Unitary orbits in a full matrix algebra

The Hilbert manifold $\Sigma$ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits $\Omega\subset \Sigma$ is studied from the topological and metric viewpoints: we seek for conditions that ensure the existence of a smooth local structure for the set $\Omega$, and we study the convexity of this set for the geodesic structures that arise when we give $\Sigma$ two Riemannian metrics.