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Curvature: A Variational Approach
About this Title
A. Agrachev, SISSA, Italy, MI RAS and IM SB RAS, Russia, D. Barilari, CNRS, CMAP École Polytechnique and Équipe INRIA GECO Saclay Île-de-France, Paris, France and L. Rizzi, SISSA, Trieste, Italy
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 256, Number 1225
ISBNs: 978-1-4704-2646-0 (print); 978-1-4704-4913-1 (online)
DOI: https://doi.org/10.1090/memo/1225
Published electronically: August 15, 2018
Keywords: Sub-Riemannian geometry,
affine control systems,
curvature,
Jacobi curves
MSC: Primary 49-02, 53C17, 49J15, 58B20
Table of Contents
Chapters
- 1. Introduction
1. Statements of the results
- 2. General setting
- 3. Flag and growth vector of an admissible curve
- 4. Geodesic cost and its asymptotics
- 5. Sub-Riemannian geometry
2. Technical tools and proofs
- 6. Jacobi curves
- 7. Asymptotics of the Jacobi curve: Equiregular case
- 8. Sub-Laplacian and Jacobi curves
3. Appendix
- A. Smoothness of value function (Theorem \[ ) \]
- B. Convergence of approximating Hamiltonian systems (Proposition )
- C. Invariance of geodesic growth vector by dilations (Lemma \[ ) \]
- D. Regularity of $C(t,s)$ for the Heisenberg group (Proposition \[ ) \]
- E. Basics on curves in Grassmannians (Lemma \[ and~\])
- F. Normal conditions for the canonical frame
- G. Coordinate representation of flat, rank 1 Jacobi curves (Proposition \[ ) \]
- H. A binomial identity (Lemma \[ ) \]
- I. A geometrical interpretation of $\dot {c}_t$
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. In particular, we 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.- A. Agrachev and D. Barilari, Sub-Riemannian structures on 3D Lie groups, J. Dyn. Control Syst. 18 (2012), no. 1, 21–44. MR 2902707, DOI 10.1007/s10883-012-9133-8
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