Covariant Derivatives on Homogeneous Spaces -- Horizontal Lifts and Parallel Transport

TL;DR

This paper explores invariant covariant derivatives on reductive homogeneous spaces, providing new characterizations and proofs related to parallel transport, horizontal lifts, and metric invariance.

Contribution

It offers a novel perspective using horizontal lifts to characterize invariant covariant derivatives and establishes a new proof of their existence and correspondence to bilinear maps.

Findings

Characterization of parallel vector fields along curves

New proof for existence of invariant covariant derivatives

One-to-one correspondence with certain bilinear maps

Abstract

We consider invariant covariant derivatives on reductive homogeneous spaces corresponding to the well-known invariant affine connections. These invariant covariant derivatives are expressed in terms of horizontally lifted vector fields on the Lie group. This point of view allows for a characterization of parallel vector fields along curves. Moreover, metric invariant covariant derivatives on a reductive homogeneous space equipped with an invariant pseudo-Riemannian metric are characterized. As a by-product, a new proof for the existence of invariant covariant derivatives on reductive homogeneous spaces and their the one-to-one correspondence to certain bilinear maps is obtained.