Curvature Properties and -Einstein -Contact Metric Manifolds

We study curvature properties in -contact metric manifolds. We give the characterization of -Einstein -contact metric manifolds with associated scalars. 1. Introduction The class of -contact manifolds [1] is of interest as it contains both the classes of Sasakian and non-Sasakian cases. The contact metric manifolds for which the characteristic vector field belongs to -nullity distribution are called contact metric manifolds. Boeckx [2] gave a classification of -contact metric manifolds. Sharma [3], Papantoniou [4], and many others have made an investigation of -contact metric manifolds. A special class of -contact metric manifolds called -contact metric manifolds was studied by authors [5, 6] and others. In this paper we study -contact metric manifolds by considering different curvature tensors on it (Table 1). We characterize -Einstein -contact metric manifolds with associated scalars by considering symmetry, -symmetry, semisymmetry, -recurrent, and flat conditions on -contact metric manifolds. The paper is organized as follows: In Section 2, we give some definitions and basic results. In Section 3, we consider conharmonically symmetric, conharmonically semisymmetric, -conharmonically flat, -conharmonically flat, and -recurrent -contact metric manifolds and we prove that such manifolds are -Einstein or -parallel or cosymplectic depending on the conditions. In Section 4, we prove that -conformally flat -contact metric manifold reduces to -contact metric manifold if and only if it is an -Einstein manifold. Further we prove conformally Ricci-symmetric and -conformally flat -contact metric manifolds are -Einstein. In Section 5, we prove that pseudoprojectively symmetric and pseudoprojectively Ricci-symmetric -contact metric manifolds are -Einstein. In Section 6 we consider Ricci-semisymmetric -contact metric manifolds and prove that such manifolds are -Einstein. In all the cases where -contact metric manifold is an -Einstein manifold, we obtain associated scalars in terms of and . Table 1: Comparison of the results for different curvature tensors in . 2. Preliminaries A dimensional -differentiable manifold is said to admit an almost contact metric structure if it satisfies the following relations [7, 8] where is a tensor field of type (1,1), is a vector field, is a 1-form, and is a Riemannian metric on . A manifold equipped with an almost contact metric structure is called an almost contact metric manifold. An almost contact metric manifold is called a contact metric manifold if it satisfies for all vector fields , . The (1,1) tensor field defined by ,