##
**An introduction to Riemann-Finsler geometry.**
*(English)*
Zbl 0954.53001

Graduate Texts in Mathematics. 200. New York, NY: Springer. xx, 431 p. (2000).

The last few decades have seen a growing interest in studying the geometry of Finsler manifolds. The book under review represents an important step in this direction. The book has three parts: Finsler manifolds and their curvature, Calculus of variations and comparison theorems, and Special Finsler spaces over the reals. The first part has four chapters and each of the other two has five chapters. Each chapter has its own references, but also a master bibliography is given at the end of the book. There are 393 helpful exercises carefully chosen by the authors.

In the first chapter, the book introduces Finsler manifolds and discusses the main properties of Minkowski norms. Explicit examples of Finsler manifolds are also given here. The second chapter deals with the Chern connection whose role in Finsler geometry is similar to that of the Levi-Civita connection in Riemannian geometry. An important tool in constructing the Chern connection (and any other classical Finsler connection) is the nonlinear connection induced by the Finsler structure. The Bianchi identities and Ricci identities with respect to the Chern connection are presented in Chapter 3. Also, flag curvature is introduced in this chapter, and Schur’s theorem is proved. In Chapter 4, it is proved that the geometry of a Finsler surface \(M\) is completely controlled by two pseudo-scalars and a scalar, all living on the 3-dimensional sphere bundle \(SM\). A Gauss-Bonnet Theorem is given in the case when \(M\) is a Landsberg surface.

Chapter 5 is the first chapter that deals with Calculus of variations. Here, using differential forms, the first and second variations of arc length are obtained. Then, geodesics, exponential mappings and Jacobi fields are investigated. More about geodesics can be found in Chapter 6. First, the Gauss lemma on radial geodesics is proved. Then the metric distance on a Finsler manifold is introduced and it is shown that short geodesics are minimizing. As the metric distance function is, in general, not symmetric in its two arguments the authors replace usual metric balls, metric spheres, geodesically complete Finsler manifolds, etc. by the so called forward metric balls, forward metric spheres, forward geodesically complete Finsler manifolds, etc., respectively. Finally, a Hopf-Rinow Theorem on forward geodesically complete Finsler manifolds is stated. Conjugate points and the index form on a Finsler manifold are studied in Chapter 7. Also in this chapter, the interplay between the Ricci scalar and the Ricci tensor on a Finsler manifold is investigated. This has paved the way for a Bonnet-Myers Theorem in Finsler geometry. Chapter 8 is devoted to the study of cut and conjugate loci. Also, shortest geodesics within homotopy classes are presented, and a Synge’s Theorem for Finsler manifolds is stated. Further study of geodesics, Jacobi fields and flag curvature takes place in Chapter 9. This leads directly to Cartan-Hadamard and Rauch’s Theorems.

The study of special Finsler spaces starts with Chapter 10, where the authors investigate Berwald spaces. Here, various characterizations of Berwald spaces, examples of Berwald spaces and Szabo’s Theorem for Berwald surfaces are given. Next, in Chapter 11, Randers spaces are studied. By using an elegant characterization of Randers spaces of Berwald type the authors construct \(y\)-global Berwald spaces and an explicit example of a 3-dimensional Berwald space that is neither Riemannian nor locally Minkowskian. Chapter 12 is entirely devoted to the study of a Finsler manifold \(M\) with constant flag curvature \(k\). Akbar-Zadeh’s Theorems for \(M\) with vanishing or negative \(k\) are given. In the particular case of a Finsler surface, a detailed computational formula for the Gaussian curvature is presented. In addition, we find here a number of interesting examples of Finsler surfaces with non-zero constant curvature. Riemannian manifolds are studied in Chapter 13, as particular cases of Finsler manifolds. The reader will find here in about 30 pages the main concepts from Riemannian geometry along with some important results of Hopf’s Theorems. The main results on the geometry of Minkowski spaces, including the theorems of Deicke and Brickell are presented in the last chapter of the book.

