# Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators

**ABSTRACT** We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n,n) for any n which is at least 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither satisfy the third Gray identity nor are they integrable.

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- Far East Journal of Mathematical Sciences 01/2014; 94(1):1-11.
- SourceAvailable from: Abdoul Salam Diallo[Show abstract] [Hide abstract]

**ABSTRACT:**The Riemannian extension of torsion free affine manifolds ( M, ∇) is an important method to produce pseudo-Riemannian manifolds. It is know that, if the manifold ( M, ∇) is a torsion-free affine two-dimensional manifold with skew symmetric tensor Ricci, then ( M, ∇) is affine Osserman manifold. In higher dimensions the skew symmetric of the tensor Ricci is a necessary but not sufficient condition for a affine connection to be Osserman. In this paper we construct affine Osserman connection with Ricci flat but not flat and example of Osserman pseudo-Riemannian metric of signature (3, 3) is exhibited.Global Journal of Advanced Research on Classical and Modern Geometries. 10/2013; 2(2):69-75. - SourceAvailable from: Abdoul Salam Diallo[Show abstract] [Hide abstract]

**ABSTRACT:**An affine manifold ( M , ∇ ) is affine Osserman if the eigenvalues of the affine Jacobi operators vanish. In the present paper, we exhibit examples of affine Osserman connections which are Ricci flat and which are not Ricci flat on R3.Far East Journal of Mathematical Sciences 12/2014; 94(1):1-11.

Page 1

arXiv:1007.2569v1 [math.DG] 15 Jul 2010

HIGHER-DIMENSIONAL OSSERMAN METRICS WITH

NON-NILPOTENT JACOBI OPERATORS

E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO

Abstract. We exhibit Osserman metrics with non-nilpotent Jacobi operators

and with non-trivial Jordan normal form in neutral signature (n,n) for any

n ≥ 3. These examples admit a natural almost para-Hermitian structure and

are semi para-complex Osserman with non-trivial Jordan normal form as well;

they neither satisfy the third Gray identity nor are they integrable.

1. Introduction

A pseudo-Riemannian manifold (M,g) is said to be Osserman if the eigenvalues

of the Jacobi operatorsgJ(X) : Y →gR(Y,X)X are constant on the unit pseudo-

sphere bundles S±(TM,g). Any isotropic space is Osserman and the converse is

true in the Riemannian (dim M ?= 16) [6, 25, 26] and Lorentzian [1, 15] settings.

However, there exist many non-symmetric Osserman pseudo-Riemannian metrics

in other signatures (cf. [17, 20] and the references therein). Since the eigenvalue

structure need not determine the conjugacy class of a self-adjoint operator in the

indefinite setting, a pseudo-Riemannian manifold is called Jordan-Osserman if the

Jordan normal form of the Jacobi operators is constant on S±(TM,g). Osserman

metrics are Einstein and thus of constant sectional curvature in dimensions 2 and

3. The special significance of the four-dimensional case relies on the fact that a

four-dimensional algebraic curvature tensor is Osserman if and only if it is Einstein

and self-dual; the classification of all four-dimensional Osserman metrics of neutral

signature (2,2) is almost complete [2, 5, 8, 10, 11, 14, 18, 19].

The situation is much more difficult in higher dimensions where only some par-

tial results are known [17]. The structure of a Jordan-Osserman algebraic curvature

tensor strongly depends on the signature (p,q) of the metric tensor. For example,

the Jacobi operators of a spacelike Jordan-Osserman algebraic curvature tensor are

necessarily diagonalizable whenever p < q [21]. In the neutral case (p = q), the

Jordan normal form can be arbitrarily complicated [22]. However in the geomet-

ric setting, less is known as with the exception of some six-dimensional examples

of Osserman metrics with non-nilpotent Jacobi operators [5], all previously known

examples of Osserman metrics have either diagonalizable or nilpotent Jacobi opera-

tors (see [17, 20, 23] and the references therein). There are, of course, other natural

operators beside the Jacobi operator that one could examine – see, for example, the

discussion of the spectral geometry of the skew-symmetric curvature operator [24].

The purpose of this paper is to investigate further the construction in [5], showing

that for any affine Osserman manifold (M,D), the cotangent bundle T∗M equipped

with the modified Riemannian extension is an Osserman manifold whose Jacobi

operators are, in general, neither diagonalizable nor nilpotent.

Key words and phrases. Affine connection, almost para-Hermitian, Einstein, Jacobi operator,

non-integrable para-complex structure, modified Riemannian extension, Osserman manifold, third

Gray identity, Walker metric.

2010 Mathematics Subject Classification. 53C50, 53B30.

Supported by projects MTM2009-07756 and INCITE09 207 151 PR (Spain).

1

Page 2

2 E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO

1.1. Affine Osserman manifolds. Let (M,D) be an n-dimensional affine mani-

fold, i.e., D is a torsion-free connection on the tangent bundle of a smooth manifold

M of dimension n. LetDR(X,Y ) := DXDY − DYDX− D[X,Y ]be the associated

curvature operator. We say that (M,D) is affine Osserman if the Jacobi operators

are nilpotent [16], i.e. 0 is the only eigenvalue ofDJ(·) on TM.

There are corresponding local notions which are important; an affine manifold

(M,D) is said to be affine Osserman at P ∈ M ifDJ(·) is nilpotent on TPM.

Similarly a pseudo-Riemannian manifold (M,g) is said to be Osserman at P ∈ M

if the eigenvalues ofgJ(·) are constant on S±(TPM,g). Clearly (M,D) is affine

Osserman if and only if (M,D) is affine Osserman at every point P ∈ M. Similarly

(M,g) is Osserman if and only if (M,g) is Osserman at every point P ∈ M and if

the eigenvalue structure and eigenvalue multiplicities are independent of P.

