The Sectional Curvature of the Tangent Bundles with General Natural Lifted Metrics

TL;DR

This paper investigates the geometric properties of tangent bundles equipped with general natural lifted metrics, specifically identifying conditions for constant sectional curvature in the tangent bundle.

Contribution

It derives conditions under which tangent bundles with general natural lifted metrics exhibit constant sectional curvature, extending understanding of their geometric structure.

Findings

Conditions for constant sectional curvature on tangent bundles are established.

Characterization of general natural lifted metrics that produce constant curvature.

Insights into the geometric structure of tangent bundles with lifted metrics.

Abstract

We study some properties of the tangent bundles with metrics of general natural lifted type. We consider a Riemannian manifold $(M,g)$ and we find the conditions under which the Riemannian manifold $(TM,G)$, where $TM$ is the tangent bundle of $M$ and $G$ is the general natural lifted metric of $g$, has constant sectional curvature.