## Abstract

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and gives necessary and sufficient conditions for a modified Riemannian extension to be Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3, 3), whose Jacobi operators have non-trivial Jordan normal form and which are not nilpotent. We present new four-dimensional results in Osserman geometry.

## 1. Introduction

Walker metrics are (generally) indecomposable pseudo-Riemannian metrics, which are not irreducible (i.e. they admit a null parallel distribution). They play a distinguished role in geometry and physics (Magid 1984; Bérard Bergery & Ikemakhen 1997; Honda & Tsukada 2004; Díaz-Ramos *et al*. 2006*b*; Derdzinski & Roter 2007; Law & Matsushita 2008; Alekseevsky *et al*. in press). Lorentzian Walker metrics have been studied extensively in physics literature as they constitute a background metric of pp-wave models (for example, Flores & Sánchez (2008) and references therein). From a purely geometric point of view, Lorentzian Walker metrics naturally appear in the investigation of the non-uniqueness of the metric for the Levi-Civita connection (Martin & Thompson 1993), a question that is trivial in the positive definite case. Moreover, Walker metrics are the underlying structure of many geometrical structures such as para-Kaehler and hypersymplectic structures (see Hitchin 1990; Cruceanu *et al*. 1996; Cortés *et al*. 2004; Ivanov & Zamkovoy 2005).

The simplest examples of non-Lorentzian Walker metrics are provided by the so-called Riemannian extensions. This construction, which relates affine and pseudo-Riemannian geometries, associates a neutral signature metric on *T*^{*}*M* to any torsion-free connection ∇ on the base manifold *M*. Riemannian extensions have been used both to understand questions in affine geometry and to solve curvature problems (e.g. Afifi 1954; Bonome *et al*. 1998; García-Río *et al*. 1999). It is a remarkable fact that Walker metrics satisfying some natural curvature conditions are locally Riemannian extensions, thus reducing the corresponding classification problem to a question in affine geometry, as shown in Derdzinski & Roter (2007), Derdzinski (2008) and Calviño-Louzao *et al*. (in press).

In this paper, we introduce a modification of the usual Riemannian extensions with special attention to the behaviour of their curvature. The geometry of modified Riemannian extensions is much less rigid than that of the Riemannian extensions, allowing the existence of many non-Ricci-flat Einstein metrics, which can be further specialized to be Osserman (i.e. the eigenvalues of the Jacobi operators are constant on the unit pseudo-sphere bundles) since their scalar curvature invariants do not vanish. In particular, we show that any paracomplex space form is locally a modified Riemannian extension where the corresponding torsion-free connection is necessarily flat (cf. theorem 2.2). This description seems to be well suited to further investigations, especially for the consideration of Lagrangian submanifolds of paracomplex space forms.

Modified Riemannian extensions turn out to be very useful in describing four-dimensional Walker geometry. Indeed, we show, in theorem 7.1, that any self-dual Walker metric is a modified Riemannian extension. As a consequence, a description of all four-dimensional Osserman metrics whose Jacobi operator has a non-zero double root of its minimal polynomial is given in theorem 7.3. This result, coupled with recent work of Derdzinski (2009), completes the classification of four-dimensional Osserman metrics. Finally, as an application of the four-dimensional results, one obtains a procedure to construct new Osserman metrics in higher dimensions, which has been an unsolved problem in the field (cf. theorem 2.3).

## 2. Summary of results

### (a) Affine geometry

Let *R*^{∇} be the curvature operator of a torsion-free connection ∇ on the tangent bundle of a smooth manifold *M* of dimension *n*; if *X* and *Y* are smooth vector fields on *M*, thenThe *Ricci tensor ρ*^{∇} is defined by contracting indicesIn contrast to the Riemannian setting, *ρ*^{∇} need not be a symmetric 2-tensor field. We refer to Bokan (1990) for further details concerning curvature decompositions. We denote the symmetric and antisymmetric Ricci tensors by

### (b) The modified Riemannian extensions

Let *Φ*∈*C*^{∞}(*S*^{2}(*T*^{*}*M*)) be a symmetric 2-tensor field and let *T*, *S* be (1, 1)-tensor fields on *M*. In §5, we will use these data to define (equation (5.2)) a neutral signature pseudo-Riemannian metric *g*_{∇,Φ,T,S} on the cotangent bundle *T*^{*}*M*, which is called the *modified Riemannian extension* and which is a *Walker metric*. The case *T*=*c* id and *S*=id is of particular importance in our treatment, and the corresponding metric will be denoted by *g*_{∇,c} if *Φ*=0 and by *g*_{∇,Φ,c} if *Φ*≠0.

