Euclidean nets under isometric embeddings
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916775759708160 |
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| author | Eilat, Matan |
| author_facet | Eilat, Matan |
| contents | Suppose that there exists a discrete subset $X$ of a complete, connected, $n$-dimensional Riemannian manifold $M$ such that the Riemannian distances between points of $X$ correspond to the Euclidean distances of a net in $\mathbb{R}^{n}$. What can then be derived about the geometry of $M$? In arXiv:2004.08621 it was shown that if $n=2$ then $M$ is isometric to $\mathbb{R}^{2}$. In this paper we show two consequential geometric properties that the manifold $M$ shares with the Euclidean space in any dimension. The first property is that $X$ is a net with respect to the Riemannian distance in $M$. The second property is that all geodesics in $M$ are distance minimizing, and there are no conjugate points in $M$. This demonstrates the possibility of inferring infinitesimal qualities from discrete data, even in higher dimensions. As a corollary we obtain that the large-scale geometry of $M$ is asymptotically Euclidean. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_19415 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Euclidean nets under isometric embeddings Eilat, Matan Metric Geometry Differential Geometry Suppose that there exists a discrete subset $X$ of a complete, connected, $n$-dimensional Riemannian manifold $M$ such that the Riemannian distances between points of $X$ correspond to the Euclidean distances of a net in $\mathbb{R}^{n}$. What can then be derived about the geometry of $M$? In arXiv:2004.08621 it was shown that if $n=2$ then $M$ is isometric to $\mathbb{R}^{2}$. In this paper we show two consequential geometric properties that the manifold $M$ shares with the Euclidean space in any dimension. The first property is that $X$ is a net with respect to the Riemannian distance in $M$. The second property is that all geodesics in $M$ are distance minimizing, and there are no conjugate points in $M$. This demonstrates the possibility of inferring infinitesimal qualities from discrete data, even in higher dimensions. As a corollary we obtain that the large-scale geometry of $M$ is asymptotically Euclidean. |
| title | Euclidean nets under isometric embeddings |
| topic | Metric Geometry Differential Geometry |
| url | https://arxiv.org/abs/2305.19415 |