On elliptic and quasiregularly elliptic manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912747022712832 |
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| author | Manin, Fedor Prywes, Eden |
| author_facet | Manin, Fedor Prywes, Eden |
| contents | In his book "Metric structures for Riemannian and non-Riemannian spaces", Gromov defined two properties of Riemannian manifolds, ellipticity and quasiregular ellipticity, and suggested that there may be a connection between the two. Since then, groups of researchers working independently have proved strikingly similar results about these two concepts. We obtain new topological obstructions to the two properties: most notably, we show that closed manifolds of both types must have virtually abelian fundamental group. We also give the first examples of open manifolds which are elliptic but not quasireguarly elliptic and vice versa. Whether there is a direct connection between these properties -- and, in particular, whether they are equivalent for closed manifolds -- remains elusive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19121 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On elliptic and quasiregularly elliptic manifolds Manin, Fedor Prywes, Eden Differential Geometry Complex Variables Metric Geometry 53C23, 30C65 In his book "Metric structures for Riemannian and non-Riemannian spaces", Gromov defined two properties of Riemannian manifolds, ellipticity and quasiregular ellipticity, and suggested that there may be a connection between the two. Since then, groups of researchers working independently have proved strikingly similar results about these two concepts. We obtain new topological obstructions to the two properties: most notably, we show that closed manifolds of both types must have virtually abelian fundamental group. We also give the first examples of open manifolds which are elliptic but not quasireguarly elliptic and vice versa. Whether there is a direct connection between these properties -- and, in particular, whether they are equivalent for closed manifolds -- remains elusive. |
| title | On elliptic and quasiregularly elliptic manifolds |
| topic | Differential Geometry Complex Variables Metric Geometry 53C23, 30C65 |
| url | https://arxiv.org/abs/2410.19121 |