The purpose of this article is to give explicit descriptions for stability groups of real rigid hypersurfaces of infinite type in C 2 . The decompositions of infinitesimal CR automorphisms are also given.

Let E(α) ⊂ C m+1 and E(β) ⊂ C n+1 be generalized pseudoellipsoids. Assume that the inequality m < n holds. They are parametrized by N -tuples of positive integers α = (α 1 , . . . , α N ) and β = (β 1 , . . . , β N ).(See introduction for the definition of a generalized pseudoellipsoid) Assume that there exists a proper holomorphic mapping between them. In this article, two facts are proved. Firstly, under the assumptions of the existence of such a mapping, certain nondegeneracy conditions of a submatrix of the Jacobian matrix and additional inequalities on dimensions, the parameters (α 1 , . . . , α N ) and (β 1 , . . . , β N ) coincide; α 1 = β 1 , . . . , α N = β N after re-ordering if necessary. Secondly, such a proper holomorphic mapping is a linear embedding up to automorphisms of a source and a target domains.

We classify proper holomorphic mappings between generalized pseudoellipsoids of different dimensions. Those domains are parametrized by the exponents. The relations among them are also obtained. Main tool is the orthogonal decomposition of a CR bundle. Such a decomposition derives the "variable-splitting" of the mapping.

Abstract. Let /: M -> M' be a real analytic CR mapping between hypersurfaces with f(p) = q, where p € M and q G M'. In this paper, the relation between the type at p and the one at q is considered. As a corollary of the type condition theorem (Theorem 1

The CR equivalence problem between CR manifolds with slice structure is studied. Let N be a connected holomorphically nondegenerate real analytic hypersurface and M(p) a finitely nondegenerate real analytic hypersurface parametrized by p ∈ N . Let M be a totality of N and M(p) with moving p in N . Assume that M and M (with a same structure as M) are CR equivalent and that N and N are also CR equivalent. Then we prove that, for any p ∈ N , there exists p ∈ N such that M(p) is CR equivalent to M( p).