Remarks on Grassmannian Symmetric Spaces
Lenka Zalabová; Vojtěch Žádník
Archivum Mathematicum (2008)
- Volume: 044, Issue: 5, page 569-585
- ISSN: 0044-8753
Access Full Article
topAbstract
topHow to cite
topZalabová, 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- Biliotti, L., On the automorphism group of a second order structure, Rend. Sem. Mat. Univ. Padova 104 (2000), 63–70. (2000) MR1809350
- 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
- Čap, A., Two constructions with parabolic geometries, Rend. Circ. Mat. Palermo (2) Suppl. 79 (2006), 11–37. (2006) Zbl1120.53013MR2287124
- Čap, A., Schichl, H., Parabolic geometries and canonical Cartan connection, Hokkaido Math. J. 29 (2000), 453–505. (2000) MR1795487
- Čap, A., Slovák, J., Parabolic Geometries, to appear in Math. Surveys Monogr., 2008.
- Čap, A., Slovák, J., Weyl Structures for Parabolic Geometries, Math. Scand. 93 (2003), 53–90. (2003) Zbl1076.53029MR1997873
- Čap, A., Slovák, J., Žádník, V., On distinguished curves in parabolic geometries, Transform. Groups 9 (2) (2004), 143–166. (2004) Zbl1070.53021MR2056534
- Čap, A., Žádník, V., On the geometry of chains, eprint arXiv:math/0504469. MR2504769
- Kobayashi, S., Nomizu, K., Foundations of Differential Geometry, vol. II, John Wiley & Sons, New York, 1969. (1969) Zbl0175.48504MR1393941
- Podesta, F., A class of symmetric spaces, Bull. Soc. Math. France 117 (3) (1989), 343–360. (1989) Zbl0697.53047MR1020111
- Sharpe, R. W., Differential geometry: Cartan’s generalization of Klein’s Erlangen program, Grad. Texts in Math. 166 (1997). (1997) Zbl0876.53001MR1453120
- Zalabová, L., Remarks on symmetries of parabolic geometries, Arch. Math. (Brno), Suppl. 42 (2006), 357–368. (2006) Zbl1164.53364MR2322422
- 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
- Zalabová, L., Symmetries of Parabolic Geometries, Ph.D. thesis, Masaryk University, 2007. (2007)
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.