Canonical sphere bundles of the Grassmann manifold

TL;DR

This paper explores the geometric and metric structures of a canonical sphere bundle over the space of projections in a Hilbert space, establishing smooth actions, minimal geodesics, and Riemannian properties, including a Hopf-Rinow theorem.

Contribution

It introduces a detailed differentiable and metric structure of the sphere bundle over projections, including homogeneous space properties and geodesic characterizations.

Findings

The sphere bundle is a homogeneous space under unitary group action.

Minimal geodesics are given by one-parameter unitary groups.

A Hopf-Rinow theorem is established for the restricted bundle.

Abstract

For a given Hilbert space $\mathcal H$, consider the space of self-adjoint projections $\mathcal P(\mathcal H)$. In this paper we study the differentiable structure of a canonical sphere bundle over $\mathcal P(\mathcal H)$ given by $$ \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \|f\|=1\, \}. $$ We establish the smooth action on $\mathcal R$ of the group of unitary operators of $\mathcal H$, therefore $\mathcal R$ is an homogeneous space. Then we study the metric structure of $\mathcal R$ by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into $\mathcal R$ by the natural action of the unitary group. Then we study the restricted bundle $\mathcal R_2^+$ given by considering only the projections in the restricted Grassmannian, locally modelled by Hilbert-Schmidt operators. Therefore we endow $\mathcal R_2^+$ with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi-Civita connection of this metric and establish a Hopf-Rinow theorem for $\mathcal R_2^+$, again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.