### (c) Einstein geometry

We have the following result.

*The modified Riemannian extension g*_{∇,Φ,c} *on the cotangent bundle of an n-dimensional affine manifold is Einstein if and only if* , *provided that c*≠0.

### (d) Para-Kaehler geometry

Our fundamental result concerning such geometry is theorem 2.2, which illustrates the importance of the modified Riemannian extensions.

*A para-Kaehler metric of non-zero constant para-holomorphic sectional curvature c is locally isometric to the cotangent bundle of an affine manifold, which is equipped with the modified Riemannian extension g*_{∇,c}, *where* ∇ *is a flat connection*.

### (e) Osserman geometry

We can use the modified Riemannian extensions to exhibit the following examples that extend previous results in signature (2, 2) (see theorem 2.1 of Díaz-Ramos *et al*. (2006*a*))—it has long been a problem in this field to build examples of Osserman manifolds that were not nilpotent and which represented a non-trivial Jordan normal form.

*Let* *and let* ∇ *be the torsion-free connection, whose only non-zero Christoffel symbol is given by* . *Let g*≔*g*_{∇,1} *on T*^{*}*M*. *Then, g is an Osserman metric of signature* (3, 3) *with eigenvalues* (0, 1, 1/4, 1/4, 1/4, 1/4), *which is neither spacelike Jordan–Osserman nor timelike Jordan–Osserman at any point*. *The Jacobi operator is neither diagonalizable nor nilpotent for a generic tangent vector*.

### (f) Notational conventions

We shall let denote an affine manifold and *R* the associated curvature operator. Similarly, we shall let denote a pseudo-Riemannian manifold, ∇^{g} denote the associated Levi-Civita connection, and *R*^{g} the associated curvature operator. If *ξ*_{i}∈*T*^{*}*N*, then the symmetric product is defined by

### (g) Outline of the paper

In §3, we establish notation and recall some basic definitions in the geometry of the curvature operator. We shall also discuss Osserman, Szabó and Ivanov–Petrova geometry. In §4, we present a brief introduction to Walker geometry and, in §5, we define the modified Riemannian extensions and establish theorem 2.1. In §6, we discuss para-Kaehler geometry and prove theorem 2.2. In §7, we present some results in four-dimensional geometry with a description of four-dimensional Osserman metrics whose Jacobi operators have a non-zero double root of its minimal polynomial, as a generalization of paracomplex space forms (cf. theorem 7.3). We conclude §8 by proving theorem 2.3 and discussing some additional results for this six-dimensional example that relates to Szabó and Ivanov–Petrova geometry. Throughout, we shall adopt the *Einstein convention* and sum over repeated indices. We shall suppress many of the technical details in the interests of brevity in giving various proofs in this paper—further details are available from the authors upon request.

## 3. The geometry of the curvature operator

### (a) Osserman geometry

Let be the Jacobi operator on a pseudo-Riemannian manifold of signature (*p*, *q*). One says that is *timelike Osserman* (respectively *spacelike Osserman*) if the eigenvalues of are constant on the pseudo-sphere bundle of unit timelike (respectively spacelike) vectors; these are equivalent concepts if *p*>0 and *q*>0 (Blažić *et al*. 1997; García-Río *et al*. 1997, 1999). Similarly, we say that is *timelike Jordan*–*Osserman* or *spacelike Jordan*–*Osserman* if the Jordan normal form of is constant on the appropriate pseudo-sphere bundles; these are, in general, not equivalent concepts.

