2018/03/02 by Andruchow, Esteban, Chiumiento, Eduardo, Larotonda, Gabriel
#22E65 #58B20 (Primary) 47B10 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1803.01057
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)∈ \mathcal P(\mathcal H)× \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 R2+ given by considering only the projections in the restricted Grassmannian, locally modelled by Hilbert-Schmidt operators. Therefore we endow \mathcal R2+ 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 R2+, again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.