Five-Dimensional Contact CR -Submanifolds in S 7 ( 1 )

Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure ( φ , ξ , η ) , we study its five-dimensional contact C R -submanifolds, which are the analogue of C R -submanifolds in (almost) Kählerian manifolds. In the case when the structure vector field ξ is tangent to M , the tangent bundle of contact C R -submanifold M can be decomposed as T ( M ) = H ( M ) ⊕ E ( M ) ⊕ R ξ , where H ( M ) is invariant and E ( M ) is anti-invariant with respect to φ . On this occasion we obtain a complete classification of five-dimensional proper contact C R -submanifolds in S 7 ( 1 ) whose second fundamental form restricted to H ( M ) and E ( M ) vanishes identically and we prove that they can be decomposed as (multiply) warped products of spheres.