## Abstract

Conformal Osserman four-dimensional manifolds are studied with special attention to the construction of new examples showing that the algebraic structure of any such curvature tensor can be realized at the differentiable level. As a consequence one gets examples of anti-self-dual manifolds whose anti-self-dual curvature operator has complex eigenvalues.

## 1. Introduction

Pseudo-Riemannian metrics of signature other than Lorentzian have received considerable attention in mathematical physics since the work of Ooguri & Vafa (1991). Applications of pseudo-Riemannian metrics in braneworld cosmology have been discussed by Sahni & Shtanov (2002). On the other hand, a central problem in differential geometry is to relate algebraic properties of the curvature tensor to the underlying geometry of the manifold. Since the full curvature tensor is a very difficult object to deal with, several curvature operators have been considered in trying to understand the geometry encoded by the curvature tensor.

A pseudo-Riemannian manifold (*M*, *g*) is said to be an *Osserman space* if the eigenvalues of the Jacobi operators are constant on the unit pseudo-sphere bundles *S*^{±}(*M*). The fact that the local isometries of any locally two-point homogeneous space act transitively on the unit pseudo-sphere bundles shows that any locally two-point homogeneous space is Osserman. The converse is known to be true in Riemannian geometry if dim*M*≠16 (Chi 1988; Nikolayevsky 2003, 2004, 2005) and it also holds for Lorentzian metrics, even under some weaker assumptions (Blažić *et al*. 1997; García-Río *et al*. 1997). The situation is, however, quite different when higher signature metrics are considered. Indeed, although some of the two-point homogeneous spaces can be recognized by some Osserman-like properties (Blažić *et al*. 2001; Bonome *et al*. 2002), a remarkable fact is the existence of many non-symmetric and even not locally homogeneous Osserman pseudo-Riemannian metrics (García-Río *et al*. 1998). A two-step strategy has been followed so far in the study of Osserman manifolds by Gilkey *et al*. (1995). The first step consists of the determination of the possible algebraic curvature tensors being Osserman, which is closely related to the existence of certain Clifford structures (Chi 1988; Bonome *et al*. 2002; Nikolayevsky 2004, 2005) and secondly to classify the manifolds with such a structure as in the work of Nikolayevsky (2003). Therefore, a fundamental aspect towards an understanding of Osserman metrics is to determine the solution of the Osserman problem at the purely algebraic level. Here, it is worth emphasizing the existence of many Osserman algebraic curvature tensors which cannot occur in an Osserman manifold (Gilkey 1994; Blažić *et al*. 2001), although they can be realized geometrically at a given point (see also Gilkey 2001; García-Río *et al*. 2002).

Although the Jacobi operator is probably the most natural operator associated to the curvature tensor, there is some important geometrical information enclosed in some other operators like the Szabó operator, the skew-symmetric curvature operator or the higher-order Jacobi operators (see Gilkey (2001) for more information). Moreover, not only the Riemann curvature tensor has been used as a starting object to define curvature operators (cf. García-Río *et al*. 1999) and indeed, recently, a conformal version of the Osserman property was studied (Blažić & Gilkey 2004; Blažić *et al.* 2003, unpublished work). A pseudo-Riemannian manifold (*M*, *g*) is said to be *conformally Osserman* if the eigenvalues of the conformal Jacobi operator (*x*)(*y*)=(*x*, *y*)*x* are constant on the unit pseudo-spheres , but allowing them to change from point to point, where denotes the Weyl tensor of the conformal structure (*M*, [*g*]). It is shown in the work of N. Blažić *et al*. (unpublished work) that such a property is a conformal invariant and that any Lorentzian or odd-dimensional Riemannian conformally Osserman manifold is locally conformally flat. (Further results on conformally Osserman 2*k*-dimensional Riemannian manifolds have been obtained in the work of Blažić & Gilkey (2004) for *k*>2.) The purpose of this work is to investigate further the geometry of four-dimensional conformally Osserman pseudo-Riemannian manifolds. Special attention is paid to the construction of a large number of examples of conformally Osserman metrics, which are neither Osserman nor in the conformal class of an Osserman metric. As a consequence we obtain that the eigenvalue structure of the conformal Jacobi operators is much richer than the corresponding one for the usual Jacobi operators.

