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Four-dimensional Osserman metrics with nondiagonalizable Jacobi operators

A complete description of Osserman four-manifolds whose Jacobi operators have a nonzero double root of the minimal polynomial is given.


Introduction
Geometric information about a pseudo-Riemannian manifold (M, g) is essentially encoded by the curvature tensor R ∈ ⊗ 4 T * M . Hence a central problem in differential geometry is to relate algebraic properties of the curvature tensor to the underlying geometry of the manifold. Due to the fact that the whole curvature tensor is so difficult to handle, the investigation usually focuses on different objects associated to the curvature tensor, the Jacobi operator being the most natural and widely investigated. A pseudo-Riemannian manifold (M, g) is said to be Osserman if the eigenvalues of the Jacobi operators are constant on the unit pseudo-sphere bundles S ± (T M ). Since the Ricci tensor is obtained by tracing the Jacobi operators, any Osserman metric is Einstein and, in particular, of constant sectional curvature in dimension 2, 3.
Dimension four is therefore the first non-trivial case for consideration in trying to determine the whole class of Osserman metrics. Moreover due to the connection between Osserman and (anti-) self-dual metrics many special features occur like the existence of pointwise Osserman metrics which are not Osserman, i.e., the eigenvalues of the Jacobi operators are still independent of the direction, but they may change from point to point. (See for example [12], [16], [19] and the references therein for more information on pointwise Osserman 4-manifolds, and [5], [17], [18], [22], [23] for the higher dimensional Riemannian and pseudo-Riemannian cases).
It has been shown in [8] and [11] (see also [2]) that any four-dimensional Osserman metric is locally isometric to a two-point homogeneous space if the signature is either Riemannian or Lorentzian. However the situation is much more complicated for neutral metrics and there exist many examples of nonsymmetric Osserman pseudo-Riemannian manifolds of neutral signature [13]. Indeed besides the partial results at [3] for Osserman (+ + −−)-metrics with diagonalizable Jacobi operators and those at [15] for locally symmetric Osserman four-manifolds, the general problem of obtaining a complete description of four-dimensional Osserman metrics of neutral signature remains open.
For any non-null vector X in the (+ + −−)-setting, the induced metric on X ⊥ is of Lorentzian signature, and hence the eigenvalue-structure does not completely characterize the Jacobi operators R X . The consideration of the Jordan normal form led to introduce the so-called Jordan-Osserman metrics as those where the Jordan normal form of the Jacobi operators is constant on S ± (T M ). Four-dimensional Jordan-Osserman metrics were initially investigated by Blažić, Bokan and Rakić [3], who considered four different possibilities according to the behavior of the Jordan normal form of the Jacobi operators as follows: (Ia): The Jacobi operators are diagonalizable, i.e., (Ib): The Jacobi operators have a complex eigenvalue, i.e., (II ): There is a double root of the minimal polynomial of R X , i.e., (III ): There is a triple root of the minimal polynomial of R X , i.e., It is shown in [3] that Jordan-Osserman metrics with diagonalizable Jacobi operators are locally isometric to a real, complex or paracomplex space form and also that Type Ib metrics cannot occur. Moreover, locally symmetric Jordan-Osserman (+ + −−)-metrics have diagonalizable Jacobi operators or they are isometric to some Type II metrics with nilpotent Jacobi operators [15]. The fact that all known examples of nonsymmetric Jordan-Osserman metrics had nilpotent Jacobi operators of degree two or three suggested that no other examples could exist (cf. [7], [12], [16]). However, very recently the authors have shown the existence of a family of Type II Jordan-Osserman metrics with non-nilpotent Jacobi operators [10]. The purpose of this work is to clarify the situation of Type II Jordan-Osserman metrics by proving the following Main Theorem Let (M, g) be a four-dimensional Type II Jordan-Osserman manifold. Then the Jacobi operators are two-step nilpotent or otherwise there exist local coordinates (x 1 , . . . , x 4 ) where the metric is given by for some functions s ij (x 1 , . . . , x 4 ) as follows and arbitrary functions P , Q, S, T , U , V depending only on the coordinates (x 3 , x 4 ), where τ = 0 denotes the scalar curvature.
The proof of the Main Theorem is based on the following facts.
Therefore, we investigate Walker metrics (i.e., those admitting a locally defined two-dimensional degenerate parallel distribution) in detail in §2, with special attention to the (anti-) self-dual Weyl curvature tensors. A complete description of self-dual Walker metrics is given in §2. 3. The integration of the Einstein equation for a self-dual Walker metric, which lets us determine all pointwise Osserman self-dual Walker metrics, is carried out in §3. This leads to (1)-(2), thus proving the Main Theorem.

