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

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

We establish the smooth action on \(\mathcal R\) of the group of unitary operators of \(\mathcal H\), and it thereby turns out that the connected components of \(\mathcal R\) are homogeneous spaces. 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 modeled 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.

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Correspondence to Eduardo Chiumiento.

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This research was supported by CONICET (PIP 2014 11220130100525), ANPCyT (PICT 2015 1505) and UNLP (10001940 11X829).

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Andruchow, E., Chiumiento, E. & Larotonda, G. Canonical sphere bundles of the Grassmann manifold. Geom Dedicata 203, 179–203 (2019). https://doi.org/10.1007/s10711-019-00431-7

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Keywords

  • Sphere bundle
  • Finsler metric
  • Riemannian metric
  • Geodesic
  • Projection
  • Flag manifold

Mathematics Subject Classification (2010)

  • 22E65
  • 47B10
  • 58B20