Hierarchy of spaces of projective connection

Abstract

We consider the bundle of projective frames over a smooth manifold, i. e. the principal bundle whose typical fiber is the projective group. The giving fundamental-group connection in this bundle transforms it into a space of general projective connection. Differential equations of compo­nents of the curvature tensor, their covariant derivatives and analogues of Bianchi identities are obtained. When the basic and fiber indices coincide, a special case is identified, called the space of a pre-Cartan projective connection. By means of glu­ing the fibers to the base of the pre-Cartan connection space we obtain the space of the Cartan projective connection. In this case the curvature ten­sor is transformed into a curvature-torsion tensor, which contains the ten­sor of affine curvature-torsion with a torsion. A hierarchy of the considered spaces is constructed, which there are a bundle of projective frames is at the beginning of the hierarchy, the space of projective connection (general, pre-Cartan and Cartan) is in the middle, a projective space is at the end of the hierarchy.

Keywords

• bundle of projective frames
• projective connection
• ana­logues of bianchi identities
• pre-cartan projective connection
• gluing
• projective cartan connection
• curvature-torsion tensor
• tensor of affine curvature-torsion