It is worth emphasizing that four-dimensional conformally Osserman metrics are equivalently either self-dual or anti-self-dual. Henceforth, we will describe a large family of self-dual (respectively, anti-self-dual) metrics whose anti-self-dual (respectively, self-dual) curvature is not necessarily diagonalizable, allowing them to have complex eigenvalues. Moreover, all these examples are naturally equipped with a symplectic structure (cf. remark 3.4).

This paper is organized as follows. Section 2 is devoted to the study of Osserman algebraic curvature tensors at a purely algebraic level. All possible Osserman algebraic curvature tensors in a four-dimensional inner product space are determined (cf. theorem 2.2) and an equivalent characterization of the Osserman property in terms of the associated Ricci tensor and the self-duality (or anti-self-duality) of the associated Weyl curvature tensor is obtained in theorem 2.4. This leads, in §3, to a nice characterization of four-dimensional conformal Osserman manifolds as those being self-dual or anti-self-dual (cf. theorem 3.1), which was previously proved by Blažić & Gilkey 2005 for Riemannian signature (see also N. Blažić & P. Gilkey 2004, unpublished work). On the basis of that result, in §4 we construct new examples of conformally Osserman manifolds for the different types of the conformal Jacobi operators, including the case of complex eigenvalues which cannot occur for the usual Osserman spaces as shown by Blažić *et al*. (2001). As a consequence we obtain that *all possibilities for a conformally Osserman algebraic curvature tensor can be realized at the differentiable level*.

## 2. Algebraic preliminaries

In this section, we restrict to a purely algebraic context and denote by (* V*,

*g*, ) an

*n*-dimensional real vector space

*endowed with a non-degenerate inner product*

**V***g*of signature (

*p*,

*q*) (

*n*=

*p*+

*q*≥3), being an

*algebraic curvature tensor*on

*, i.e. ∈⊗*

**V**^{4}(

**V**^{*}) satisfies the symmetries,The

*Jacobi operator*associated to is the self-adjoint map on

*characterized byLet*

**V***S*

^{+}(

*,*

**V***g*) and

*S*

^{−}(

*,*

**V***g*) be the pseudo-spheres of space-like and time-like unit vectors, i.e. . (

*,*

**V***g*, ) is said to be

*space-like*(respectively,

*time-like*)

*Osserman*if the eigenvalues of are constant on

*S*

^{+}(

*,*

**V***g*) (respectively, on

*S*

^{−}(

*,*

**V***g*)). Now, observe that the same argument as in García-Río

*et al*. (1999) or proceeding as in Gilkey (2001) shows the equivalence between space-like and time-like Osserman conditions. Hence, from now on we just refer to (

*,*

**V***g*, ) as an Osserman algebraic curvature tensor if any of these conditions is satisfied. (Note that the eigenvalues of the Jacobi operators (

*x*) may change from time-like to space-like directions although they remain constant on each of

*S*

^{−}(

*,*

**V***g*) and

*S*

^{+}(

*,*

**V***g*).)

Now, for any basis {*e*_{i} : *i*=1, …, *n*} put *g*_{ij}=*g*(*e*_{i}, *e*_{j}) while (*g*^{ij}) denotes the inverse matrix. Then the associated -Ricci tensor and -scalar curvature are given byAlso, recall that the -sectional curvature of a non-degenerate plane *π*=〈{*x*, *y*}〉 is given byMoreover, the -sectional curvature is constant *κ* if and only if =*κ*_{ϕ}, for *ϕ*=*id*, whereFurther, note that the space of all algebraic curvature tensors is spanned by the _{ϕ}s above (Gilkey 2001; Fiedler 2003; Díaz-Ramos & García-Río 2004).

