Isoholonomic problems and some applications

We study the problem of finding the shortest loops with a given holonomy. We show that the solutions are the trajectories of particles in Yang-Mills potentials (Theorem 4), or, equivalently, the projections of Kaluza-Klein geodesics (Theorem 2). Applications to quantum mechanics (Berry's phase, Sect. 3) and the optimal control of deformable bodies (Sect. 6) are touched upon.

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