About differential equations of the curvature tensors of a fundamental group and affine connections

Abstract

The principal bundle is considered, the base of which is an n-dimen­sional smooth manifold, and the typical fiber is an r-fold Lie group. Struc­ture equations for the forms of the fundamental group and affine connec­tions are given, each of which contains the corresponding compo­nents of the curvature tensor. For each connection, an approach is shown that al­lows to find the differential equations for the components of the curvature tensor of the corresponding connection in a faster way than by differentia­ting the expressions of these objects in terms of the connection objects and their Pfaffian derivatives. The method consists in successively solv­ing cubic equations, first by Laptev’s lemma, then by Cartan’s lem­ma. Taking into account the comparisons modulo basic forms, we ob­tain already known results (see [3]). Thus, differential equations are deri­ved for the components of the curvature tensor of the first-order fun­da­mental-group connection, as well as for the components of the curva­ture tensor of the affine connection.

Keywords

• structure equations of laptev
• fundamental-group connec­tion
• affine connection
• connection object
• curvature tensor