If (* V*,

*g*, ) is an Osserman algebraic curvature tensor, then the -Ricci tensor satisfies =(/

*n*)

*g*, which shows that any Osserman algebraic curvature tensor in a three-dimensional vector space

*is of constant -sectional curvature. Therefore, the first non-trivial case is that of dim*

**V***=4. Further, note that proceeding as in (Blažić*

**V***et al*. 1997; García-Río

*et al*. 1997), if

*g*is of Lorentzian signature, then (

*,*

**V***g*, ) is Osserman if and only if is constant, and thus the space of Osserman algebraic curvature tensors reduces to span {

_{id}}. The same result holds true for Osserman algebraic curvature tensors in an odd-dimensional Riemannian space (Chi 1988; Gilkey

*et al*. 1995).

The above motivates a further analysis of the four-dimensional case with inner product of Riemannian or neutral signatures. First of all note that since (*x*)*x*=0, one often restricts the Jacobi operator (*x*) to *x*^{⊥}. The different possibilities for the structure of the Jacobi operators, as well as an algebraic description of Osserman algebraic curvature tensors in the neutral signature setting is given in the following result, which is essentially obtained in the same way as in (Blažić *et al*. 2001; García-Río *et al*. 2002). It provides a characterization of the four-dimensional Osserman algebraic curvature tensors at the algebraic level.

*Let* *be an algebraic curvature tensor on a four-dimensional vector space V equipped with an inner product g of neutral signature. Then,* (

*,*

**V***g*, )

*is Osserman if and only if one of the following holds:*

*Type Ia: the Jacobi operators are diagonalizable,*(2.1)*which occurs if and only if there exists an orthonormal basis*{*e*_{1},*e*_{2},*e*_{3},*e*_{4}}*for***V**where the non-vanishing components of*are given by*(2.2)*Type Ib: there is a complex eigenvalue for the Jacobi operators,*(2.3)*which occurs if and only if there exists an orthonormal basis*{*e*_{1},*e*_{2},*e*_{3},*e*_{4}}*for***V**such that the non-vanishing components of*are those given by*(2.4)*Type II: the Jacobi operators can be written in the form*(2.5)*which is equivalent to the existence of an orthonormal basis*{*e*_{1},*e*_{2},*e*_{3},*e*_{4}}*for***V**, where the non-vanishing components of*are given by*(2.6)*Type III: the Jacobi operators can be written in the form*(2.7)*if and only if there exists an orthonormal basis*{*e*_{1},*e*_{2},*e*_{3},*e*_{4}}*for***V**such that the non-vanishing components of*are given by*(2.8)

Note that if *g* is of Riemannian signature, then the Jacobi operators are always diagonalizable, which shows that only type Ia may occur in the Riemannian setting. Further, note that types II and III correspond to the existence of a double and triple root for the minimal polynomial of (*x*), respectively.

We finish this section with a characterization of four-dimensional Osserman algebraic curvature tensors which will provide its usefulness in constructing examples in the next sections. An specific feature of dimension four comes from the properties of the Hodge star operator, which acts on the space of two-forms *Λ*=〈{*e*^{i}∧*e*^{j} : *i*, *j*∈{1,2,3,4}, *i*<*j*}〉 bywhere *ϵ*_{i}=*g*(*e*_{i}, *e*_{i}). Since ★^{2}=*id* for any inner product of definite or neutral signature, it induces a splitting *Λ*=*Λ*^{+}⊕*Λ*^{−}, where *Λ*^{+} and *Λ*^{−} denote the spaces of self-dual and anti-self-dual two-forms,Recall that for any algebraic curvature tensor , its associated Weyl tensor is defined by(2.9)for all *x*, *y*, *z*, *v*∈* V*. Next put for the restriction of to the spaces . Then (

*,*

**V***g*, ) is said to be a

*self-dual*(respectively,

*anti-self-dual*) algebraic curvature tensor if (respectively, ); see for example Kamada & Machida (1997). Finally, observe that the induced inner products on are positive definite if

*g*is Riemannian, but they are Lorentzian if

*g*is of neutral signature. Therefore, the structure of the self-dual and anti-self-dual curvatures may follow any of the types Ia, Ib, II or III as in theorem 2.2.