1.2. Riemannian extensions. Let N := T∗M be the cotangent bundle of an

n-dimensional manifold M, let σ : N → M be the natural projection, and let

Z(N) be the zero section. If x = (x1,...,xn) are local coordinates on M, let

x′= (x1′,...,xn′) be the associated dual coordinates on the fiber where we expand

a 1-form ω as ω = xi′dxi; we shall adopt the Einstein convention and sum over

repeated indices henceforth. The following natural distribution will play a crucial

role in our analysis:

Y := Span{∂x1′,...,∂xn′} = ker(σ∗).

For each vector field X = Xi∂xion M, the evaluation map ιX(P,ω) = ω(XP)

defines a function on N which, in local coordinates, is given by

ιX(xi,xi′) = xi′Xi.

Vector fields on N are characterized by their action on functions ιX; the complete

lift XCof a vector field X on M to N is characterized by the identity

XC(ιZ) = ι[X,Z],for all Z ∈ C∞(TM).

Moreover, since a (0,s)-tensor field on N is characterized by its evaluation on com-

plete lifts of vector fields on M, for each tensor field S of type (1,1) on M, we define

a 1-form ιS on N which is characterized by the identity

(ιS)(XC) = ι(SX).

Let (M,D) be an affine manifold. The Riemannian extension gDis the pseudo-

Riemannian metric on N of neutral signature (n,n) characterized by the identity:

gD(XC,YC) = −ι(DXY + DYX).

If u and v are cotangent vectors, let u ◦ v :=1

2(u ⊗ v + v ⊗ u). Expand

D∂xi∂xj=DΓijℓ∂xℓ

to define the Christoffel symbolsDΓ of D. One then has:

gD= 2dxi◦ dxi′− 2xk′DΓijkdxi◦ dxj.

Riemannian extensions were originally defined by Patterson and Walker [27] and

further investigated in relating pseudo-Riemannian properties of N with the affine

structure of the base manifold (M,D). Moreover, Riemannian extensions were also

considered in [16] in relation to Osserman manifolds (see also [9]).

The modified Riemannian extension is the neutral signature metric on N defined

by (see [5] for a more general construction)

gN:= ιId◦ιId+gD.

In a system of local coordinates one has

(1) gN= 2dxi◦ dxi′+ {xi′xj′ − 2xk′DΓijk}dxi◦ dxj.

Page 3

HIGHER-DIMENSIONAL OSSERMAN METRICS3

The manifold (N,gN) is a Walker manifold where the parallel degenerate distribu-

tion is in this instance given by Y [29]. There is a canonical almost para-Hermitian

structure J, i.e. a linear map of TN so that J2= Id and J∗gN= −gN, which will

play a crucial role in our analysis. In local coordinates, it is given by

(2)

J : ∂xi→ ∂xi− {xi′xj′ − 2xk′DΓijk}∂xj′

and

J : ∂xi′→ −∂xi′.

The case that D is flat is of particular interest. Let˜CP be para-complex projec-

tive space of constant para-holomorphic sectional curvature +1. Then [5]:

Theorem 1.1. If D is flat, then (N,gN) is isomorphic to˜CP.

Para-complex projective space˜CP is Jordan-Osserman with diagonalizable Ja-

cobi operators. Let g˜CP(ξ,ξ) = ±1. Then the eigenvalues ofg˜CPJ(ξ) are ±(0,1,1

with multiplicities (1,1,2n − 2), respectively. If (M,D) is affine Osserman, then

gN can be viewed as a deformation of g˜CP. This introduces Jordan normal form

into the Jacobi operator, but does not change the eigenvalue structure in the affine

Osserman context:

4)

Theorem 1.2. Let (M,D) be an affine manifold.

(1) If (M,D) is affine Osserman at P ∈ M, then (N,gN) is Osserman at any

Q ∈ σ−1(P). The eigenvalues ofgNJ(·) on S±(TQN,gN) are ±(0,1,1

multiplicities (1,1,2n − 2), respectively.

(2) If (M,D) is affine Osserman, then (N,gN) is Osserman.

4) with

Let P(N) be the bundle over N of non-degenerate J-invariant tangent 2-planes.

If π ∈ P(N), choose ξ ∈ S+(π,gN) and, following [28], define the para-complex

Jacobi operator to be:

gNJ(π) :=gNJ(ξ) −gNJ(Jξ);

this operator is independent of the particular ξ chosen. Higher order Jacobi op-

erators of this nature were first considered by Stanilov and Videv [28] in the real

setting. One says that (N,gN,J) is semi para-complex Osserman ifgNJ(·) has con-

stant eigenvalues on P(N); if additionallygNJ(π) commutes with J for all π ∈ P(N),

then (N,gN,J) is said to be para-complex Osserman – this implies D is flat by

Theorem 1.6 so this condition is not particularly interesting in the setting we are

considering.

Theorem 1.3. Let (M,D) be an affine manifold. Let π ∈ P(N). The eigenvalues

ofgNJ(π) are (1,1

2) with multiplicities (2,2n−2), respectively, and any Jordan block

forgNJ(π) has size at most 2 × 2; (N,gN,J) is semi para-complex Osserman.

These examples provide genuinely new phenomena. The following result shows

that Jordan normal form ofgNJ can be quite complicated; it also shows that (N,gN)

need not be Jordan-Osserman:

Theorem 1.4. Let r ≥ 2 and let U be an r×r lower triangular matrix. There exists

an affine Osserman manifold (M,D) of dimension r + 1, there exists Q ∈ Z(N),

and there exist ξi∈ S+(TQN,gN) for i = 1,2 so that:

(1)gNJ(ξ1) is diagonalizable.

