Local isometric immersions of pseudospherical surfaces described by a class of third order differential equations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913873778442240 |
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| author | Guo, Mingyue Shi, Zhenhua |
| author_facet | Guo, Mingyue Shi, Zhenhua |
| contents | We discuss a specific type of pseudospherical surfaces defined by a class of third order differential equations, of the form $u_t - u_{xxt} = λu^2 u_{xxx} + G(u, u_x, u_{xx})$, and poses a question about the dependence of the triples $\{a,b,c\}$ of the second fundamental form in the context of local isometric immersion in $\mathbb{E}^3$. It is demonstrated that the triples $\{a,b,c\}$ of the second fundamental form are not influenced by a jet of finite order of $u$. Instead, they are shown to rely on a jet of order zero, making them universal and not reliant on the specific solution chosen for $u$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local isometric immersions of pseudospherical surfaces described by a class of third order differential equations Guo, Mingyue Shi, Zhenhua Differential Geometry Mathematical Physics We discuss a specific type of pseudospherical surfaces defined by a class of third order differential equations, of the form $u_t - u_{xxt} = λu^2 u_{xxx} + G(u, u_x, u_{xx})$, and poses a question about the dependence of the triples $\{a,b,c\}$ of the second fundamental form in the context of local isometric immersion in $\mathbb{E}^3$. It is demonstrated that the triples $\{a,b,c\}$ of the second fundamental form are not influenced by a jet of finite order of $u$. Instead, they are shown to rely on a jet of order zero, making them universal and not reliant on the specific solution chosen for $u$. |
| title | Local isometric immersions of pseudospherical surfaces described by a class of third order differential equations |
| topic | Differential Geometry Mathematical Physics |
| url | https://arxiv.org/abs/2501.01444 |