Remarks on Grassmannian Symmetric Spaces

Lenka Zalabová; Vojtěch Žádník

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 5, page 569-585
  • ISSN: 0044-8753

Abstract

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The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for | 1 | -graded parabolic geometries and for almost Grassmannian structures, in particular. As an application of two general constructions with parabolic geometries, we present an example of non-flat Grassmannian symmetric space. Next we observe there is a distinguished torsion-free affine connection preserving the Grassmannian structure so that, with respect to this connection, the Grassmannian symmetric space is an affine symmetric space in the classical sense.

How to cite

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Zalabová, Lenka, and Žádník, Vojtěch. "Remarks on Grassmannian Symmetric Spaces." Archivum Mathematicum 044.5 (2008): 569-585. <http://eudml.org/doc/250304>.

@article{Zalabová2008,
abstract = {The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for $|1|$-graded parabolic geometries and for almost Grassmannian structures, in particular. As an application of two general constructions with parabolic geometries, we present an example of non-flat Grassmannian symmetric space. Next we observe there is a distinguished torsion-free affine connection preserving the Grassmannian structure so that, with respect to this connection, the Grassmannian symmetric space is an affine symmetric space in the classical sense.},
author = {Zalabová, Lenka, Žádník, Vojtěch},
journal = {Archivum Mathematicum},
keywords = {parabolic geometries; Weyl structures; almost Grassmannian structures; symmetric spaces; parabolic geometry; Weyl structure; almost Grassmannian structure; symmetric space},
language = {eng},
number = {5},
pages = {569-585},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Remarks on Grassmannian Symmetric Spaces},
url = {http://eudml.org/doc/250304},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Zalabová, Lenka
AU - Žádník, Vojtěch
TI - Remarks on Grassmannian Symmetric Spaces
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 5
SP - 569
EP - 585
AB - The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for $|1|$-graded parabolic geometries and for almost Grassmannian structures, in particular. As an application of two general constructions with parabolic geometries, we present an example of non-flat Grassmannian symmetric space. Next we observe there is a distinguished torsion-free affine connection preserving the Grassmannian structure so that, with respect to this connection, the Grassmannian symmetric space is an affine symmetric space in the classical sense.
LA - eng
KW - parabolic geometries; Weyl structures; almost Grassmannian structures; symmetric spaces; parabolic geometry; Weyl structure; almost Grassmannian structure; symmetric space
UR - http://eudml.org/doc/250304
ER -

References

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  14. Zalabová, L., Symmetries of Parabolic Geometries, Ph.D. thesis, Masaryk University, 2007. (2007) 

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