Projectively and affinely invariant PDEs on hypersurfaces

• Published in: arXiv.org
• Main Authors: Alekseevsky, Dmitri V, Manno, Gianni, Moreno, Giovanni
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 23.03.2024
• Subjects: Dependent variables; Hyperspaces; Independent variables; Invariants; Lie groups; Mathematical analysis; Mathematics - Differential Geometry; Riemann manifold
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1907.06283

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.