Group actions on manifolds and smooth ambient homogeneity
A subset \(K\) of a smooth manifold \(M\) is called smoothly homogeneous if for every pair of points \(x,y\in K\) there exist neighborhoods \(O_x,O_y\subset M\) and a diffeomorphism \(h:O_x\widetilde{\rightarrow} O_y\) such that \(h(O_x\cap K)=O_y\cap K\) and \(h(x)=y\). The main result of the paper is the following Theorem. Let \(K\) be a locally compact subset of a manifold \(M\). Then \(K\) is a smooth submanifold if and only if \(K\) is smoothly homogeneous. Unfortunately, only a sketch of proof is provided.
scientific article; zbMATH DE number 1058643
scientific article