Real hypersurfaces in S^6(1) equipped with structure Jacobi operator satisfying L_X l = ∇_X l

Abstract

It is well known that the sphere $S^6(1)$ 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 $\xi = −JN$ is said to be characteristic. The Jacobi operator with respect to $\xi$ is called structure Jacobi operator and is denoted by $l = R(·,\xi)\xi$, where R is the curvature tensor on M. We investigate real hypersurfaces in nearly Kähler sphere $S^6(1)$ whose Lie derivative of structure Jacobi operator coincides with the covariant derivative of it and show that such submanifolds do not exist.