On nearly K\"{a}hler and K\"{a}hler-Codazzi type manifolds

Abstract

Nearly K\"{a}hler and K\"{a}hler-Codazzi type manifolds are defined in a very similar way. We prove that nearly K\"{a}hler type manifolds have sense just in Hermitian and para-Hermitian contexts, and that K\"{a}hler-Codazzi type manifolds reduce to K\"{a}hler type manifolds in all the four Hermitian, para-Hermitian, Norden and product Riemannian geometries.