Some Riemannian properties of SU_n endowed with a bi-invariant metric

We study some properties of SU(n) endowed with the Frobenius metric f, which is, up to a positive constant multiple, the unique bi-invariant Riemannian metric on SU(n). In particular we express the distance between P, Q in SU(n) in terms of eigenvalues of P*Q; we compute the diameter of (SU(n), f) and we determine its diametral pairs; we prove that the set of all minimizing geodesic segments with endpoints P, Q can be parametrized by means of a compact connected submanifold of su(n), diffeomorphic to a suitable complex Grassmannian depending on P and Q.