Canonical sphere bundles of the Grassmann manifold
For a given Hilbert space H, consider the space of self-adjoint projections P(H). In this paper we study the differentiable structure of a canonical sphere bundle over P(H) given by R={(P,f)∈P(H)×H:Pf=f,‖f‖=1}.We establish the smooth action on R of the group of unitary operators of H, and it thereby...
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2019
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00465755_v_n_p_Andruchow http://hdl.handle.net/20.500.12110/paper_00465755_v_n_p_Andruchow |
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| Sumario: | For a given Hilbert space H, consider the space of self-adjoint projections P(H). In this paper we study the differentiable structure of a canonical sphere bundle over P(H) given by R={(P,f)∈P(H)×H:Pf=f,‖f‖=1}.We establish the smooth action on R of the group of unitary operators of H, and it thereby turns out that the connected components of R are homogeneous spaces. Then we study the metric structure of 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 R by the natural action of the unitary group. Then we study the restricted bundle R2+ given by considering only the projections in the restricted Grassmannian, locally modeled by Hilbert–Schmidt operators. Therefore we endow 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 R2+, again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds. © 2019, Springer Nature B.V. |
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