Artículo
Canonical sphere bundles of the Grassmann manifold
Fecha de publicación:
11/2019
Editorial:
Springer
Revista:
Geometriae Dedicata
ISSN:
0046-5755
Idioma:
Inglés
Tipo de recurso:
Artículo publicado
Clasificación temática:
Resumen
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.
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Articulos(IAM)
Articulos de INST.ARG.DE MATEMATICAS "ALBERTO CALDERON"
Articulos de INST.ARG.DE MATEMATICAS "ALBERTO CALDERON"
Citación
Andruchow, Esteban; Chiumiento, Eduardo Hernan; Larotonda, Gabriel Andrés; Canonical sphere bundles of the Grassmann manifold; Springer; Geometriae Dedicata; 203; 1; 11-2019; 179-203
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