Boundary values of diffeomorphisms of simple polytopes, and controllability
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arXiv
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| Format: | Preprint |
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2024
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| author | Glöckner, Helge Grong, Erlend Schmeding, Alexander |
| author_facet | Glöckner, Helge Grong, Erlend Schmeding, Alexander |
| contents | We consider the Lie group of smooth diffeomorphisms Diff$(M)$ of a simple polytope $M$ in the euclidean space. Simple polytopes are special cases of manifolds with corners. The geometric setting allows to study in particular, the subgroup of face respecting diffeomorphisms and its Lie theoretic properties. We find a canonical Lie group structure for the quotient of the diffeomorphism by the subgroup Diff$^{\partial,id}(M)$ of maps that equal the identity on the boundary, turning the canonical quotient homomorphism Diff$(M)\rightarrow $Diff$(M)/$Diff$^{\partial,id}(M)$ into a smooth submersion. We also show that the identity component of the diffeomorphism group is generated by the exponential image, by proving general controllability results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_05444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundary values of diffeomorphisms of simple polytopes, and controllability Glöckner, Helge Grong, Erlend Schmeding, Alexander Group Theory Differential Geometry 58D05 (primary), 22E65, 34H05, 52B08, 52B70, 53C17 (secondary) We consider the Lie group of smooth diffeomorphisms Diff$(M)$ of a simple polytope $M$ in the euclidean space. Simple polytopes are special cases of manifolds with corners. The geometric setting allows to study in particular, the subgroup of face respecting diffeomorphisms and its Lie theoretic properties. We find a canonical Lie group structure for the quotient of the diffeomorphism by the subgroup Diff$^{\partial,id}(M)$ of maps that equal the identity on the boundary, turning the canonical quotient homomorphism Diff$(M)\rightarrow $Diff$(M)/$Diff$^{\partial,id}(M)$ into a smooth submersion. We also show that the identity component of the diffeomorphism group is generated by the exponential image, by proving general controllability results. |
| title | Boundary values of diffeomorphisms of simple polytopes, and controllability |
| topic | Group Theory Differential Geometry 58D05 (primary), 22E65, 34H05, 52B08, 52B70, 53C17 (secondary) |
| url | https://arxiv.org/abs/2407.05444 |