Geometry of conformally flat harmonic nearly trans-Sasakian manifolds

Conformally flat harmonic nearly trans-Sasakian manifolds are studied. In the space of adjoint G-structure, components of the Weyl tensor of the conformal curvature are calculated, identities for this tensor are found, and also some conformal invariants of harmonic nearly trans-Sasakian manifold are provided. An exhaustive description of the class W1 of such manifolds is obtained. It is proved that this class is locally equivalent to the product of a complex Euclidean space and a real line. Complete local characteristic of harmonic nearly trans-Sasakian manifolds of the class W6, which are Einstein manifolds, is obtained. Full classification of conformally flat harmonic nearly trans-Sasakian manifolds is given. In particular, it is proved that a conformally flat harmonic nearly trans-Sasakian manifold is either a space of constant negative curvature, or an image of the product of a six-dimensional sphere, equipped with a Kahler structure, and a real line under the canonical concircular transformation. Conformally flat harmonic nearly trans-Sasakian manifold of characteristic zero is locally equivalent to the product of a complex Euclidean space Cn equipped with the standard Hermitian metric and the real line R.