Calculus of Variations and Geometric Measure Theory

A. Agrachev - D. Barilari - L. Rizzi

Curvature: a variational approach

created by barilari on 25 Jun 2013
modified by rizzi1 on 17 Aug 2018


Published Paper

Inserted: 25 jun 2013
Last Updated: 17 aug 2018

Journal: Memoirs of the AMS
Volume: 256
Number: 1225
Pages: 142
Year: 2018
Doi: 10.1090/memo/1225

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.