The curvature discussed in this paper is a far reaching generalisation of the Riemannian sectional curvature. We give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optimal control problems, including Riemannian, sub-Riemannian, Finsler and sub-Finsler spaces. Special attention is paid to the sub-Riemannian (or Carnot–Carathéodory) metric spaces. Our construction of curvature is direct and naive, and similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.

Curvature: a variational approach / Agrachev, Andrey; Barilari, D.; Rizzi, L.. - In: MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0065-9266. - 256:1225(2018), pp. 1-116. [10.1090/memo/1225]

Curvature: a variational approach

Agrachev, Andrey;Rizzi L.
2018-01-01

Abstract

The curvature discussed in this paper is a far reaching generalisation of the Riemannian sectional curvature. We give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optimal control problems, including Riemannian, sub-Riemannian, Finsler and sub-Finsler spaces. Special attention is paid to the sub-Riemannian (or Carnot–Carathéodory) metric spaces. Our construction of curvature is direct and naive, and similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.
2018
256
1225
1
116
http://cvgmt.sns.it/media/doc/paper/2172/Curvature-memoir-revised3.pdf
http://arxiv.org/abs/1306.5318v5
Agrachev, Andrey; Barilari, D.; Rizzi, L.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/14389
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