1. The Osserman Conditions in
Semi-Riemannian Geometry
chapter we introduce the basic notation and terminology which t...
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1. The Osserman Conditions in
Semi-Riemannian Geometry
chapter we introduce the basic notation and terminology which throughout this book. In Section 1.1, we give the definition of Jacobi
In this
are
used
op-
erator and its relation to curvatures. In Section
1.2,
we
define the timelike and
spacelike Osserman conditions at a point and show their equivalence. Also by using this equivalence, we give the definition of Osserman condition at a point. In Section 1.3, we give the definitions of semi-Riemannian pointwise and global Osserman manifolds and a sufficient condition for a pointwise Osserman manifold to be globally Osserman is studied under some assumptions on the number of different eigenvalues of the Jacobi operators. In this section, we also give some model semi-Riemannian global Osserman manifolds. Throughout this chapter, let (M,g) be a semi-Riemannian manifold of dimension n > 2 and index v. That is, g is a metric tensor on M with v. n signature (v,,q), where q =
1.1 The Jacobi First
we
fix
some
Operator
notation and
Definition 1. 1.1. Let vector
z
E
TpM
-
(M, g)
terminology. be
a
semi-Riemannian
manifold. A
nonzero
is called:
a) timelike if g(z, z) < 0, b) spacelike if g(z, z) > 0, c) null if g(z, z) 0, d) nonnull if g (z, z) i4 0. =
Also
a
Let
nonnull vector
z
is called
a
unit vector
and S,, (M) be the sets TpM, respectively. That is,
S; (M), SP+ (M)
nonnull vectors in
S;(M) S+(M) P Sp(M)
=
Iz
E
TpMlg(z,z)
=
-1},
=
Iz
E
TpMlg(z,z)
=
1},
=
Iz
E
TpM1 I g(z, z) 1= 1}
E. García-Río et al: LNM 1777, pp. 1 - 20, 2002 © Springer-Verlag Berlin Heidelberg 2002
if I g(z, z) 1= of unit
=
timelike, spacelike and
Sp- (M)
U
Sp+(M).
1. The Osserman Conditions in Semi-Riemannian
2
(M, g)
Note that if not
(that is,
is not Riemannian
v
Geometry
$ 0, n)
then
Sp (M)
is
compact.
Definition 1. 1. 2. Let
U
a) S_(M)
(M, g)
Sp (M)
be
Iz
=
semi-Riemannian
a
TMlg(z,z)
E
manifold,
-1}
=
Then
is called the unit
PEM
of (M,g).
timelike bundle
+
U
b) S+ (M)
S., (M)
Iz
E
TMlg(z, z)
1}
=
is called the unit space-
PEM
like bundle
of (M, g).
U
c) S(M)
Iz
Sp(M)
TMI I g(z, z) 1= 1}
E
is called the unit
PEM
of (M, g).
nonnull bundle Now
we are
ready
(M, g) be nection of (M, g). Let
a
to define the Jacobi
We define the curvature tensor R of V
R(X, Y) Z where Let
z
X, Y,
TpM
E
operator.
semi-Riemannian manifold and V the Levi-Civita
Z E FTM
=
are
Vx Vy Z
-
VyVx Z
vector fields
on
-
con-
by
V[X,y] Z,
M and
is the Lie bracket.
and let
R(., z)z: TpM be the linear map defined by (R(., identities, since g(R(x, z)z, z) = 0,
z)z)x we
-+
TpM
R(x, z)z.
=
Note
that, by
curvature
have
R(., z)z: TpM
--+ z
where z-L is the
orthogonal space to spanfz}. Now using the linear map R( z)z, we define the Jacobi operator with respect to z. Note that, if z E S(M) then z' is a nondegenerate hyperspace in TpM (that is, the restriction of g to zJ- is nondegenerate), where z E Sp(M), -
,
Definition 1.1.3. Let
S(M). z'
(M,g)
Then the restriction Rz
is called the Jacobi
be
x
Now
E
se
z'
of
manifold
the linear
and
z
E
map'R(.,z)z
to
z, that is
R(x, z)z,
=
z
state
Proposition
S(M),
-+
operator with respect to
R,x where
semi-Riemannian
a
zI
:
some
properties of the Jacobi operator.