Now we state the following.

*Let* *be an algebraic curvature tensor on a four-dimensional vector space V with inner product g. Then,* (

*,*

**V***g*, )

*is Osserman if and only if it is self-dual (or anti-self-dual) and*=(/4)

*g. Moreover,*

*the types of the self-dual curvature*

*(respectively,*

*anti-self-dual curvature*

*) are in one to one correspondence with the different types of the Jacobi operators in*

*theorem 2.2*.

Note that any algebraic curvature tensor on a four-dimensional vector space decomposes as(2.10)where denotes the trace-free -Ricci tensor, =−(/4)*g*. Further, in order to express the self-dual and anti-self-dual curvature tensors, a basis of the space of two-forms is constructed as follows. Let {*e*_{i}} be an orthonormal basis of * V* and consider the induced basis on the spaces of self-dual and anti-self-dual two-forms given by , whereand , , . Thus, with respect to the above basis the self-dual and anti-self-dual Weyl operators have the matrix form(2.11)Now we proceed as in García-Río

*et al*. (2002). If is an Osserman algebraic curvature tensor, then specialize the orthonormal basis {

*e*

_{i}} above as given in theorem 2.2 to get after a straightforward calculation that or vanishes and, moreover, the structure of the self-dual or anti-self-dual part of the Weyl tensor corresponds to the structure of the Jacobi operators . Conversely, if is assumed to be self-dual and =(/4)

*g*, then equation (2.10) becomes(2.12)from where it follows that is Osserman proceeding as in (Gilkey

*et al*. 1995; Alekseevsky

*et al*. 1999; García-Río

*et al*. 2002). ▪

## 3. Conformally Osserman manifolds and (anti-) self-dual structures

Let (*M*, *g*) be an *n*-dimensional pseudo-Riemannian manifold with Levi-Civita connection ∇ and denote by *R* the Riemann curvature tensor taken with the sign convention *R*(*X*, *Y*)*Z*=∇_{[X,Y]}*Z*−[∇_{X}, ∇_{Y}]*Z*, for all vector fields . In what follows we will restrict to the algebraic curvature tensor field given by the Weyl tensor associated to the Riemann curvature tensor, which is defined as in equation (2.9). Following Blažić & Gilkey (2004) we refer to the associated Jacobi operators as the *conformal Jacobi operators* and (*M*, *g*) is said to be *conformally Osserman* if (*T*_{p}*M*, *g*_{p}, _{p}) is Osserman at each point *p*∈*M*. Note that the eigenvalues of are allowed to change from point to point and, moreover, that the conformal Osserman property only depends on the conformal class of the metric [*g*].

Now, a characterization of four-dimensional conformal Osserman manifolds is obtained from theorem 2.4 as follows.

*Let* (*M*, *g*) *be a four-dimensional pseudo-Riemannian manifold. Then* (*M*, *g*) *is conformally Osserman if and only if it is self-dual or anti-self-dual.*

Since the Weyl tensor is trace-free, and , from where it follows that the algebraic Weyl curvature tensor constructed by equation (2.9) from the Weyl tensor _{R}, coincides with the Weyl tensor . Now the result follows immediately from theorem 2.4. ▪

As an immediate application of previous theorem, examples of conformally Osserman (++−−)-metrics are obtained from the work of Dunajski (2002). Our purpose in what remains of this paper is to use theorem 3.1 above to construct new examples of conformally Osserman metrics. Two facts should be emphasized as concerns those examples:

There exist conformal Osserman manifolds which are not in the conformal class of any Osserman metric.

All the possible algebraic structures of the conformal Jacobi operators in theorem 2.2 can be realized at the differentiable level.

