Free $n$-distributions: holonomy, sub-Riemannian structures, Fefferman constructions and dual distributions

Details Free $n$-distributions: holonomy, sub-Riemannian structures, Fefferman constructions and dual distributions Free $n$-distributions: holonomy, sub-Riemannian structures, Fefferman constructions and dual distributions

2007-06-29

arxiv.org/abs/0706.4441v1

Mathematics

Differential Geometry

First Draft

Scientific paper

This paper analyses the parabolic geometries generated by a free $n$-distribution in the tangent space of a manifold. It shows that certain holonomy reductions of the associated normal Tractor connections, imply preferred connections with special properties, along with Riemannian or sub-Riemannian structures on the manifold. It constructs examples of these holonomy reductions in the simplest cases. The main results, however, lie in the free 3-distributions. In these cases, there are normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively. There is also a fascinating construction of a `dual' distribution when the holonomy reduces to $G_2'$.

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