Lectures on Finsler geometry. (English) Zbl 0974.53002

Shen, Zhongmin

Singapore: World Scientific. xiv, 307 p. (2001).

This book is intended to provide basic material for Finsler geometry including Riemannian geometry. In particular, the first four chapters cover basics on regular metrics and regular measures on manifolds. In the next chapters, the author introduces the notions of geodesics, of Chern connection, various non-Riemann quantities with their geometric meanings, and he studies metric properties of Finsler spaces, geometric structure of Finsler spaces and Finsler \(m\) spaces. Moreover, he applies Morse theory to the canonical energy functional on the loop space. Finally, in the last chapter, the author considers a Finsler space as a point of a space of Finsler spaces equipped with the Gromov-Hausdorff distance.

Reviewer: P.Tamia-Dimopoulou (Thessaloniki)

MSC:

53-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C60	Global differential geometry of Finsler spaces and generalizations (areal metrics)
58B20	Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
53C20	Global Riemannian geometry, including pinching

Keywords:

Finsler geometry; Riemannian geometry; regular metrics; regular measures; Chern connection; Morse theory; energy functional; loop space; space of Finsler spaces; Gromov-Hausdorff distance