(2) Relative to a suitable basis for TQN,

gNJ(ξ2) = 0 · Id1⊕1 · Id1⊕(1

4· Idr+U) ⊕ (1

4· Idr+Ut).

There also are non-trivial examples in the para-complex setting:

Theorem 1.5. Let n ≥ 3. There exists an affine Osserman manifold (M,D) of

dimension n so that (N,gN,J) is not Jordan semi para-complex Osserman, and so

that the para-complex Jacobi operators are not always diagonalizable.

Page 4

4E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO

One says that an almost para-Hermitian manifold (A,gA,J) satisfies the third

Gray identity if

(3)

gAR(X,Y,Z,W) =gAR(JX,JY,JZ,JW)for allX,Y,Z,W .

An almost para-Hermitian manifold (A,gA,J) is integrable if there exist local coor-

dinates (u1,...,un,v1,...,vn) centered at any given point of A so that

J∂ui= ∂vi

and

J∂vi= ∂ui

or, equivalently [7], if the Nijenhuis tensor NJvanishes where

(4)NJ(X,Y ) := [X,Y ] − J[JX,Y ] − J[X,JY ] + [JX,JY ].

Theorem 1.6. Let (M,D) be an affine manifold. The following conditions are

equivalent:

(1) (M,D) is flat.

(2) (N,gN,J) is integrable.

(3) (N,gN,J) satisfies the third Gray identity.

(4) JgNJ(π) =gNJ(π)J for all π ∈ P(N).

Theorem 1.2 was first discovered in low dimensions using a computer assisted

calculation; subsequently the general case was derived. Here is a brief outline to

the paper. In Section 2, we prove Theorem 1.2 and Theorem 1.3. In Section 3,

we construct various examples to demonstrate Theorem 1.4 and Theorem 1.5. We

conclude the paper in Section 4 by establishing Theorem 1.6.

2. The eigenvalue structure

We begin our discussion with a technical result. Although well known, we include

the proof to keep our discussion as self-contained as possible. If D is an arbitrary

connection on TM, the torsion tensor T ∈ Λ2(T∗M) is defined by:

T (X,Y ) := DXY − DYX − [X,Y ].

Lemma 2.1. Let D be an arbitrary connection on TM. Let P ∈ M. The following

conditions are equivalent:

(1) There exist local coordinates x = (x1,...,xn) centered at P soDΓ(P) = 0.

(2) The torsion tensor T = 0 vanishes at P.

Proof. Let x = (x1,...,xn) be a system of local coordinates on M. The torsion

tensor T vanishes at P if and only ifDΓijk(P) =DΓjik(P). In particular, if there

exists a coordinate system whereDΓ(P) = 0, then necessarily T vanishes at P.

Thus Assertion (1) implies Assertion (2). Conversely, assume that Assertion (2)

holds. Define a new system of coordinates by setting:

zi= xi+1

2aijkxjxk

where aijk= aikj remains to be chosen. As ∂xj= ∂zj+ aljixi∂zl,

D∂xi∂xj(0) = D∂zi∂zj(0) + alji∂zl(0).

Assertion (1) now follows by setting alij:=DΓijl; the fact that alij= aljiis exactly

the assumption that D is torsion-free at P.

?

Let (M,D) be an affine manifold, let Q ∈ N = T∗M, and let P := σQ ∈ M. As

D is torsion-free, we may apply Lemma 2.1 to make a change of coordinates on M

so thatDΓ(P) = 0. Letg˜CPR be the curvature tensor of the metric

g˜CP:= 2dxi◦ dxi′+ xi′xj′dxi◦ dxj.

This metric is not invariantly defined but depends on the coordinates chosen.

We note gN(Q) = g˜CP(Q). We set2R :=gNR−g˜CPR. Let {gNJ,g˜CPJ,2J,DJ,gDJ}

be the associated Jacobi operators.

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HIGHER-DIMENSIONAL OSSERMAN METRICS5

Lemma 2.2. Let (M,D) be an affine manifold, let Q ∈ N, and let P = σQ.

Let ξ ∈ TQN, and let a = σ∗(ξ) ∈ TPM. Then we have, relative to the natural

coordinate frame {∂x1,...,∂xn,∂x1′,...,∂xn′} for TN, that:

2J(ξ) =

?

DJ(a)

⋆

0

DJ(a)t

?

.

Proof. DecomposegNR =g˜CPR+gDR+ER whereER is an additional term measuring

the interactions between the metrics g˜CPand gD in the combined metric gN of

Equation (1). By [4] (page 54 Equation (3.6)),gDJ(ξ) has the form given in the

Lemma. We will complete the proof by showing that the additional interaction

terms define a Jacobi operator withEJ(ξ) : Span{∂xi} → Y, i.e.

EJ(ξ) =

?

0

⋆

0

0

?

.

We use dimensional analysis. Define:

Deg(xi) = −1,

Deg(xi′) = +1,

Deg(DΓ⋆⋆⋆) = +1.

Deg(dxi) = −1,

Deg(dxi′) = +1,

Deg(∂xi) = +1,

Deg(∂xi′) = −1,

We consider the rescaling xi→ c−1xi; this induces a dual rescaling xi′ → cxi′. If

Θ is a tensor of degree k, then Θ → ckΘ under this rescaling. For example, the

metrics g˜CP, gD, and gN are all homogeneous of degree 0; thus they are invariant

under this rescaling.

It is clear that the Christoffel symbols of the first kind decouple:

gNΓ⋆⋆⋆=g˜CPΓ⋆⋆⋆+gDΓ⋆⋆⋆.

