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

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

### Abstract

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.