Canonical connections on paracontact manifolds

TL;DR

This paper explores the properties of canonical connections on paracontact manifolds, linking torsion to paraSasakian structures, and characterizing $ ext{η}$-Einstein manifolds and special connections with skew-symmetric torsion.

Contribution

It introduces the canonical paracontact connection, analyzes its torsion, and characterizes conditions for special geometric structures and transformations on paracontact manifolds.

Findings

Torsion of the canonical connection obstructs paraSasakian structure.

$ ext{η}$-Einstein manifolds have constant scalar curvature.

Conditions for existence of connections with skew-symmetric torsion.

Abstract

The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $\mathcal{D}$-homothetic transformation is determined as a special gauge transformation. The $\eta$-Einstein manifold are defined, it is prove that their scalar curvature is a constant and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with a $\mathcal{D}$-homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing.