Four‐dimensional contact C R ‐submanifolds in S 7 ( 1 )

The analogue of C R ‐submanifolds in (almost) Kählerian manifolds is the concept of contact C R ‐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 ) ⊕ R ξ , 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 C R ‐submanifolds inS 5 ( 1 )andS 7 ( 1 )for which the second fundamental form restricted to H ( M ) and E ( M ) vanishes identically.