Real Hypersurfaces in S⁶(1) with a Condition on the Structure Jacobi Operator

Abstract

t is well known that the sphere S⁶(1) admits an almost complex structure J which is nearly K¨ahler. 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. The Jacobi operator with respect to ξ is called structure Jacobi operator and is denoted by l = R(·, ξ)ξ, where R is the curvature tensor on M . The study of hypersurfaces of almost Hermitian manifolds by means of their Jacobi operators has been highly active in recent years. Specially, many recent results answer the question of the existence of hypersurfaces with a structure Jacobi operator that satisfies conditions related to their parallelism.

We investigate real hypersurfaces in nearly K¨ahler sphere S⁶(1) whose Lie derivative of structure Jacobi operator coincides with the covariant derivative of it, i.e. LX l = ∇X l, ∀X ∈ T M , and show that such submanifolds do not exist.