Symetrie parabolických geometrií

Mgr. Lenka Zalabová, Ph.D.

Disertační práce

Symmetries of Parabolic Geometries

Abstract:

We introduce and discuss symmetries for the so called parabolic geometries. Our motivation comes from affine locally symmetric spaces. The manifold with affine connection is locally symmetric if and only if the torison vanishes and the curvature is covariantly constant. These geometries can be understood as the special case of reductive Cartan geometries. The parabolic geometries represent another special case of the general Cartan geometries. They are of second order and never reductive. We are interested in $|1|$--graded geometries. In this case, the definition of the symmetry is a generalization of the clasical one and follows the intuitive idea. We show an analogy of the results from the affine locally symmetric spaces and we get more curvature restrictions, which come from the general theory of parabolic geometries. Many types of symmetric $|1|$--graded geometries have to be locally flat. There are also some `interesting' types, which can carry a symmetry in the point and still allow some nonzero curvature at this point. Our main tool to study these examples is the theory of Weyl structures. We study the Weyl curvature of $|1|$--graded geometries to get more restriction for the `interesting' ones. Finally we show some corollaries in the projective and conformal geometries.

Keywords

Cartan geometries, parabolic geometries, Weyl structures, symmetric spaces

Jazyk práce: angličtina

Datum vytvoření / odevzdání či podání práce: 1. 6. 2007

Obhajoba závěrečné práce

• Obhajoba proběhla 25. 6. 2007
• Vedoucí: prof. RNDr. Jan Slovák, DrSc.
• Oponent: prof. RNDr. Ivan Kolář, DrSc., prof. Dimitri Alekseevsky

Masarykova univerzita

Přírodovědecká fakulta

Doktorský studijní program / obor:
Matematika / Geometrie, topologie a globální analýza