Curvature: A Variational Approach
Jan 2019 · American Mathematical Soc.
Ebook
142
Pages
About this ebook
The curvature discussed in this paper is a far reaching generalization of the Riemannian sectional curvature. The authors 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. The authors' construction of curvature is direct and naive, and similar to the original approach of Riemann. In particular, they extract geometric invariants from the asymptotics of the cost of optimal control problems. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.
The curvature discussed in this paper is a far reaching generalization of the Riemannian sectional curvature. The authors 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-Rie...
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About the author
A. Agrachev: SISSA, Trieste, Italy and Sobolev Institute of Mathematics, Novosibirsk, Russia,
D. Barilari: École Polytechnique, Paris, France and INRIA GECO Saclay-Ile-de-France, Paris, France,
L. Rizzi: SISSA, Trieste, Italy
A. Agrachev: SISSA, Trieste, Italy and Sobolev Institute of Mathematics, Novosibirsk, Russia,
D. Barilari: École Polytechnique, Paris, France and INRIA GECO Saclay-Ile-de-France, Paris, France,
L. Rizzi: SISSA, Trieste, Italy
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