The geometry of modified Riemannian extensions

TL;DR

This paper explores the geometric properties of modified Riemannian extensions, characterizing when they are Einstein, and constructs specific Osserman manifolds with unique Jacobi operator properties, advancing understanding in differential geometry.

Contribution

It provides necessary and sufficient conditions for modified Riemannian extensions to be Einstein and constructs new Osserman manifolds with non-trivial Jordan normal form.

Findings

Every paracomplex space form is locally isometric to a modified Riemannian extension.

Conditions for a modified Riemannian extension to be Einstein are established.

Constructed Riemannian extension Osserman manifolds with non-trivial Jordan normal form.

Abstract

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and give necessary and sufficient conditions so that a modified Riemannian extension is 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.