Synopses & Reviews
"In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So yardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe? It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one. David Bao is Professor of Mathematics and of the Honors College, at the University of Houston. He obtained his Ph.D. from the University of California at Berkeley in 1983, with Jerry Marsden as his advisor. Before coming to Houston, he did two years of post-doctoral studies at the Institute for Advanced Study in Princeton, New Jersey. Besides differential geometry, he is passionately curious about the ways cats and goldfish think. Shiing-Shen Chern is Professor Emeritus of Mathematics at the University of California at Berkeley, and Director Emeritus of the Mathematical Sciences Research Institute. He is also Distinguished Visiting Professor Emeritus at the University of Houston. Chern received his D.Sc. in 1936, as a student of W. Blaschke. He carried out his post-doctoral studies under E. Cartan. Chern has garnered a good number of distinctions to date. These include the Chauvenet Prize (1970), National Medal of Science (1975), the Humboldt Award (1982), the Steele Prize (1983), and the Wolf Foundation Prize (1983-84). Zhongmin Shen is Associate Professor of Mathematics at Indiana University Purdue University Indianapolis (IUPUI). He earned his Ph.D. from the State University of New York at Stony Brook in 1990 under Detlef Gromoll. He spent 1990-91 at the Mathematical Sciences Research Institute at Berkeley, and 1991-93 as a Hildebrandt Assistant Professor at the University of Michigan at Ann Arbor."
Review
"Das Buch ist sehr gut strukturiert und stellt die doch umfangreiche Materie klar dar. Es wendet sich an Studenten höherer Semester und erlaubt einen guten Einstieg in das weite Gebiet der Finsler-Geometrie. Für alle Interessierten bietet das Werk einen klaren Zugang, der auch die geschichtliche Entwicklung von der Euklidischen über die Riemannsche zur Finslerschen Geometrie deutlich macht." Internationale Mathematische Nachrichten, Nr. 187, August 2001
Review
"This book offers the most modern treatment of the topic and will attract both graduate students and a broad community of mathematicians from various related fields." EMS Newsletter, Issue 41, September 2001
Synopsis
This book focuses on the elementary but essential problems in Riemann-Finsler Geometry, which include a repertoire of rigidity and comparison theorems, and an array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. "This book offers the most modern treatment of the topic ..." EMS Newsletter.
Description
Includes bibliographical references (p. [419]-425) and index.
Table of Contents
From the contents: Finsler methods and the fundamentals of Minkowski norms.- The Chern connection.- Curvature and Schur's Lemma.- Finsler surfaces and a generalized Gauss-Bonnet theorem.- Variations of arc length, Jacobi fields, and the effect of curvature.- The Gauss lemma and the Hopf-Rinow theorem.- The index form and the Bonnet-Myers theorem.- The cut and conjugate loci, and Synge's theorem.- The Cartan-Hadamard theorem and Rauch's first theorem.- Berwald spaces and Szabo's theorem for Berwald spaces.- Randers spaces and a theorem from the Japanese school.- Constant flag curvature spaces, and ther Andar-Zadeh theorem.- Riemannian manifolds and two theorems of Hopf's.- Minkowski spaces, the theorems of Dickie and Brickell.