Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

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Let M and N be two connected smooth manifolds, where M is compact and oriented and N is Riemannian. Let E be the Fréchet manifold of all embeddings of M in N, endowed with the canonical weak Riemannian metric. Let ∼ be the equivalence relation on E defined by f ∼ g if and only if f = g ◦ φ for some...

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Autor principal: Salvai, Marcos Luis
Formato: article
Lenguaje:Inglés
Publicado: 2021
Materias:
Manifold of embeddings
Geodesic
Symmetric space
Reflective submanifold
Acceso en línea:http://hdl.handle.net/11086/20321
http://dx.doi.org/10.1007/s00605-014-0642-2
Aporte de:
Repositorio Digital Universitario (UNC) de Universidad Nacional de Córdoba
id I10-R141-11086-20321
record_format dspace
institution Universidad Nacional de Córdoba
institution_str I-10
repository_str R-141
collection Repositorio Digital Universitario (UNC)
language Inglés
topic Manifold of embeddings
Geodesic
Symmetric space
Reflective submanifold
spellingShingle Manifold of embeddings
Geodesic
Symmetric space
Reflective submanifold
Salvai, Marcos Luis
Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space
topic_facet Manifold of embeddings
Geodesic
Symmetric space
Reflective submanifold
description Let M and N be two connected smooth manifolds, where M is compact and oriented and N is Riemannian. Let E be the Fréchet manifold of all embeddings of M in N, endowed with the canonical weak Riemannian metric. Let ∼ be the equivalence relation on E defined by f ∼ g if and only if f = g ◦ φ for some orientation preserving diffeomorphism φ of M. The Fréchet manifold S = E/∼ of equivalence classes, which may be thought of as the set of submanifolds of N diffeomorphic to M and is called the nonlinear Grassmannian (or Chow manifold) of N of type M, inherits from E a weak Riemannian structure. Its geodesics, although they are not good from the metric point of view, are distinguished curves and have proved to be useful in various situations. We consider the following particular case: N is a compact irreducible symmetric space and M is a reflective submanifold of N (that is, a connected component of the set of fixed points of an involutive isometry of N). Let C be the set of submanifolds of N which are congruent to M. We prove that the natural inclusion of C in S is totally geodesic.
format article
author Salvai, Marcos Luis
author_facet Salvai, Marcos Luis
author_sort Salvai, Marcos Luis
title Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space
title_short Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space
title_full Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space
title_fullStr Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space
title_full_unstemmed Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space
title_sort some totally geodesic submanifolds of the nonlinear grassmannian of a compact symmetric space
publishDate 2021
url http://hdl.handle.net/11086/20321
http://dx.doi.org/10.1007/s00605-014-0642-2
work_keys_str_mv AT salvaimarcosluis sometotallygeodesicsubmanifoldsofthenonlineargrassmannianofacompactsymmetricspace
bdutipo_str Repositorios
_version_ 1764820391912538115

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