**Memoirs of the American Mathematical Society**

2018;
142 pp;
Softcover

MSC: Primary 49; 53; 58;

**Print ISBN: 978-1-4704-2646-0
Product Code: MEMO/256/1225**

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

**Electronic ISBN: 978-1-4704-4913-1
Product Code: MEMO/256/1225.E**

List Price: $78.00

AMS Member Price: $46.80

MAA Member Price: $70.20

# Curvature: A Variational Approach

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*A. Agrachev; D. Barilari; L. Rizzi*

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.

#### Table of Contents

# Table of Contents

## Curvature: A Variational Approach

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Part 1 . Statements of the results 916
- Chapter 2. General setting 1118
- Chapter 3. Flag and growth vector of an admissible curve 1926
- Chapter 4. Geodesic cost and its asymptotics 2936
- Chapter 5. Sub-Riemannian geometry 4148
- 5.1. Basic definitions 4148
- 5.2. Existence of ample geodesics 4653
- 5.3. Reparametrization and homogeneity of the curvature operator 4956
- 5.4. Asymptotics of the sub-Laplacian of the geodesic cost 5057
- 5.5. Equiregular distributions 5360
- 5.6. Geodesic dimension and sub-Riemannian homotheties 5663
- 5.7. Heisenberg group 5865
- 5.8. On the “meaning” of constant curvature 6370

- Part 2 . Technical tools and proofs 6774
- Part 3 . Appendix 113120
- Appendix A. Smoothness of value function (Theorem 2.19) 115122
- Appendix B. Convergence of approximating Hamiltonian systems (Proposition 5.15) 119126
- Appendix C. Invariance of geodesic growth vector by dilations (Lemma 5.20) 121128
- Appendix D. Regularity of 𝐶(𝑡,𝑠) for the Heisenberg group (Proposition 5.51) 123130
- Appendix E. Basics on curves in Grassmannians (Lemma 3.5 and 6.5) 125132
- Appendix F. Normal conditions for the canonical frame 127134
- Appendix G. Coordinate representation of flat, rank 1 Jacobi curves (Proposition 7.7) 129136
- Appendix H. A binomial identity (Lemma 7.8) 131138
- Appendix I. A geometrical interpretation of 𝑐_{𝑡} 135142
- Bibliography 137144
- Index 141148

- Back Cover Back Cover1154