##
**A comprehensive introduction to sub-Riemannian geometry. From the Hamiltonian viewpoint. With an appendix by Igor Zelenko.**
*(English)*
Zbl 1487.53001

Cambridge Studies in Advanced Mathematics 181. Cambridge: Cambridge University Press (ISBN 978-1-108-47635-5/hbk; 978-1-108-67732-5/ebook). xviii, 745 p. (2020).

This is a very well-written text that presents a thorough introduction to sub-Riemannian geometry. The results are well organized and carefully presented throughout the whole text. Actually, this textbook offers more than resources for specialists, it is also intended to students. It does a great presentation on more classical topics from differential and Riemannian geometry. The authors even give a possible selection of chapters that could serve as the basis of three possible graduate-level courses: one on Riemannian geometry; one at the introductory level on sub-Riemannian geometry; and one at an advanced level on sub-Riemannian geometry.

Sub-Riemannian geometry can be seen as a generalization of the Riemannian one, the main difference being that one “cannot move, receive or send information in all directions”. Thus, in a nutshell, a sub-Riemannian structure on a smooth manifold has a fixed admissible subspace in any tangent space, which has some Euclidean setup. The admissible paths are those whose velocities are admissible, and these generate a suitable notion of distance between points.

These spaces turned out to be extremely rich in applications, and one obvious field for such is the optimal control theory. Therefore, the authors have a “control theoretic” mindset when presenting the results. In particular, they use the language of Hamiltonian dynamics throughout.

After an introduction, the book is divided into 21 chapters and an appendix. These chapters could be roughly categorized into three main parts. Chapters 1–5 represent Part one, these are presenting introductory concepts from the theory of surfaces, sub-Riemannian structures (such as the distance, characterization of geodesics, cf. Rashevskii-Chow theorem, Filippov’s theorem and the Pontryagin maximum principle), symplectic geometry, Hamiltonian systems (as action-angle coordinates, geodesic flows, etc.).

Part 2 consists of Chapters 6–13. Some more classical results from functional analysis, operator calculus and Lie groups are presented in Chapters 6–7. The following chapter deals with notions such as the endpoint and exponential maps, cut and conjugate points. Chapter 9 presents in particular an interesting study on the Grushin plane. Chapter 10 discusses so-called nonholonomic tangent spaces, while Chapter 11 studies general analytic properties of the sub-Riemannian distance function. Chapter 12 turns to abnormal geodesics and Chapter 13 is devoted to some explicite calculations of the sub-Riemannian optimal synthesis for model spaces.

The remaining part from Chapter 14 to the end of the book can be seen as Part 3. These chapters are particularly devoted to the study of curvature notions and their applications (these include in particular the study of Jacobi curves; Levi-Civita connection; the Riemannian curvature and its symplectic meaning; Poisson manifolds; integrability of sub-Riemannian geodesic flows on 3D Lie groups, etc.). The last two chapters address the question of defining a canonical volume in sub-Riemannian geometry and the construction of the sub-Riemannian Laplace operator and the associated heat equation.

The theoretical results are accompanied by a good number of examples, by lots of suitable exercises and by bibliographical notes at the end of each chapter.

Overall, I find this text as an excellent resource for a broad range of topics in Riemannian and sub-Riemannian geometry. I am strongly convinced that this will become one of the main references for people interested in these topics, ranging from students to specialists.

Sub-Riemannian geometry can be seen as a generalization of the Riemannian one, the main difference being that one “cannot move, receive or send information in all directions”. Thus, in a nutshell, a sub-Riemannian structure on a smooth manifold has a fixed admissible subspace in any tangent space, which has some Euclidean setup. The admissible paths are those whose velocities are admissible, and these generate a suitable notion of distance between points.

These spaces turned out to be extremely rich in applications, and one obvious field for such is the optimal control theory. Therefore, the authors have a “control theoretic” mindset when presenting the results. In particular, they use the language of Hamiltonian dynamics throughout.

After an introduction, the book is divided into 21 chapters and an appendix. These chapters could be roughly categorized into three main parts. Chapters 1–5 represent Part one, these are presenting introductory concepts from the theory of surfaces, sub-Riemannian structures (such as the distance, characterization of geodesics, cf. Rashevskii-Chow theorem, Filippov’s theorem and the Pontryagin maximum principle), symplectic geometry, Hamiltonian systems (as action-angle coordinates, geodesic flows, etc.).

Part 2 consists of Chapters 6–13. Some more classical results from functional analysis, operator calculus and Lie groups are presented in Chapters 6–7. The following chapter deals with notions such as the endpoint and exponential maps, cut and conjugate points. Chapter 9 presents in particular an interesting study on the Grushin plane. Chapter 10 discusses so-called nonholonomic tangent spaces, while Chapter 11 studies general analytic properties of the sub-Riemannian distance function. Chapter 12 turns to abnormal geodesics and Chapter 13 is devoted to some explicite calculations of the sub-Riemannian optimal synthesis for model spaces.

The remaining part from Chapter 14 to the end of the book can be seen as Part 3. These chapters are particularly devoted to the study of curvature notions and their applications (these include in particular the study of Jacobi curves; Levi-Civita connection; the Riemannian curvature and its symplectic meaning; Poisson manifolds; integrability of sub-Riemannian geodesic flows on 3D Lie groups, etc.). The last two chapters address the question of defining a canonical volume in sub-Riemannian geometry and the construction of the sub-Riemannian Laplace operator and the associated heat equation.

The theoretical results are accompanied by a good number of examples, by lots of suitable exercises and by bibliographical notes at the end of each chapter.

Overall, I find this text as an excellent resource for a broad range of topics in Riemannian and sub-Riemannian geometry. I am strongly convinced that this will become one of the main references for people interested in these topics, ranging from students to specialists.

Reviewer: Alpár R. Mészáros (Durham)

### MathOverflow Questions:

How to find equations of a sub-Riemannian problemProof of Rashevskii-Chow theorem

### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53C17 | Sub-Riemannian geometry |

53Dxx | Symplectic geometry, contact geometry |

53C20 | Global Riemannian geometry, including pinching |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |