On some classes of almost contact metric manifolds

Abstract

In [1] J. Berndt and L. Vanhecke introduced two classes ((£-and 93-spaces) of Riemannian manifolds which include the class of locally symmetric spaces using the properties of Jaoobi operators along geodesies. They provided some characterizations of (£-and 23-spaces and gave the classificationsfor dimensions two and three. For further developments on the two spaces, we refer to [2], [3] and [8]. Further, T. Takahashi ([19]) introduced the notion of a (Sasakian) locally ^-symmetric space which may be considered as the analogue in the almost contact metric case of locally Hermitian symmetric spaces. Also he gave examples and equivalent properties of Sasakian locally ^-symmetric spaces. For further results about the Sasakian locally ^-symmetric spaces, we refer to [5], [6]. In the present paper, we introduce in an analogous way as in [1] four classes of almost contact metric manifolds involving Sasakian locally ^-symmetric spaces. In section 2, we recall definitionsand several elementary properties of an almost contact, a contact, a if-contact metric manifold and a Sasakian manifold. In sections 3 and 4 we give the definitionsof a %^,-space, a %%-space, a %&-space and a ^-space which are almost contact metric analogues of a (£-space or a $J$-spacein the Riemannian case. We may observe that a Sasakian manifold is a £(£-spaceand at the same time a f^-space. Also we prove that a Sasakian manifold is locally ^-symmetric if and only if it is a SDS-space and at the same time a RSJ3-space. In section 5, we show that the tangent sphere bundle of a 2-dimensional Riemannian manifold is a f$-space if and only if the base manifold is flat or of constant curvature 1. Furthermore, we give some examples of almost contact metric R(S-spaces and c^-spaces. In section 6, we consider real hypersurfaces of a complex projective space CPn with FubiniStudy metric and determine £S)8-hypersurfacesof CPn. We also show that a homogeneous real hypersurface of CPn is a fK-space, and moreover, we give