The well adapted connection of a $(J^{2}=\pm 1)$-metric manifold

TL;DR

This paper investigates the well adapted connection on $(J^{2}= ext{±}1)$-metric manifolds, providing explicit formulas and characterizations of its relation to other fundamental connections across different geometries.

Contribution

It introduces a unified approach to the well adapted connection for all four geometries and characterizes its coincidence with Levi Civita and Chern connections.

Findings

Existence of the well adapted connection for all four geometries.

Explicit formula for the connection as a derivation law.

Conditions for coincidence with Levi Civita and Chern connections.

Abstract

In this paper, we study the well adapted connection attached to a $(J^{2}=\pm 1)$-metric manifold, proving it exists for any of the four geometries and obtaining a explicit formula as a derivation law. Besides we characterize the coincidence of the well adapted connection with the Levi Civita and the Chern connections.