Conformally flat tangent bundles with general natural lifted metrics

TL;DR

This paper investigates when the tangent bundle of a Riemannian manifold, equipped with a general natural lifted metric, is conformally flat, establishing that the base must have constant sectional curvature and deriving conditions on the metric.

Contribution

It provides necessary conditions for the tangent bundle to be conformally flat and characterizes the form of the natural lifted metric that achieves this property.

Findings

Base manifold must have constant sectional curvature.

Derived explicit expressions for the natural lifted metric G.

Identified conditions under which (TM,G) is conformally flat.

Abstract

We study the conditions under which the tangent bundle $(TM,G)$ of an $n$-dimensional Riemannian manifold $(M,g)$ is conformally flat, where $G$ is a general natural lifted metric of $g$. We prove that the base manifold must have constant sectional curvature and we find some expressions for the natural lifted metric $G$, such that the tangent bundle $(TM,G)$ become conformally flat.