Communications in Contemporary Mathematics ( IF 1.278 ) Pub Date : 2021-01-07 , DOI: 10.1142/s0219199720500893
Dmitri V. Alekseevsky; Jan Gutt; Gianni Manno; Giovanni Moreno

Let $M=G/H$ be an $(n+1)$-dimensional homogeneous manifold and $Jk(n,M)=:Jk$ be the manifold of $k$-jets of hypersurfaces of $M$. The Lie group $G$ acts naturally on each $Jk$. A $G$-invariant partial differential equation of order $k$ for hypersurfaces of $M$ (i.e., with $n$ independent variables and $1$ dependent one) is defined as a $G$-invariant hypersurface $ℰ⊂Jk$. We describe a general method for constructing such invariant partial differential equations for $k≥2$. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup $H(k−1)$ of the $(k−1)$-prolonged action of $G$. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space $𝔼n+1$ and in the conformal space $𝕊n+1$. Our method works under some mild assumptions on the action of $G$, namely: A1) the group $G$ must have an open orbit in $Jk−1$, and A2) the stabilizer $H(k−1)⊂G$ of the fiber $Jk→Jk−1$ must factorize via the group of translations of the fiber itself.

$中号=G/H$$（ñ+1个）$维均匀流形和 $Ĵķ（ñ，中号）=：Ĵķ$ 成为 $ķ$的超表面喷射 $中号$。谎言集团$G$ 自然地作用于每个 $Ĵķ$。一种$G$不变偏微分方程 $ķ$ 适用于 $中号$ （即 $ñ$ 自变量和 $1个$ 相依者）被定义为 $G$不变超曲面 $ℰ⊂Ĵķ$。我们描述了构造此类不变偏微分方程的一般方法$ķ≥2$。该问题简化为在一定向量空间中对曲面的描述，该曲面相对于稳定性子组的线性作用是不变的$H（ķ-1个）$$（ķ-1个）$-的长时间动作 $G$。我们采用这种方法来描述欧氏空间中超曲面的不变偏微分方程$𝔼ñ+1个$ 在保形空间 $𝕊ñ+1个$。我们的方法在一些温和的假设下对$G$，即：A1）组 $G$ 必须有一个开放的轨道 $Ĵķ-1个$，以及A2）稳定器 $H（ķ-1个）⊂G$ 纤维的 $Ĵķ→Ĵķ-1个$ 必须通过光纤本身的平移组进行分解。

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