An Introduction to Riemann-Finsler Geometry

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Beschreibung

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.

Zusammenfassung

PRELIMINARY TEXT. DO NOT USE. Finsler geometry is a metric generalization of Riemannian geometry and has become a comparatively young branch of differential geometry. Although Finsler geometry has its genesis in Riemann's 1854 "Habilitationsvortrag," its systematic study was not initiated until 1918 by Finsler, and the fundamentals were not completely formulated until the mid-thirties. Later, however, the field underwent a rapid development by mathematicians and physicists of many countries. The main purpose of this book is to study the metric geometry of Finsler manifolds. Portions of the book generalize some standard concepts from Riemannian geometry to the Finsler setting, while other

Inhaltsverzeichnis

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.

Erscheinungsjahr:	2000
Fachbereich:	Geometrie
Genre:	Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Buch
Seiten:	460
Reihe:	Graduate Texts in Mathematics
Inhalt:	xx 435 S. 20 s/w Illustr. 20 s/w Zeichng.
ISBN-13:	9780387989488
ISBN-10:	038798948X
Sprache:	Englisch
Ausstattung / Beilage:	HC runder Rücken kaschiert
Einband:	Gebunden
Autor:	Bao, D. Shen, Z. Chern, S. -S.
Auflage:	2000
Hersteller:	Springer US Springer New York Graduate Texts in Mathematics
Maße:	241 x 160 x 31 mm
Von/Mit:	D. Bao (u. a.)
Erscheinungsdatum:	17.03.2000
Gewicht:	0,852 kg

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