Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

Abstract

We treat m-dimensional real submanifolds M of complex space forms M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.