Conjugate Connections and their Applications on Pure Metallic Metric Geometries

TL;DR

This paper explores the geometric properties of metallic pseudo-Riemannian manifolds, focusing on conjugate connections, tensor interactions, and new characterizations involving the Tachibana operator, advancing understanding of their structure.

Contribution

It introduces new characterizations of locally metallic pseudo-Riemannian manifolds using Codazzi couplings and Tachibana operator, and establishes conditions for non-integrable metallic manifolds to be quasi metallic.

Findings

Characterization of locally metallic pseudo-Riemannian manifolds

Conditions for non-integrable metallic manifolds to be quasi metallic

Introduction of metallic-like pseudo-Riemannian manifolds

Abstract

Let $\left( M,J,g\right) $ be a metallic pseudo-Riemannian manifold equipped with a metallic structure $J$ and a pseudo-Riemannian metric $g$. The paper deals with interactions of Codazzi couplings formed by conjugate connections and tensor structures. The presence of Tachibana operator and Codazzi couplings presented a new characterization for locally metallic pseudo-Riemannian manifold. Also, a necessary and sufficient condition a non-integrable metallic pseudo-Riemannian manifold is a quasi metallic pseudo Riemannian manifold is derived. Finally, it is introduced metallic-like pseudo-Riemannian manifolds and presented some results concerning them.