Geometry of Ricci tensor of harmonic nearly trans–Sasakian manifolds

Abstract

In this paper, we study the geometry of the Ricci tensor of a harmonic nearly trans–Sasakian manifold. On the space of associated $G$-structure we introduce fundamental identities of harmonic nearly trans–Sasakian manifolds. We prove that Ricci–flat harmonic nearly trans–Sasakian manifolds are closely cosymplectic. We obtain conditions, which ensure that harmonic nearly trans–Sasakian manifolds are Einstein and $\eta$–Einstein manifolds. We obtain identities for the Ricci tensor of harmonic nearly trans–Sasakian manifolds. We provide local characterizations for the following harmonic nearly trans-Sasakian manifolds: Einstein manifolds; manifolds, the Ricci tensor of which is parallel, $\eta$–parallel, the Codazzi tensor, the Killing tensor, and satisfies the three selected identities.