On Eisenhart's type theorem for sub-Riemannian metrics on step $2$ distributions with $\mathrm{ad}$-surjective Tanaka symbols
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911765839740928 |
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| author | Lin, Zaifeng Zelenko, Igor |
| author_facet | Lin, Zaifeng Zelenko, Igor |
| contents | The classical result of Eisenhart states that if a Riemannian metric $g$ admits a Riemannian metric that is not constantly proportional to $g$ and has the same (parameterized) geodesics as $g$ in a neighborhood of a given point, then $g$ is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step $2$ graded nilpotent Lie algebras, called $\mathrm{ad}$-surjective, and extend the Eisenhart theorem to sub-Riemannian metrics on step 2 distributions with $\mathrm{ad}$-surjective Tanaka symbols. The class of ad-surjective step 2 nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_14218 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Eisenhart's type theorem for sub-Riemannian metrics on step $2$ distributions with $\mathrm{ad}$-surjective Tanaka symbols Lin, Zaifeng Zelenko, Igor Differential Geometry Optimization and Control 53C17, 58A30, 58E10, 53A15, 37J39, 35N10, 17B70 The classical result of Eisenhart states that if a Riemannian metric $g$ admits a Riemannian metric that is not constantly proportional to $g$ and has the same (parameterized) geodesics as $g$ in a neighborhood of a given point, then $g$ is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step $2$ graded nilpotent Lie algebras, called $\mathrm{ad}$-surjective, and extend the Eisenhart theorem to sub-Riemannian metrics on step 2 distributions with $\mathrm{ad}$-surjective Tanaka symbols. The class of ad-surjective step 2 nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case. |
| title | On Eisenhart's type theorem for sub-Riemannian metrics on step $2$ distributions with $\mathrm{ad}$-surjective Tanaka symbols |
| topic | Differential Geometry Optimization and Control 53C17, 58A30, 58E10, 53A15, 37J39, 35N10, 17B70 |
| url | https://arxiv.org/abs/2308.14218 |