The book is ideal for a first graduate course in Finsler geometry. Students will learn both the rigorous foundations for the subject as well as meet a large number of interesting examples. The beginner will find the book helpful, while those who already know something about Finsler geometry will appreciate Bao-Chern-Shen’s approach.

In the first chapter, the book introduces Finsler manifolds and discusses the main properties of Minkowski norms. Explicit examples of Finsler manifolds are also given here. The second chapter deals with the Chern connection whose role in Finsler geometry is similar to that of the Levi-Civita connection in Riemannian geometry. An important tool in constructing the Chern connection (and any other classical Finsler connection) is the nonlinear connection induced by the Finsler structure. The Bianchi identities and Ricci identities with respect to the Chern connection are presented in Chapter 3. Also, flag curvature is introduced in this chapter, and Schur’s theorem is proved. In Chapter 4, it is proved that the geometry of a Finsler surface \(M\) is completely controlled by two pseudo-scalars and a scalar, all living on the 3-dimensional sphere bundle \(SM\). A Gauss-Bonnet Theorem is given in the case when \(M\) is a Landsberg surface.

Chapter 5 is the first chapter that deals with Calculus of variations. Here, using differential forms, the first and second variations of arc length are obtained. Then, geodesics, exponential mappings and Jacobi fields are investigated. More about geodesics can be found in Chapter 6. First, the Gauss lemma on radial geodesics is proved. Then the metric distance on a Finsler manifold is introduced and it is shown that short geodesics are minimizing. As the metric distance function is, in general, not symmetric in its two arguments the authors replace usual metric balls, metric spheres, geodesically complete Finsler manifolds, etc. by the so called forward metric balls, forward metric spheres, forward geodesically complete Finsler manifolds, etc., respectively. Finally, a Hopf-Rinow Theorem on forward geodesically complete Finsler manifolds is stated. Conjugate points and the index form on a Finsler manifold are studied in Chapter 7. Also in this chapter, the interplay between the Ricci scalar and the Ricci tensor on a Finsler manifold is investigated. This has paved the way for a Bonnet-Myers Theorem in Finsler geometry. Chapter 8 is devoted to the study of cut and conjugate loci. Also, shortest geodesics within homotopy classes are presented, and a Synge’s Theorem for Finsler manifolds is stated. Further study of geodesics, Jacobi fields and flag curvature takes place in Chapter 9. This leads directly to Cartan-Hadamard and Rauch’s Theorems.

The study of special Finsler spaces starts with Chapter 10, where the authors investigate Berwald spaces. Here, various characterizations of Berwald spaces, examples of Berwald spaces and Szabo’s Theorem for Berwald surfaces are given. Next, in Chapter 11, Randers spaces are studied. By using an elegant characterization of Randers spaces of Berwald type the authors construct \(y\)-global Berwald spaces and an explicit example of a 3-dimensional Berwald space that is neither Riemannian nor locally Minkowskian. Chapter 12 is entirely devoted to the study of a Finsler manifold \(M\) with constant flag curvature \(k\). Akbar-Zadeh’s Theorems for \(M\) with vanishing or negative \(k\) are given. In the particular case of a Finsler surface, a detailed computational formula for the Gaussian curvature is presented. In addition, we find here a number of interesting examples of Finsler surfaces with non-zero constant curvature. Riemannian manifolds are studied in Chapter 13, as particular cases of Finsler manifolds. The reader will find here in about 30 pages the main concepts from Riemannian geometry along with some important results of Hopf’s Theorems. The main results on the geometry of Minkowski spaces, including the theorems of Deicke and Brickell are presented in the last chapter of the book.

The book is ideal for a first graduate course in Finsler geometry. Students will learn both the rigorous foundations for the subject as well as meet a large number of interesting examples. The beginner will find the book helpful, while those who already know something about Finsler geometry will appreciate Bao-Chern-Shen’s approach.

Reviewer: Aurel Bejancu (Safat)

### MSC:

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

58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |

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

53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |

53C20 | Global Riemannian geometry, including pinching |

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |