Curvature: A Variational Approach

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.