On geometry of frame bundles

TL;DR

This paper introduces new Riemannian metrics on the frame bundle of a manifold derived from tangent bundle metrics, and computes their Levi-Civita connection and curvature, enriching the geometric understanding of frame bundles.

Contribution

It constructs novel Riemannian metrics on frame bundles from tangent bundle metrics and calculates their Levi-Civita connection and curvature.

Findings

New Riemannian metrics on frame bundles derived from tangent bundle metrics

Explicit formulas for Levi-Civita connection of these metrics

Curvature properties of the constructed metrics

Abstract

Let $(M,g)$ be a Riemannian manifold, $L(M)$ its frame bundle. We construct new examples of Riemannian metrics on $L(M)$, which are obtained from Riemannian metrics on the tangent bundle $TM$. We compute the Levi--Civita connection and curvatures of these metrics.