David Bao, Shiing-Shen Chern, Zhongmin Shen

An Introduction to Riemann-Finsler Geometry

• Gebundenes Buch

Produktbeschreibung

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 ardsticks 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.

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Produktdetails

• Graduate Texts in Mathematics 200
• Verlag: Springer / Springer New York / Springer, Berlin
• Artikelnr. des Verlages: 978-0-387-98948-8
• 2000.
• Seitenzahl: 460
• Erscheinungstermin: 17. März 2000
• Englisch
• Abmessung: 241mm x 160mm x 31mm
• Gewicht: 850g
• ISBN-13: 9780387989488
• ISBN-10: 038798948X
• Artikelnr.: 08773368

Inhaltsangabe

One Finsler Manifolds and Their Curvature.- 1 Finsler Manifolds and the Fundamentals of Minkowski Norms.- 2 The Chern Connection.- 3 Curvature and Schur's Lemma.- 4 Finsler Surfaces and a Generalized Gauss-Bonnet Theorem.- Two Calculus of Variations and Comparison Theorems.- 5 Variations of Arc Length, Jacobi Fields, the Effect of Curvature.- 6 The Gauss Lemma and the Hopf-Rinow Theorem.- 7 The Index Form and the Bonnet-Myers Theorem.- 8 The Cut and Conjugate Loci, and Synge's Theorem.- 9 The Cartan-Hadamard Theorem and Rauch's First Theorem.- Three Special Finsler Spaces over the Reals.- 10 Berwald Spaces and Szabó's Theorem for Berwald Surfaces.- 11 Randers Spaces and an Elegant Theorem.- 12 Constant Flag Curvature Spaces and Akbar-Zadeh's Theorem.- 13 Riemannian Manifolds and Two of Hopf's Theorems.- 14 Minkowski Spaces, the Theorems of Deicke and Brickell.

Rezensionen

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
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.

"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