Four-dimensional Walker metrics
A Walker manifold is a triple (M, g, D) where M is an n-dimensional manifold, g an indefinite metric and D an r-dimensional parallel null distribution. Of special interest are those manifolds admitting a field of null planes of maximum dimensionality (r = n 2 ). Since the dimension of a null plane is r ≤ n 2 , the lowest possible case of a Walker metric is that of (+ + −−)-manifolds admitting a field of parallel null two-planes.
Observe that there is a tight connection between Walker structures and Osserman metrics. First of all, it is a matter of fact that all known examples of Osserman metrics with nondiagonalizable Jacobi operators (i.e., Types II and III) are realized as Walker metrics. On the other hand, Walker metrics appear as the underlying structure of several specific pseudo-Riemannian structures. For instance, the metric tensor of any para-Kähler [9] (and hence any hypersymplectic [20]) structure is necessarily of Walker type. The same occurs for the underlying metric of real hypersurfaces in indefinite space forms whose shape operator is nilpotent [21].

The canonical form of a Walker metric
For our purposes it is convenient to use a special coordinate system associated to any Walker metric. So, let g be a four-dimensional pseudo-Riemannian metric admitting a two-dimensional parallel null distribution. A canonical form for such a metric has been obtained by Walker [24] showing the existence of suitable coordinates (x 1 , . . . , x 4 ) where the metric expresses as for some functions a, b and c depending on the coordinates (x 1 , . . . , x 4 ). As a matter of notation, throughout this work we denote by ∂ i the coordinate vectors, i = 1, . . . , 4. Also, h i1...ir means partial derivatives ∂h ∂xi 1 ...∂xi r , for any function h(x 1 , . . . , x 4 ). Now, a straightforward calculation from (3) shows that the Levi Civita connection is given by and the Riemann curvature tensor R(X, U )Y } and τ = trace ρ be the Ricci tensor and the scalar curvature, respectively. Then and τ = a 11 + b 22 + 2c 12 .
Further, let W denote the Weyl conformal curvature tensor given by
Let {e 1 , e 2 , e 3 , e 4 } be an orthonormal basis and, as a convention, assume that e 1 and e 2 are spacelike while e 3 and e 4 are timelike vectors. Now, local bases of the spaces of self-dual and anti-self-dual two-forms can be constructed as Here observe that the Hodge star operator satisfies where ε i = g(e i , e i ). Further note that E ± 1 , E ± = −1, and therefore with respect to the above bases the self-dual and anti-self-dual Weyl curvature operators W ± : Λ 2 ± −→ Λ 2 ± have the following matrix form: where W ± ij = W (E ± i , E ± j ) and W (e i ∧ e j , e k ∧ e l ) = W (e i , e j , e k , e l ). Next, for a Walker metric (3) an orthonormal basis can be specialized by using the canonical coordinates as follows: Now, a long but straightforward calculation using (5), (6) and (7) shows that the components of W − in (11) are given by while the components of W + are determined by W + 11 , W + 12 and the scalar curvature as follows: the expressions of W + 11 and W + 12 being W + 12 = 1 4 (−2ca 11 − ba 12 + 2a 14 + ab 12 − 2b 23 + ac 11 − 2cc 12 − 2c 13 − bc 22 + 2c 24 ).
Remark 1 It is important to recognize here that the connection between Einstein (anti-) self-dual and pointwise Osserman manifolds at Fact A goes further to the Jordan normal forms of the nonzero part of the Weyl curvature tensor W ± and the Jacobi operators (cf. [12]). So pointwise Osserman manifolds whose Jacobi operators are of Type Ia, Ib, II or III correspond to self-dual (or antiself-dual) Einstein manifolds whose self-dual (or anti-self-dual) Weyl curvature tensor is of Type Ia, Ib, II or III, respectively (see also [4]).
The relations in (14) among the components of W + show that and, as a consequence, the eigenvalues of W + are given by Since the induced metric on Λ 2 + is of Lorentzian signature the structure of W + is determined by its Jordan normal form, which may correspond to Type Ia or type II/III, depending on whether W + is diagonalizable or not. A straightforward calculation shows that from where we have the following: (i) If τ = 0, then W + has nonzero eigenvalues τ 6 , − τ 12 , − τ 12 , and τ 2 + 12τ W + 11 + 48 W + On the other hand, observe from (17) that any anti-self-dual Walker metric has vanishing scalar curvature and hence Einstein anti-self-dual Walker metrics are Ricci flat.

Explicit form of Self-dual Walker metrics
Recall that our main purpose is to obtain a description of non Ricci flat Type II Jordan-Osserman four-manifolds. Then, as a consequence of Remark 1, we restrict our analysis to self-dual Walker metrics. Thus, in what remains of this section we will give a complete description of self-dual Walker metrics by integrating the PDE system given by (13).

