Curvature

Curvature

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-Rieman...

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Bibliographic Details
Main Author: Agrachev, A.
Other Authors: Barilari, D., Rizzi, L.
Format: eBook
Language:English
Published: Providence : American Mathematical Society, 2019.
Series:Memoirs of the American Mathematical Society Ser.
Subjects:
Curvature.
Riemannian manifolds.
Geometry, Differential.
Curvature
Geometry, Differential
Riemannian manifolds
Online Access:Click for online access
Table of Contents:
  • 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 definitionChapter 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 curvaturePart 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 frame8.2. 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

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