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

Abstract

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 CR-submanifolds, which are the analogue of CR-submanifolds in (almost) Kählerian manifolds. In the case when the structure vector field ξ is tangent to M, the tangent bundle of contact CR-submanifold M can be decomposed as T(M) = H(M) ⊕ E(M) ⊕ Rξ, 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 CR-submanifolds in S7(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.