Projections with fixed difference: a Hopf-Rinow theorem

TL;DR

This paper establishes that the set of orthogonal projection pairs with fixed difference forms a smooth homogeneous manifold with a natural geometric structure, allowing for unique minimal geodesics between generic pairs.

Contribution

It proves that these projection pairs form a Riemannian-like manifold with explicit geometric properties and geodesic completeness, extending classical geometric concepts to operator projections.

Findings

The set is a homogeneous smooth manifold.

Any two generic pairs are connected by a minimal geodesic.

The manifold admits a natural reductive structure and Finsler metric.

Abstract

The set $D_{A_0}$, of pairs of orthogonal projections $(P,Q)$ in generic position with fixed difference $P-Q=A_0$, is shown to be a homogeneus smooth manifold: it is the quotient of the unitary group of the commutant $\{A_0\}'$ divided by the unitary subgroup of the commutant $\{P_0, Q_0\}'$, where $(P_0,Q_0)$ is any fixed pair in $D_{A_0}$. 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 $D_{A_0}$ are joined by a geodesic of minimal length. Given a base pair $(P_0,Q_0)$, pairs in an open dense subset of $D_{A_0}$ can be joined to $(P_0,Q_0)$ by a {\it unique} minimal geodesic.