Euler-Poincaré Reduction in Principal Fibre Bundles and the Problem of Lagrange

Marco Castrillón, Pedro L. García and César Rodrigo
Departamento de Geometría y Topología
Univ. Complutense de Madrid (Spain), and
Departamento de Matemáticas
Univ. de Salamanca (Spain)

To appear in PROCEEDINGS OF THE 9th DGA CONFERENCE, Prague 2004.

Abstract

We compare Euler-Poincaré reduction in principal fibre bundles, as a constrained variational problem on the connections of this fibre bundle and constraint defined by the vanishing of the curvature of the connection, with the corresponding Problem of Lagrange. Under certain cohomological condition we prove the equality of the sets of critical sections of both problems with the one obtained by application of the Lagrange multiplier rule. We compute the corresponding Cartan form and characterize critical sections as the set of holonomic solutions of the Cartan equation and, in particular, under a certain regularity condition for the problem, we prove the holonomy of any solution of this equation.

Procceedings of the 9 International Congress on Differential Geometry and Applications (to appear). 14 pages

Remark: Due to some error of the editors, this work and a few others, finally didn't appear with the Procceedings. We are now looking for a solution to this inconvenience.