A functional characterization of isometries between non-reversible Finsler manifolds
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913635418243072 |
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| author | M, Francisco Venegas |
| author_facet | M, Francisco Venegas |
| contents | We provide a functional characterization of isometries between non-reversible Finsler manifolds, in the form of a generalization of the Myers-Nakai Theorem for Riemannian manifolds. We show that, since non-reversible Finsler manifolds are a fundamentally asymmetric object, such a result can not be obtained by means of a symmetric function space, and we define the appropriate asymmetric structure needed to describe all possible isometries between this class of manifolds. The result is based on the ideas used in a previous generalization for reversible Finsler manifolds proved in \cite{GJR-13}, in which the normed algebra of $C^1$-smooth Lipschitz functions is used. To reflect the quasi-metric structure of non-reversible Finsler manifolds, this normed algebra had to be modified to include the cone of smooth semi-Lipschitz functions, resulting in a partial loss of the normed space and algebra structures. In order to achieve the desired result, we define new algebraic/quasi-metric structures to model the behavior of the aforementioned function space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06788 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A functional characterization of isometries between non-reversible Finsler manifolds M, Francisco Venegas Functional Analysis Differential Geometry 54E40, 54C65 We provide a functional characterization of isometries between non-reversible Finsler manifolds, in the form of a generalization of the Myers-Nakai Theorem for Riemannian manifolds. We show that, since non-reversible Finsler manifolds are a fundamentally asymmetric object, such a result can not be obtained by means of a symmetric function space, and we define the appropriate asymmetric structure needed to describe all possible isometries between this class of manifolds. The result is based on the ideas used in a previous generalization for reversible Finsler manifolds proved in \cite{GJR-13}, in which the normed algebra of $C^1$-smooth Lipschitz functions is used. To reflect the quasi-metric structure of non-reversible Finsler manifolds, this normed algebra had to be modified to include the cone of smooth semi-Lipschitz functions, resulting in a partial loss of the normed space and algebra structures. In order to achieve the desired result, we define new algebraic/quasi-metric structures to model the behavior of the aforementioned function space. |
| title | A functional characterization of isometries between non-reversible Finsler manifolds |
| topic | Functional Analysis Differential Geometry 54E40, 54C65 |
| url | https://arxiv.org/abs/2405.06788 |