Projections with fixed difference: a Hopf-Rinow theorem

Abstract

The set D_A0 , of pairs of orthogonal projections (P,Q) in generic position with fixed difference P−Q=A_0, is shown to be a homogeneous smooth manifold: it is the quotient of the unitary group of the commutant {A_0}′ divided by the unitary subgroup of the commutant {P0,Q0}′, where (P0,Q0) is any fixed pair in D_A0. Endowed with a natural reductive structure (a linear connection) and the quotient Finsler metric of the operator norm, it behaves as a classic Riemannian space: any two pairs in D_A0 are joined by a geodesic of minimal length. Given a base pair (P0,Q0), pairs in an open dense subset of DA0 can be joined to (P0,Q0) by a unique minimal geodesic.