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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