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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Detalles Bibliográficos
Publicado: 2019
Materias:
Finsler metric
Flag manifold
Geodesic
Projection
Riemannian metric
Sphere bundle
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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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00465755_v_n_p_Andruchow
record_format dspace
spelling paper:paper_00465755_v_n_p_Andruchow2023-06-08T15:05:32Z Canonical sphere bundles of the Grassmann manifold Finsler metric Flag manifold Geodesic Projection Riemannian metric Sphere bundle 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. 2019 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
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Finsler metric
Flag manifold
Geodesic
Projection
Riemannian metric
Sphere bundle
spellingShingle Finsler metric
Flag manifold
Geodesic
Projection
Riemannian metric
Sphere bundle
Canonical sphere bundles of the Grassmann manifold
topic_facet Finsler metric
Flag manifold
Geodesic
Projection
Riemannian metric
Sphere bundle
description 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.
title Canonical sphere bundles of the Grassmann manifold
title_short Canonical sphere bundles of the Grassmann manifold
title_full Canonical sphere bundles of the Grassmann manifold
title_fullStr Canonical sphere bundles of the Grassmann manifold
title_full_unstemmed Canonical sphere bundles of the Grassmann manifold
title_sort canonical sphere bundles of the grassmann manifold
publishDate 2019
url 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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