On some classes of contact metric manifolds

Abstract

We define four new classes Open image in new window of contact metric manifoulds using Tanaka connection Open image in new window and Jacobi operators. We prove that a contact metric manifold with the structure vector field ξ belonging to thek-nullity distribution is contact metric locally ϕ-symmetric (in the sense of D. B. Blair) if and only if the manifold is a Open image in new window and Open image in new window space. Also, we prove that a 3-dimensional contact metric Open image in new window and Open image in new window is locally ϕ-symmetric (in the sense of D. E. Blair) and give counter-examples of the converse.