Some relationships between the geometry of the tangent bundle and the geometry of the Riemannian base manifold

TL;DR

This paper investigates how the curvature of the tangent bundle relates to the curvature of the original Riemannian manifold, providing formulas that connect their geometric properties.

Contribution

It computes the curvature tensor of the tangent bundle with a natural metric and establishes relationships between the geometries of the tangent bundle and the base manifold.

Findings

Derived explicit formulas for the curvature tensor of the tangent bundle.

Identified conditions linking the curvature properties of the tangent bundle and the base manifold.

Provided insights into how the tangent bundle's geometry reflects the base manifold's curvature.

Abstract

We compute the curvature tensor of the tangent bundle of a Riemannian manifold endowed with a natural metric and we get some relationships between the geometry of the base manifold and the geometry of the tangent bundle.