Canonical sphere bundles of the Grassmann manifold

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 $$\mathcal{R}=\{(P,f)\in \mathcal{P}(\mathcal{H})\times \mathcal{H}:Pf=f,\Vert f\Vert =1\}.$$ 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.

arXiv comment26 pages