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 <

skew-adjoint and (p2

no

spacelike

+

i:

p.

e+q

e

+

vectors it follows that

1.2.4, where Q(-, (-,4i(-)), is spacelike Jordansimple calculation of R12 and RI? shows that R17 is

in Remark

Osserman. However,

e+1

0 with ker 4i

Since ker iP contains

q

a

e"

e1

(This also shows Jordan-Osserman manifolds).

not timelike Jordan-Osserman.

be extended to

q

7

e ,eq-+j,...'e;}. P R1? defined

+

-

that Theorem 1.2.1 cannot

1.3 Semi-Riemannian Pointwise and Global Osserman

Manifolds

generalize the Osserman condition to the by giving the definitions of pointwise and global

In this section in two ways

we

whole manifold Osserman

man-

ifolds. Definition 1.3. 1. Let

(M, g)

Remark 1. 3. 1. Note that if

be

a

semi-Riemannian

if (M, g)

is called pointwise Osserman

(M, g)

is

a

manifold.

Then

(M, g)

is Osserman at each p E M.

semi-Riemannian

pointwise Osserman

manifold then by Proposition (M, g) is Einstein at each p, E M. Hence, > dimM and 3 connected if M is then, by Schur Lemma, (M, g) is an Einstein 1.2. 1,

manifold,

that

is, Ric

=

Definition 1.3.2. Let is called

dent

globally

of z

E S-

or z

this

on

(M, g)

Osserman

(M)

Throughout

Ag

E

book,

if

M, where A be

a

E R.

manifold. polynomial of R_

semi-Riemannian

the characteristic

Then is

(M, g)

indepen-

S+ (M). we

also call

a

global Osserman condition, no ambiguity.

for

con-

venience, Osserman condition whenever there is Remark 1.3.2. Let

(M, g)

be

a

semi-Riemannian

globally Osserman manifold.

polynomial of Rz is independent of Z E S- (M) if if the characteristic polynomial of R,, is independent of z E S+ (M)

Note that the characteristic

and

only

-

1. The Osserman Conditions in Semi-Riemannian

10

(See

proof of

the

polynomial

1.2.1.)

Theorem

nomial of R,, for all

of R, for all

Also recall that the characteristic

z

S+(M).

E

Remark 1. 3.3. A semi-Riemannian manifold

isotropic there is

poly-

S- (M) may be different from the characteristic

E

z

Geometry

(M, g)

is said to be

locally

if for each

point p E M and each x, y E TpM with g(x, x) = g(y, y), local isometry of (M, g) of a neighborhood of p which fixes p and

a

exchanges

and y.

x

(Cf. [142].) Clearly,

a

locally isotropic semi-Riemannian

manifold is Osserman. Next

we

analyze the relation

(M, g)

conditions. Let

be

Jacobi operators, there

fk (p, Z)

are

=

between the

pointwise and global Osserman

semi-Riemannian manifold. Associated to the

a

functions

fk defined by

g(Z' Z)k traceR (k),

k

Z

=

1, 2,3,.

(1.2)

..

(k) M, z E Sp(M) and Rz is the k th power of the Jacobi operator and these functions, f, Rz. Among f2 have a special significance. Indeed, a semi-Riemannian manifold (M, g) is Einstein if f, (p, z) is constant on S(M), and is called 2-stein if f, and f2 are independent of z E Sp(M) at each p E M. Note that (M,g) is pointwise Osserman if and only if the functions fic depend only on the point p for each k. Also the global Osserman condition is equivalent to the constancy of functions fk on S(M) for each k. The following result about the relation between the pointwise and global Osserman conditions is proved in [78] for Riemannian manifolds. Here, we present its semi-Riemannian version which is essentially obtained by following

where p E

the

same

steps.

Theorem 1.3.1. Let serman

(i) (ii)

manifold

such

(M, g)

be

connected semi-Riemannian pointwise Os-

a

that,

the Jacobi operators have

only

one

eigenvalue and dim

M >

3,

or

the Jacobi operators have exactly two distinct eigenvalues, which are either complex or, real with constant multiplicities, at every p E M and dim M > 4.

