Mathematical Physics Seminar
Monday 18.I.2010
Time: 16:00-17:00
Room: 0089
Instytut Matematyki, ul. Łojasiewicza 6, 30-348 Kraków
Ben Warhurst
(Università degli Studi di Milano-Bicocca, Italy)
Symmetry and conformality in sub-Riemannian geometry
Abstract:
Subriemannian geometries are manifolds with a "horizontal bundle" equipped with an inner
product. The horizontal bundle is a bracket generating sub-bundle of the tangent bundle which
implies that the manifold is connected by horizontal paths, that is curves whose tangents lie in
the horizontal bundle. The subriemannian distance is the infimum of lengths of all horizontal
curves, where the length is the integral of the lengths of the tangents measured by the inner
product.
Symmetries such as isometries or conformal maps must necessarily be contact maps, i.e.
diffeomorphisms which preserve the horizontal bundle, and typically form finite-dimensional
Lie groups. The prototypical models of subriemannian geometries are called Carnot groups, i.e.
the underlying manifold is a symply connected stratified nilpotent Lie group. The rigidity
problem for Carnot groups is to classify those which have finite-dimensional families of
symmetries versus those which have infinite dimensional families of symmetries. In this talk, I
will discuss a complete solution to the rigidity problem, as well as a Liouville type theorem on
conformal maps.
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