Cohomogeneity one RCD-spaces
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909819240185856 |
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| author | Corro, Diego Núñez-Zimbrón, Jesús Santos-Rodríguez, Jaime |
| author_facet | Corro, Diego Núñez-Zimbrón, Jesús Santos-Rodríguez, Jaime |
| contents | We study $\mathsf{RCD}$-spaces $(X,d,\mathfrak{m})$ with group actions by isometries preserving the reference measure $\mathfrak{m}$ and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that when $X$ is non-collapsed the slices are homeomorphic to metric cones over homogeneous spaces with $\mathrm{Ric} \geq 0$. As a consequence we obtain complete topological structural results (also in the collapsed case) and a regular orbit representation theorem. Conversely, we show how to construct new $\mathsf{RCD}$-spaces from a cohomogeneity one group diagram, giving a complete description of $\mathsf{RCD}$-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed $\mathsf{RCD}$-spaces of essential dimension at most $4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomogeneity one RCD-spaces Corro, Diego Núñez-Zimbrón, Jesús Santos-Rodríguez, Jaime Metric Geometry Differential Geometry 53C20, 53C21, 53C23 We study $\mathsf{RCD}$-spaces $(X,d,\mathfrak{m})$ with group actions by isometries preserving the reference measure $\mathfrak{m}$ and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that when $X$ is non-collapsed the slices are homeomorphic to metric cones over homogeneous spaces with $\mathrm{Ric} \geq 0$. As a consequence we obtain complete topological structural results (also in the collapsed case) and a regular orbit representation theorem. Conversely, we show how to construct new $\mathsf{RCD}$-spaces from a cohomogeneity one group diagram, giving a complete description of $\mathsf{RCD}$-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed $\mathsf{RCD}$-spaces of essential dimension at most $4$. |
| title | Cohomogeneity one RCD-spaces |
| topic | Metric Geometry Differential Geometry 53C20, 53C21, 53C23 |
| url | https://arxiv.org/abs/2405.09448 |