1.1.1. Let
(M,g)
be
Then the Jacobi operator R,
a :
semi-Riemannian
z_L
-+
z'
is
a
manifold and self-adjoint map.
z
Spacelike Osserman Conditions
1.2 The Timelike and
Proof. Let x, y E
z.
Then
g(Rzx, y)
by
curvature
at
a
Point
identities,
=
g(R(x, z)z, y)
=
g(R(z, y)x, z)
=
g(R(y, z)z, x)
=
g(R-.y, x). 0
self-adjoint.
Hence Rz is
Remark 1. 1. 1. Let
3
(M, g)
be
semi-Riemannian
a
the Jacobi operator. Then note
manifold,
z
E S (M) and
R,
that,
n-1
g(xi, xi)g(Rzxi, xi)
traceRz n-1
g(xi, xi)g(R(xi, z)z, xi)
Ric(z, z), where unit
x
Ixi, E zj-,
Xn-l} is an orthonormal basis for z'. Also for every nonnull spanjx, z} is a nondegenerate plane in TpM (that is, the P is nondegenerate), where z E TpM, and the curvature to g
P
restriction of
r.(P)
of P is
=
given by
x(p)
=
g(R(x, z)z, x) g(x, X)g(z, Z) g(x, Z),
x) g(x, XWZ' Z) g (R, x,
-
-
Hence, in Riemannian geometry, the eigenvalues of Rx represent the extremal values of the sectional curvatures of all
1.2 The Timelike and a
planes containing
Spacelike
x.
Osserman Conditions at
Point
Let
(M, g)
be
a
semi-Riemannian manifold and
z
E
S(M)
-
Then the Jacobi
1
operator R, : z_L _4 Z is a self-adjoint linear map. But in general, since 1 has an induced indefinite inner product, R, may not be diagonalizable. z
why we state Osserman conditions in terms of the characteristic polynomial of Rz rather than its eigenvalues as in Riemannian geometry. As we will remark later, both statements of Osserman conditions in terms of characteristic polynomials and eigenvalues of Rz coincide in Riemannian
That is
geometry. Definition 1.2.1. Let
a) (M, g) of R,,
(M, g)
be asemi-Riemannian
is called timelike Osserman at p is
independent of z
E
S;-(M).
if
manifold
and p E M.
the characteristic
polynomial
4
1. The Osserman Conditions in Semi-Riemannian
b) (M, g) of R ,
Geometry
is called
spacelike Osserman at p, if the characteristic polynomial independent of z E S+(M). P
is
Remark 1.2.1. It is important to note that the Osserman condition could equivalently stated in terms of the constancy of the (possibly complex)
be
of the Jacobi, operators, counted with
eigenvalues
multiplicities.
The fact that the Jacobi operators are, in general, nondiagonalizable in the semi-Riemannian setting motivated the study of their normal forms (see
Section
4.2). Here it is worth to emphasize the role played by the minimal polynomial of the Jacobi operators, since they may have nonconstant -roots even if the manifold is assumed to be Osserman as pointed out in the examples in sections 4.1 And 5.1. Such
a behaviour does not affect Riemannian nor Lorentzian Osserman manifolds since their Jacobi operators are completely determined by the knowledge of the corresponding eigenvalues.
Now we show that (M, g) being timelike Osserman at'p is equivalent (M, g) being spacelike Osserman at p. Note in the proof, however, that
to
this
equivalence does not imply that the characteristic polynomial of R, independent of z E S(M).
Theorem 1.2.1. M. Then
(M, g)
[60]
Let
(M, g)
be
a
is
semi-Riemannian manifold and p E if and only if (M, g) is spacelike
is timelike Osserman at p
Osserman at p. Proof. Let -
-
(M, g)
be timelike Osserman at p and f, (t) = tn- 1 + an-2 tn-2 + polynomial of R,, for all z E S - (M), where.
+ a, t + ao be the characteristic
-
an-2 i
z'
.
C
.
.
,
a,, ao E R. Then the characteristic
is also
TpM
polynomial'of R( all
z
E
S;- (M)
-,
independent
z)z
is then
of
h., (t)
z
polynomial of R( z) z : TpM -+ S-(M). In fact, the characteristic
E .
