zbMATH — the first resource for mathematics

Parabolic symmetric spaces. (English) Zbl 1188.53026
The author considers on a manifold \(M\) a parabolic geometry of type \((G,P)\) for a semisimple Lie group \(G\) and a parabolic subgroup \(P\). This geometry is assumed to be \(|1|\)-graded: \({\mathfrak g}= \text{Lie}(G)\) admits a grading \({\mathfrak g}={\mathfrak g}_{-1}\oplus {\mathfrak g}_0\oplus {\mathfrak g}_1\) such that \({\mathfrak p}= \text{Lie}(P)={\mathfrak g}_0\oplus {\mathfrak g}_1\). For this geometry a symmetry centered at \(x\in M\) is a diffeomorphism \(s_x\) of \(M\) such that \(s_x(x)=x\), \(T_xs_x=- \text{id}\) on \(T_xM\), and \(s_x\) is covered by an automorphism of the geometry. The author proves that, if there is such a symmetry \(s_x\) for each \(x\in M\), and if the system of symmetries is smooth, then the parabolic geometry is homogeneous. Furthermore if
\[ s_x\circ s_y\circ s_x=s_{s_x(y)}\quad (x,y\in M), \]
and under an additional assumption, the author establishes the existence of an affine connection \(\nabla \) on \(M\) such that \((M,\nabla )\) is an affine symmetric space.

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53A40 Other special differential geometries
53C05 Connections (general theory)
53C35 Differential geometry of symmetric spaces
Full Text: DOI arXiv
[1] Cahen, M., Schwachhöfer, L.J.: Special Symplectic Connection. J. Diff. Geom. (in press) · Zbl 1190.53019
[2] Čap A.: Two constructions with parabolic geometries. Rend. Circ. Mat. Palermo (2) Suppl. 79, 11–37 (2006) · Zbl 1120.53013
[3] Čap A., Gover R.: Tractor bundles for irreducible parabolic geometries. SMF, Séminaires et Congrès 4, 129–154 (2000) · Zbl 0996.53012
[4] Čap A., Gover R.: Tractor calculi for parabolic geometries. Trans. Am. Math. Soc. 354, 1511–1548 (2002) · Zbl 0997.53016
[5] Čap A., Schichl H.: Parabolic geometries and canonical Cartan connection. Hokkaido Math. J. 29, 453–505 (2000) · Zbl 0996.53023
[6] Čap A., Slovák J.: Weyl structures for parabolic geometries. Math. Scand. 93, 53–90 (2003) · Zbl 1076.53029
[7] Čap, A., Slovák, J.: Parabolic geometries I: background and general theory. Mathematical Surveys and Monographs, AMS, vol. 154, 628 pp · Zbl 1183.53002
[8] Čap, A., Žádník, V.: On the geometry of chains. J. Differ. Geom. (in press)
[9] Čap A., Slovák J., Souček V.: Invariant operators on manifolds with almost Hermitian symmetric structures, II. Normal Cartan connections. Acta Math. Univ. Comen. 66(2), 203–220 (1997) · Zbl 1024.53007
[10] Čap A., Slovák J., Žádník V.: On distinguished curves in parabolic geometries. Transform. Groups 9(2), 143–166 (2004)
[11] Hammerl M.: Homogeneous Cartan geometries. Arch. Math. (Brno) 43(suppl.), 431–442 (2007) · Zbl 1199.53021
[12] Helgason S.: Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics. Academic Press, New York (1978) · Zbl 0451.53038
[13] Kobayashi S., Nomizu K.: Foundations of Differential Geometry, vol. II. Wiley, New York (1969) · Zbl 0175.48504
[14] Kolář I., Michor P.W., Slovák J.: Natural Operations in Differential Geometry, pp. 434. Springer-Verlag, Berlin (1993) · Zbl 0782.53013
[15] Kowalski O.: Generalized Symmetric Spaces, Lecture Notes in Mathematics, vol. 805. Springer-Verlag, Berlin (1980) · Zbl 0431.53042
[16] Loos O.: Symmetric Spaces I: General Theory, Math Lecture Note Series. W.A. Benjamin, Inc., New York (1969) · Zbl 0175.48601
[17] Podesta F.: A Class of Symmetric Spaces. Bull. Soc. Math. Fr. 117(3), 343–360 (1989)
[18] Sharpe R.W.: Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program. Graduate Texts in Mathematics 166. Springer-Verlag, Berlin (1997) · Zbl 0876.53001
[19] Yamaguchi K.: Differential systems associated with simple graded Lie algebras. Adv. Stud. Pure Math. 22, 413–494 (1993) · Zbl 0812.17018
[20] Zalabová L.: Remarks on symmetries of parabolic geometries. Arch. Math. (Brno) 42(suppl.), 357–368 (2006) · Zbl 1164.53364
[21] Zalabová, L.: Symmetries of almost Grassmannian geometries. In: Differential Geometry and its Applications, Proceedings of 10th International Conference, Olomouc, pp. 371–381 (2007) · Zbl 1165.53021
[22] Zalabová, L.: Symmetries of parabolic geometries. Ph.D. thesis, Masaryk University, Brno (2007) · Zbl 1187.53036
[23] Zalabová, L.: Symmetries of parabolic geometries. Differ. Geom. Appl. (in press). doi: 10.1016/j.difgeo.2009.03.001 · Zbl 1187.53036
[24] Zalabová L., Žádník V.: Remarks on Grassmannian symmetric spaces. Arch. Math. (Brno) 44(suppl.), 569–585 (2008) · Zbl 1212.53054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.