SinceDΓ vanishes at P, we must have at least one ∂xiderivative ofDΓ in computing

gNR; thus any variable involvingDΓ has degree at least +2. In raising indices in

the metric gN rather than in the metric gD, we must take into consideration the

xi′xj′dxi◦dxjterm; thus interactions of this form involvingDΓ contribute terms of

degree at least 2 + 2 = 4 toER. We also have interaction terms which are bilinear

ing˜CPΓ⋆⋆⋆ andgDΓ⋆⋆⋆ after an index is raised in each factor. Such terms are at

least quadratic in {xi′} and linear in ∂xiDΓ and consequently have total degree at

least +4. We therefore conclude that any monomial of the interaction tensorER has

total degree at least +4. The degree ofERw1w2w3w4is ±1 ± 1 ± 1 ± 1; such a term

has degree at most +4 and the degree is exactly +4 if and only if w1∈ {1,...,n},

w2∈ {1,...,n}, w3∈ {1,...,n}, and if w4∈ {1′,...,n′}. ConsequentlyER defines

a Jacobi operator mapping Span{∂xi} to Y.

?

Let ξ ∈ S+(TQN,gN) and let ξ1:= Jξ ∈ S−(TQN,gN). Let Eλ(ξ) (resp. Eλ(ξ1))

be the eigenspaces ofg˜CPJ(ξ) (resp.g˜CPJ(ξ1)) for the eigenvalue λ ∈ {0,1,1

for λ ∈ {0,−1,−1

4}). Set

4} (resp.

(5)

E0(ξ) = ξ · R = E−1(ξ1),

E 1

E1(ξ) = ξ1· R = E0(ξ1),

4(ξ) = {E0(ξ) ⊕ E1(ξ)}⊥= {E−1(ξ1) ⊕ E0(ξ1)}⊥= E−1

S(ξ) := Y ∩ E 1

4(ξ1),

4(ξ),U(ξ) := E0(ξ) ⊕ Y .

We then have TQN = E0(ξ) ⊕ E1(ξ) ⊕ E 1

4(ξ).

Lemma 2.3. Let (M,D) be an affine manifold. Let Q ∈ N. Let ξ ∈ S+(TQN,gN).

(1) Y = (ξ1− ξ) · R + S(ξ).

(2)2J(ξ)Y ⊂ S(ξ).

(3)g˜CPJ(ξ)U(ξ) ⊂ U(ξ) and2J(ξ)U(ξ) ⊂ U(ξ).

Page 6

6E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO

Proof. Equation (2) implies ξ1− ξ ∈ Y. We can choose an orthonormal basis for

E 1

4(ξ) of the form

{e+

1,...,e+

n−1,Je+

iare timelike. We prove Assertion (1) by

1,...,Je+

n−1}

where the e+

noting that we have the following basis for Y:

iare spacelike and the Je+

{ξ − ξ1,e+

1− Je+

1,...,e+

n−1− Je+

n−1}.

Suppose that Assertion (2) fails. We argue for a contradiction. Choose η ∈ Y

so that2J(ξ)η / ∈ S(ξ). By Lemma 2.2,2J(ξ)η ∈ Y. By Assertion (1), there exists

c ?= 0 so that

2J(ξ)η = c(ξ − ξ1) + η1

forη1∈ S(ξ).

Thus cξ ∈ E1(ξ)+Range(2J(ξ))+E 1

establishes Assertion (2). To prove Assertion (3), express:

4(ξ) ⊂ E0(ξ)⊥which is false; this contradiction

(6)U(ξ) = E0(ξ) ⊕ Y = ξ · R ⊕ (ξ1− ξ) · R ⊕ S(ξ) = ξ · R ⊕ ξ1· R ⊕ S(ξ).

Asg˜CPJ(ξ)ξ = 0, asg˜CPJ(ξ)ξ1= ξ1, and as S(ξ) ⊂ E 1

As2J(ξ)ξ = 0 and as2J(ξ)Y ⊂ Y,2J(ξ) preserves U(ξ) as well.

4(ξ),g˜CPJ(ξ) preserves U(ξ).

?

We examine the eigenvalue structure:

Lemma 2.4. Let (M,D) be an affine manifold. Let Q ∈ N. Let ξ ∈ S+(TQN,gN).

Assume (M,D) is affine Osserman at P = σ(Q). If there is 0 ?= η ∈ TQN ⊗RC

withgNJ(ξ)η = µη, then:

(1) If η ?∈ U(ξ) ⊗RC, then µ =1

4.

(2) If η ∈ U(ξ) ⊗RC and if η ?∈ S(ξ) ⊗RC, then µ = 0 or µ = 1.

(3) If η ∈ S(ξ) ⊗RC, then µ =1

4.

(4) Spec{gNJ(ξ)} ⊂ {0,1,1

4}.

Proof. By Lemma 2.2,2J(ξ) is nilpotent since (M,D) is affine Osserman at P. By

Lemma 2.3, U(ξ) is preserved byg˜CPJ(ξ) and by2J(ξ). Thus, there are induced

operatorsg˜CP˜ J(ξ),2˜ J(ξ), andgN˜ J(ξ) =g˜CP˜ J(ξ) +2˜ J(ξ) on the quotient space:

V(ξ) := {TQN/U(ξ)} ⊗RC.

If η / ∈ U(ξ) ⊗RC, then ˜ η ∈ V(ξ), ˜ η ?= 0 andgN ˜ J(ξ)˜ η = µ˜ η. By Equation (6),

V(ξ) = {E 1

4(ξ)/S(ξ)} ⊗RC.

