Properties of modified Riemannian extensions

TL;DR

This paper investigates the properties of modified Riemannian extensions on cotangent bundles, focusing on conditions for Kähler-Norden structures and analyzing curvature properties of associated connections.

Contribution

It introduces conditions under which the cotangent bundle with a modified Riemannian extension forms a Kähler-Norden manifold and explores curvature characteristics of various connections.

Findings

Conditions for Kähler-Norden structures on cotangent bundles.

Curvature properties of Levi-Civita and other metric connections.

Characterization of the modified Riemannian extension $ ilde{g}_{ abla,c}$.

Abstract

Let $M$ be an $n-$dimensional differentiable manifold with a symmetric connection $\nabla $ and $T^{\ast}M$ be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension $% \widetilde{g}_{\nabla,c}$ on $T^{\ast}M$ defined by means of a symmetric $% (0,2)$-tensor field $c$ on $M.$ We get the conditions under which $T^{\ast}M $ endowed with the horizontal lift $^{H}J$ of an almost complex structure $J$ and with the metric $\widetilde{g}_{\nabla,c}$ is a K\"{a}hler-Norden manifold. Also curvature properties of the Levi-Civita connection and another metric connection of the metric $\widetilde{g}_{\nabla,c}$ are presented.