### (b) Szabó geometry

A pseudo-Riemannian manifold is said to be *Szabó* if the *Szabó operator* has constant eigenvalues on *S*^{±}(*TN*) (Gilkey *et al*. 2003). Any Szabó manifold is locally symmetric in the Riemannian (Szabó 1991) and the Lorentzian (Gilkey & Stavrov 2002) setting, but the higher signature case provides examples with nilpotent Szabó operators (cf. Gilkey *et al*. (2003) and references therein).

### (c) Ivanov–Petrova geometry

For any oriented non-degenerate 2-plane *π*, the *skew-symmetric curvature operator* of the Levi-Civita connection is defined by is a skew-adjoint operator, which is independent of the oriented basis {*X*, *Y*} of *π*. is said to be spacelike (respectively, timelike or mixed) *Ivanov*–*Petrova* if the eigenvalues of are constant on the appropriate Grassmannian (Ivanov & Petrova 1998; Zhang 2000). If *p*≥2 and *q*≥2, these are equivalent conditions (Gilkey 2001) so one simply says the metric is Ivanov–Petrova in this setting.

## 4. Walker geometry

A *Walker manifold* is a triple (*N*, *g*, ), in which *N* is an *n*-dimensional manifold, *g* is a pseudo-Riemannian metric on *N* and is an *r*-dimensional parallel null distribution (*r*>0).

### (a) Geometrical contexts

Walker metrics appear as the underlying structure of several specific pseudo-Riemannian structures. For instance, indecomposable metrics that are not irreducible play a distinguished role in investigating the holonomy of indefinite metrics. Those metrics are naturally equipped with a Walker structure (for example Bérard Bergery & Ikemakhen (1997) and references therein). Einstein hypersurfaces in indefinite real space forms with two-step nilpotent shape operators (Magid 1984) are Walker. Similarly, locally conformally flat manifolds with nilpotent Ricci operators are Walker manifolds (Honda & Tsukada 2004). Also, non-trivial conformally symmetric manifolds (i.e. neither symmetric nor locally conformally flat) may only occur in the pseudo-Riemannian setting and they are Walker manifolds (Derdzinski & Roter 2007).

### (b) Neutral signature Walker manifolds

Of special interest are those manifolds admitting a field of parallel null planes of maximum dimension *r*=*n*/2. This is the case of para-Kaehler (Ivanov & Zamkovoy 2005) and hypersymplectic structures (Hitchin 1990). Note that, in contrast to the non-degenerate case where the existence of a parallel plane field leads to a local de Rham decomposition, complementary distributions to a parallel degenerate plane field are not necessarily parallel (even not integrable). Moreover, the existence of a parallel complementary plane field is indeed equivalent to the existence of a para-Kaehler structure (Bérard Bergery & Ikemakhen 1997). Note that any four-dimensional Osserman manifold of neutral signature whose Jacobi operators have a non-zero double root of their minimal polynomial is necessarily Walker with a parallel field of planes of maximal dimensionality (Blažić *et al*. 2001; Díaz-Ramos *et al*. 2006*b*).

### (c) Walker coordinates

Walker (1950; see also the discussion in Derdzinski & Roter (2006)) constructed simplified local coordinates in this setting. We shall restrict our attention to a neutral signature (*p*, *p*). Let (*N*^{n}, *g*, ) be a Walker manifold of signature (*p*, *p*), where . There are local coordinates (*x*^{1}, …, *x*^{p}, *x*_{1′}, …, *x*_{p′}), so that(4.1)Here, *B* is a symmetric matrix of functions depending on the coordinates (*x*^{1}, …, *x*^{p}, *x*_{1′}, …, *x*_{p′}), and the parallel degenerate distribution is given by

### (d) The Christoffel symbols of ∇^{g}

We sum over 1≤*s*≤*p* as follows:and

### (e) The Riemann curvature tensor of ∇^{g}

We sum over 1≤*s*≤*p*, 1≤*t*≤*p* as follows:

and

## 5. Modified Riemannian extensions

### (a) The geometry of the cotangent bundle

We refer to Yano & Ishihara (1973) for further details concerning the material of this section. Let *T*^{*}*M* be the cotangent bundle of an *n*-dimensional manifold *M* and let *π*:*T*^{*}*M*→*M* be the projection. Let , where *p*∈*M* and denote a point of *T*^{*}*M*. Local coordinates (*x*^{i}) in a neighbourhood *U* of *M* induce coordinates (*x*^{i}, *x*_{i′}) in *π*^{−1}(*U*), where we decompose

For each vector field *X* on *M*, define a function byWe may expand *X*=*X*^{j}∂_{j} and express

*Let* *be smooth vector fields on T*^{*}*M*. *Then*, *if and only if* *for all smooth vector fields X*∈*C*^{∞}(*TM*).

Let *X*∈*C*^{∞}(*TM*) be a vector field on *M*. The *complete lift X*^{C} is, by lemma 5.1, characterized by the identityWe then have and consequently lemma 5.2.

*A* (0, *s*)-*tensor field on T*^{*}*M is characterized by its evaluation on complete lifts of vector fields on M*.

Let *T* be a tensor field of type (1, 1) on *M*, i.e. *T*∈*C*^{∞}(end(*TM*)). We define a 1-form *ιT*∈*C*^{∞}(*T*^{*}(*T*^{*}*M*)), which is characterized by the identity(5.1)

### (b) The Riemannian extension

Let ∇ be a torsion-free affine connection on *M*. The *Riemannian extension g*_{∇} is the pseudo-Riemannian metric *g*_{∇} on *N*≔*T*^{*}*M* of neutral signature (*n*, *n*) characterized by the identityLet give the Christoffel symbols of the connection ∇. Then,

Riemannian extensions were originally defined by Patterson & Walker (1952) and further investigated in Afifi (1954), thus relating pseudo-Riemannian properties of *T*^{*}*M* with the affine structure of the base manifold (*M*, ∇). Moreover, Riemannian extensions were also considered in García-Río *et al*. (1999) in relation to Osserman manifolds (see also Derdzinski 2008).

### (c) The modified Riemannian extension

Let *Φ*∈*C*^{∞}(*S*^{2}(*T*^{*}*M*)) be a symmetric (0, 2)-tensor field on *M*, and let *T*, *S*∈*C*^{∞}(end(*TM*)) be tensor fields of type (1, 1) on *M*. The modified Riemannian extension is the neutral signature metric on *T*^{*}*M* defined byIn a system of local coordinates, one has(5.2)The case where *T*=*c* id and *S*=id is important and plays a central role in our treatment. More precisely, ifthen one has, in a system of local coordinates, that(5.3)These metrics are Walker metrics on *T*^{*}*M*, where the tensor *B*_{ij}(*x*, *x*′) of equation (4.1) is a quadratic function of *x*′ (and affine if *c*=0). The distribution is parallel and null, and the scalar curvature is a suitable multiple (depending on the dimension) of the parameter *c*.

The modified Riemannian extensions *g*_{∇,Φ,0} have been used in Bonome *et al*. (1998) to construct Kaehler and para-Kaehler Osserman metrics with one-side bounded (para) holomorphic sectional curvature.

### (d) Proof of theorem 2.1

Let and let *τ*^{g} be the scalar curvature. The trace-free Ricci tensor can then be determined to beTheorem 2.1 now follows. ▪

