Different definitions of homogeneity of real hypersurfaces in \(\mathbb{C}^2\)
The article discusses different notions of homogeneity for germs of CR hypersurfaces. A (germ of a) CR manifold is called (locally) ``homogenenous'' if there is a (local) Lie group acting (locally) transitively. It is called ``weakly homogeneous'' iff the (local) group of all CR automorphisms acts (locally) transitively. Clearly both notions coincide if the group of all (CR) automorphisms happens to be a Lie group. In the global case it is well-known that this is the case for a CR hypersurface if the Levi form is non-degenerate at one point [see e.g. \textit{N. Tanaka}, J. Math. Soc. Japan 14, 397-429 (1962; Zbl 0113.06303)]. In the article under review the local case is studied, i.e. germs of CR hypersurfaces are regarded. The author proves the following: Let \((M,0)\) be the germ of a nonspherical CR-Hypersurface in \({\mathbb C}^2\) with non-degenerate Levi form. Assume that \((M,0)\) is locally weakly homogeneous. Then \(M\) itself can (locally) endowed with the structure of a local Lie group. This implies in particular that the notions ``homogeneous'' and ``weakly homogeneous'' coincide for germs of CR-hypersurfaces in \({\mathbb C}^2\). The proof is based on the Chern Moser normal form [see \textit{S. S. Chern} and \textit{J. K. Moser}, Acta Math. 133, 219-271 (1974; Zbl 0302.32015)] and estimates obtained by \textit{V. K. Beloshapka} and \textit{A. G. Vitushkin} [see Math. USSR, Izv. 19, 241-259 (1982; Zbl 0501.32011)].
scientific article; zbMATH DE number 1407600