The manifold of finite rank projections in the space L(H)

TL;DR

This paper explores the geometric structure of the manifold of finite rank projections in the algebra of bounded operators on a Hilbert space, defining connections, geodesics, and distances using Jordan-Banach triple theory.

Contribution

It introduces a natural Levi-Civita connection on the manifold of finite rank projections and characterizes its geodesics and metric properties.

Findings

Defined a Levi-Civita connection on the manifold

Identified geodesics in the manifold

Computed the Riemannian distance

Abstract

Given a complex Hilbert space H and the von Neumann algebra L(H) of all bounded linear operators on H, we study the Grassmann manifold M of all projections in L(H) that have a fixed finite rank r. We take the Jordan-Banach triple theory approach which allows us to define a natural Levi-Civita connection on M. We identify the geodesics, compute the Riemann distance and prove some properties of M