Theorem 2 A Walker metric (3) is self-dual if and only if the defining func-
where capital, calligraphic and Greek letters are all arbitrary smooth functions depending only on the coordinates (x 3 , x 4 ).
Proof. First of all observe that using (13) the self-duality can be initially characterized by means of the following five PDEs: So (19) is obtained as the solution of (20). We proceed.

The first step. First and second equations in (20) translate into
and hence the third equation in (20) The second step. In this step we make a long but straightforward use of the fourth equation in (20), which by (21) and (22) means First, appropriate successive differentiations in the above equation lead to A 1111 (x 1 , x 3 , x 4 ) = 0, E 111 (x 1 , x 3 , x 4 ) = 0 and F 111 (x 1 , x 3 , x 4 ) = 0, and hence By (24), condition (23) reduces to from where C 222 (x 2 , x 3 , x 4 ) = 6G(x 3 , x 4 ), which implies that The final form of (25) is given by and as a consequence Collecting together the information in (24), (26) and (27), at the end of this step we have that (21) and (22) transform into The third step. To finish the determination of the defining functions we deal with the last equation in (20), which under the expressions in (28) and differentiating by x 1 and x 2 leads to and hence that equation reduces to It follows that B 1111 (x 1 , x 3 , x 4 ) = 0, from where and (30) takes the form Thus, by (29), (31) and (33), the expression of a in (28) transforms into which finishes the process for this defining function. At this point, (32) has the form from where D 2222 (x 2 , x 3 , x 4 ) = 0, and hence Then, (35) reduces to Thus, (29), (36) and (37) show that b and c in (28) take the desired form finishing the proof.
Remark 3 A four-dimensional Walker metric is said to be strict if it admits two orthogonal parallel null vector fields rather than a parallel two-dimensional null distribution. It follows from the work of Walker that any strict Walker metric is given by (3) for any functions a, b and c depending only on the coordinates (x 3 , x 4 ). Now (6) and Theorem 2 imply that any strict Walker metric is Ricci flat and self-dual, and hence Osserman. Moreover, Remark 1 shows that the Jacobi operators are identically zero or otherwise they are two-step nilpotent (depending on whether W + 11 = 2c 34 − a 44 − b 33 vanishes or not).

Proof of the Main Theorem
Recall that our purpose is to obtain a local description of Type II Jordan-Osserman metrics whose Jacobi operators have nonzero eigenvalues. In such a case, the eigenvalues must be in a ratio 1 : 1 4 and the underlying metric is a Walker metric (cf. Facts B and C). Therefore in order to achieve the desired result, only Osserman metrics on Walker manifolds deserve further consideration and moreover, it immediately follows from Remark 1 that we may restrict to those being self-dual. In what follows we obtain a complete description of selfdual Einstein Walker metrics in Theorem 4, from where our main result is derived.
where capital letters are arbitrary functions depending only on (x 3 , x 4 ). (ii) The scalar curvature vanishes and the metric tensor is given by where P , Q, S, T , U , V , ξ, η and γ are smooth functions depending only on (x 3 , x 4 ) satisfying Proof. Let ρ 0 = ρ − τ 4 g be the trace-free Ricci tensor. Then the Einstein equations for a general Walker metric (3) are as follows.
As a consequence of the previous theorem and Remark 1, we have the following characterization of Jordan-Osserman Walker metrics.
Theorem 5 Let (M, g) be a Jordan-Osserman Walker 4-manifold. Then one of the following holds: (i) If the Jacobi operators are diagonalizable, then (M, g) is either flat or locally isometric to a paracomplex space form.
(ii) If the Jacobi operators are non diagonalizable, then either (ii.1) the Jacobi operators are two-step or three-step nilpotent (ii.2) the metric g is given by (1)-(2).
Proof. Four-dimensional Jordan-Osserman manifolds with diagonalizable Jacobi operators have been classified at [3], showing that they correspond to real, complex or paracomplex space forms. Next note that real and complex space forms do not support a Walker metric unless they are flat. Indeed, any space of constant curvature is locally conformally flat and hence vanishing of W + shows that any such a Walker metric must be flat. Analogously, Kähler metrics of constant holomorphic sectional curvature have zero Bochner tensor, which shows that W + = 0 [6], [14]. Hence, no Kähler metric of constant holomorphic sectional curvature may be Walker unless it is flat.
On the other hand, if the Jacobi operators are nondiagonalizable, they must be of Type II or III, since Type Ib cannot occur [3]. Then, since anti-self-dual Jordan-Osserman Walker metrics have vanishing scalar curvature, the corresponding Jacobi operators are either two-step or three-step nilpotent. The only remaining case is that of self-dual Jordan-Osserman Walker metrics, which corresponds to the Main Theorem, thus finishing the proof. Then (R 4 , g, J) is a para-Kähler manifold of constant paraholomorphic sectional curvature α.