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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.

##### MSC:
 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
##### Keywords:
parabolic geometry; symmetric space
Full Text:
##### References:
 [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
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