Consequently,g˜CP˜ J(ξ) =1

gN˜ J(ξ) has only the eigenvalue1

To prove Assertion (2), suppose there exists 0 ?= η ∈ U(ξ) ⊗RC such that

η / ∈ S(ξ) ⊗RC andgNJ(ξ)η = µη. By Lemma 2.3, S(ξ) is preserved byg˜CPJ(ξ) and

2J(ξ). Thus there are induced operators that we again denote byg˜CP˜ J(ξ),2˜ J(ξ),

andgN˜ J(ξ) =g˜CP˜ J(ξ) +2˜ J(ξ) on the quotient space:

4Id. Since2˜ J(ξ) is nilpotent andgN˜ J(ξ) =1

4. Thus µ =1

4Id+2˜ J(ξ),

4. This establishes Assertion (1).

W(ξ) := {U(ξ)/S(ξ)} ⊗RC.

Since ˜ η ?= 0, µ is an eigenvalue ofgN˜ J(ξ). By Equation (6), W(ξ) =˜ξ · R ⊕˜ξ1· R.

By Lemma 2.3,2J(ξ)ξ = 0 and2J(ξ)ξ1=2J(ξ)(ξ1−ξ) ∈ S(ξ) and thus2˜ J(ξ) = 0.

Sinceg˜CP˜ J(ξ)˜ξ = 0 andg˜CP˜ J(ξ)˜ξ1 =˜ξ1 we havegN˜ J(ξ)˜ξ = 0 andgN˜ J(ξ)˜ξ1 =˜ξ1.

Thus µ ∈ {0,1}. Assertion (2) follows.

To prove Assertion (3), we note thatg˜CPJ(ξ) =1

nilpotent and preserves S(ξ). Assertion (4) follows from Assertions (1)-(3).

4Id on S(ξ) and that2J(ξ) is

?

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HIGHER-DIMENSIONAL OSSERMAN METRICS7

Proof of Theorem 1.2. Let (M,D) be an affine manifold which is affine Osserman

at P ∈ M. Choose local coordinates on M soDΓ(P) = 0. Let D0 be the flat

torsion-free connection defined on a neighborhood of P whose Christoffel symbols

vanish in these coordinates. Set

Dε:= εD + (1 − ε)D0

to define a 1-parameter family of metrics gε

g0

P. Thus Lemma 2.4 implies Spec{gε

tiplicities are unchanged during this perturbation as well. Taking ε = 0 yields the

desired multiplicities and establishes Assertion (1) of Theorem 1.2 for ξ spacelike.

We now use results of [17] to see that spacelike Osserman implies timelike Osserman

and to relate the eigenvalues and eigenvalue multiplicities on S+(TQN,gN) to the

eigenvalues and eigenvalue multiplicities on S−(TQN,gN); alternatively, of course,

one could simply proceed directly as well. This proves Assertion (1) of Theorem

1.2; Assertion (2) follows from Assertion (1).

Ninterpolating between g1

N= gN and

N= g˜CP. SinceDεR(P) = ε·DR(P), all the connections Dεare affine Osserman at

NJ(ξ)} ⊂ {0,1,1

4} for all ε so the eigenvalue mul-

?

Proof of Theorem 1.3. Let (M,D) be an affine manifold, let Q ∈ N, let π ∈ PQ(N),

and let ξ ∈ S+(π,gN). Let a = σ∗ξ = σ∗J(ξ). By Lemma 2.2,

?

⋆

?

⋆π

2J(ξ) =

DJ(a)0

DJ(a)t

?

,

2J(Jξ) =

?

DJ(a)

⋆1

0

DJ(a)t

?

,

2J(π) =2J(ξ) −2J(Jξ) =

00

0

?

.

Thus Range(2J(π)) ⊂ Y,2J(π)Y = 0, and2J(π) is nilpotent. Choose a basis

{e1,e2,f1,...,f2n−2} for TQN so Span{e1,e2} is the +1 eigenspace ofg˜CPJ(π) and

so Span{f1,...,f2n−2} is the +1

(gNJ(π) − Id)ei=2J(π)ei∈ Yand

2eigenspace ofg˜CPJ(π). We compute:

(gNJ(π) −1

2Id)fi=2J(π)fi∈ Y .

By Equation (5),

g˜CPJ(π) = Id on (ξ − Jξ) · R

and

g˜CPJ(π) =1

2Id on S(ξ).

Consequently by Lemma 2.3,g˜CPJ(π)Y ⊂ Y. Since2J(π) = 0 on Y, this implies

gNJ(π)Y ⊂ Y. We may now conclude:

Range{(gNJ(π) − Id) · (gNJ(π) −1

2Id)} ⊂ Y .

Since2J(π) = 0 on Y,gNJ(π) =g˜CPJ(π) on Y and consequently by Equation (5)

and Lemma 2.3,

(gNJ(π) − Id) · (gNJ(π) −1

(gNJ(π) − Id)2·(gNJ(π) −1

2Id)Y = {0}

2Id)2= {0}.

so

Consequently Spec{gNJ(π)} ⊂ {1

blocks. As in the proof of Theorem 1.2, we set Dε:= εD+(1−ε)D0to construct a 1-

parameter family of semi para-complex Osserman metrics gε

g1

N= g˜CP. Since the eigenvalues are unchanged, the eigenvalue

multiplicities are unchanged. Consequently1

and 1 is an eigenvalue of multiplicity 2.

2,1} andgNJ(π) has only 1 × 1 or 2 × 2 Jordan

Ninterpolating between

N= gN and g0

2is an eigenvalue of multiplicity 2n−2

?

