Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in S6(1)

Abstract

It is well known that the sphere S6(1) admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector field ξ=−JN is said to be characteristic or the Reeb vector field. The Jacobi operator with respect to ξ is called the structure Jacobi operator, and is denoted by l=R(·,ξ)ξ, where R is the curvature tensor on M. The study of Riemannian submanifolds in different ambient spaces by means of their Jacobi operators has been highly active in recent years. In particular, many recent results deal with questions around the existence of hypersurfaces with a structure Jacobi operator that satisfies conditions related to their parallelism. In the present paper, we study the parallelism of the structure Jacobi operator of real hypersurfaces in the nearly Kähler sphere S6(1). More precisely, we prove that such real hypersurfaces do not exist.