-
,
tn + an-2 tn-1 +
+ a, t2 + aot for
-
First note
that, at p E M, the metric tensor g and the curvature tensor R and extend g analytic functions on TpM. Now complexify TpM to TCM P and R to be complex linear gC and R'c on TCM respectively. Next note that are
P
V
=
1-y
E
define RC
T'cMlg'c(-y, -y) 0 0} P :
VCM
-+
P
is
h.,(t)
=
7 E V
by
R'C (a, 7)-y a
91 (-Y, -Y)
tn +An-2 (,,)tn-1 +,
polynomial RSY VCM. Since R'C and P of
)
connected open dense subset of VCM and P
TCM for each P
R,cy Let
a
.+A, (,y)t2 +Ao(-y)t be the characteristic An-2) A,, AO are functions on analytic on T'CM clearly An-2) A,, AO are
for each 7 E V, where
g'c
are
...,
P
...,
analytic on T'CM. P Furthermore, let U- and U+ be the sets of timelike and spacelike TpM, respectively. We haveVnTpM U- U U+.
also
in
=
vectors
Spacelike Osserman Conditions
1.2 The Timelike and
at
a
Point
Sp (M) C V. Then note that gc- (z, z) Rr- is the complex linear and hence, gc(z,z)RC and R(.,z)z have the extension of R(.,z)z to VCM P same characteristic polynomials. Now since (M, g) is timelike Osserman at p, it follows that the characteristic polynomial of g'c(z, z)RC is independent of z E S (M) C V. Thus, since the coefficients of the characteristic polynomial of gC (z, z) RC are analytic, they are constant for all 7 E V. Hence the coeffiLet
E
z
z
z
z
cients
A,2,
...,
A,, Ao
of the characteristic
polynomial
RE,
constant for
are
are all y E V. This immediately implies that the coefficients of g'C (z, z) R'C z of characteristic the coefficients and of polyE z hence, independent
Sp+(M)
nomial of
R(.,z)z
Thus the characteristic independent of z E S+(M). P That of E z is, (M,g) is spacelike independent Sp+(M).
are
polynomial of Rz is Osserman at p. The
converse
by Theorem 1.2.1, generality. Now
of
Definition 1.2.2. Let is called
Osserman
is obtained in the
the
(M, g)
at p, E M
following
be
a
same
definition
can
semi-Riemannian
if (M, g)
0
way.
without loss
begiven
manifold. Then (M, g) spacelike Osserman
is both timelike and
at p.
Remark 1.2.2. Note that the above definition does not mean that the characteristic polynomial of Rz is independent of z E Sp(M).
that, being Osserman for a semi-Riemannian manifold point simplifies the geometry at that point. Next
a
we
show
Lemma 1.2.1.
h be
a
[44]
form
bilinear
Let on
(V, (,))
be
an
indefinite
inner
product
at
space and let
V.
a) If h(u, u) 0 for every null u E V then h A(,), where A E R. b) If I h(x, x) I< d E R for every unit timelike vector x E V (or for every unit spacelike vector x E V) then h A(,), where A E R. timelike unit R E (resp., spacelike) vector x E V for every C) If h(x, x) :5 d, and h(y, y) ! d2 E R for every unit spacelike (resp., timelike) vector A(,), where A E R. y E V then h =
=
=
=
Proof. See for
Proposition
example [441 and [101].
(M, g) be a semi-Riemannian manifold. If (M, g) then (M, g) is Einstein at p E M, that is, Ric Ag
1.2.1. Let
Osserman at p E M M, where A E R.
=
is at
p E
Proof. Consider
(M,g)
as
timelike Osserman at p. Then the characteris-
n-1
tic
polynomial fz(t)
T ait'
of the Jacobi operator R_. is
independent of
i=O z
E
S;- (M),
where a,,-,
=
1. In
particular,
since
traceRz
=
-an-2 and
The Osserman Conditions in Semi-Riemannian
1.