In order to give some motivation for the metrics to be considered in what follows, recall that an specific feature of pseudo-Riemannian metrics is related to the local reducibility/decomposability of such structures as shown by Wu (1964). It is a matter of fact that many striking differences between the Riemannian and pseudo-Riemannian situations come from the existence of parallel degenerate distributions, which do not lead to local decompositions of the manifold in the indefinite setting. Among such metrics, an interesting family was investigated by Walker (1950): pseudo-Riemannian manifolds admitting a parallel degenerate distribution of maximal rank, which include some metrics on tangent and cotangent bundles as special cases García-Río *et al*. 1999. It was shown by Walker that any such four-dimensional metric can be locally expressed in adapted coordinates (*x*_{1}, …, *x*_{4}) bywhere *I*_{2} represents the 2×2-identity matrix andfor arbitrary functions *a*(*x*_{1}, …, *x*_{4}), *b*(*x*_{1}, …, *x*_{4}) and *c*(*x*_{1}, …, *x*_{4}). For simplicity, in what follows we will consider those metrics above with *a*≡*b*≡0, i.e.(3.1)since the single function *c*(*x*_{1}, …, *x*_{4}) suffices to provide the desired examples corresponding to all the possibilities in theorem 2.2.

Next, in order to decide on the conformally Osserman property of metrics (3.1), we analyse the self-duality and the anti-self-duality of such a Walker metric in view of theorem 3.1. First, we determine the curvature tensor, the Ricci tensor and the scalar curvature of a Walker manifold as above. As a matter of notation, in what follows we write *h*_{i}=(∂*h*/∂*x*_{i}), *h*_{ij}=(∂*h*/∂*x*_{i}∂*x*_{j}) for any function *h*(*x*_{1}, …, *x*_{4}), and ∂_{i}=(∂/∂*x*_{i}) (*i*, *j*=1, …, 4). After a long but straightforward calculation we get that the non-vanishing components of the (0,4)-curvature tensor are determined by(3.2)From equation (3.2) we obtain that the non-vanishing components of the Ricci tensor are characterized by(3.3)and the scalar curvature is given by(3.4)Now, for a Walker metric (3.1) let {*e*_{1}, *e*_{2}, *e*_{3}, *e*_{4}} be an orthonormal basis, whereand(3.5)(Here, note that *e*_{1} and *e*_{2} are space-like, while *e*_{3} and *e*_{4} are time-like vectors.) Then, using equations (3.2), (3.3) and (3.4), a long but straightforward calculation shows that the non-vanishing components of ^{−} and ^{+} are given by(3.6)and(3.7)Now, we have the following characterization of self-dual metrics.

*A Walker metric* *(3.1)* *is self-dual if and only if the defining function c*(*x*_{1}, *x*_{2}, *x*_{3}, *x*_{4}) *is of the form*(3.8)*for any functions P*(*x*_{3}, *x*_{4})*, Q*(*x*_{3}, *x*_{4}) *and S*(*x*_{3}, *x*_{4}). *Moreover, in such a case*, *the characteristic polynomial of* ^{+} *reduces to p _{λ}*(

^{+})=−

*λ*

^{3}

*, and the minimal polynomial is characterized as follows:*

*m*(_{λ}^{+})=*λ*^{3}*at those points where c*_{13}−*c*_{24}≠0.*m*(_{λ}^{+})=*λ or m*(_{λ}^{+})=*λ*^{2}*at those points where c*_{13}−*c*_{24}=0*, depending on whether the function**vanishes or not.*

Note from equation (3.6) that a Walker metric (3.1) is self-dual if and only if the defining function *c*(*x*_{1}, *x*_{2}, *x*_{3}, *x*_{4}) satisfies(3.9)First, *c*_{12}=0 implies that . But and , so it follows thatfrom where , with , which shows equation (3.8).

On the other hand, the characterization of ^{+} given by equation (3.7) lets us get the general expression of the characteristic polynomial of ^{+}, which is given by(3.10)Now, if the metric is self-dual, then *c*_{12}=0 and, therefore, equation (3.10) reduces to *p*_{λ}(_{+})=−*λ*^{3}. Finally, it is straightforward to show that under the conditions given in equation (3.9) one hasandfrom where the characterization of the minimal polynomial of ^{+} is obtained. ▪

Proceeding in an analogous way, anti-self-dual metrics are characterized as follows.