3. Examples

Throughout Section 3, we will take M = Rnfor some n and we will consider

a point Q ∈ Z(N) so thatDΓ(P) = 0 where P = σ(Q).

computations greatly. We will list only the possibly non-zero components of various

tensors up to the obvious Z2symmetries.

This simplifies the

Page 8

8E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO

Lemma 3.1. Let (M,D) be an affine manifold, let Q ∈ Z(N), and let P = σ(Q).

Assume thatDΓ(P) = 0.

(1) The possibly non-zero curvatures ofgDR(Q) aregDRijkl′(Q) =DRijkl(P).

(2) The non-zero curvatures ofg˜CPR(Q) are:

g˜CPRi,i′,i′,i(Q) = −1,

g˜CPRi,j′,i′,j(Q) =g˜CPRi,i′,j′,j(Q) = −1

2for i ?= j .

(3)gNR(Q) =g˜CPR(Q) +gDR(Q).

Proof. Let (u1,...,un) be coordinates on a pseudo-Riemannian manifold (U,gU).

Expand gU= gabdua◦dub. Suppose that the 1-jets of the functions gabvanish at a

point S of U. We then have

gURabcd(S) =1

2{∂ua∂ucgbd+ ∂ub∂udgac− ∂ua∂udgbc− ∂ub∂ucgad}(S).

We apply this observation to the setting at hand. Since x′andDΓ vanish at Q

and P, respectively, the 1-jets of gD, of g˜CP, and of gN vanish at Q. We establish

Assertion (1) by computing:

gDRijkl′(Q) =1

2{∂xj∂xℓ′(−2xh′DΓikh) − ∂xi∂xℓ′(−2xh′DΓjkh)}(Q)

DΓjkl− ∂xj

= {∂xi

DΓikl}(P) =DRijkl(P).

The proof of Assertion (2) and of Assertion (3) is similar.

?

Proof of Theorem 1.4. Let r ≥ 2 and let M := Rr+1. Let {x0,...,xr} be the usual

coordinates on M and {x0′,...,xr′} the dual fiber coordinates on T∗M. We let

indices a,b,c,d range from 1 through r and indices i,j,k,l range from 0 through

r. Let Uabbe a lower triangular matrix, i.e. Uab= 0 for b ≤ a. Let θ = θ(x0) be

a smooth function of 1 variable. Define a torsion-free connection D on TM with

non-zero Christoffel symbols:

DΓ0ab=DΓa0b= θUab.

The curvature is given by

DRijkl= ∂xi{DΓjkl} − ∂xj{DΓikl} + {DΓicl}{DΓjkc} − {DΓjcl}{DΓikc}.

Without loss of generality, we suppose i < j. The first term can play a role only

if i = k = 0. The second term plays no role. The third can play a role only if

i = k = 0. The final term plays no role. Thus possibly non-zero curvatures are:

DR0a0b= ∂x0{DΓa0b} + {DΓ0cb}{DΓa0c} = ∂x0θ · Uab+ θ2· UcbUac.

Let X ∈ TPM. As we must have 0 < a < b in the above relation,

DJ(X)∂xi∈ Span{∂xi+1,...,∂xr}.

Consequently (M,D) is affine Osserman. Assume θ(0) = 0 and ∂x0θ(0) = −1. We

set P = 0 and take Q = (0,0). We may then apply Lemma 3.1 to see:

gNR(Q) =g˜CPR(Q) +gDR(Q),

gNR(∂xi,∂xi′,∂xi′,∂xi)(Q) = −1,

gNR(∂xi,∂xj′,∂xi′,∂xj)(Q) =gNR(∂xi,∂xi′,∂xj′,∂xj)(Q) = −1

2

(i ?= j),

gNR(∂xi,∂xj,∂xk,∂xd′)(Q) =DRijkd(P).

First, take ξ1 :=

exhibits trivial Jordan normal form; the curvature of D plays no role. Next, we

consider ξ2:=

1

√2(∂x1+ ∂x1′). ThengNJ(ξ1) =g˜CPJ(ξ1) is diagonalizable and

1

√2(∂x0+ ∂x0′) and Jξ2=

1

√2(∂x0− ∂x0′). Then:

Page 9

HIGHER-DIMENSIONAL OSSERMAN METRICS9

g˜CPJ(ξ2)ξ2= 0,

g˜CPJ(ξ2)∂xa=1

g˜CPJ(ξ2)Jξ2= Jξ2,

g˜CPJ(ξ2)∂xa′=1

Proof of Theorem 1.5. Let n ≥ 3, let M = Rn, let P = 0, and let Q = (0,0). Let

θ = θ(x1) be a smooth function of 1 variable. Let θ1:= ∂x1θ; we suppose θ(0) = 0

and θ1(0) ?= 0. Let D be the affine connection whose only non-zero Christoffel

symbol isDΓ223= θ.Since θ = θ(x1) and since the only non-zero covariant

derivative is D∂x2∂x2= θ∂x3,DJ is nilpotent and (M,D) is affine Osserman as the

only non-zero curvature is:

gDJ(ξ2)ξ2= 0,

gDJ(ξ2)∂xa= Uab∂xb,

gDJ(ξ2)Jξ2= 0,

gDJ(ξ2)∂xa′= Uba∂xb′.

4∂xa,

4∂xa′,

?

DR(∂x1,∂x2)∂x2= θ1∂x3.

By Lemma 3.1,

gNR(∂xi,∂xi′,∂xi′,∂xi)(Q) = −1,

gNR(∂xi,∂xj′,∂xi′,∂xj)(Q) =gNR(∂xi,∂xi′,∂xj′,∂xj)(Q) = −1

gNR(∂x3′,∂x2,∂x2,∂x1)(Q) = θ1.

