Generalized principal logarithms and Riemannian properties of a class of subgroups of \(\mathbf {U_n}\) endowed with the Frobenius bi-invariant metric

We study the geometric-differential properties of a wide class of closed subgroups of \(U_n\) $U_n$ 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 \(P_0\) $P_0$ and \(P_1\) $P_1$ by means of the set of generalized principal logarithms of \(P_0^*P_1\) $P_0^{\ast }{P}_1$ 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 \(\mathfrak {u}_n\) ${\mathfrak{u}}_n$ diffeomorphic to suitable (and explicitly determined) homogeneous spaces.