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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todo:paper_00465755_v_n_p_Andruchow2023-10-03T14:52:18Z Canonical sphere bundles of the Grassmann manifold Andruchow, E. Chiumiento, E. Larotonda, G. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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 Andruchow, E. Chiumiento, E. Larotonda, G. 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. |
| format |
JOUR |
| author |
Andruchow, E. Chiumiento, E. Larotonda, G. |
| author_facet |
Andruchow, E. Chiumiento, E. Larotonda, G. |
| author_sort |
Andruchow, E. |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_00465755_v_n_p_Andruchow |
| work_keys_str_mv |
AT andruchowe canonicalspherebundlesofthegrassmannmanifold AT chiumientoe canonicalspherebundlesofthegrassmannmanifold AT larotondag canonicalspherebundlesofthegrassmannmanifold |
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1807316341285191680 |