Then

(M,g)

is

globally Osserman.

Before proving this theorem, we need f, and f2 as follows.

a

technical result

on

the constancy

of the functions

Lemma 1.3.1. Let serman

(M,g)

be

a

connected semi-Riemannian pointwise Os-

manifold.

a) If dimM b) If dimM

then

=

n

> 3

=

n

> 4 then

f, f2

is constant

on

M.

is constant

on

M.

1.3 Semi-Riemannian Pointwise and Global Osserman Manifolds

(a)

Proof.

Since trace R_.

=

Ric(z, z)

Ric(z, z) for all

z

E TM. Hence since

(b)

n

>

Z

E

S(M),

we

have

(1.3)

fi (p)g(z, z)

=

3, the function fi

lei,..., en}

Let p E M and

for all

11

be

is constant

on

orthonormal basis for

an

M.

Then

TpM.

n

f2 (p)g (z, z)'

traceR(')

=

R(ej, z, z, ej )2 6ei 6ej

Z

(1.4)

for any z E Sp(M), where R(ej, z, z, ej) = g(R(ei, z)z, ej) and 6ei = g(ei, ej) for i = 1, 2,..., n. Therefore, if x, y E TpM and x + ay, where u = 1, then (1.4) shows that n

f2 (P) g (X

+ Ory,

+

X

ory)

2

R(ej, x

+ ory,

+ ay, ej

x

)26ei -ej,

and

f2 (p) jg (x, x) =

1:

+ g (y,

y)

+

JR(ej, x, x, ej)

2og (x, Y) }2 +

R(ej, y, y, ej)

+

uR(ei, x, y, ej)

i,j=l

+aR(ei, y, x, ej) 12Sei Eej

*

adding those corresponding

above and

Now, after linearizing the expression to o, 1, one has =

f2 (P)

jg(X, X)2

+

g(y, 02

+

4g(x, y)2

+

2g(x, x)g(y, Y)

n

=

I: I R(ej, x, x, ej)2 + R(ej, y, y, ej )2 + R(ei, x, y, ej )2 + R(ej, x, y, ej)2 +2R(ei, x, x, ej)R(ei, y, y, ej)

+

2R(ei, x, y, ej)R(ej, x, y, ei)} 6ei6ei

-

n

Once more,

applying (1.4)

to

x

and y and

using

E

R(ej, x, y, ej )26ei -ej

i,j=l n

R(ej, x, y, ej )2 6ei6ej,

it follows that

n

f2 (p)

f g (x, x) g (y, y)

+

2g (x, y)21

=1:

JR(ej, x, x, ej)R(ei, y, y, ej)

i,j=l

+R(ei, x, y, ej)R(ej, x, y, ei) Putting y k 1,...'n, =

=

ek in the above we

obtain

+

R(ej, x, y, ej )21

expression, multiplying by

-e,

6ei -ej

and

-

adding for

1. The Osserman Conditions in Semi-Riemannian

12

n

Geometry

n

f2 (P) E

f g (x, x),-,,

2g (x, ek)

+

2

1eek

1:

=

k=1

f R(ej, x, x, ej)R(ei, ek, ek, ej)

i,j,k=l

+R(ei, x, ek, ej)R(ej, x, ek, ej)

+

R(ej, x, ek, ej)21

ei6ej6ek

-

(1.5) Moreover, since n

1:

R(ei,x,x,ej)R(ei,eklek,ej)6ei-'ej-ek

i,j,k=l n

ER(ej,x,x,ej)Ric(ej,ej)6ej6ej

=

)

i,j=l

and n

E fg(X) X)Fek

+

29(x, ek )21

Ee ,

=

ng(x, x)

+

2g(x, x)

(n

=

+

2)g(x, x)

,

k=1

(1.5)

becomes n

f2 (p) (n

+

2) g (x, x)=

1: R(ej, x, x, ej)Ric(ei, ej)6ei 6ej

n

+

E JR(ei,X,ek,ej)R(ej,X,e-k,ei)+R(ei,x,ek,ej )21

-ej -ej

eel,

i,j,k=i

(1.6) Now, Ric

since dimM

=

f1g, and

=

n

>

2,

it follows from

(a)

that

(M, g)

is Einstein with

thus

n

R(ej, x, x, ej)Ric(ei, ej)6ei-'ej

=

fRic(x, x)

