preprint
Inserted: 15 sep 2020
Last Updated: 15 sep 2020
Year: 2019
Abstract:
Let $M$ be an $(n+1)$-dimensional manifold, let a group $G$ act transitively on $M$ and let $J^k(n,M)$ denote the space of $k$-jets of hypersurfaces of $M$. We make the following two assumptions on the action of $G$. First, there exists a hypersurface $\mathcal{S}_F\subset M$, referred to as a fiducial hypersurface, such that the $G$-orbit of the $k-1^{\textrm{st}}$ jet of $\mathcal{S}_F$ at a point $o\in M$ is open in $J^{k-1}(n,M)$. Second, there are no open $G$-orbits in $J^{k}(n,M)$. Then, starting from such an $\mathcal{S}_F$, we construct a family of (scalar) $G$-invariant $k^{\textrm{th}}$ order PDEs in $n$ independent variables and 1 dependent one. We show that the solutions to these equations are a natural analogue of the Weingarten hypersurfaces in (semi-)Riemannian manifolds. The cases when $k=2$ or $k=3$ are carefully examined. In particular, we find convenient coordinates to locally describe the so-obtained equations.