Riemannian geometry of finite rank positive operators

ESTEBAN ANDRUCHOW, ALEJANDRO VARELA

DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Elsevier

Lugar: Amsterdam; Año: 2005 vol. 23 p. 305 - 326

0926-2245

A riemannian metric is introduced in the infinite dimensional manifold $\Sigma_n$ of positive operators with rank $n<\infty$ on a Hilbert space $H$.  The geometry of this manifold is studied and related to the geometry of the submanifolds $\Sigma_p$ of positive operators with range equal to the range of a projection $p$ (rank of $p$ $=n$), and ${\cal P}_p$ of selfadjoint projections in the connected component of $p$. It is shown that these spaces are complete in the geodesic distance.