Geodesic
Local Generalized Symmetries and Locally Symmetric Parabolic Geometries
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017)

Voir la notice de l'article provenant de la source Math-Net.Ru

We investigate (local) automorphisms of parabolic geometries that generalize geodesic symmetries. We show that many types of parabolic geometries admit at most one generalized geodesic symmetry at a point with non-zero harmonic curvature. Moreover, we show that if there is exactly one symmetry at each point, then the parabolic geometry is a generalization of an affine (locally) symmetric space.
Keywords: parabolic geometries; generalized symmetries; generalizations of symmetric spaces; automorphisms with fixed points; prolongation rigidity; geometric properties of symmetric parabolic geometries. 
Jan Gregorovič; Lenka Zalabová. Local Generalized Symmetries and Locally Symmetric Parabolic Geometries. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a31/
@article{SIGMA_2017_13_a31,
     author = {Jan Gregorovi\v{c} and Lenka Zalabov\'a},
     title = {Local {Generalized} {Symmetries} and {Locally} {Symmetric} {Parabolic} {Geometries}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a31/}
}
TY  - JOUR
AU  - Jan Gregorovič
AU  - Lenka Zalabová
TI  - Local Generalized Symmetries and Locally Symmetric Parabolic Geometries
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2017
VL  - 13
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a31/
LA  - en
ID  - SIGMA_2017_13_a31
ER  - 
%0 Journal Article
%A Jan Gregorovič
%A Lenka Zalabová
%T Local Generalized Symmetries and Locally Symmetric Parabolic Geometries
%J Symmetry, integrability and geometry: methods and applications
%D 2017
%V 13
%U http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a31/
%G en
%F SIGMA_2017_13_a31

[1] Besse A. L., Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, Berlin, 1987 | DOI | MR | Zbl

[2] Bieliavsky P., Falbel E., Gorodski C., “The classification of simply-connected contact sub-Riemannian symmetric spaces”, Pacific J. Math., 188 (1999), 65–82 | DOI | MR | Zbl

[3] Čap A., “Correspondence spaces and twistor spaces for parabolic geometries”, J. Reine Angew. Math., 582 (2005), 143–172, arXiv: math.DG/0102097 | DOI | MR | Zbl

[4] Čap A., Slovák J., Parabolic geometries, v. I, Mathematical Surveys and Monographs, 154, Background and general theory, Amer. Math. Soc., Providence, RI, 2009 | DOI | MR | Zbl

[5] Derdzinski A., Roter W., “Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds”, Tohoku Math. J., 59 (2007), 565–602, arXiv: math.DG/0604568 | DOI | MR | Zbl

[6] Gregorovič J., “General construction of symmetric parabolic structures”, Differential Geom. Appl., 30 (2012), 450–476, arXiv: 1207.0190 | DOI | MR | Zbl

[7] Gregorovič J., Geometric structures invariant to symmetries, FOLIA Mathematica, 18, Masaryk University, Brno, 2012, arXiv: 1207.0193 | DOI

[8] Gregorovič J., “Local reflexion spaces”, Arch. Math. (Brno), 48 (2012), 323–332, arXiv: 1207.0189 | DOI | MR | Zbl

[9] Gregorovič J., “Classification of invariant AHS-structures on semisimple locally symmetric spaces”, Cent. Eur. J. Math., 11 (2013), 2062–2075, arXiv: 1301.5123 | DOI | MR | Zbl

[10] Gregorovič J., Zalabová L., “Symmetric parabolic contact geometries and symmetric spaces”, Transform. Groups, 18 (2013), 711–737 | DOI | MR | Zbl

[11] Gregorovič J., Zalabová L., “Notes on symmetric conformal geometries”, Arch. Math. (Brno), 51 (2015), 287–296, arXiv: 1503.02505 | DOI | MR | Zbl

[12] Gregorovič J., Zalabová L., “On automorphisms with natural tangent actions on homogeneous parabolic geometries”, J. Lie Theory, 25 (2015), 677–715, arXiv: 1312.7318 | MR | Zbl

[13] Gregorovič J., Zalabová L., “Geometric properties of homogeneous parabolic geometries with generalized symmetries”, Differential Geom. Appl., 49 (2016), 388–422, arXiv: 1411.2402 | DOI | MR | Zbl

[14] Helgason S., Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, Amer. Math. Soc., Providence, RI, 2001 | DOI | MR | Zbl

[15] Kaup W., Zaitsev D., “On symmetric Cauchy–Riemann manifolds”, Adv. Math., 149 (2000), 145–181, arXiv: math.CV/9905183 | DOI | MR | Zbl

[16] Kobayashi S., Nomizu K., Foundations of differential geometry, v. II, Interscience Tracts in Pure and Applied Mathematics, 15, John Wiley Sons, Inc., New York–London–Sydney, 1969 | MR

[17] Kowalski O., Generalized symmetric spaces, Lecture Notes in Mathematics, 805, Springer-Verlag, Berlin–New York, 1980 | DOI | MR | Zbl

[18] Kruglikov B., The D., “The gap phenomenon in parabolic geometries”, J. Reine Angew. Math., 723 (2017), 153–215, arXiv: 1303.1307 | DOI | MR | Zbl

[19] Loos O., “Spiegelungsräume und homogene symmetrische Räume”, Math. Z., 99 (1967), 141–170 | DOI | MR | Zbl

[20] Loos O., “An intrinsic characterization of fibre bundles associated with homogeneous spaces defined by Lie group automorphisms”, Abh. Math. Sem. Univ. Hamburg, 37 (1972), 160–179 | DOI | MR | Zbl

[21] Podestà F., “A class of symmetric spaces”, Bull. Soc. Math. France, 117 (1989), 343–360 | DOI | MR | Zbl

[22] Reynolds R. F., Thompson A. H., “Projective-symmetric spaces”, J. Austral. Math. Soc., 7 (1967), 48–54 | DOI | MR | Zbl

[23] Zalabová L., “Symmetries of parabolic geometries”, Differential Geom. Appl., 27 (2009), 605–622, arXiv: 0901.0626 | DOI | MR | Zbl

[24] Zalabová L., “Parabolic symmetric spaces”, Ann. Global Anal. Geom., 37 (2010), 125–141, arXiv: 0908.0839 | DOI | MR | Zbl

[25] Zalabová L., “Symmetries of parabolic contact structures”, J. Geom. Phys., 60 (2010), 1698–1709, arXiv: 1003.5443 | DOI | MR | Zbl