preprint · arXiv (Cornell University) · 2020

Connections and Frame Bundle Reductions

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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 this paper

APA 7

Lachièze-Rey, M. (2020). Connections and Frame Bundle Reductions. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2002.01410

MLA 9

Lachièze-Rey, Marc. “Connections and Frame Bundle Reductions.” arXiv (Cornell University), 2020, https://doi.org/10.48550/arxiv.2002.01410.

Chicago

Lachièze-Rey, Marc. 2020. “Connections and Frame Bundle Reductions.” arXiv (Cornell University). https://doi.org/10.48550/arxiv.2002.01410.

Harvard

Lachièze-Rey, M. (2020) ‘Connections and Frame Bundle Reductions’, arXiv (Cornell University). Available at: https://doi.org/10.48550/arxiv.2002.01410.

BibTeX

@misc{lachiezerey2020connections,
  title = {{Connections and Frame Bundle Reductions}},
  author = {Lachièze-Rey, Marc},
  journal = {arXiv (Cornell University)},
  year = {2020},
  doi = {10.48550/arxiv.2002.01410}
}

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