Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups

Abstract

We formulate Euler-Poincaré equations on the Lie group Aut(P) of automorphisms of a principal bundle P. The corresponding flows are referred to as EP A ut flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle P, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing Δ-like momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group Aut vol(P) of volume-preserving bundle automorphisms. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group. © American Institute of Mathematical Sciences.