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Foundations of Finsler geometry and special Finsler spaces. (English) Zbl 0594.53001
Shiga-Ken 520, Japan: Kaiseisha Press. VI, 344 p. (1986).
By this book the author succeeded in giving to the mathematical community a comprehensive study of Finsler geometry based on the theory of fiber bundles. In order to make the book self-contained the author arranged in the first chapter most of the required background material. Chapter II is devoted to the study of Finsler connections by means of Finsler bundles. It is noteworthy that such a study is performed independent of Finsler metrics, so it has a large area of applications. The case of Finsler connections related to Finsler metrics is investigated in Chapter III. More precisely, the special connections of Cartan, Berwald, Rund and Hashiguchi are defined axiomatically and studied. In chapter IV, by means of the nonlinear connection (a component of a Finsler connection) a special linear connection on the tangent bundle is constructed, whose torsion and curvature tensors give all torsions and curvatures of a Finsler connection. Special Finsler spaces as for instance Berwald spaces, Landsberg spaces, Einstein-Finsler spaces, Finsler spaces of scalar curvature are studied in chapter V. Also in this chapter the most important properties of two and three dimensional Finsler spaces are given. Chapters VI and VII deal with various transformations of Finsler spaces and parallel displacements and geodesics respectively.
In fact the topics from the last three chapters are still under research. The book will be of primary interest to research workers in Finsler geometry and students of mathematics. It will also be of value for physicists.
Reviewer: A.Bejancu

MSC:
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)