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Abstract: Lie contact structures generalize the classical Lie sphere geometry oforiented hyperspheres in the standard sphere. They can be equivalentlydescribed as parabolic geometries corresponding to the contact grading oforthogonal real Lie algebra. It follows the underlying geometric structure canbe interpreted in several equivalent ways. In particular, we show this is givenby a split-quaternionic structure on the contact distribution, which iscompatible with the Levi bracket. In this vein, we study the geometry ofchains, a distinguished family of curves appearing in any parabolic contactgeometry. Also to the system of chains there is associated a canonicalparabolic geometry of specific type. Up to some exceptions in low dimensions,it turns out this can be obtained by an extension of the parabolic geometryassociated to the Lie contact structure if and only if the latter is locallyflat. In that case we can show that chains are never geodesics of an affineconnection, hence, in particular, the path geometry of chains is alwaysnon-trivial. Using appropriately this fact, we conclude that the path geometryof chains allows to recover the Lie contact structure, hence, in particular,transformations preserving chains must preserve the Lie contact structure.



Author: Vojtech Zadnik

Source: https://arxiv.org/



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