Canonical sphere bundles of the Grassmann manifold

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Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.

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Autores principales: Andruchow, Esteban, Chiumiento, Eduardo Hernán, Larotonda, Gabriel Andrés
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: Springer 2024
Materias:
Finsler metric
Flag manifold
Geodesic
Projection
Riemannian metric
Sphere bundle
Acceso en línea:http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1806
Aporte de:
Repositorio Institucional Digital de Acceso Abierto (UNGS) de Universidad Nacional de General Sarmiento
id I71-R177-UNGS-1806
record_format dspace
spelling I71-R177-UNGS-18062024-12-23T13:21:47Z Canonical sphere bundles of the Grassmann manifold Andruchow, Esteban Chiumiento, Eduardo Hernán Larotonda, Gabriel Andrés Finsler metric Flag manifold Geodesic Projection Riemannian metric Sphere bundle Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina. Fil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina. Fil: Larotonda, Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto Argentino de Matemática "Alberto P. Calderón"; Argentina. 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. 2024-12-23T13:21:47Z 2024-12-23T13:21:47Z 2019 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion Andruchow, E., Chiumiento, E. y Larotonda, G. (2019). Canonical sphere bundles of the Grassmann manifold. Geometriae Dedicata, 203(1), 179-203. 0046-5755 http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1806 eng http://dx.doi.org/10.1007/s10711-019-00431-7 info:eu-repo/semantics/restrictedAccess https://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Springer Geometriae Dedicata. Feb. 2019; 203(1): 179-203 https://link.springer.com/journal/10711/articles?link_id=G_Geometriae_1972-1999_Springer&filter-by-volume=203&sortBy=newestFirst
institution Universidad Nacional de General Sarmiento
institution_str I-71
repository_str R-177
collection Repositorio Institucional Digital de Acceso Abierto (UNGS)
language Inglés
orig_language_str_mv eng
topic Finsler metric
Flag manifold
Geodesic
Projection
Riemannian metric
Sphere bundle
spellingShingle Finsler metric
Flag manifold
Geodesic
Projection
Riemannian metric
Sphere bundle
Andruchow, Esteban
Chiumiento, Eduardo Hernán
Larotonda, Gabriel Andrés
Canonical sphere bundles of the Grassmann manifold
topic_facet Finsler metric
Flag manifold
Geodesic
Projection
Riemannian metric
Sphere bundle
description Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina.
format Artículo
Artículo
publishedVersion
author Andruchow, Esteban
Chiumiento, Eduardo Hernán
Larotonda, Gabriel Andrés
author_facet Andruchow, Esteban
Chiumiento, Eduardo Hernán
Larotonda, Gabriel Andrés
author_sort Andruchow, Esteban
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
publisher Springer
publishDate 2024
url http://repositorio.ungs.edu.ar:8080/xmlui/handle/UNGS/1806
work_keys_str_mv AT andruchowesteban canonicalspherebundlesofthegrassmannmanifold
AT chiumientoeduardohernan canonicalspherebundlesofthegrassmannmanifold
AT larotondagabrielandres canonicalspherebundlesofthegrassmannmanifold
_version_ 1824528660365836288

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