  Metrics in Grassmannians, or distances between subspaces of same dimension,
have many applications. However, usual extensions to the Total Grassmannian of
subspaces of different dimensions lack useful properties or give little
information. Dimensional asymmetries call for the use of asymmetric metrics,
which arise naturally in this space. Their geometry reflects containment
relations of subspaces, and minimal geodesics link subspaces of distinct
dimensions. In particular, the Fubini-Study metric extends as an asymmetric
angle with useful properties, that can be computed in arbitrary bases or via
Grassmann algebra. It has a nice geometric interpretation, as does its sine, an
asymmetric Binet-Cauchy metric.
