On twisted Riemannian extensions associated with Szab\'o metrics

TL;DR

This paper explores the geometric properties of twisted Riemannian extensions related to Szabó metrics, linking affine and pseudo-Riemannian geometries, and investigates the spectral geometry of the Szabó operator.

Contribution

It introduces a study of twisted Riemannian extensions associated with Szabó metrics, highlighting their geometric structure and spectral properties.

Findings

Establishes a connection between affine and pseudo-Riemannian geometries via twisted extensions.

Analyzes the spectral geometry of the Szabó operator on manifolds and their cotangent bundles.

Provides new insights into the geometric and spectral characteristics of Szabó metrics.

Abstract

Let $M$ be an $n$-dimensional manifold with a torsion free affine connection $\nabla$ and let $T^*M$ be the cotangent bundle. In this paper, we consider some of the geometrical aspect of a twisted Riemannian extension which provide a link between the affine geometry of $(M,\nabla)$ and the neutral signature pseudo-Riemannian geometry of $T^*M$. We investigate the spectral geometry of the Szab\'o operator on $M$ and on $T^*M$.