Reductions of the object and the affine torsion tensor of a centroprojective connection on a distribution of planes

Abstract

The projective group is represented as a bundle of centroprojective frames. This bundle is endowed with a centroprojective connection and becomes the space of this centroprojective connection. Structure equations of this space are found, which include the affine torsion tensor and the centroprojective curvature tensor containing the affine curvature subtensor. A distribution of planes in projective space and its associated principal bundle (which has two simplest and two simple (in the sense of [1]) quotient principal bundles) are considered. On the associated bundle, a group connection is defined. The object of the centroprojective connection is reduced to the object of the group connection. The object of the group connection contains the objects of the flat and normal linear connections, the centroprojective subconnection, and the affine-group connection as subobjects. The torsion object of the affine-group connection is determined. It is proved that it forms a tensor, which contains the torsion tensor of the normal linear connection as a subtensor. It is shown that the affine torsion tensor of the centroprojective connection reduces to the torsion tensor of the affine-group connection.