=,

f,2g(x, x)

i,j==l

Hence

(1.6)

f2 (p) (n

becomes

+

2)g(x, x)

-

f,2g(x, x)

n

JR(x,ei,ej,ek)R(x,ej,ei,ek)+R(x,ei,ej,'ek )21 EeiEej,-,-k

-

i,j,k=l

(1-7) To

simplify (1.7),

let J? be the bilinear form defined

by

n

2 (t,

W)

=

1:

R(t,ei,ej,ek)R(w,eiej,ek)EeiCejEek

i,j,k=l

Now, by the first Bianchi identity,

we

have

(1.8)

1.3 Semi-Riemannian Pointwise and Global Osserman Manifolds

13

n

E

R(x;ei,ej,ek)R(x,ej,ei,ek)'FeiEei6ek

i,j,k=l n

R(x,ei,ej,ek)I-R(ej,ei,x,ek)-R(ei,x,ej,ek)}-eiEejEek n

1:

R(X,ei,ek,ej)R(x,ek,ei,ej)6eiSej6eA,

i,j,k=l n

+

1:

R(x, ej, ej, ek )2 6ej-'ej-ek

i,j,k=l n

-

1:

R(x, ej, ek, ej)R(x, ek, ej, ej),ei Eej Eek

+

fl(Xi X)

)

and thus n

E

R(x,ei,ej,ek)R(x,ej,ei,ek)Eei,ej-ek

+

2)g(x, x)

-

2S? (X, X)

.

(1.9)

it follows that

Now, from (1-7), (1-8) and (1.9),

f2 (p) (n

=

f,2g(x, x)

=

IS? (X, X) + 0 (X, X)

2

and thus

S?(x, x) where F

:

M

--*

=

F(p)g(x, x)

R is the function defined

(1.10)

,

by

2

F(q)

=

3

(f2(q)(n+2)-fj2),

qEM.

that, from (1.10),

Note here

S?

=

F(p)g

(1.12)

,

n

and therefore

ES?(ei, ei)-ei

n

Yf?(ej,ej)6ei i=1

and thus

=

nF(p).

On the other hand, from

(1.8),

n

1: R(ej,ej,ek,ej)2-ej-ej6eh6ej7 i,j,k,l=l

(1.12) yields JJRJ 12g

S? n

(1.13)

14

1. The Osserman Conditions in Semi-Riemannian

Geometry

n

E R(ej,ej,ek)eI)'--ei6ej6ek6ei-

JJRJ 12

where

i,j,k,l=l

Also

covariant differentiation in

by using

(1.8),

we

obtain

n

(Veb S?) (e, eb)-eb b=1

n

n

=E

E

b=1

I(VebR)(e,.,ei,ej,ek)R(e-b,ei,ej,ek)

i,j,k=l

+R(e, ej, ej, ek) (Veb R) (eb, ej, ej, ek) I Sei 6ej 6e,

Seb

n

1: (VebR)(e,,,ei,ej,ek)R(eb,ei,ej,ek)-ej-ej,ekEeb ij, k, b= 1 n

n

E

+

R(ea, ej, ej, ek)-ej

ej

E (Ve, R) (eb, ej, ej, ek),Feb

6e

j,k=

and, by

b=1

the second Bianchi

identity,

n

n

E(VeA(eb,

ej, ej,

ek)Eeb

b=1

since

E(VebR)(ej, ek, eb) ei)6eb b=1

=

.Now,

=

-

(M, g)

(Vej Ric) (ei, ek)

is

+

(Ve, Ric) (ei, ej)

=

0.

