Canonical connections on paracontact manifolds

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 η-Einstein manifold are defined, it is proved 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 \({\mathcal{D}}\) -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.