*A Walker metric* *(3.1)* *is anti-self-dual if and only if the defining function c*(*x*_{1}, *x*_{2}, *x*_{3}, *x*_{4}) *is of the form*(3.11)*for any functions ξ*(*x*_{1}, *x*_{4}) *and η*(*x*_{2}, *x*_{3})*, and functions P*(*x*_{3}, *x*_{4})*, Q*(*x*_{3}, *x*_{4}) *and S*(*x*_{3}, *x*_{4}) *satisfying**Moreover*, *in such a case,* ^{−} *has eigenvalues* 0*,* ±(1/2)(−*c*_{11}*c*_{22})^{1/2} *and the minimal polynomial is characterized as follows:*

*m*(_{λ}^{−})=−*p*(_{λ}^{−})*if there are three different eigenvalues*(*c*_{11}*c*_{22}≠0).*m*(_{λ}^{−})=*λ or m*(_{λ}^{−})=*λ*^{3}*if zero is the unique eigenvalue*(*c*_{11}*c*_{22}=0)*, depending on whether c*_{11}*and c*_{22}*vanish simultaneously or not,**respectively.*

First note that, from equation (3.7), the anti-self-duality is equivalent to(3.12)Since *c*_{12}=0, we have . Now, , and therefore,where *P*_{3}−*Q*_{4}=0. Then, putting , from *cc*_{13}−*c*_{34}=0 we obtain *cP*_{3}−*x*_{1}*P*_{34}−*x*_{2}*Q*_{34}−*S*_{34}=0, which shows the first part of the result since *P*_{3}−*Q*_{4}=0 implies that *Q*_{34}=*P*_{33}.

Now, we analyse the characteristic and the minimal polynomials of ^{−}. From equation (3.6) we get the general expression of the characteristic polynomial of ^{−}, which is given by(3.13)If the manifold is anti-self-dual then *c*_{12}=0 and this lets us reduce equation (3.13) toNow, if *c*_{11}*c*_{12}≠0 then there are three distinct eigenvalues and, therefore, *m*_{λ}(^{−})=−*p*_{λ}(^{−}). On the other hand, it is straightforward to show that under the conditions given in equation (3.12) we haveandwhich shows the different possibilities for the minimal polynomial when *p*_{λ}(^{−})=−*λ*^{3} (i.e. if *c*_{11}*c*_{22}=0). ▪

An almost complex structure on a Walker 4-manifold is said to be *proper* if it induces a positive *π*/2-rotation on the degenerate parallel field spanned by *∂*_{1}, *∂*_{2}. Such almost complex structures are completely determined by the metric as follows:(3.14)Moreover, it was shown by Matsushita (2004) that *J* above defines a symplectic structure if and only if(3.15)which is Kählerian if and only if(3.16)Next, note that since we assume *a*≡0, *b*≡0, any metric given by equation (3.1) defines an almost Kähler structure which is not Kähler, unless *c*_{1}=*c*_{2}=0. Further, observe that the natural orientation defined by *J* is the opposite to the one given in equation (3.5).

## 4. Examples of conformally Osserman manifolds

On the basis of the results in previous section, our purpose in what follows is to construct some explicit examples of conformally Osserman metrics. Special emphasis will be made on those examples being *strictly* conformally Osserman, i.e. conformally Osserman metrics which are neither Osserman nor in the conformal class of an Osserman metric. Therefore, first of all we obtain the necessary and sufficient conditions for a Walker metric (3.1) to be Osserman as follows.