Einstein,

n

1: (VebS?) (e, eb)-eb b=1

(VebR) (ea, ej, ej, ek)R(eb, ej, ej, ek)-ej 6ej Eek Eeb i,j,k,b=l

and

again, by the second Bianchi identity, n

1:

(VebR)(ea,ei,ej,ek)R(eb,ei,ej,ek)Sei -ej 6ek Eeb

i,j,k,b=l n

1:

(VeaR) (ei, eb, ej, ek)R(ei, eb, ej, ek)Eei Eej Eek Eeb

i,j,k,b=l n

1: i,j,k,b=l

(Vej R) (ea, eb, ej, ek)R(ei, eb, ej, ek)Sej6ej6ek-eb

)

1.3 Semi-Riemannian Pointwise and Global Osserman Manifolds

reduces to

(1.14)

Thus

15

n

E (Ve,, f?) (ea, eb)Eeb b=1 n

1

E

2

(VeaR)(ei,eb,ej,ek)R(ei,eb,ej,ek)6ei6ejEek-eb

i,j,k,b=l

Now, by differentiating the expression of JJRJ 12,

obtain

we

n

Vea JJRJ 12

=

E

2

(Vea R) (ei, eb, ej, ek)R(ei, eb, ej, ek)Eei -ej -ek 6eb

i,j,k,b=l

and thus the

previous expression reduces

to

1

J:(VebQ)(ea,eb)-'eb

=

4

V,. I I RI 12

(1.15)

b=1

J:(VebS?)(ea,eb)I'eb

Also, (1.13) implies that

=

n

Eg(ea, eb)6ebVeb I IRI 12, b=1

b=1

and hence

VealIRI 12

(Veb 0) (ea, eb)''eb

(1.16)

n

b=1

Thus, for

0 4, (1.15) and (1.16) imply that VeaJJRJ 12 and therefore JJRJ 12 must be constant. Now, by (1.12) and

since dimM n,

a

n

>

f2(P)

=

=

=

(1.13) 1

and

hence, f2

Now

(

n+2

3 2n

JJRJ 12

+

2)

fl

0

is constant.

ready to prove the mentioned result pointwise and global Osserman conditions.

we are

tween the

Proof of Theorem 1.3.1. First note that if

(M, g)

about the relation be-

is timelike Osserman at

p E M with single eigenvalue A(p) of R, then this eigenvalue is necessarily real. Also by the proof of Theorem 1.2.1, (M,g) is spacelike Osserman at p E M with

(n

-

single eigenvalue -A(p)

1)A(p)g(z, z)

for all

follows that A is constant

z

on

of R_.. Then since

Sp(M)

E

Ric(z, z)

=

at each p E M and dimM

M and hence

(M, g)

is

globally

traceR, n > 3, it =

=

Osserman.

Now suppose that (M, g) is timelike Osserman at every p E M with two distinct complex eigenvalues of Rz. Then since the coefficients of the characteristic

polynomial

of R,

are

to each other with the same

real, these eigenvalues are complex conjugate a(p) + ifl(p) and multiplicities, say, w(p) =

16

The Osserman Conditions in Semi-Riemannian

1.

0(p)

=

a(p)

-

constant

are

and traceRZ2

at each P E M. Then

ift) M

on =

(n

and,

-

Geometry

by Lemma 1.3.1, traceR,, (n

in the above case,

as

1) (a (p)'

-

O(P)2)

for every

=

z

Sp(M),

E

since

f,

and

f2

1)a(p)g(z, z),

-

it follows that

a

0 are constant on M and hence, (M, g) is globally Osserman. Finally suppose (M, g) is timelike Osserman at every p E M with two distinct real eigenvalues of R, say, A(p) and p(p) with constant multiplicities m,x and m,,, respectively. Then by Lemma 1.3.1, since f, and f2 are constant on M and, as in the first case, traceR, (A(p)m,\ + [z(p)m,,)g(z, z), and

and

=

traceR 2Z

A(P)2M,\

=

constant

are

/,t(P)2,Mjj

for every z E Sp(M), it follows that A and p M and hence (M, g) is globally Osserman.

on

Here note that

+

we

cannot deduce the

constancy of X and p

on

M without

assuming that they have constant multiplicities at every point. Because there is no reason that the multiplicity should not change from point to point. Next

give

we

examples of semi-Riemannian globally Osserman

some

man-

model spaces. The necessary background for the following examples may be found in [101], [114], [146]. Also, examples of strictly pointwise Osserman manifolds are presented below (cf. Remark 1.3.4.)

