Some type of semisymmetry on two classes of almost Kenmotsu manifolds

Abstract

Abstract The object of the present paper is to study some types of semisymmetry conditions on two classes of almost Kenmotsu manifolds. It is shown that a (k, µ)-almost Kenmotsu manifold satisfying the curvature condition Q · R = 0 is locally isometric to the hyperbolic space ℍ2n+1(−1). Also in (k, µ)-almost Kenmotsu manifolds the following conditions: (1) local symmetry (∇R = 0), (2) semisymmetry (R·R = 0), (3) Q(S, R) = 0, (4) R·R = Q(S, R), (5) locally isometric to the hyperbolic space ℍ2n+1(−1) are equivalent. Further, it is proved that a (k, µ)′-almost Kenmotsu manifold satisfying Q · R = 0 is locally isometric to ℍn+1(−4) × ℝn and a (k, µ)′--almost Kenmotsu manifold satisfying any one of the curvature conditions Q(S, R) = 0 or R · R = Q(S, R) is either an Einstein manifold or locally isometric to ℍn+1(−4) × ℝn. Finally, an illustrative example is presented.