Nilpotent Spacelike Jorden Osserman pseudo-Riemannian manifolds

Abstract

Pseudo-Riemannian manifolds of balanced signature which are both spacelike and timelike Jordan Osserman nilpotent of order 2 and of order 3 have been constructed previously. In this short note, we shall construct pseudo-Riemannian manifolds of signature (2s,s) for any s (which is at least 2) which are spacelike Jordan Osserman nilpotent of order 3 but which are not timelike Jordan Osserman. Our example and techniques are quite different from known previously both in that they are not in neutral signature and that the manifolds constructed will be spacelike but not timelike Jordan Osserman.