Canonical sphere bundles of the Grassmann manifold

Mostrar todas las versiones(2)

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 thereb...

Descripción completa

Guardado en:
Detalles Bibliográficos
Autores principales: Andruchow, Esteban, Chiumiento, Eduardo Hernán, Larotonda, Gabriel
Formato: Articulo Preprint
Lenguaje:Inglés
Publicado: 2019
Materias:
Matemática
Acceso en línea:http://sedici.unlp.edu.ar/handle/10915/129631
Aporte de:
SEDICI (UNLP) de Universidad Nacional de La Plata
id I19-R120-10915-129631
record_format dspace
institution Universidad Nacional de La Plata
institution_str I-19
repository_str R-120
collection SEDICI (UNLP)
language Inglés
topic Matemática
spellingShingle Matemática
Andruchow, Esteban
Chiumiento, Eduardo Hernán
Larotonda, Gabriel
Canonical sphere bundles of the Grassmann manifold
topic_facet Matemática
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 R+2 given by considering only the projections in the restricted Grassmannian, locally modeled by Hilbert–Schmidt operators. Therefore we endow 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 R+2, again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.
format Articulo
Preprint
author Andruchow, Esteban
Chiumiento, Eduardo Hernán
Larotonda, Gabriel
author_facet Andruchow, Esteban
Chiumiento, Eduardo Hernán
Larotonda, Gabriel
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
publishDate 2019
url http://sedici.unlp.edu.ar/handle/10915/129631
work_keys_str_mv AT andruchowesteban canonicalspherebundlesofthegrassmannmanifold
AT chiumientoeduardohernan canonicalspherebundlesofthegrassmannmanifold
AT larotondagabriel canonicalspherebundlesofthegrassmannmanifold
bdutipo_str Repositorios
_version_ 1764820454645694466

Ejemplares similares