ifolds

as some

Example 1.3.1. A semi-Riemannian manifold (Mn,g) of ignature (v,,q) a real space form if (M, g) is of constant sectional curvature. Hence (M, g) is a real space form then the curvature tensor of (M, g) is given by

is

called

R(x, y) z where x, y,

z

E

TpM

and

c

if

cR0 (x, y) z,

=

E R. Then the Jacobi

operator of

z

E

S(M)

is

given by Rz Thus the characteristic z

E

the

S(M)

vectors

(cf.

cg(z, z) id.

polynomial of R.,

is

f., (t)

=

(t

-

cg(z, Z))n-1

for all

real space form is globally Osserman. (Note that of the Jacobi operators change sign from timelike to spacelike

and hence

eigenvalues

=

Remark

a

1.2.2).)

A complete and

simply

connected real space form

is isometric to either of the

P ,' (R)

=

according

SO'(n

+

1) /SO'(n),

R.,

to the sectional curvature

,

or

H,,n (R)

=

SO'+' (n

+

being positive, negative

1) /SO'(n) or

zero

(see

[142].) Example 1.3.2. Let (M, J) plex structure J (i.e., J is

be

almost

complex manifold with almost comfield on M satisfying j2 -id.) A semi-Riemannian metric tensor g of signature (2v, 2-q) is said to be Her0 for all X, Y E FTM. Also (M, g, J) is mitian if 9(JX, Y) + g(X, JY) said to be a Kdhler manifold if J is a complex structure and the 2-form a

an

(1, I)-tensor =

=

1.3 Semi-Riemannian Pointwise and Global Osserman Manifolds

17

g(X, JY) is closed. This couple of conditions can be equivalently 0, where V is the Levi-Civita connection of 9. by VJ A plane P is called holomorphic if it remains invariant by the complex structure (JP C P), and the holomorphic sectional curvature is defined to be the restriction of the sectional curvature to nondegenerate holomorphic planes. A Kdhler manifold (M, g, J) is called a complex space form if (M, g, J) is of constant holomorphic sectional curvature. Hence, the curvature tensor of (M, g, J) is given by [4], P(X, Y)

=

described

=

c

R(x, y)z where x, y,

TpM

E

z

and

=

4

[RO (x, y)z

-

Rj (x, y)z]

E R. Then the Jacobi

c

,

operator of

z

E S

(M)

is

given by

cg(z, z) id

on

spanjJz},

on

(spanjz, Jz})--L

Rz c

49 (z,

z) id

n

zJ-.

fz (t) cg(z, z)) (t (t cg(z, z))n-2 for all z E S(M) and hence a complex space form is globally Osserman. The model spaces of nonzero constant holomorphic sectional curvature are given by the symmetric spaces Thus the characteristic

P,n(C)

=

SU'(n

+

polynomial

1)/U'(n)

of R.

and

is

H,,n(C)

=

=

-

-

SU'+'(n + 1)/U'(n)

(see [142].) Generalizing the form of the curvature tensor of a complex form, an almost Hermitian manifold (M, g, J) is called a generalized, complex space form if its curvature tensor satisfies

Remark 1.3.4. space

R(x, y)z

=

f RO(x, y)z

+

hR'(x, y)z,

f h : M -+ R are smooth functions. Generalized complex space forms pointwise Osserman manifolds with at most two distinct eigenvalues. It is shown in [140] that generalized complex space forms are complex space forms

where

,

are

if dim M > 6

4-dimensional

(which also follows from Theorem 1.3. L) Yet the existence of generalized complex space forms which are not complex space

forms is shown in Note that

form is

a

[113].

generalized complex

pointwise Osserman but

nonconstant functions

on

space form which is not a complex space globally Osserman, since f and h are

not

investigate the semi-Riemannian coming chapters as well as the special role

M. We will further

pointwise Osserman manifolds played by generalized complex

in

space forms.

