 (1973), 629-641. [7] Constancy of holomorphic sectional curvature in almost Hermitian manifolds, Kodai Math. Sem. Rep. 25 (1973), 190-201. Tanno, S. and Takahashi Toshio [1] Some hypersurfaces of a sphere, Tehoku Math. J. 22 (1970), 212-219.
Tashiro, Y. and Tachibana, S.
[1] On Pubinian and C-Bibinian manifolds, Kodai Math. Sem. Rep. 15 (1963), 176-183.
490
Thomas, T. Y.
[1] On closed spaces of constant mean curvature, Amer. J. Math. 58 (1936), 702-704. [2] Extract from a letter by E. Cartan concerning my note: On
closed spaces of constant mean curvature, Amer. J. Math. 59 (1937), 793-794.
Verheyen, P: and Verstraelen, L. [1] Conformally flat totally real sutmanifolds of complex projective spaces, Soocbow J. Math. 6 (1980), 137-143. Waerden, R. [1] Untermannigfaltigkeiten mit paralleler zweiter Fundamental form in Euklidishen Raumen and Spharen, Manuscripts Math. 10 (1973), 91-102. Walker, A. G. [1] Connexions for parallel distributions in the large I, II,
Quart. J. Math. 6 (1955), 301-308; 9 (1958), 221-231. Watanabe, Y. and Takamatsu, K. [1] On a K-space of constant holomorphic sectional curvature,
Kodai Math. Sm. Rep. 25 (1973), 297-306. Watson, B.
[1] Almost Hermitian submersion, J. Differential Geometry 11 (1976), 147-165.
Wells, Jr., R. 0. [1] Canpact real sutrnanifolds of a complex manifold with non-
degenerate holcmorphic tangent bundles, Math. Ann. 179 (1969), 123-129.
[2] Function theory on differentiable manifolds, Contribution to analysis, Academic Press, 1974, 407-441. Weyl, H.
[1] Reine Infinitesimalygeometrie, Math. Z. 26 (1918), 384-411.
[2] Zur Infiniteimalgeametrie: Einordnung der projektiven and der konformen Auffassung, Gottingen Nachr. 1921, 99-112. Willmore, T. J.
[1] Connexion for system of parallel distributions, Quart. J Math. 7 (1956), 269-276.
491
Yamaguchi, S., Chiman, G. and Matsumoto, M. [1] On a special almost Tachibana space, Tensor N. S. 24 (1972), 351-354.
Yamaguchi f S. and Kon, M. [1] Kaehler manifolds satisfying the axiom of anti-invariant 2-spheres, Geometriae Dedicata 7 (1978), 403-406.
Yano, K. [1] On harmonic and Killing vector fields, Ann. of Math. 55 (1952), 38-45.
[2] The Theory of Lie Derivatives and its Applications, North-
Holland, Amsterdam, 1957. [3] Affine connections in an almost product space, Kodai Math. Sem. Rep. 11 (1959), 1-24.
[4] On a structure defined by a tensor field f of type (1,1) satisfying f3 + f = 0, Tensor N. S. 14 (1963), 99-109. [5] Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, New York, 1965.
[6] Integral Formulas in Riemannian Geometry, Marcel Dekker, Inc, New York, 1970.
[7] Manifolds and sub
