Calculus of Variations and Geometric Measure Theory
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A. Agrachev - D. Barilari - L. Rizzi

Curvature: a variational approach

created by barilari on 25 Jun 2013
modified by rizzi1 on 15 May 2017


Accepted Paper

Inserted: 25 jun 2013
Last Updated: 15 may 2017

Journal: Memoirs of the AMS
Year: 2013

ArXiv: 1306.5318 PDF


The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.


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