Some Riemannian properties of $\mathbf{S}{\mathbf{U}}_{\mathbf{n}}$ endowed with a bi-invariant metric

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