*A Walker metric* *(3.1)* *is Osserman if and only if the defining function c*(*x*_{1}, *x*_{2}, *x*_{3}, *x*_{4}) *is of one of the following forms:*

*for any function S*(*x*_{3},*x*_{4}).*for any function S*(*x*_{3},*x*_{4})*and any real constant α.**for any function S*(*x*_{3},*x*_{4})*and any real constant α.**for any function S*(*x*_{3},*x*_{4})*and functions A*(*x*_{3})*and B*(*x*_{4})*satisfying that*

Recall from theorem 2.4 that a metric (3.1) is Osserman at a given point if and only if it is Einstein and self-dual or anti-self-dual at that point. Note from equations (3.3) and (3.4) that a metric (3.1) is Einstein if and only if(4.1)On the other hand, the metric must be self-dual or anti-self-dual and, in both cases, *c*_{12}=0 (see the proofs of theorems 3.2 and 3.3). Hence *c*_{11}=*c*_{22}=*c*_{12}=0, which is nothing but equation (3.9), and thus the Osserman condition implies that *c*(*x*_{1}, *x*_{2}, *x*_{3}, *x*_{4}) must be of the form(4.2)for some functions *P*(*x*_{3}, *x*_{4}), *Q*(*x*_{3}, *x*_{4}) and *S*(*x*_{3}, *x*_{4}) satisfying(4.3)Now, the first two equations above imply thatand Thus, expressions in (*i*)–(*iv*) easily follow just considering the last equation in (4.3). ▪

Observe from remark 3.4 that those metrics in theorem 4.1 are Einstein almost-Kähler. Moreover, they are Kähler only in case (*i*).

In order to analyse the Jordan normal form of the Jacobi operators corresponding to the Osserman metrics in previous theorem, a distinction should be made among metrics (*i*)–(*iii*) and those given by (*iv*). Let be any non-null tangent vector. Then the Jacobi operator *J*_{R}(*x*) takes the formfor any metric (*i*)–(*iii*), while in case (*iv*) it becomeswhereThis shows that all Osserman metrics in theorem 4.1 have nilpotent Jacobi operators. Moreover, in order to decide the degree of nilpotency, note that *J*_{R}(*x*)^{2}=0 for any metric (*i*)–(*iii*). In case (*iv*), since is unitary at any point it suffices to analyse the Jacobi operator associated to that vector, and we obtainOn the other hand, in vanishes if and only if , and with this additional condition one can compute to obtainwhereSummarizing the above, we have that the characteristic polynomial of the Jacobi operators *J*_{R}(*x*) for any Osserman metric in theorem 4.1 is always *λ*^{4}, while the minimal polynomial behaves as follows:

In cases (

*i*), (*ii*) and (*iii*),*m*_{λ}(*J*_{R}(*x*))=*λ*or*λ*^{2}depending on whether*S*_{34}vanishes or not.In case (

*iv*),*m*_{λ}(*J*_{R}(*x*))=*λ*^{3}if 4′*B*′−^{2}*B*^{2}≠0, while*m*_{λ}(*J*_{R}(*x*))=*λ*or*λ*^{2}if 4′*B*′−^{2}*B*^{2}=0, depending on whether*AB*(2+*x*_{4}) +2′(2+*x*_{3}*B*) and vanish simultaneously or not.

Next we will give some simple examples of conformally Osserman metrics corresponding to the different possibilities in theorem 2.2. Note that none of them is Osserman as an application of theorem 4.1. Further, observe that although the conformal Osserman property does not require the global constancy of the eigenvalues of the conformal Jacobi operators (in order to be a conformal property) that is indeed the case in the examples below.

*Conformally Osserman metrics of type Ia*. For the special choice of the characteristic polynomial of (*x*) is given by*p*_{λ}((*x*))=*λ*^{4}−(1/4)λ^{2}and, therefore, its eigenvalues are 0, 0 and ±1/2.*Conformally Osserman metrics of type Ib*. For the special choice of the characteristic polynomial of (*x*) is given by*p*_{λ}((*x*))=*λ*^{4}+(1/4)λ^{2}and, therefore, its eigenvalues are 0, 0 and ±(1/2)*i*.*Conformally Osserman metrics of type II*. For the special choice of*c*(*x*_{1},*x*_{2},*x*_{3},*x*_{4})=*x*_{1}*x*_{4}+*x*_{3}*x*_{4}the characteristic polynomial of (*x*) is given by*p*_{λ}((*x*))=*λ*^{4}and its minimal polynomial is*m*_{λ}((*x*))=*λ*^{2}.*Conformally Osserman metrics of type III*. For the special choice of , the characteristic polynomial of (*x*) is given by*p*_{λ}((*x*))=*λ*^{4}and its minimal polynomial is*m*_{λ}((*x*))=*λ*^{3}.

