On anti-Hermitian metric connections

Abstract

Presented by the Editorial BoardIt is a remarkable fact that anti-Kähler and its twin metrics share the same Levi-Civita connection. Such torsion-free metric connection also emphasizes the importance of antiHermitian metric connections with torsion in the study of anti-Hermitian geometry. With the objective of defining new types of anti-Hermitian metric connections, in the present paper we consider classes of anti-Hermitian manifolds associated with these connections.© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é C'est un fait remarquable que les métriques anti-kählériennes et leurs métriques jumelles possèdent la même connexion de Levi-Civita. De telles connexions métriques sans torsion mettent aussi en relief l'importance des connexions métriques anti-hermitiennes avec torsion dans l'étude de la géométrie anti-hermitienne. Dans le but de définir de nouveaux types de connexions métriques anti-hermitiennes, nous considérons dans la présente note des classes de variétés anti-hermitiennes associées à ces connexions.

On nearly Kähler and Kähler-Codazzi type manifolds

“…Proof. If ∇ g satisfies (∇ g X J)Y − (∇ g Y J)X = 0, for all vector fields X, Y on M , then, taking into account (10), one concludes that the torsion tensor T 0 vanishes, and therefore, (M, J, g) is Kähler type manifold (see Theorem 3.1).…”

“…In the case α = ε = −1, manifolds satisfying the Codazzi equation are called anti-Kähler-Codazzi manifolds, while in the case α = ε = 1, this kind of manifolds are called para-Kähler-Norden-Codazzi manifolds. First, in the Norden case, they were introduced in [12] and have been intensively studied (see also [10,11]). Afterward, the class of manifolds characterized by the Codazzi equation were extended without changes in [8] to the other αε = 1 case, the product Riemannian case.…”

Metallic Kähler and Nearly Metallic Kahler Manifolds

“…In this section, by employing the method proposed in [4] for anti-Hermitian manifolds we search for linear connections with torsion on an almost metallic Hermitian manifold (M 2k , g, J M ). We will be calling these connections linear connections of the first type and of the second type, respectively.…”

“…Following the method from [4], we have the following definition. i.e., the linear connection of the first type is given by ∇ = ∇ + 1 3q J M (∇J M ).…”