Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 032, 33 pages      arXiv:1607.01965      https://doi.org/10.3842/SIGMA.2017.032

Local Generalized Symmetries and Locally Symmetric Parabolic Geometries

Jan Gregorovič a and Lenka Zalabová b
a) E. Čech Institute, Mathematical Institute of Charles University, Sokolovská 83, Praha 8 - Karlín, Czech Republic
b) Institute of Mathematics and Biomathematics, Faculty of Science, University of South Bohemia in České Budĕjovice, Branišovská 1760, České Budĕjovice, 370 05, Czech Republic

Received August 29, 2016, in final form May 18, 2017; Published online May 23, 2017

Abstract
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.

Key words: parabolic geometries; generalized symmetries; generalizations of symmetric spaces; automorphisms with fixed points; prolongation rigidity; geometric properties of symmetric parabolic geometries.

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References

  1. Besse A.L., Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 10, Springer-Verlag, Berlin, 1987.
  2. Bieliavsky P., Falbel E., Gorodski C., The classification of simply-connected contact sub-Riemannian symmetric spaces, Pacific J. Math. 188 (1999), 65-82.
  3. Čap A., Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math. 582 (2005), 143-172, math.DG/0102097.
  4. Čap A., Slovák J., Parabolic geometries. I. Background and general theory, Mathematical Surveys and Monographs, Vol. 154, Amer. Math. Soc., Providence, RI, 2009.
  5. Derdzinski A., Roter W., Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds, Tohoku Math. J. 59 (2007), 565-602, math.DG/0604568.
  6. Gregorovič J., General construction of symmetric parabolic structures, Differential Geom. Appl. 30 (2012), 450-476, arXiv:1207.0190.
  7. Gregorovič J., Geometric structures invariant to symmetries, FOLIA Mathematica, Vol. 18, Masaryk University, Brno, 2012, arXiv:1207.0193.
  8. Gregorovič J., Local reflexion spaces, Arch. Math. (Brno) 48 (2012), 323-332, arXiv:1207.0189.
  9. Gregorovič J., Classification of invariant AHS-structures on semisimple locally symmetric spaces, Cent. Eur. J. Math. 11 (2013), 2062-2075, arXiv:1301.5123.
  10. Gregorovič J., Zalabová L., Symmetric parabolic contact geometries and symmetric spaces, Transform. Groups 18 (2013), 711-737.
  11. Gregorovič J., Zalabová L., Notes on symmetric conformal geometries, Arch. Math. (Brno) 51 (2015), 287-296, arXiv:1503.02505.
  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.
  13. Gregorovič J., Zalabová L., Geometric properties of homogeneous parabolic geometries with generalized symmetries, Differential Geom. Appl. 49 (2016), 388-422, arXiv:1411.2402.
  14. Helgason S., Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, Vol. 34, Amer. Math. Soc., Providence, RI, 2001.
  15. Kaup W., Zaitsev D., On symmetric Cauchy-Riemann manifolds, Adv. Math. 149 (2000), 145-181, math.CV/9905183.
  16. Kobayashi S., Nomizu K., Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, Vol. 15, John Wiley & Sons, Inc., New York - London - Sydney, 1969.
  17. Kowalski O., Generalized symmetric spaces, Lecture Notes in Mathematics, Vol. 805, Springer-Verlag, Berlin - New York, 1980.
  18. Kruglikov B., The D., The gap phenomenon in parabolic geometries, J. Reine Angew. Math. 723 (2017), 153-215, arXiv:1303.1307.
  19. Loos O., Spiegelungsräume und homogene symmetrische Räume, Math. Z. 99 (1967), 141-170.
  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.
  21. Podestà F., A class of symmetric spaces, Bull. Soc. Math. France 117 (1989), 343-360.
  22. Reynolds R.F., Thompson A.H., Projective-symmetric spaces, J. Austral. Math. Soc. 7 (1967), 48-54.
  23. Zalabová L., Symmetries of parabolic geometries, Differential Geom. Appl. 27 (2009), 605-622, arXiv:0901.0626.
  24. Zalabová L., Parabolic symmetric spaces, Ann. Global Anal. Geom. 37 (2010), 125-141, arXiv:0908.0839.
  25. Zalabová L., Symmetries of parabolic contact structures, J. Geom. Phys. 60 (2010), 1698-1709, arXiv:1003.5443.

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