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

TL;DR

This paper investigates the properties of nearly Kähler and Kähler-Codazzi manifolds, revealing their specific contexts and showing that Kähler-Codazzi manifolds simplify to Kähler manifolds across various geometric settings.

Contribution

It establishes the conditions under which nearly Kähler manifolds exist and demonstrates that Kähler-Codazzi manifolds reduce to Kähler manifolds in multiple geometries.

Findings

Nearly Kähler manifolds exist only in Hermitian and para-Hermitian contexts.

Kähler-Codazzi manifolds simplify to Kähler manifolds in all studied geometries.

The paper clarifies the geometric contexts of these manifold types.

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.