Four-dimensional contact CR-submanifolds in S7(1)

Abstract

The analogue of CR -submanifolds in (almost) Kählerian manifolds is the concept of contact CR -submanifolds in Sasakian manifolds. These are submanifolds for which the structure vector field ξ is tangent to the submanifold and for which the tangent bundle of M can be decomposed as T(M)=H(M)⊕E(M)⊕Rξ, where H(M) is invariant with respect to the endomorphism φ and E(M) is antiinvariant with respect to φ. The lowest possible dimension for M in which this decomposition is non trivial is the dimension 4. In this paper we obtain a complete classification of four-dimensional contact CR -submanifolds in S5(1) and S7(1) for which the second fundamental form restricted to H(M) and E(M) vanishes identically.