A pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

Let p be a point of a Lorentzian manifold M. We show that if M is spacelike Osserman at p, then M has constant sectional curvature at p ; similarly, if M is timelike Osserman at p, then M has constant sectional curvature at p. The reverse implications are immediate. The timelike case and 4-dimensional spacelike case were first studied in [3] ; we use a different approach to this case.

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Abstract. We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also study similar questions related to the skew-symmetric curvature operator defined by the Weyl conformal curvature tensor.

Abstract. Let M be a Riemannian manifold with the Jacobi operator, which has constant eigenvalues, independent on the unit vector X ∈ TpM and the point p ∈ M . Osserman conjectured that these manifolds are flat or rankone locally symmetric spaces (∇R = 0). It is known that for a general pseudo-Riemannian manifold, the Osserman-type conjecture is not true and 4-dimensional Kleinian Jordan-Osserman manifolds are curvature homogeneous. We show that the length of the first covariant derivative of the curvature tensor is isotropic, i.e. ∇R = 0. For known examples of 4-dimensional Osserman manifolds of signature (− − ++) we check also that ∇R = 0. By the presentation of a class of examples we show that curvature homogeneity and ∇R = 0 do not imply local homogeneity; in contrast to the situation in the Riemannian geometry, where it is unknown if the Osserman condition implies local homogeneity. §0. Introduction Let (M, g) be a 4-dimensional Kleinian (neutral) manifold, i.e. a pseudo-Riemannian manifold with a metric g of signature (− − ++). We denote its curvature tensor by R. The Jacobi operator R X : Y → R(Y, X)X is a symmetric endomorphism of T p M and K X is its restriction to X ⊥ in T p M . For Riemannian manifolds, Osserman [16], based on joint results with Sarnak [17], has conjectured that if the eigenvalues of the Jacobi operator K X are independent of the choice of unit vectors X ∈ T p M and of the choice p ∈ M , then either M is locally a rank-one symmetric space or M is flat. We have generalized in [3] the Osserman-type condition in the pseudo-Riemannian setup in terms of the Jordan form of K X , that is equivalent, especially for 4-dimensions, to the conditions in terms of the constancy of the minimal polynomial for K X . Namely, M is spacelike (resp. timelike) Jordan-Osserman at p if the Jordan form of K X is independent of X ∈ T p M , g(X, X) = 1 (resp. g(X, X) = −1). If M is spacelike (resp. timelike) Jordan-Osserman at every p ∈ M , one says M is pointwise spacelike (resp. timelike) Jordan-Osserman. If the Jordan form of K X is independent of p ∈ M , then M is spacelike (resp. timelike) Jordan-Osserman.

The main goal is to classify 4-dimensional real Lie algebras g which admit a para-hypercomplex structure. This is a step toward the classification of Lie groups admitting the corresponding left-invariant structure and therefore possessing a neutral, left-invariant, anti-self-dual metric. Our study is related to the work of Barberis who classified real, 4-dimensional simply-connected Lie groups which admit an invariant hypercomplex structure.

We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.