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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2405.06998 |
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| _version_ | 1866909200250044416 |
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| author | Bryant, Robert L. |
| author_facet | Bryant, Robert L. |
| contents | A symmetric quadratic form $g$ on a surface~$M$ is said to be locally Hessianizable if each $p\in M$ has an open neighborhood~$U$ on which there exists a local coordinate chart $(x^1,x^2):U\to\mathbb{R}^2$ and a function $f:U\to\mathbb{R}$ such that, on $U$, we have $$ g = \frac{\partial^2 f}{\partial x^i\partial x^j}\,\mathrm{d} x^i\circ\mathrm{d} x^j. $$ In this article, I show that, when $g$ is nondegenerate and smooth, it is always smoothly locally Hessianizable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hessianizability of surface metrics Bryant, Robert L. Differential Geometry 53B05, 58A15 A symmetric quadratic form $g$ on a surface~$M$ is said to be locally Hessianizable if each $p\in M$ has an open neighborhood~$U$ on which there exists a local coordinate chart $(x^1,x^2):U\to\mathbb{R}^2$ and a function $f:U\to\mathbb{R}$ such that, on $U$, we have $$ g = \frac{\partial^2 f}{\partial x^i\partial x^j}\,\mathrm{d} x^i\circ\mathrm{d} x^j. $$ In this article, I show that, when $g$ is nondegenerate and smooth, it is always smoothly locally Hessianizable. |
| title | Hessianizability of surface metrics |
| topic | Differential Geometry 53B05, 58A15 |
| url | https://arxiv.org/abs/2405.06998 |