Ric(z, z) dent of
S (M)
E
Z
it follows that
traceR,
=
Thus, if
Ric(z, z)
Geometry
-a,,-2 and hence is
=
indepen-
-
g is indefinite then
X E. R, and if g is definite then A E K Remark 1. 2.3. Note that if
by Lemma 1.2.1-(b), Ric by polarization identity, Ric
=
=
Ag at p, where Ag at p, where 0
(M, g)
is
Riemannian manifold
(that is, v S(M), (M, g) is Osserman at p E M if and only if the eigenvalues (counting with multiplicities) of R., are independent of z E S(M). 0, n) then
since
a
R;, is diagonalizable for every
In what remains of this
section,
Riemannian manifolds which
are
we
=
E
z
will construct
Osserman at
a
examples of semigiven point. Since the notion
of
algebraic curvature maps plays an essential role (cf. Remark 1.2.5), we begin by recalling some basic facts about such maps. Let V be a vector space. A quadrilinear map F: V x V x V x V -+ R is called an algebraic curvature map
if it satisfies
F(x, y, z, w) F(x, y, z, w) F(x, y, z, w) for all x, y, z,
V
E
w
=
=
+
-F(y, x, z, w) -F(x, y, w, z), F(z, w, x, y), F(y, z, x, w) + F(z, x, y, w) 0, =
=
Furthe more, if : V x V
.
is
curvature tensor F
tensor with
defined
respect
to
an
by F(x, y, z, w)
=
inner
x
V
an
inner
product
on
V then the
V of F with respect to (,) is defined for each x, y, z E V by (P(x, y)z, w) = F(x, y, z, w), where w E V. Also a trilinear map P : V x V x V -+ V is called an algebraic curvature
algebraic
product (, )
(-P(x, y)z, w)
is
-+
on
an
V if F
:
algebraic
V
x
V
x
V
x
V
R
-+
curvature map, where
x,y,z,w E V.
Let
(V, (,))
algebraic
P, P
be
inner
an
1
z
: z
of
P
for
a
applies
a
to
timelike
or
spacelike unit
z
algebraic curvature tensors.)
curvature tensors
are as
RO
:
V
:
V
x
V
x
V
-+
V be
an
polynomial of P,
E V.
Basic
(Note
is
independent
that Theorem 1.2.1 also
examples of Osserman algebraic
follows:
Example 1.2.1. Let (V, (,)) be curvature tensor
P
on
if the characteristic
is called Osserman
of either
space and
product
V with respect to (,). Then the Jacobi operator nonnull unit z E V is defined by F,, x = F (x, z) z and
curvature tensor
x
V
x
an
V
RO (x, y)z
inner -+
=
V
product by
(y, z)x
-
space and define
an
algebraic
(x, z)y.
Note that RI is Osserman with Jacobi operator RO Z nonnull unit vector in V.
=
(z, z)id,
where
z
is
a
Example 1.2.2. A complex structure on a vector space V is a linear map -id. An inner product (,) on (V, J) is said to be J : V -* V satisfying j2 =
Spacelike Osserman Conditions
1.2 The Timelike and
Hermitian if
is called
(x, Jy) RJ
curvature tensor
:
V
RJ (x, y) z RJ
Note that
x
=
a
Point
0 for all x, y E V and the triplet product space. Furthermore,, define an
(Jx, y)
+
Hermitian inner
a
at
V
=
x
V
V
-+
(Jx, z) Jy
-
(V, J, algebraic
by
(Jy, z) Jx
y) Jz.
+ 2 (Jx,
is Osserman with Jacobi operator
-3(z,z)id
on
spanfJz},
0
on
(span I Jz})
RJ z
where
z
is
a
-1
n
z
nonnull unit vector in V.
Example 1.2.3. A product structure on a vector space V is a linear map J id. It induces a decomposition of V into V V -4 V satisfying P V(+) (D to of J the eigenvalues 1. corresponding eigenspaces V(-), where V() are direct is sum if V a decomposition of Y then the Conversely, V(+) ED V(-) defined J V V linear map J : by 7r(-) Iis a product structure ir(+) the V are -+ where projections. For the special case, on V, V() 7r() J on V is called a paracomplex structure the product dimV(-), dimV(+) =
=
=
=
-
=
structure
on
An inner Hermitian if
V.
product (,)
(x, Jy)
Hermitian inner tensor
RJ: V
x
+
(Jx, y)
product
V
V
x
RJ (x, y) z Note that
RJ
paracomplex
on a
-+
=
=
space.
V
z
is
a
(Jx, z) Jy
-
(Jy, z) Jx
+ 2 (Jx,
y) Jz.
operator
3(z,z)id
on
spanfJz},
0
-on
(spanfjzF)-L
n
zi-,
nonnull unit vector in V.
