Description

Book Synopsis
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.

Table of Contents
  • 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.

    Curvature A Variational Approach

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      A Paperback by A. Agrachev, D. Barilari, L. Rizzi

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        Publisher: MP-AMM American Mathematical
        Publication Date: Publication Date: 5/30/2019 12:00:00 AM
        ISBN13: 9781470426460, 978-1470426460
        ISBN10: 1470426463

        Description

        Book Synopsis
        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.

        Table of Contents
        • 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.

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