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

Abstract

We study some properties of \(SU_n\) endowed with the Frobenius metric \(\phi \), 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 \((SU_n, \phi )\) 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 \(\mathfrak {su}_n\), diffeomorphic to a suitable complex Grassmannian depending on P and Q.