On the geometry of chains

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ČAP Andreas ŽÁDNÍK Vojtěch

Year of publication 2009
Type Article in Periodical
Magazine / Source Journal of Differential Geometry
MU Faculty or unit

Faculty of Education

Field General mathematics
Keywords parabolic contact geometries; path geometries; chains
Description The chains studied here form a canonical family of paths in manifolds endowed with a parabolic contact structure. Both the parabolic contact structure and the system of chains can be encoded as Cartan geometries. The aim of this paper is to study the relation between these two Cartan geometries for Lagrangean contact and almost CR structures by a general method of extending Cartan geometries. We show the canonical Cartan geometry associated to the family of chains can be obtained in that way if and only if the original parabolic contact structure is torsion free. In that case, the Cartan curvature associated to the family of chains is carefully analyzed. In the end, this allows to prove the underlying torsion free parabolic contact structure can be (almost) recovered just from its chains. In particular, this leads to a very conceptual proof of the fact that chain preserving contact diffeomorphisms are either isomorphisms or anti-isomorphisms of the structure.
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