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arXiv · 2308.07089

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

Markus Schlarb
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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.

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Markus Schlarb. 2023-08-14. Covariant Derivatives on Homogeneous Spaces -- Horizontal Lifts and Parallel Transport. https://arxiv.org/abs/2308.07089

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