Example 1.3.3. An almost quaternionic manifold is a manifold M equipped a 3-dimensional vector bundle Q of (1, 1)-tensor fields on M such that there exists a local basis jJj, J2, J3} for Q -satisfying Jj2 1,'2, 3, -id, i with

=

=

1. The Osserman Conditions in Semi-Riemannian

18

and

Geometry

Ji Jj Jk, where (i, j, k) is a cyclic permutation of (1, 2, 3). Such a local I Jj, J2, J3} is called a canonical local basis for Q and Q is referred as an almost quaternionic structure on M. A semi-Riemannian metric tensor g of signature (4v, 477) is said to be adapted to the almost quaternionic structure =

basis

if

g(OX, Y) + g(X, OY) 0 for all 0 E Q and X, Y E FTM. (M, g, Q) be an almost quaternionic manifold and f J1, J2, J3} be a canonical local basis for Q. Then for each i 1, 2,3, 4ii(X, Y) g(X, JjY), Q

=

Let

=

where

X, Y E rTM, is a locally defined two-form such that 0 (fil A!Pl + !P2 A!P2 + 4 3 NP3 gives rise to a globally defined 4-form on M. A quaternionic metric structure (g, Q) is said to be Kdhlerian if S? is parallel (or equivalently, if Q is parallel) with respect to the Levi-Civita connection V of g (cf. [83],

[1281, [118].) Let (M, g, Q) determines

be

a

quaternionic Kdhler manifold. Then

any vector

x

E

4-dimensional subspace Q(x) spanjx, Jlx, J2x, j3xj which remains invariant under the action of the quaternionic structure. We call it the Q-section determined by x. If the sectional curvature of planes in

TpM

a

=

Q(x) is a constant c(x), where x E TpM is nonnull, we call this constant c(x) the quaternionic sectional curvature of (M, g) with respect to x. A quaternionic Kdhler manifold (M, g, Q) is called a quaternionic space form

if

(M, g, Q)

is of constant

curvature tensor is

quaternionic sectional

curvature. Hence its

given by c

R(x,y)z

=

[R

3 0

(X, Y)z

-

E Rji (x, y)z j=1

I

where x, y, z E TpM, c E R and jJj, J2, J3} is a canonical local basis for Then the Jacobi operator with respect to z E S(M) is given by

cg(z, z) id

on

spanjjjz, J2z, J3z},

cg(z, z) id

on

(span I J, z, j2z, j3z})-Ln z-L.

Q.

Rz

Thus the characteristic

cg(z, z))n-4

for all

z

Osserman. A nonflat

E

polynomial of R.,

S(M)

is

fz(t)

=

(t

-

Cg (Z'

Z)) 3 (t

and hence

quaternionip

a quaternionic space form is globally space form is isometric to any of the model

spaces

Pn (Q)

=

Sp'(n + 1) ISp'(n) x Sp(l),

H,,n (Q)

=

Sp'+' (n + 1) ISp'(n) x Sp(l)

(see [142].) In addition to the well-known

examples of semi-Riemannian Osserman above, there are some other examples which have no Riemannian analog. However, they may be considered as a kind of real version of both complex and quaternionic manifolds. manifolds described

1.3 Semi-Riemannian Pointwise and Global Osserman Manifolds

19

Example 1.3-4. A para-Kdhler manifold is a symplectic manifold locally diffeomorphic to a product of Lagrangian submanifolds. Such a product induces a decomposition of the tangent bundle TM into a Whitney sum of Lagrangian L W. By generalizing this definition, an subbundles L and L', that is, TM almost para-Hermitian manifold is defined to be an almost symplectic manifold (M, S?) whose tangent bundle splits into a Whitney sum of Lagrangian subbundles. This definition implies that the (1, l)-tensor'field J defined by J id, on M 7rL' is an almost paracomplex structure, that is, JI 7rL where for Y all such that f2(JX, JY) E FTM, X, irL and 7r'L f2(X, Y) S? induces 2-form and The TM L of onto the are L', respectively. projections M defined field a nondegenerate (0, 2)-tensor by g(X, Y) Q(X, JY), g on where X, Y E PTM. Now, by using the relation between the almost paracomplex and the almost sympl ctic structures on M, it follows that g defines a semi-Riemannian metric tensor of signature (n, n) on M and moreover, 0, where X, Y E FTM The special significance of g(JX, Y) + g(X, JY) the para-Kdhler condition is equivalently stated in terms of the parallelizability of the paracomplex structure with respect to the Levi-Civita conection of 0 [42]. g, that is, VJ A plane P is called paraholomorphic if it is left invariant by the action of paracomplex structure J, that is, JP C P. Now the paraholomorphic sectional, curvature H is defined by the restriction of the sectional curvature to paraholomorphic nondegenerate planes. A para-Kiffiler manifold (M, g, J) is called a paracomplex space form if (M, g, J) is of constant paraholomorphic sectional curvature. Hence, the curvature tepsor of (M, g, J) is given by [54] =

