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Principal bundle structure of matrix manifolds


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Título : Principal bundle structure of matrix manifolds
Autor : Billaud Friess, Marie
Falcó Montesinos, Antonio
Nouy, Anthony
Materias: Topología diferencial.Differential topology.Variedades (Matemáticas)Grassmann, Variedades de.Grassmann manifolds.Geometría diferencial.Geometry, Differential.Manifolds (Mathematics)
Editorial : MDPI
Citación : Billaud-Friess, M., Falcó, A. & Nouy, A. (2021). Principal bundle structure of matrix manifolds. Mathematics, vol. 9, i. 14 (15 jul.), art. 1669. DOI: https://doi.org/10.3390/math9141669
Resumen : In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r < k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(k􀀀r) r. Then, we define an atlas for the set Mr(Rk r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk k. Finally, we define an atlas for the setMr(Rn m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn) Gr(Rm) and typical fibre GLr. The atlas ofMr(Rn m) is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the setMr(Rn m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn m, seen as the union of manifoldsMr(Rn m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.
Descripción : Este artículo se encuentra disponible en la siguiente URL: https://www.mdpi.com/2227-7390/9/14/1669
Este artículo de investigación pertenece al número especial "Differential Geometry: Structures on Manifolds and Their Applications".
URI : http://hdl.handle.net/10637/13579
Derechos: http://creativecommons.org/licenses/by/4.0/deed.es
ISSN : 2227-7390 (Electrónico)
Fecha de publicación : 15-jul-2021
Centro : Universidad Cardenal Herrera-CEU
Aparece en las colecciones: Dpto. Matemáticas, Física y Ciencias Tecnológicas





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