## 6. Para-Kaehler manifolds

A para-Kaehler manifold is a symplectic manifold *N* admitting two transversal Lagrangian foliations (see Cruceanu *et al*. 1996; Ivanov & Zamkovoy 2005). Such a structure induces a decomposition of the tangent bundle *TN* into the Whitney sum of Lagrangian subbundles *L* and *L*′, i.e. *TN*=*L*⊕*L*′. By generalizing this definition, an almost para-Hermitian manifold is defined to be an almost symplectic manifold (*N*, *Ω*), whose tangent bundle splits into the Whitney sum of the Lagrangian subbundles. This definition implies that the (1, 1)-tensor field *J* defined by *J*=*π*_{L}−*π*_{L′} is an almost paracomplex structure, i.e. *J*^{2}=id on *N*, such that *Ω*(*JX*, *JY*)=−*Ω*(*X*, *Y*) for all vector fields *X*, *Y* on *N*, where *π*_{L} and *π*_{L′} are the projections of *TN* onto *L* and *L*′, respectively. The 2-form *Ω* induces a non-degenerate (0, 2)-tensor field *g* on *N* defined by *g*(*X*, *Y*)=*Ω*(*X*, *JY*), where *X*,*Y* are vector fields on *N*. Now, the relationship between the almost paracomplex and the almost symplectic structures on *N* shows that *g* defines a pseudo-Riemannian metric of signature (*n*, *n*) on *N*, and one has that *g*(*JX*, *Y*)+*g*(*X*, *JY*)=0, where *X*, *Y* are vector fields on *N*. We refer to Cruceanu *et al*. (1996) for further details on paracomplex geometry.

The special significance of the para-Kaehler condition is equivalently stated in terms of the fact that the paracomplex structure is parallel with respect to the Levi-Civita connection of *g*, i.e. ∇^{g}*J*=0. The ± eigenspaces of the paracomplex structure *J* are null distributions. Moreover, since *J* is parallel in the para-Kaehler setting, the distributions are parallel. This shows that any para-Kaehler structure (*g*, *J*) has, necessarily, an underlying Walker metric.

### (a) Proof of theorem 2.2

Choose an affine manifold (*M*, ∇), with a flat connection ∇ on *M*. We normalize the choice of coordinates so the associated Christoffel symbols vanish. Consider the modified Riemannian extensionLet be the symplectic form. Now, the associated almost para-Hermitian structure *J*, defined by *Ω*(*X*, *Y*)=*g*(*JX*, *Y*), is given by (sum over *j*)for *i*=1, …, *n*, and a direct calculation shows that the Nijenhuis tensorAs *J* is integrable, and as *Ω* is closed, (*T*^{*}*M*, *g*, *J*) is a para-Kaehler manifold.

We finish the proof showing that (*T*^{*}*M*, *g*, *J*) has constant para-holomorphic sectional curvature *c*. First, a straightforward calculation from the expressions for the curvature given in §4*e* shows that the non-null components of the curvature tensor of the Levi-Civita connection are determined by (we do not sum over repeated indices)adding the symmetry condition , and where *i*, *j*, *k*, *h* are different indices in {1,…,*n*}. Now, for any vector field , a long, but straightforward, calculation from the above relationships shows that (sum over *i* and *j*)Thus, *R*^{g} (*JX*, *X*)=*cg* (*X*, *X*)*JX*, showing that (*T*^{*}*M*, *g*, *J*) has constant para-holomorphic sectional curvature *c*. Now, the fact that para-Kaehler manifolds of constant para-holomorphic sectional curvature have a unique local-isometry type finishes the proof. ▪

## 7. Four-dimensional geometry

### (a) Self-dual Walker metrics

In the particular case of *n*=4, we choose suitable coordinates , where the Walker metric takes the form(7.1)for some functions *a*, *b* and *c* depending on the coordinates .

Considering the Riemann curvature tensor as an endomorphism of *Λ*^{2}(*N*), we have the following *O*(2, 2)-decomposition:where *W*^{g} denotes the Weyl conformal curvature tensor; *τ*^{g} the scalar curvature; and the traceless Ricci tensor,The Hodge star operator associated with any metric of signature (2, 2) induces a further splitting , where denotes the ±1-eigenspace of the Hodge star operator, i.e.Correspondingly, the curvature tensor decomposes asRecall that a pseudo-Riemannian 4-manifold is called *self-dual* (respectively *anti-self-dual*) if (respectively ).

Self-dual Walker metrics have been previously investigated in Brozos-Vázquez *et al*. (2006) and Díaz-Ramos *et al*. (2006*b*; see also Davidov & Muskarov 2006), where a local description in Walker coordinates of such metrics was obtained. As a further application, self-dual Walker metrics can be completely described in terms of the modified Riemannian extension as follows.

