Сurvature-torsion tensor for Cartan connection

Abstract

A Lie group containing a subgroup is considered. Such a group is a principal bundle, a typical fiber of this principal bundle is the subgroup and a base is a homogeneous space, which is obtained by factoring the group by the subgroup. Starting from this group, we constructed structure equations of a space with Cartan connection, which generalizes the Cartan point projective connection, Akivis’s linear projective connection, and a plane projective connection. Structure equations of this Cartan connec­tion, containing the components of the curvature-torsion object, allowed: 1) to show that the curvature-torsion object forms a tensor containing a torsion tensor; 2) to find an analogue of the Bianchi identities such that the curvature-torsion tensor and its Pfaff derivatives satisfy this analogue; 3) to obtain the conditions for the transformation of Pfaffian derivatives of the curvature-torsion tensor into covariant derivatives with respect to the Cartan connection.

Keywords

• cartan connection
• cartan projective connection
• cartan connection curvature-torsion tensor
• analogue of bianchi identities
• covar­iant derivatives with respect to cartan connection