2

(i ?= j),

Let ξ :=

0,gNJ(ξ) =g˜CPJ(ξ) andgNJ(πξ) =g˜CPJ(πξ) are diagonalizable. Next consider

η :=1

Jη =1

1

√2(∂x1+∂x1′) ∈ S+(TQN,gN); Jξ =

1

√2(∂x1−∂x1′). As2J(ξ) =2J(Jξ) =

2(∂x1+ ∂x3+ ∂x1′+ ∂x3′) ∈ S+(TQN,gN),

2(∂x1+ ∂x3− ∂x1′− ∂x3′) ∈ S−(TQN,gN).

The only non-trivial components are provided by:

gDJ(η)∂x2=1

2θ1∂x2′,

gDJ(Jη)∂x2= −1

2θ1∂x2′,

gDJ(πη)∂x2= θ1∂x2′.

Since π2:= Span{∂x2,∂x2′} is contained both in the

in the1

exhibit non-trivial Jordan normal form.

1

4eigenspace ofg˜CPJ(η) and

2eigenspace ofg˜CPJ(πη), this analysis shows that bothgNJ(η) andgNJ(πη)

?

4. The third Gray identity, integrability, flatness, and para-complex

Osserman

In Section 4.1, we show (M,D) is flat implies (N,gN,J) is integrable and satisfies

the third Gray identity. In Section 4.2, we show (N,gN,J) is integrable implies

(M,D) is flat. In Section 4.3, we show (N,gN,J) satisfies the third Gray identity

implies (M,D) is flat. In Section 4.4, we show JgNJ(π) =gNJ(π)J for all π ∈ P(N)

if and only if (N,gN,J) satisfies the third Gray identity. This will complete the

proof of Theorem 1.6.

4.1. Flat geometry. If (M,D) is flat, then (N,gN,J) is isomorphic to˜CP by

Theorem 1.1;˜CP is integrable and satisfies the third Gray identity.

4.2. Integrability. Let (M,D) be an affine manifold. Suppose that the Nijenhuis

tensor NJof Equation (4) vanishes for the manifold (N,gN,J). Let P ∈ M. Choose

local coordinates on M so thatDΓ(P) = 0. Let Q ∈ σ−1(P). Then:

J∂xi= ∂xi− {xi′xa′ − 2xb′DΓiab}∂xa′,

J∂xj= ∂xj− {xj′xc′ − 2xd′DΓjcd}∂xc′,

[∂xi,∂xj] = 0,

J[J∂xi,∂xj] = 2xb′∂xjDΓiab∂xa′,

J[∂xi,J∂xj] = −2xb′∂xiDΓjab∂xa′,

[J∂xi,J∂xj]Q= {2xb′∂xiDΓjab− 2xb′∂xjDΓiab}Q∂xa′

+{xi′xa′∂xa′(xj′xc′) − xj′xa′∂xa′(xi′xc′)}Q∂xc′.

Page 10

10E. CALVI˜NO-LOUZAO, E. GARC´IA-R´IO, P. GILKEY, AND R. V´AZQUEZ-LORENZO

NJ(∂xi,∂xj)(Q) = 4xb′DRijab(P)∂xa′.

Since NJ = 0, we conclude (M,D) is flat. We note that NJ always vanishes on

Z(N). Thus for this computation it is necessary to take Q arbitrary.

4.3. The third Gray identity. Suppose that (N,gN,J) satisfies the third Gray

identity which is given in Equation (3). Let Q ∈ Z(N) and let P = σ(Q). Choose

coordinates on M soDΓ(P) = 0. We apply Lemma 3.1. Since˜CP satisfies the

third Gray identity, we concludegDR satisfies the third Gray identity at Q. Thus

gDR(∂xi,∂xj,∂xk,∂xl′)(Q) =gDR(J∂xi,J∂xj,J∂xk,J∂xl′)(Q)

gDR(∂xi,∂xj,∂xk,−∂xl′)(Q) = −gDR(∂xi,∂xj,∂xk,∂xl′)(Q).

ConsequentlygDR(Q) = 0. By Lemma 3.1, this impliesDR(P) = 0. Since Q, and

hence P, was arbitrary, (M,D) is flat.

=

4.4. The commutation relation JgAJ(·) =gAJ(·)J. The third Gray identity in

the complex setting is crucial – see, for example, the discussion in [3, 12, 13]; a purely

algebraic computation shows that this condition is equivalent to the condition that

gAJ(π) commutes with the almost complex structure for every complex 2-plane

π. This computation extends to show that an almost para-Hermitian manifold

(A,gA,J) satisfies the third Gray identity if and only ifgAJ(π)J = JgAJ(π) for all

π ∈ P(A). This completes the proof of Theorem 1.6.

?

References

[1] N. Blaˇ zi´ c, N. Bokan, and P. Gilkey, A note on Osserman Lorentzian manifolds, Bull. London

Math. Soc. 29 (1997), 227–230.

[2] N. Blaˇ zi´ c, N. Bokan, and Z. Raki´ c, Osserman pseudo-Riemannian manifolds of signature

(2,2), J. Aust. Math. Soc. 71 (2001), 367–395.

[3] M. Brozos-V´ azquez, E. Garc´ ıa-R´ ıo, and P. Gilkey, Relating the curvature tensor and the

complex Jacobi operator of an almost Hermitian manifold, Adv. Geom. 8 (2008), 353–365.

[4] M. Brozos-V´ azquez, E. Garc´ ıa-R´ ıo, P. Gilkey, S. Nikˇ cevi´ c, and R. V´ azquez-Lorenzo, The

geometry of Walker manifolds, Synthesis Lectures on Mathematics and Statistics 5, Morgan

& Claypool Publ., 2009.

