Projectively and affinely invariant PDEs on hypersurfaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2019
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| _version_ | 1866914724969447424 |
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| author | Alekseevsky, Dmitri V. Manno, Gianni Moreno, Giovanni |
| author_facet | Alekseevsky, Dmitri V. Manno, Gianni Moreno, Giovanni |
| contents | In [Alekseevsky, Gutt, Manno, Moreno: "A general method to construct invariant PDEs on homogeneous manifolds", Communications in Contemporary Mathematics (2021)] the authors have developed a method for constructing $G$-invariant PDEs imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie groups $G$. In the present paper the method is applied to the case when $G=\mathsf{PGL}(n+1)$ or $G=\mathsf{Aff}(n+1)$ and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ or affine $\mathbb{A}^{n+1}$ space, respectively. The paper's main result is that projectively or affinely invariant PDEs with $n$ independent and one unknown variables are in one-to-one correspondence with $\mathsf{CO}(d,n-d)$-invariant hypersurfaces of the space of trace-free cubic forms in $n$ variables. Local descriptions are also provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_06283 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Projectively and affinely invariant PDEs on hypersurfaces Alekseevsky, Dmitri V. Manno, Gianni Moreno, Giovanni Differential Geometry 53C30, 58J70, 35A30, 58A20 In [Alekseevsky, Gutt, Manno, Moreno: "A general method to construct invariant PDEs on homogeneous manifolds", Communications in Contemporary Mathematics (2021)] the authors have developed a method for constructing $G$-invariant PDEs imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie groups $G$. In the present paper the method is applied to the case when $G=\mathsf{PGL}(n+1)$ or $G=\mathsf{Aff}(n+1)$ and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ or affine $\mathbb{A}^{n+1}$ space, respectively. The paper's main result is that projectively or affinely invariant PDEs with $n$ independent and one unknown variables are in one-to-one correspondence with $\mathsf{CO}(d,n-d)$-invariant hypersurfaces of the space of trace-free cubic forms in $n$ variables. Local descriptions are also provided. |
| title | Projectively and affinely invariant PDEs on hypersurfaces |
| topic | Differential Geometry 53C30, 58J70, 35A30, 58A20 |
| url | https://arxiv.org/abs/1907.06283 |