Randers metrics with compatible linear connections: a coordinate-free approach
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915202973302784 |
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| author | Oláh, Márk Vincze, Csaba |
| author_facet | Oláh, Márk Vincze, Csaba |
| contents | A Randers space is a differentiable manifold equipped with a Randers metric. It is the sum of a Riemannian metric and a one-form on the base manifold. The compatibility of a linear connection with the metric means that the parallel transports preserve the Randers norm of tangent vectors. The existence of such a linear connection is not guaranteed in general. If it does exist then we speak about a generalized Berwald Randers metric. In what follows we give a necessary and sufficient condition for a Randers metric to be a generalized Berwald metric and we describe some distinguished compatible linear connections. The method is based on the solution of constrained optimization problems for tensors that are in one-to-one correspondence to the compatible linear connections. The solutions are given in terms of explicit formulas by choosing the free tensor components to be zero. Throughout the paper we use a coordinate-free approach to keep the geometric feature of the argumentation as far as possible. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_13665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Randers metrics with compatible linear connections: a coordinate-free approach Oláh, Márk Vincze, Csaba Differential Geometry 53C60, 58B20 A Randers space is a differentiable manifold equipped with a Randers metric. It is the sum of a Riemannian metric and a one-form on the base manifold. The compatibility of a linear connection with the metric means that the parallel transports preserve the Randers norm of tangent vectors. The existence of such a linear connection is not guaranteed in general. If it does exist then we speak about a generalized Berwald Randers metric. In what follows we give a necessary and sufficient condition for a Randers metric to be a generalized Berwald metric and we describe some distinguished compatible linear connections. The method is based on the solution of constrained optimization problems for tensors that are in one-to-one correspondence to the compatible linear connections. The solutions are given in terms of explicit formulas by choosing the free tensor components to be zero. Throughout the paper we use a coordinate-free approach to keep the geometric feature of the argumentation as far as possible. |
| title | Randers metrics with compatible linear connections: a coordinate-free approach |
| topic | Differential Geometry 53C60, 58B20 |
| url | https://arxiv.org/abs/2503.13665 |