Finsler metrics - a global approach. With applications to geometric function theory.

*(English)*Zbl 0837.53001
Lecture Notes in Mathematics. 1591. Berlin: Springer-Verlag. ix, 177 p. (1994).

The authors present a global theory of both real and complex Finsler metrics. The main idea is to replace the Finsler metric defined on the tangent bundle by a Riemannian metric on a suitable vector subbundle of the tangent bundle of second order, and then to use the standard tools of Riemannian geometry in the study. In the real case, the above idea was used by the reviewer for developing a theory of submanifolds of a Finsler manifold [cf. the reviewer, Finsler geometry and applications, Ellis Horwood, New York (1990; Zbl 0702.53001)].

The material in this book is carefully and clearly presented in three chapters. The first chapter deals with real Finsler geometry. The authors introduce the necessary definitions and objects leading to a coordinate- free definition of the classical Cartan connection. Then the first and the second variation of the length integral, the exponential map, Jacobi fields, conjugate points and the Morse index form are discussed. Finsler versions of the Hopf-Rinow, Cartan-Hadamard and Bonnet theorems are proved.

The second chapter is devoted to the geometry of complex Finsler metrics. It is for the first time in literature that this theory is presented in a book. Following the ideas from the first chapter, but in the complex setting, the authors succeed in defining the Chern-Finsler connection for complex Finsler metrics, which is the main tool in this study. The Kähler-Finsler metrics and the holomorphic curvature of complex Finsler manifolds are studied. Finally, the Cartan connection and the Chern- Finsler connection are compared.

In the third chapter there are brought together, in a unified form, the results and applications that motivated the work of the authors. Here we find a study of the function theory on Kähler-Finsler manifolds with constant nonpositive holomorphic curvature (from a differential geometric point of view) and a study of manifolds on which there exists a Monge- Ampere foliation (from a complex analysis point of view). Finally, a characterization of strongly convex circular domains is given in terms of differential geometric properties of the Kobayashi metric.

I recommend the book to everyone who appreciates the beauty of both real and complex Finsler geometry or who uses differential geometry for global approaches of certain problems of real and complex analysis.

The material in this book is carefully and clearly presented in three chapters. The first chapter deals with real Finsler geometry. The authors introduce the necessary definitions and objects leading to a coordinate- free definition of the classical Cartan connection. Then the first and the second variation of the length integral, the exponential map, Jacobi fields, conjugate points and the Morse index form are discussed. Finsler versions of the Hopf-Rinow, Cartan-Hadamard and Bonnet theorems are proved.

The second chapter is devoted to the geometry of complex Finsler metrics. It is for the first time in literature that this theory is presented in a book. Following the ideas from the first chapter, but in the complex setting, the authors succeed in defining the Chern-Finsler connection for complex Finsler metrics, which is the main tool in this study. The Kähler-Finsler metrics and the holomorphic curvature of complex Finsler manifolds are studied. Finally, the Cartan connection and the Chern- Finsler connection are compared.

In the third chapter there are brought together, in a unified form, the results and applications that motivated the work of the authors. Here we find a study of the function theory on Kähler-Finsler manifolds with constant nonpositive holomorphic curvature (from a differential geometric point of view) and a study of manifolds on which there exists a Monge- Ampere foliation (from a complex analysis point of view). Finally, a characterization of strongly convex circular domains is given in terms of differential geometric properties of the Kobayashi metric.

I recommend the book to everyone who appreciates the beauty of both real and complex Finsler geometry or who uses differential geometry for global approaches of certain problems of real and complex analysis.

Reviewer: A.Bejancu (Iaşi)

##### MSC:

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

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

32E05 | Holomorphically convex complex spaces, reduction theory |

32W20 | Complex Monge-Ampère operators |