*A four-dimensional Walker metric is self-dual if and only if it is locally isometric to the cotangent bundle T*^{*}*Σ of an affine surface* (*Σ*, ∇), *with metric tensor**where X*, *T*, ∇ *and Φ are a vector field,* *a* (1, 1)-*tensor field,* *a torsion-free affine connection and a symmetric* (0, 2)-*tensor field on Σ, respectively*.

It follows from Díaz-Ramos *et al*. (2006*b*) that the metric of equation (7.1) is self-dual if and only if the functions *a*, *b*, *c* have the form(7.2)where all capital, calligraphic and Greek letters stand for arbitrary smooth functions depending only on the coordinates (*x*^{1},*x*^{2}).

For a vector field on *Σ*, we haveand henceandNext, let *T* be a (1,1)-tensor field on *Σ* with componentsIt follows from the definition of *ιT* in equation (5.1) thatand thereforeandNow the result follows from equation (7.2). ▪

Note that the total space of a cotangent bundle is always canonically oriented by its symplectic structure and hence, as a direct consequence of equation (7.2), any modified Riemannian extension *g*_{∇,Φ,0} is necessarily a self-dual Walker metric, where self-duality refers to that canonical orientation. Anti-self-dual Walker metrics are more difficult to describe and only partial results are available (cf. Dunajski 2002; Díaz-Ramos *et al*. 2006*b*). Recently, the authors have showed that Ivanov–Petrova self-dual Walker 4-metrics correspond to modified Riemannian extensions *g*_{∇,Φ,0} (Calviño-Louzao *et al*. in press), and the same was shown by Derdzinski (2008) for Ricci-flat self-dual Walker 4-metrics. For the sake of completeness, we recall the following (Derdzinski 2008, theorem 6.1; see also Díaz-Ramos *et al*. 2006*b*, theorem 3.1), which strengthens the main result in García-Río *et al*. (1999).

*A four-dimensional Ricci-flat self-dual Walker metric is locally isometric to the cotangent bundle T*^{*}*Σ of an affine surface* (*Σ*, ∇) *equipped with the modified Riemannian extension* , *where* ∇ *is a torsion-free connection with skew-symmetric Ricci tensor and Φ is an arbitrary symmetric* (0, 2)-*tensor field on Σ*.

*Moreover, the self-dual Weyl curvature operator W*_{+} *is three-step nilpotent whenever the base manifold* (*Σ*, ∇) *is non-flat, in which case* ∇ *expresses in adapted coordinates* (*x*^{1}, *x*^{2}) *by**for an arbitrary function φ satisfying* ∂_{12}*φ*≠0. *If the base manifold* (*Σ*, ∇) *is flat, then W*_{+} *is two-step nilpotent if and only if* *and vanishes otherwise*.

### (b) Osserman 4-metrics with non-diagonalizable Jacobi operators

Let be a pseudo-Riemannian manifold of signature (2, 2). Then, for each non-null vector *X*, the induced metric on *X*^{⊥} is of Lorentzian signature and thus the Jacobi operator , viewed as an endomorphism of *X*^{⊥}, corresponds to one of the following possibilities (Blažić *et al*. 2001):Type I(a) Osserman metrics correspond to real, complex and paracomplex space forms, type I(b) Osserman metrics do not exist (Blažić *et al*. 2001), and types II and III Osserman metrics with non-nilpotent Jacobi operators have recently been classified in Díaz-Ramos *et al*. (2006*b*) and Derdzinski (2009), respectively. Furthermore, note that any type II Osserman metric whose Jacobi operators have non-zero eigenvalues is necessarily a Walker metric. Moreover, since four-dimensional Osserman metrics are locally Einstein and self-dual, type II Osserman metrics can be described by means of modified Riemannian extensions as follows.

*A four-dimensional type II Osserman metric whose Jacobi operator has non-zero eigenvalues is locally isometric to the cotangent bundle T*^{*}*Σ of an affine surface* (*Σ*, ∇), *with metric tensor**where τ*≠0 *denotes the scalar curvature of* (*T*^{*}*Σ*, *g*); ∇ *is an arbitrary non-flat connection on Σ*; *and Φ is the symmetric part of the Ricci tensor of* ∇.