Recall that a metric is in the conformal class of a metric *g* if and only if for some positive function *Ψ*. Next, let for a Walker metric *g* as in equation (3.1). A long but straightforward calculation from the components , and of the Ricci tensor of shows that if is Einstein, then the conformal factor must be of the formfor some functions *ϕ*, *ψ* and *ξ*. Now assuming *g* to be any of the metrics (Ia), (Ib) or (III) above, it follows from the component that both functions *ϕ* and *ψ* must vanish, and thus *Ψ*=*ξ*(*x*_{3}, *x*_{4}). Moreover, a conformal deformation of a metric (3.1) with *Ψ*=*ξ*(*x*_{3}, *x*_{4}) is Einstein if and only if(4.4)As a consequence, we obtain the non-existence of Osserman metrics in the conformal class of the conformal Osserman metrics (Ia), (Ib) and (III) above.

Next we give some simple examples showing that the normal Jordan form of the conformal Jacobi operators may change from point to point and, moreover, that the eigenvalues of the conformal Jacobi operators may also change (even from real to complex eigenvalues).

The characteristic polynomial is

*p*_{λ}((*x*))=*λ*^{4}while the minimal polynomial is ,*λ*^{2}or*λ*, depending on the point considered. For the special choice of the minimal polynomial is*m*_{λ}((*x*))=*λ*^{3}at any point with*x*_{4}≠0,*m*_{λ}((*x*))=*λ*^{2}at those points with*x*_{4}=0,*x*_{3}≠0 and*m*_{λ}((*x*))=*λ*at points with*x*_{3}=*x*_{4}=0.The characteristic polynomial is

*p*_{λ}((*x*))=*λ*^{4}while the minimal polynomial is or*λ*^{2}, depending on the point considered. For the special choice of the minimal polynomial is*m*_{λ}((*x*))=λ^{3}at any point with*x*_{4}≠0 and*m*_{λ}((*x*))=*λ*^{2}at those points with*x*_{4}=0.The characteristic polynomial is

*p*_{λ}((*x*))=*λ*^{4}while the minimal polynomial is*m*_{λ}((*x*))=*λ*^{3}or*λ*, depending on the point considered. For the special choice of the minimal polynomial is*m*_{λ}((*x*))=*λ*^{3}at any point with*x*_{3}≠0 and*m*_{λ}((*x*))=*λ*at those points with*x*_{3}=0.The characteristic polynomial is

*p*_{λ}((*x*))=*λ*^{4}while the minimal polynomial is*m*_{λ}((*x*))=*λ*^{2}or*λ*, depending on the point considered. For the special choice of*c*(*x*_{1},*x*_{2},*x*_{3},*x*_{4})=*x*_{1}*x*_{3}+*x*_{2}*x*_{4}the minimal polynomial is*m*_{λ}((*x*))=*λ*^{2}at any point with*x*_{1}*x*_{3}+*x*_{2}*x*_{4}≠0 and*m*_{λ}((*x*))=*λ*at those points with*x*_{1}*x*_{3}+*x*_{2}*x*_{4}=0.The conformal Jacobi operators have real eigenvalues at each point, but changing from point to point. For the special choice of we have and, therefore, its eigenvalues are 0, 0 and .

The conformal Jacobi operators have complex eigenvalues at each point, but changing from point to point. For the special choice of we have and, therefore, its eigenvalues are 0, 0 and .

The conformal Jacobi operators have real or complex eigenvalues depending on the point considered. For the special choice of we have and, therefore, its eigenvalues are 0, 0 and ±(3/2)(

*x*_{1}*x*_{2})^{1/2}.

## Acknowledgements

Supported by projects BFM 2003-02949 and PGIDIT04PXIC20701PN(Spain).

## Footnotes

- Received July 21, 2005.
- Accepted November 22, 2005.

- © 2006 The Royal Society