Some remarks on geodesics in gauge groups and harmonic maps (Q1110147)

Let \(P\to M\) be a smooth principal SU(N)-bundle over a compact Riemannian manifold M. Consider the ``gauge group'' \({\mathcal G}\) of smooth automorphisms of P. Every connection A on P induces a weak right invariant Riemannian metric on \({\mathcal G}\), via right translation of the inner product: \(<u,v>=\int_{M}(d_ Au,d_ Av)\) on the Lie algebra \({\mathfrak g}\) of \({\mathcal G}\). In this paper, the author studies the Euler equations for geodesics in \({\mathcal G}\) (or in associated spaces of connections) and shows the existence and uniqueness of solutions for the Cauchy problem. After defining ``harmonic elements'' of \({\mathcal G}\) with respect to the fixed connection A: they generalize harmonic maps \(M\to U(N)\), the author gives two families of examples of closed geodesics in \({\mathcal G}\), connecting certain classes of such harmonic elements to the identity. Finally, the case of Riemann surfaces is studied.