[5] E. Calvi˜ no-Louzao, E. Garc´ ıa-R´ ıo, P. Gilkey, and R. V´ azquez-Lorenzo, The geometry of

modified Riemannian extensions, Proc. R. Soc. A 465 (2009), 2023–2040.

[6] Q. S. Chi, A curvature characterization of certain locally rank-one symmetric spaces, J. Diff.

Geom. 28 (1988), 187–202.

[7] V. Cort´ es, C. Mayer, T. Mohaupt, and F. Saueressig, Special geometry of Euclidean super-

symmetry I: vector multiplets, arXiv:hep-th/0312001.

[8] A. Derdzinski, Non-Walker Self-Dual Neutral Einstein Four-Manifolds of Petrov Type III, J.

Geom. Anal. 19 (2009), 301–357.

[9] A. Derdzinski, Connections with skew-symmetric Ricci tensor on surfaces, Results Math. 52

(2008), 223–245.

[10] J. C. D´ ıaz-Ramos, E. Garc´ ıa-R´ ıo, and R. V´ azquez-Lorenzo, New examples of Osserman met-

rics with nondiagonalizable Jacobi operators, Differential Geom. Appl. 24 (2006), 433–442.

[11] J. C. D´ ıaz-Ramos, E. Garc´ ıa-R´ ıo, and R. V´ azquez-Lorenzo, Four-dimensional Osserman met-

rics with nondiagonalizable Jacobi operators, J. Geom. Anal. 16 (2006), 39–52.

[12] A. Di Scala, and L. Vezzoni, Gray Identities, Canonical connection, and integrability,

arXiv:0802.2163v3.

[13] A. Di Scala, J. Lauret, and L. Vezzoni, Quasi-Kaehler Chern-Flat manifolds and complex

2-step nilpotent Lie algebras, arXiv:0911.56552v2.

[14] E. Garc´ ıa-R´ ıo, P. Gilkey, M. E. V´ azquez-Abal, and R. V´ azquez-Lorenzo, Four-dimensional

Osserman metrics of neutral signature, Pacific J. Math. 244 (2010), 21–36.

[15] E. Garc´ ıa-R´ ıo, D. N. Kupeli, and M. E. V´ azquez-Abal, On a problem of Osserman in

Lorentzian geometry, Differential Geom. Appl. 7 (1997), 85–100.

[16] E. Garc´ ıa-R´ ıo, D. N. Kupeli, M. E. V´ azquez-Abal, and R. V´ azquez-Lorenzo, Affine Osserman

connections and their Riemann extensions, Differential Geom. Appl. 11 (1999), 145–153.

[17] E. Garc´ ıa-R´ ıo, D. N. Kupeli, and R. V´ azquez-Lorenzo, Osserman manifolds in semi-

Riemannian geometry, Lecture Notes in Math. 1777, Springer, Berlin, 2002.

[18] E. Garc´ ıa-R´ ıo, M. E. V´ azquez-Abal, and R. V´ azquez-Lorenzo, Nonsymmetric Osserman

pseudo-Riemannian manifolds, Proc. Amer. Math. Soc. 126 (1998), 2771–2778.

Page 11

HIGHER-DIMENSIONAL OSSERMAN METRICS11

[19] E. Garc´ ıa-R´ ıo, and R. V´ azquez-Lorenzo, Four-dimensional Osserman symmetric spaces,

Geom. Dedicata 88 (2001), 147–151.

[20] P. Gilkey, Geometric properties of natural operators defined by the Riemannian curvature

tensor, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.

[21] P. Gilkey, and R. Ivanova, Spacelike Jordan-Osserman algebraic curvature tensors in the

higher signature setting, Differential Geometry, Valencia, 2001, 179–186, World Sci. Publ.,

River Edge, NJ, 2002.

[22] P. Gilkey, and R. Ivanova, The Jordan normal form of Osserman algebraic curvature tensors,

Results Math. 40 (2001), 192–204.

[23] P. Gilkey, R. Ivanova, and I. Stavrov, Jordan Szab´ o algebraic covariant derivative curvature

tensors, Recent advances in Riemannian and Lorentzian geometries (Baltimore, MD, 2003),

65–75, Contemp. Math., 337, Amer. Math. Soc., Providence, RI, 2003.

[24] S. Ivanov, and I. Petrova, Riemannian manifold in which the skew-symmetric curvature op-

erator has pointwise constant eigenvalues, Geom. Dedicata 70 (1998), 269–282.

[25] Y. Nikolayevsky, Osserman manifolds of dimension 8, Manuscripta Math. 115 (2004), 31–53.

[26] Y. Nikolayevsky, Osserman conjecture in dimension ?= 8,16, Math. Ann. 331 (2005), 505–522.

[27] E. M. Patterson, and A. G. Walker, Riemann extensions, Quart. J. Math., Oxford Ser. (2) 3

(1952), 19–28.

[28] G. Stanilov, and V. Videv, On a generalization of the Jacobi operator in the Riemannian

geometry, Annuaire Univ. Sofia Fac. Math. Inform. 86 (1992), 27–34.

[29] A. G. Walker, Canonical form for a Riemannian space with a parallel field of null planes,

Quart. J. Math. Oxford (2) 1 (1950), 69–79.

C-L, G-R, V-L: Department of Geometry and Topology, Faculty of Mathematics,

University of Santiago de Compostela, 15782 Santiago de Compostela, Spain

E-mail address: estebcl@edu.xunta.es, eduardo.garcia.rio@usc.es,

ravazlor@edu.xunta.es

G: Mathematics Department, University of Oregon, Eugene, Oregon 97403, USA

E-mail address: gilkey@uoregon.edu

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