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Handbook of Finsler geometry. Vols. 1 and 2. (English) Zbl 1057.53001
Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1557-7/set; 1-4020-1555-0/v.1; 1-4020-1556-9/v.2). 1472 p. (2003).

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Finsler geometry is the most general among those geometries which satisfy certain highly natural conditions: it is built on the arc length independent of orientation preserving parameter transformations, and its normed tangent vector spaces satisfy the triangle inequality. Finsler geometry is a very natural generalization of Riemann geometry. According to S. S. Chern it is Riemann geometry without the quadratic restriction. Its first period from 1918 included the time of foundation and was characterized by applying tensor calculus in local components. Results of this period are gathered in the book of H. Rund: The Differential Geometry of Finsler Spaces, which was a basic monograph in its time (1959; Zbl 0087.36604). About this time started a new period characterized by the use of component free expressions and global notions. In the last fifty year many papers and more books appeared on Finsler geometry. So it became difficult to obtain a good overlook on the subject. The present Handbook aims to fill this gap. It is no encyclopedia. Rather it reflects the personal field of interest, approach and view of the individual authors. Problems and themes are not strictly separated. Thus several notions and problems are enlightened from different points of view. This way of reduction needs a little more place, but has many advantages. The range of the themes embraces classical and modern fields in local and global approach, even problems of probability, statistics and finance. Also computer algebra adapted to Finsler geometry is discussed.
Parts of the book appear in alphabetic order of the authors. Part 1 is “Complex Finsler geometry” (3–79) written by Tadashi Aikou. Part 2 is “KCC theory of a system of second order differential equations” (83–174) by P. L. Antonelli and I. Bucataru. Part 3 is “Fundamentals of Finslerian diffusion with applications” (177–355) by P. L. Antonelli and T. J. Zastawniak. Part 4 written by D. Hrimiuc and H. Shimada treats “Symplectic transformations of the geometry of \(T^*M\); \(\mathcal L\)-duality” (359–442). Part 5 by L. Kozma is devoted to “Holonomy structures in Finsler geometry” (445–488). Part 6 written by Brad Lackey has the title “On the Gauss-Bonnet-Chern theorem in Finsler geometry” (491–509), while part 7 by the same author is titled “The Hodge theory of Finsler-type geometries” (513–554). Part 8: “Finsler geometry in the 20th century” (557–966) is the work of M. Matsumoto. Part 9 is “The geometry of Lagrange spaces” (969–1122) by Radu Miron, Mihai Anastasiei and Ioan Bucataru. Part 10: “Symbolic Finsler geometry” (1125–1179) is the work of S. F. Rutz and R. Portugal. Part 11, authored by József Szilasi is “A setting for spray and Finsler geometry” (1183–1426).
Each Part contains a Bibliography, and the book is provided with a detailed common Index of 11 pages. The authors wrote well understandable parts. So the book is easily readable also with little knowledge on Finsler geometry.
The articles of this volume will be reviewed individually.
Indexed articles:
Aikou, Tadashi, Complex Finsler geometry, 3-79 [Zbl 1105.53016]
Antonelli, P. L.; Bucataru, I., KCC theory of a system of second order differential equations, 83-174 [Zbl 1105.53017]
Antonelli, P. L.; Zastawniak, T. J., Fundamentals of Finslerian diffusion with applications, 177-355 [Zbl 1105.53018]
Hrimiuc, D.; Shimada, H., Symplectic transformation of the geometry of \(T^*M\); \({\mathcal L}\)-duality, 359-442 [Zbl 1105.53023]
Kozma, L., Holonomy structures in Finsler geometry, 445-488 [Zbl 1105.53039]
Lackey, Brad, On the Gauss-Bonnet-Chern theorem in Finsler geometry, 491-509 [Zbl 1105.53055]
Lackey, Brad, The Hodge theory of Finsler-type geometries, 513-554 [Zbl 1105.53056]
Matsumoto, M., Finsler geometry in the 20th-century, 557-966 [Zbl 1105.53019]
Miron, Radu; Anastasiei, Mihai; Bucataru, Ioan, The geometry of Lagrange spaces, 969-1122 [Zbl 1105.53042]
Rutz, S. F.; Portugal, R., Symbolic Finsler geometry, 1125-1179 [Zbl 1105.53020]
Szilasi, József, A setting for spray and Finsler geometry, 1183-1426 [Zbl 1105.53043]

53-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to differential geometry
53-06 Proceedings, conferences, collections, etc. pertaining to differential geometry
00B15 Collections of articles of miscellaneous specific interest
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)