Introduction to modern Finsler geometry (Q2790419)

``Introduction to Modern Finsler Geometry'', by Yi-Bing Shen and Zhongmin Shen, is the result of more than 20 years of studies devoted to Finsler geometry, both authors having a long experience in this field. In the last years substantial progress has been made in Finsler geometry. Two of the main monographs related to this subject are [\textit{D. Bao} et al., An introduction to Riemann-Finsler geometry. New York, NY: Springer (2000; Zbl 0954.53001)] and [\textit{Z. Shen}, Lectures on Finsler geometry. Singapore: World Scientific (2001; Zbl 0974.53002)].NEWLINENEWLINEThe present book is a continuation of the cited monographs, an update of the ``classical theory'' of Finsler geometry with all new global results obtained by the two authors and other important researchers. The first part contains five chapters and it is devoted to the foundations of Riemann-Finsler Geometry. It starts with some preliminaries about differentiable manifolds and all geometric objects used in the book: Finsler metrics, the Chern connection, different types of curvatures, Legendre transformation, volume measures, Schur theorem, global rigidity results, navigation problem. The second part, composed by six chapters, contains further studies: projective transformations, comparison theorems, fundamental group, minimal immersions, harmonic maps, Einstein metric, conformal transformations and conformal vector fields, the Finsler Laplacian and its first eigenvalue, etc. Each chapter ends with some exercises and a final appendix provides Maple programs on the computations of geometric Finsler quantities. The result is a comprehensive book, based on many years of teaching experience. It may be used by students who want to start their studies in Finsler Geometry, but also by experienced researchers.