Remark 1.2.4. It has been recently shown curvature map vature maps
is called para-
by
z
where
(V, J)
triplet (V, J, (,)) is called a paraFurthermore, define an algebraic curvature
is Osserman with Jacobi
RJ
vector space
0 and the
can
be
RO (x, y,
z,
expressed
v)
=
as
a
that any
[52], [72], [71]
algebraic algebraic curO(x, z) 0(y, w) defined by some
linear combination of
0(y, z)o(x, w)
-
bilinear forms
0. symmetric Equivalently, any algebraic curvature map can be expressed as a linear combination of algebraic curvature maps RQ(x,y,z,v) f?(y,z)J?(x,v) bilinear defined some skew-symmetric by 2f?(x,y)R(z,v) f2(x,z)f?(y,v) =
-
-
forms fl.
Using the Osserman algebraic
curvature tensors
given
in
Examples
1.2.1
and 1.2.2, a large family of Osserman algebraic curvature tensors can be constructed by considering Clifford module structures. Due to the important
1. The Osserman Conditions in Semi-Riemannian
8
between the Osserman
relationship
algebraic
Geometry
curvature tensors and Clifford
module structures, we recall here their definition and refer to Theorem 2.1.1 for necessary and sufficient conditions for their existence. Definition 1.2.3. Let V be module structure C
V with
i'j
1,
=
set
a -
-
on,
an
n-dimensional vector space. A real Clifflv)v < n is a collection of linear maps Ji on
where
V,
of generators f Jj,..., Jj
such that
JiJj
+
JjJi
=
-25ij for
V.
-,
(That is, C determines an anti-commutative family of complex structures on V.) Note here the existence of Ji-Hermitian inner products with respect to all complex structures in a Cliff(v)-module structure. Denote such an inner product by (,). Then one has,
[66] Suppose there is a Clifflv)-module structure on Rn and of generators IJ,,..., J,} such that JjJj + JjJj -26ij. If Ao,
Theorem 1.2.2.
consider ...,
A,
a
are
set
=
arbitrary
real
numbers,
then the trilinear map R: V
x
V
x
V.-4 V
defined by V
R
=
AoRO
+
1:(Ai Ao)Rj' -
3
i=1
is
algebraic
Osserman
an
curvature tensor with
R ,Jjx where Z, y
=
Aijix,
nonnull unit vectors
are
on
Rxy
=
Rn with y
Aoy, orthogonal
to
f x J, x, ,
...
jVXj. that, for any of the algebraic curvature maps R defined may construct (semi)-Riemannian metrics whose curvature tensor coincides with R at a given point. For this, let F be an algebraic curvature Remark 1.2.5. Note
above,
one
map
on
basis
on
Rn and put Fijkl. = F(ej, e5j, ek, el), where Next, define a Riemannian metric tensor
Rn.
centered at the
je,j on
is
an
orthonormal
the unit ball B C Rn
g
origin by
=
1: ij
Jjj
+
1: Fijkl Xk x'
dx'dxj
+
O(X3)
kj
Now, the theory of normal coordinates shows that the curvature tensor of g coincides with F at the origin. By following this procedure, examples of Riemannian manifolds which are Osserman at a given point, yet whose curvature tensors do not correspond to a rank-one symmetric space, are constructed by Gilkey in [66] using the algebraic curvature maps defined in Theorem 1.2.2. Remark 1.2.6. It is worth to empasize here the different roles played by eigenvalue structure (Osserman property) and the Jordan form (Jordan-
the
Osserman
property)
of the Jacobi operators
(cf.
Definition
4.2.1). Indeed,
1.3 Semi-Riemannian Pointwise and Global Osserman Manifolds
examples of Osserman manifolds which are not JordanOsserman (see 4.1) as well as algebraic curvature tensors which are Jordan-Osserman but not Osserman (Remark 5.1.1). Furthermore, it is now Osserman of exhibit to algebraic curvature tensors which examples possible timelike Jordan-Osserman. Following but not Jordan-Osserman are spacelike basis of (V, orthonormal be an let Suppose e...... le 1...... e-, [73], e+} 1 q P define linear and a and even map P by (q 2q) q p, > q
there
are
many
section
=
4i(e2+j_j)
=
e 2ij
4i (e 2i-,) j,
=
-e 2i,
P(e,--) P is
Then
=
P(e2+j)
=
-e 2ij_j
4i(ej;) 2i
=
e 2ij -,
e2+j,
+
e2+j,
-
0,
e2+-l,
e2+i
q q