  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 $Q$.
