Sub-Riemannian Geometry and Optimal Transport by Ludovic Rifford

Author:Ludovic Rifford
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

where is a smooth function and equipped with a metric making an orthonormal family. In a sufficiently small neighborhood of the origin , singular curves are given by the horizontal paths which are contained in the Martinet set

that is of the form

with . Such curves are locally minimizing.

Theorem 2.23

There is such that for every the horizontal path given by

minimizes the length among all horizontal paths joining to .

Proof

We need to show that among all controls with steering the origin to , we have . There is such that is included in . If , then any minimizing geodesic joining to is contained in . As a matter of fact, we know that . Let be upper bounds for on Let be a competitor for . We get easily

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.