The Osserman Conditions in Semi-Riemannian Geometry

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