Connections and Frame Bundle Reductions
Abstract
In general relativity, the gravitational potential is represented by the Levi-Civita connection, the only symmetric connection preserving the metric. On a differentiable manifold, a metric identifies with an orthogonal structure, defined as a Lorentz reduction of the frame bundle. The Levi-Civita connection appears as the only symmetric connection preserving the reduction. This paper presents generalization of this process to other aproaches of gravitation: Weyl structure with Weyl connections, teleparallel structures with Weitzenbock connections, unimodular structure, similarly appear as frame bundle reductions, with preserving connections. To each subgroup H of the linear group GL correspond reduced structures, or H-structures. They are subbundles of the frame bundle (with GL as principal group), with H as principal group. A linear connection in a manifold M is a principal connection on the frame bundle. Given a reduction, the corresponding preserving connections on M are the linear connections which preserve it. I also show that the time gauge used in the 3+1 formalism for general relativity similarly appears as the result of a bundle reduction.
Cite
@article{arxiv.2002.01410,
title = {Connections and Frame Bundle Reductions},
author = {M. Lachieze-Rey},
journal= {arXiv preprint arXiv:2002.01410},
year = {2020}
}