Remarks on Grassmannian Symmetric Spaces

Lenka Zalabová; Vojtěch Žádník

Archivum Mathematicum (2008)

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

Abstract

top
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

top

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

top
  1. Biliotti, L., On the automorphism group of a second order structure, Rend. Sem. Mat. Univ. Padova 104 (2000), 63–70. (2000) MR1809350
  2. Čap, A., 10.1515/crll.2005.2005.582.143, J. Reine Angew. Math. 582 (2005), 143–172. (2005) Zbl1075.53022MR2139714DOI10.1515/crll.2005.2005.582.143
  3. Čap, A., Two constructions with parabolic geometries, Rend. Circ. Mat. Palermo (2) Suppl. 79 (2006), 11–37. (2006) Zbl1120.53013MR2287124
  4. Čap, A., Schichl, H., Parabolic geometries and canonical Cartan connection, Hokkaido Math. J. 29 (2000), 453–505. (2000) MR1795487
  5. Čap, A., Slovák, J., Parabolic Geometries, to appear in Math. Surveys Monogr., 2008. 
  6. Čap, A., Slovák, J., Weyl Structures for Parabolic Geometries, Math. Scand. 93 (2003), 53–90. (2003) Zbl1076.53029MR1997873
  7. Čap, A., Slovák, J., Žádník, V., On distinguished curves in parabolic geometries, Transform. Groups 9 (2) (2004), 143–166. (2004) Zbl1070.53021MR2056534
  8. Čap, A., Žádník, V., On the geometry of chains, eprint arXiv:math/0504469. MR2504769
  9. Kobayashi, S., Nomizu, K., Foundations of Differential Geometry, vol. II, John Wiley & Sons, New York, 1969. (1969) Zbl0175.48504MR1393941
  10. Podesta, F., A class of symmetric spaces, Bull. Soc. Math. France 117 (3) (1989), 343–360. (1989) Zbl0697.53047MR1020111
  11. Sharpe, R. W., Differential geometry: Cartan’s generalization of Klein’s Erlangen program, Grad. Texts in Math. 166 (1997). (1997) Zbl0876.53001MR1453120
  12. Zalabová, L., Remarks on symmetries of parabolic geometries, Arch. Math. (Brno), Suppl. 42 (2006), 357–368. (2006) Zbl1164.53364MR2322422
  13. Zalabová, L., Symmetries of almost Grassmannian geometries, Proceedings of 10th International Conference on Differential Geometry and its Applications, Olomouc, 2007, pp. 371–381. (2007) MR2462807
  14. Zalabová, L., Symmetries of Parabolic Geometries, Ph.D. thesis, Masaryk University, 2007. (2007) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.