=

=

-

=

=

.

=

R(x, y)z where x, y,

z

E

TpM

and

c

=

-C [RO(x, y)z

Rj(x, y)z],

+

4

E R. Then the Jacobi

operator of

z

E

S(M)

is

given by

cg(z, z) id

on

span I Jz},

z) id llg(z, 4

on

(spanIz, iz})-L

n

z-L.

polynomial of Rz is fz(t) (t cg(z,z))(t S(M) and hence a paracomplex space form is globally cP(Z, Z))n-2 Osserman. Nonflat complete and simply connected paracomplex space forms are isometric to the symmetric spaces SL(n, R)/SL(n 1, R) x R (see [62].)

Thus the chara'cteristic for all

z

=

-

-

E

-

Example to be

a

exists

a

1.3.5. A

paraquaternionic

3-dimensional bundle of local basis

JJ1, J2, J3}

j2i where el

=

62

=

signature (2n, 2n) if it satisfies

=

-63

for

Q

on a

fields

manifold M is defined on

M such that there

Q satisfying

eiid, =

structure

(1, 1)-tensor

J1J2

=

-J2J1

A-

1. A semi-Riemannian metric tensor g on M of adapted to the paraquaternionic structure Q

is said to be

g(JiX, Y)

+

g(X, JiY)

=

0 for all

X, Y

E FTM and any local

20

1. The Osserman Conditions in Semi-Riemannian

basis for

Geometry

Q. Moreover, (M, g, Q) is called a paraquaternionic Kdhler manifold Q is parallel with respect to the Levi-Civita connection of g (cf.

if the bundle

[14], [62].) Let E

TpM

(M, g, Q)

be

determines

a

a

paraquaternionic Kdhler manifold. Then any

4-dimensional

subspace Q(x)

=

vector

x

spanjx, Jlx, j2x, j3x}

which remains invariant under the action of the paraquaternionic structure. We call it the Q-section determined by x. Note that restriction of the

/the

metric tensor g to any Q-section is indefinite of signature (2,2) or totally degenerate, where the latter case occurs if and only if the Q-section is generated by a null vector. If the sectional curvature of nondegenerate planes in

Q(x)

is a constant c(x), where x E TpM is nonnull, we call this constant c(x) the paraquaternionic sectional curvature of (M, g) with respect to x. A paraquaternionic Khhler manifold (M, g, Q) is called a paraquaternionic

form if (M, g, Q) is of constant paraquaternionic Hence, its curvature tensor satisfies space

sectional curvature.

3 c

R(x, y)z

R'(x, y)z +

=

4

where 61 = 62 = -63 x, y, z E with respect to z E S(M) is given

=

cg(z,z)id f

49 (z,z)id

Thus the characteristic

cg(Z' Z))n-4 :i

for all

z

E

globally Osserman. Complete and simply isometric to the

and

E R. Then the Jacobi

qpanjjjz,J2z,J3z

on

(spanfjjz,j2z, j3zl)j-

polynomial of R,

S(M)

c

operator

by

on

-

Rz

TpM

E EiRji (x, y)z

and hence

a

is

fz(t)

=

(t

n

zj-.

-

paraquaternionic

Cg(Z,Z))3(t space form is

connected nonflat paraquaternionic space forms

symmetric

spaces

Sp(n; R)ISp(l; R)

x

Sp(n

-

1; R).

are