Distinguished connections on $(J^{2}=\pm 1)$-metric manifolds

TL;DR

This paper investigates various linear connections on $(J^{2}= ext{±}1)$-metric manifolds, characterizing their properties, coincidences, and introducing a unified family of canonical connections across different geometric structures.

Contribution

It introduces a unified framework for canonical connections on $(J^{2}= ext{±}1)$-metric manifolds, extending existing families to new geometric contexts and characterizing when these connections are adapted.

Findings

The first canonical and well adapted connections form a one-parameter family of adapted connections.

All studied connections are canonical when they exist and are adapted.

The paper extends known families of connections to almost Norden and almost product Riemannian manifolds.

Abstract

We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of $(J^2=\pm1)$-metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among connections. We prove that the first canonical and the well adapted connections define a one-parameter family of adapted connections, named canonical connections, thus extending to almost Norden and almost product Riemannian manifolds the families introduced in almost Hermitian and almost para-Hermitian manifolds. We also prove that every connection studied in this paper is a canonical connection, when it exists and it is an adapted connection.