Some Riemannian properties of \(\mathbf {SU_n}\) endowed with a bi-invariant metric

We study some properties of \(SU_n\) $S{U}_n$ endowed with the Frobenius metric \(\phi \) $\varphi$ , which is, up to a positive constant multiple, the unique bi-invariant Riemannian metric on \(SU_n\) $S{U}_n$ . In particular we express the distance between \(P, Q \in SU_n\) $P,Q\in S{U}_n$ in terms of eigenvalues of \(P^*Q\) $P^{\ast }Q$ ; we compute the diameter of \((SU_n, \phi )\) $(S{U}_n,\varphi )$ 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 \(\mathfrak {su}_n\) ${\mathfrak{su}}_n$ , diffeomorphic to a suitable complex Grassmannian depending on P and Q.