Generalized principal logarithms and Riemannian properties of a class of subgroups of U_n endowed with the Frobenius bi-invariant metric

We study the geometric-differential properties of a wide class of closed subgroups of U_n endowed with a natural bi-invariant metric. For each of these groups, we explicitly express the distance function, the diameter, and, above all, we parametrize the set of minimizing geodesic segments with arbitrary endpoints P_0 and P_1 by means of the set of generalized principal logarithms of P_0*P_1 in the Lie algebra of the group. We prove that this last set is a non-empty disjoint union of a finite number of compact submanifolds of u_n diffeomorphic to suitable (and explicitly determined) homogeneous spaces.