Nearly trans-Sasakian manifolds of constant holomorphic sectional curvature

Abstract

The geometry of nearly trans-Sasakian manifolds of constant holomorphic sectional curvature is studied in this paper. It is proved that a harmonic nearly trans-Sasakian Einstein manifold is a manifold of non-positive scalar curvature, and, in the case of zero scalar curvature, it is locally equivalent to the product of a Ricci-flat nearly Kählerian manifold and the real line. Expressions for the Riemannian curvature and Ricci tensors are obtained. We show that a harmonic nearly trans-Sasakian manifold is a space of a constant curvature k=−χ2 if and only if it is canonically concircular to the manifold Cn×R equipped with the canonical cosymplectic structure. It is also proved that there are no harmonic nearly trans-Sasakian manifolds of constant positive curvature. The full classification of harmonic nearly trans-Sasakian manifolds of constant Φ-holomorphic sectional curvature is given.