On pseudo-conformal transformations of hypersurfaces

Abstract

Given a real differentiable hypersurface $S_{i},$ $i=1,2$, of a complex manifold $M_{i}$ , we say that a mapping $f$ of $S_{1}$ into $S_{2}$ is pseudo-conformal if $f$ extends to a holomorphic mapping of a neighborhood of $S_{1}$ in $M_{1}$ into that of $S_{2}$ in $M_{2}$ . $S_{1}$ is called pseudo-conformally equivalent to $S_{2}$ by $f$ if moreover $f$ is bijec- tive and $f^{-1}$ is also pseudo-conformal. In this paper we shall consider pseudo- conformal transformations of a compact hypersurface $S$ , which is by definition pseudo-conformally equivalent to itself by these transformations. The set of all the pseudo-conformal transformations of $S$ forms a group, which becomes, with the natural topology, a Lie transformation group under some hypothesis (cf. Theorem 5 and Corollary in Our aim is to classify all compact hypersurfaces admitting transitive pseudo-conformal transformation groups. The obtained results are shown in Theorems 1 and 2 (at the beginning of Section 2).