Projectively and affinely invariant PDEs on hypersurfaces. (English) Zbl 07930421
Summary: In Communications in Contemporary Mathematics24 3, (2022),the authors have developed a method for constructing \(G\)-invariant partial differential equations (PDEs) imposed on hypersurfaces of an \((n+1)\)-dimensional homogeneous space \(G/H\), under mild assumptions on the Lie group \(G\). In the present paper, the method is applied to the case when \(G=\mathsf{PGL}(n+1)\) (respectively, \(G=\mathsf{Aff}(n+1)\)) and the homogeneous space \(G/H\) is the \((n+1)\)-dimensional projective \(\mathbb{P}^{n+1} \) (respectively, affine \(\mathbb{A}^{n+1} )\) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with \(n\) independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of trace-free cubic forms in \(n\) variables with respect to the group \(\mathsf{CO}(d,n-d)\) of conformal transformations of \(\mathbb{R}^{d,n-d} \).
MSC:
| 53C30 | Differential geometry of homogeneous manifolds |
| 58J70 | Invariance and symmetry properties for PDEs on manifolds |
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
| 58A20 | Jets in global analysis |
References:
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