## Dmitri V. Alekseevsky - J. Gutt - G. Manno - G. Moreno

# Invariant PDEs on homogeneous manifolds via the affine structure of the
bundles of jet spaces

created by moreno on 15 Sep 2020

[

BibTeX]

*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.