On the Transformation Group of a Real Hypersurface

Abstract

The group of biholomorphic transformations leaving fixed a strongly pseudoconvex real hypersurface in a complex manifold is a Lie group.In this paper it is shown that the Chern-Moser invariants must vanish if this group is noncompact and the hypersurface is compact.Also considered are transformation groups of flat hypersurfaces and intransitive groups.Introduction.The purpose of this paper is to study the group of structure preserving, or pseudoconformal, transformations of a real hypersurface M (of dimension 2n + 1) in a complex n + 1 manifold, or more generally, of a manifold with the same C-R structure.We will always assume that the C-R structure is integrable and that the Levi form is nondegenerate.It follows from the work of Chern and Moser [2] that this group is a Lie transformation group.There is some similarity between the geometry of M and the conformai geometry of a Riemannian manifold.Over M there is a principal fibre bundle B (the pseudoconformal bundle), a connection on B, and curvature invariants the most important of which is the fourth order curvature tensor Sg^ (see [2]).These have as analogues in the Riemannian case the bundle of conformai frames the conformai connection, and the Weyl conformai curvature tensor.If n > 2, S vanishes if and only if M is pseudoconformally flat, i.e., locally equivalent to the standard sphere S2n+l in C+I [8].Locally the structure of M is given by a real one-form 9, n complex one-forms 9a, and their complex conjugates 9" -9a.They satisfy 0/\9x /\---/\9n/\9X /\-• • Afl">0 and the equation dO = igaß9a f\0t mod 9.Here and throughout this paper Greek indices run from 1 to n and the summation convention is used.The nondegenerate hermitian matrix (gaß) is