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Analysis and PDE

Analysis Seminar: Mauricio Godoy

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Speaker: Mauricio Antonio Godoy Molina, Postdoc, Mathematical Department, UiB.

Title: Riemannian and Sub-Riemannian geodesic flows

Abstract: Sub-Riemannian (sR) geometry is very different from Riemannian (R) geometry in many senses, and not just an extra "sub-" in the name. Besides striking differences between their metric structures, many of the geometric invariants for R and for sR manifolds that might look quite similar in spirit, sometimes have completely unrelated behaviors. This is indeed the case for the (R and sR) geodesic flows, although in some well-studied situations (e.g., certain kinds of Lie group actions) the extra structure implies very nice relations between these flows. The goal of this talk is to show that the geodesic flows of a sR metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions. As a consequence we can characterize sub-Riemannian geodesics as the horizontal lifts of projections of Riemannian geodesics. This talked is based on a joint preprint with E. Grong available at http://arxiv.org/abs/1502.06018.