The geometry of modified Riemannian extensions

Abstract

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and gives necessary and sufficient conditions for a modified Riemannian extension to be Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3, 3), whose Jacobi operators have non-trivial Jordan normal form and which are not nilpotent. We present new four-dimensional results in Osserman geometry.