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.
Highlights
Let M be a Riemannian submanifold of the seven-dimensional unit sphere
Having in mind the behaviour of the endomorphism φ, submanifolds in the Sasakian manifolds carrying a φ-invariant distribution such that its orthogonal complement is φ-anti-invariant, are called contact CR-submanifolds. This notion is the odd-dimensional analogue of CR-submanifolds in Kählerian manifolds, introduced by Bejancu in [1], who requested the existence of a differentiable holomorphic distribution such that its orthogonal complement is a totally real distribution
In this paper we continue our study of certain contact CR-submanifolds in seven-dimensional unit sphere, which we started in [4] for the case of four-dimensional submanifolds and continued in [5], where we presented several examples of four and five-dimensional contact CR-submanifolds of S7 (1), which are of product and warped product type
Summary
Let M be a Riemannian submanifold of the seven-dimensional unit sphere. It is well-known that possesses the almost contact structure ( φ, ξ, η ), which is contact and Sasakian. It is interesting to investigate totally geodesic submanifolds, that is, those submanifolds for which all geodesics—when the induced Riemannian metric is considered—are geodesics on the ambient manifold This property is equivalent to the vanishing of the second fundamental form. On this occasion we study those five-dimensional contact CR-submanifolds in a Sasakian sphere S7 (1) which are close to be totally geodesic, namely those whose second fundamental form restricted to both φ-invariant and φ-anti-invariant distributions vanishes identically. In [5] we presented several examples of four and five-dimensional contact CR-submanifolds of product and warped product type of seven-dimensional unit sphere, which are nearly totally geodesic, minimal and which satisfy the equality sign in some Chen type inequalities
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