Previous |  Up |  Next


parabolic geometries; Weyl structures; almost Grassmannian structures; symmetric spaces
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.
[1] Biliotti, L.: On the automorphism group of a second order structure. Rend. Sem. Mat. Univ. Padova 104 (2000), 63–70. MR 1809350
[2] Čap, A.: Correspondence spaces and twistor spaces for parabolic geometries. J. Reine Angew. Math. 582 (2005), 143–172. DOI 10.1515/crll.2005.2005.582.143 | MR 2139714 | Zbl 1075.53022
[3] Čap, A.: Two constructions with parabolic geometries. Rend. Circ. Mat. Palermo (2) Suppl. 79 (2006), 11–37. MR 2287124 | Zbl 1120.53013
[4] Čap, A., Schichl, H.: Parabolic geometries and canonical Cartan connection. Hokkaido Math. J. 29 (2000), 453–505. MR 1795487
[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. MR 1997873 | Zbl 1076.53029
[7] Čap, A., Slovák, J., Žádník, V.: On distinguished curves in parabolic geometries. Transform. Groups 9 (2) (2004), 143–166. MR 2056534 | Zbl 1070.53021
[8] Čap, A., Žádník, V.: On the geometry of chains. eprint arXiv:math/0504469. MR 2504769
[9] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. vol. II, John Wiley & Sons, New York, 1969. MR 1393941 | Zbl 0175.48504
[10] Podesta, F.: A class of symmetric spaces. Bull. Soc. Math. France 117 (3) (1989), 343–360. MR 1020111 | Zbl 0697.53047
[11] Sharpe, R. W.: Differential geometry: Cartan’s generalization of Klein’s Erlangen program. Grad. Texts in Math. 166 (1997). MR 1453120 | Zbl 0876.53001
[12] Zalabová, L.: Remarks on symmetries of parabolic geometries. Arch. Math. (Brno), Suppl. 42 (2006), 357–368. MR 2322422 | Zbl 1164.53364
[13] Zalabová, L.: Symmetries of almost Grassmannian geometries. Proceedings of 10th International Conference on Differential Geometry and its Applications, Olomouc, 2007, pp. 371–381. MR 2462807
[14] Zalabová, L.: Symmetries of Parabolic Geometries. Ph.D. thesis, Masaryk University, 2007.
Partner of
EuDML logo