Grassmannian

The Grassmannian is the smooth manifold whose points are the -dimensional linear subspaces of an -dimensional space . It is the homogeneous space , the Stiefel manifold of orthonormal -frames modulo change of frame, of dimension . It also serves as the state space over which an active space, a reduced density matrix, is transported. Choosing a -dimensional subspace amounts to choosing which degrees of freedom to keep, so the geometry of reduced-density-matrix transport and of Grassmann-based compression lives on this manifold.

Example of the tautological bundle

Over sits the tautological bundle , also called universal. Its fibre over a subspace is itself, a rank- subbundle of the trivial bundle . Every rank- bundle is a pullback of along a map into a Grassmannian, which is the classifying property.

Property of the canonical connection

Orthogonal projection onto the moving subspace splits , and projecting the trivial flat connection onto gives the canonical connection . Parallel transport carries a vector along a path of subspaces by continually reprojecting it, keeping it “as unrotated as possible”. This connection is not flat, since its curvature is the second fundamental form, so transport around a loop of subspaces returns a rotated vector, its holonomy.