论文标题
一种在均匀流形上构建不变PDE的一般方法
A general method to construct invariant PDEs on homogeneous manifolds
论文作者
论文摘要
令$ m = g/h $为$（n+1）$ - 尺寸均质歧管，$ j^k（n，m）=：j^k $是$ m $ $ k $ jets $ k $ jets of $ m $ $ $。 Lie Group $ G $自然地对每个$ J^k $起作用。 $ g $ invariant的订单$ k $的$ m $ $ m $（即带有$ n $ n $自变量和$ 1 $依赖性的）被定义为$ g $ -Invariant hyverface $ \ mathcal $ \ nathcal {e} e} \ subset j^k $。我们描述了一种用于$ k \ geq 2 $的不变PDE的一般方法。该问题减少了在特定矢量空间中的超曲面的描述，相对于$（k-1）$的稳定性子组$ h^{（k-1）$的线性动作是不变的 - $ g $的延长动作。我们应用这种方法来描述Euclidean Space中的Hypersurfaces的不变PDE $ \ MATHBB {E}^{n+1} $以及在保形空间$ \ Mathbb {s}^{n+1} $中。我们的方法在一些$ g $的动作中有一些轻微的假设，即： A1）组$ g $必须在$ j^{k-1} $中具有开放轨道，并且 A2）稳定器$ h^{（k-1）} \子集的子集g $ $ j^k \ to j^{k-1} $必须通过光纤本身的翻译组进行分解。
Let $M = G/H$ be an $(n+1)$-dimensional homogeneous manifold and $J^k(n,M)=:J^k$ be the manifold of $k$-jets of hypersurfaces of $M$. The Lie group $G$ acts naturally on each $J^k$. A $G$-invariant PDE of order $k$ for hypersurfaces of $M$ (i.e., with $n$ independent variables and $1$ dependent one) is defined as a $G$-invariant hypersurface $\mathcal{E} \subset J^k$. We describe a general method for constructing such invariant PDEs for $k\geq 2$. 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 PDEs for hypersurfaces in the Euclidean space $\mathbb{E}^{n+1 }$ and in the conformal space $\mathbb{S}^{n+1}$. Our method works under some mild assumptions on the action of $G$, namely: A1) the group $G$ must have an open orbit in $J^{k-1}$, and A2) the stabilizer $H^{(k-1)}\subset G$ of the fibre $J^k\to J^{k-1}$ must factorize via the group of translations of the fibre itself.
