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-Caratheodory) 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.
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Höhe: 254 mm
Breite: 178 mm
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ISBN-13
978-1-4704-2646-0 (9781470426460)
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Schweitzer Klassifikation
A. Agrachev, SISSA, Trieste, Italy, and Sobolev Institute of Mathematics, Novosibirsk, Russia.
D. Barilari, Ecole Polytechnique, Paris, France, and INRIA GECO Saclay-Ile-de-France, Paris, France.
L. Rizzi, SISSA, Trieste, Italy.
Introduction
Part 1. Statements of the results: General setting
Flag and growth vector of an admissible curve
Geodesic cost and its asymptotics
Sub-Riemannian geometry
Part 2. Technical tools and proofs: Jacobi curves
Asymptotics of the Jacobi curve: equiregular case
Sub-Laplacian and Jacobi curves
Part 3. Appendix: Appendix A. Smoothness of value function (Theorem $2.19$)
Appendix B. Convergence of approximating Hamiltonian systems (Proposition 5.15)
Appendix C. Invariance of geodesic growth vector by dilations (Lemma $5.20$)
Appendix D. Regularity of $C(t,s)$ for the Heisenberg group (Proposition $5.51$)
Appendix E. Basics on curves in Grassmannians (Lemma $3.5$ and $6.5$)
Appendix F. Normal conditions for the canonical frame
Appendix G. Coordinate representation of flat, rank 1 Jacobi curves (Proposition $7.7$)
Appendix H. A binomial identity (Lemma $7.8$)
Appendix I. A geometrical interpretation of $\dot c_t$
Bibliography
Index.