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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.


Documento: Artículo
Título:Canonical sphere bundles of the Grassmann manifold
Autor:Andruchow, E.; Chiumiento, E.; Larotonda, G.
Filiación:Instituto Argentino de Matemática, ‘Alberto P. Calderón’, CONICET, Saavedra 15 3er. piso, Buenos Aires, 1083, Argentina
Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J.M. Gutierrez 1150, Los Polvorines, 1613, Argentina
Departamento de Matemática, FCE-UNLP, Calles 50 y 115, La Plata, 1900, Argentina
Departamento de Matemática, FCEyN-UBA, Ciudad Universitaria, Ciudad Autónoma de Buenos Aires, 1428, Argentina
Palabras clave:Finsler metric; Flag manifold; Geodesic; Projection; Riemannian metric; Sphere bundle
Título revista:Geometriae Dedicata
Título revista abreviado:Geom. Dedic.


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---------- APA ----------
Andruchow, E., Chiumiento, E. & Larotonda, G. (2019) . Canonical sphere bundles of the Grassmann manifold. Geometriae Dedicata.
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Andruchow, E., Chiumiento, E., Larotonda, G. "Canonical sphere bundles of the Grassmann manifold" . Geometriae Dedicata (2019).
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Andruchow, E., Chiumiento, E., Larotonda, G. "Canonical sphere bundles of the Grassmann manifold" . Geometriae Dedicata, 2019.
---------- VANCOUVER ----------
Andruchow, E., Chiumiento, E., Larotonda, G. Canonical sphere bundles of the Grassmann manifold. Geom. Dedic. 2019.