BLAZIC, N., BOKAN, N., and RAKIC, Z.: Nondiagonalizable timelike (spacelike) Osserman (2,2) manifolds

Saitama Mathematical Journal 1998(Vol. 16) pp.15-22


Recently it was shown that the Osserman conjecture in the pseudo-Riemannian setup can not hold for some signatures (see [4,8]). The geometry of manifolds satisfying the Osserman type conditions depends on the algebraic structure of Jordan form of the corresponding Jacobi operator. For four dimensional manifolds of signature (2,2) we naturally distinguish three types of the Osserman manifolds, depending on degree of the minimal polynomial of the Jacobi operator. The main goal in this paper is to understand symmetric type properties of locally non-symmetric Osserman manifolds of signature (2,2). We establish that Ricci flat Osserman manifolds of type II are semisymmetric. Particular attention is dedicated to the Ricci flat manifolds since all known examples of Osserman (2,2) maniforlds satisfies that condition. Diagonalizable Osserman manifolds of signater (2,2) (type I) are rank-one locally symmetric spaces.
1991 Mathematics Subject Classification: 53B30, 53C50.