Nonsymmetric Osserman pseudo-Riemannian manifolds

Abstract

Examples of Osserman pseudo-Riemannian manifolds with metric of any signature (p, q), p, q > 1 which are not locally symmetric are exhibited.

Conformally Osserman four-dimensional manifolds whose conformal Jacobi operators have complex eigenvalues

“…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).…”

Nilpotent Szabó, Osserman and Ivanov-Petrova pseudo-Riemannian manifolds

“…Such manifolds are necessarily Osserman. Osserman nilpotent manifolds of orders 2 and 3 have been constructed previously [2,7,6,8]. These manifolds need not be homogeneous, thus the question Osserman raised has a negative answer in the higher signature setting.…”

The geometry of a bi-Lagrangian manifold

“…On the other hand, para-Kähler space forms are Osserman manifolds, i.e., the eigenvalues of the Jacobi operator are constant (see [35]; in that paper one can find examples of neutral manifolds which are nonsymmetric. A complete monograph about this topic is [34]).…”