Curvature : a Variational Approach.

Cover -- Title page -- Chapter 1. Introduction -- 1.1. Structure of the paper -- 1.2. Statements of the main theorems -- 1.3. The Heisenberg group -- Part 1 . Statements of the results -- Chapter 2. General setting -- 2.1. Affine control systems -- 2.2. End-point map -- 2.3. Lagrange multipliers rule -- 2.4. Pontryagin Maximum Principle -- 2.5. Regularity of the value function -- Chapter 3. Flag and growth vector of an admissible curve -- 3.1. Growth vector of an admissible curve -- 3.2. Linearised control system and growth vector -- 3.3. State-feedback invariance of the flag of an admissible curve -- 3.4. An alternative definition -- Chapter 4. Geodesic cost and its asymptotics -- 4.1. Motivation: a Riemannian interlude -- 4.2. Geodesic cost -- 4.3. Hamiltonian inner product -- 4.4. Asymptotics of the geodesic cost function and curvature -- 4.5. Examples -- Chapter 5. Sub-Riemannian geometry -- 5.1. Basic definitions -- 5.2. Existence of ample geodesics -- 5.3. Reparametrization and homogeneity of the curvature operator -- 5.4. Asymptotics of the sub-Laplacian of the geodesic cost -- 5.5. Equiregular distributions -- 5.6. Geodesic dimension and sub-Riemannian homotheties -- 5.7. Heisenberg group -- 5.8. On the "meaning" of constant curvature -- Part 2 . Technical tools and proofs -- Chapter 6. Jacobi curves -- 6.1. Curves in the Lagrange Grassmannian -- 6.2. The Jacobi curve and the second differential of the geodesic cost -- 6.3. The Jacobi curve and the Hamiltonian inner product -- 6.4. Proof of Theorem -- 6.5. Proof of Theorem -- Chapter 7. Asymptotics of the Jacobi curve: Equiregular case -- 7.1. The canonical frame -- 7.2. Main result -- 7.3. Proof of Theorem 7.4 -- 7.4. Proof of Theorem -- 7.5. A worked out example: 3D contact sub-Riemannian structures -- Chapter 8. Sub-Laplacian and Jacobi curves -- 8.1. Coordinate lift of a local frame.

8.2. Sub-Laplacian of the geodesic cost -- 8.3. Proof of Theorem -- 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 ( , ) 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 _{ } -- Bibliography -- Index -- Back Cover.

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.

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