It follows, after some straightforward calculations as in theorem 7.1, that any type II Osserman metric of non-zero scalar curvature *τ* is obtained by the above modified Riemannian extension of a torsion-free connection ∇ given byjust considering Díaz-Ramos *et al*. (2006*b*, theorem 3.1). ▪

The first examples of non-Ricci-flat type II Osserman metrics were given in Díaz-Ramos *et al*. (2006*a*) as follows. Let with usual coordinates (*x*^{1}, *x*^{2}, *x*^{3}, *x*^{4}) and define a metric by(7.3)where *k* is a non-zero constant and *f*(*x*^{4}) is an arbitrary function.

Now, an easy calculation shows that equation (7.3) is nothing else than the modified Riemannian extension of the torsion-free connection ∇ given by , whose Ricci tensor is given byThis is neither symmetric nor skew-symmetric. We symmetrize to see that

Four-dimensional type II Osserman metrics have been studied intensively in recent years. The existence of many nilpotent examples seemed to suggest that the family of Osserman metrics with two-step nilpotent Jacobi operators was larger than the non-nilpotent one. However, this appears not to be true (see theorems 7.2 and 7.3).

## 8. Six-dimensional geometry

Let *p*=3. We consider on the torsion-free connection ∇ with the only non-zero Christoffel symbol given by . This connection is Ricci flat, but not flat. We set *Φ*=0 and take *c*=1 in equation (5.3) to define a metric *g*=*g*_{∇,1}, where the tensor *B* of equation (4.1) is given by

### (a) Proof of theorem 2.3

The curvature tensor of the Levi-Civita connection is given by

The eigenvalues of the Jacobi operator of an Osserman metric change sign when passing from timelike to spacelike directions. Thus, for the purpose of studying the Osserman property, it is convenient to consider the operator associated to each non-null vector *v*, whose eigenvalues must be constant if and only if the manifold is Osserman.

We now determine the Jacobi operator. Let be a non-null vector, where {∂_{i}, ∂_{i′}} denotes the coordinate basis. The associated Jacobi operator can be expressed, with respect to the coordinate basis, as follows:with

The characteristic polynomial of the Jacobi operator is now seen to beand therefore *g* is Osserman with eigenvalues {0, 1, 1/4, 1/4, 1/4, 1/4}. Now, for any unit vector *v* setFor the particular choice of the unit vectorswe have *g*(*v*, *v*)=*ϵ* and , and a straightforward calculation shows that , while . Therefore, at any point, diagonalizes while is not diagonalizable, and hence the metric is neither spacelike nor timelike Jordan–Osserman at any point. ▪

We make the following observations concerning the metric of theorem 2.3.

The Jacobi operator associated to a unit vector

*v*is either diagonalizable, has a single 2×2 Jordan block, or has a 3×3 Jordan block or has two 2×2 Jordan blocks, depending on the point and the vector*v*considered; all possibilities can arise.Observe that the eigenvalues of the Jacobi operator of a pseudo-Riemannian Osserman metric change sign from spacelike to timelike vectors, and thus they are all zero for null vectors (cf. Gilkey 2001; García-Río

*et al*. 2002), which shows that any Osserman metric is null Osserman. Hence, the metric of theorem 2.3 is null Osserman; moreover, proceeding as above, one checks that the null Jacobi operators can be two-step nilpotent, three-step nilpotent or four-step nilpotent, changing even at a fixed point, and therefore the metric is not pointwise null Jordan–Osserman.The metric of theorem 2.3 is timelike and spacelike nilpotent Szabó, it is not symmetric and it is not Jordan–Szabó.

A straightforward calculation shows that metric of theorem 2.3 is not Ivanov–Petrova.

## Acknowledgments

Supported by projects MTM2006-01432 and PGIDIT06PXIB207054PR (Spain).

## Footnotes

- Received January 27, 2009.
- Accepted March 5, 2009.

- © 2009 The Royal Society