Lectures on Finsler geometry

Zhongmin Shen

In 1854, B. Riemann introduced the notion of curvature for spaces with a family of inner products. There was no significant progress in the general case until 1918, when P. Finsler studied the variation problem in regular metric spaces. Around 1926, L. Berwald extended Riemann's notion of curvature to regular metric spaces and introduced an important non-Riemannian curvature using his connection for regular metrics. Since then, Finsler geometry has developed steadily. In his Paris address in 1900, D. Hilbert formulated 23 problems, the 4th and 23rd problems being in Finsler's category. Finsler geometry has broader applications in many areas of science and will continue to develop through the efforts of many geometers around the world. Usually, the methods employed in Finsler geometry involve very complicated tensor computations. Sometimes this discourages beginners. Viewing Finsler spaces as regular metric spaces, the author discusses the problems from the modern geometry point of view. The book begins with the basics on Finsler spaces, including the notions of geodesics and curvatures, then deals with basic comparison theorems on metrics and measures and their applications to the Levy concentration theory of regular metric measure spaces and Gromov's Hausdorff convergence theory.

「Nielsen BookData」より

In 1854, B. Riemann introduced the notion of curvature for spaces with a family of inner products. There was no significant progress in the general case until 1918, when P. Finsler studied the variation problem in regular metric spaces. Around 1926, L. Berwald extended Riemann's notion of curvature to regular metric spaces and introduced an important non-Riemannian curvature using his connection for regular metrics. Since then, Finsler geometry has developed steadily. In his Paris address in 1900, D. Hilbert formulated 23 problems, the 4th and 23rd problems being in Finsler's category. Finsler geometry has broader applications in many areas of science and will continue to develop through the efforts of many geometers around the world. Usually, the methods employed in Finsler geometry involve very complicated tensor computations. Sometimes this discourages beginners. Viewing Finsler spaces as regular metric spaces, the author discusses the problems from the modern geometry point of view. The book begins with the basics on Finsler spaces, including the notions of geodesics and curvatures, then deals with basic comparison theorems on metrics and measures and their applications to the Levy concentration theory of regular metric measure spaces and Gromov's Hausdorff convergence theory.

「Nielsen BookData」より

[目次]

  • Finsler Spaces
  • Finsler m Spaces
  • Co-Area Formula
  • Isoperimetric Inequalities
  • Geodesics and Connection
  • Riemann Curvature
  • Non-Riemannian Curvatures
  • Structure Equations
  • Finsler Spaces of Constant Curvature
  • Second Variation Formula
  • Geodesics and Exponential Map
  • Conjugate Radius and Injectivity Radius
  • Basic Comparison Theorems
  • Geometry of Hypersurfaces
  • Geometry of Metric Spheres
  • Volume Comparison Theorems
  • Morse Theory of Loop Spaces
  • Vanishing Theorems for Homotopy Groups
  • Spaces of Finsler Spaces.

「Nielsen BookData」より

[目次]

  • Finsler Spaces
  • Finsler m Spaces
  • Co-Area Formula
  • Isoperimetric Inequalities
  • Geodesics and Connection
  • Riemann Curvature
  • Non-Riemannian Curvatures
  • Structure Equations
  • Finsler Spaces of Constant Curvature
  • Second Variation Formula
  • Geodesics and Exponential Map
  • Conjugate Radius and Injectivity Radius
  • Basic Comparison Theorems
  • Geometry of Hypersurfaces
  • Geometry of Metric Spheres
  • Volume Comparison Theorems
  • Morse Theory of Loop Spaces
  • Vanishing Theorems for Homotopy Groups
  • Spaces of Finsler Spaces.

「Nielsen BookData」より

この本の情報

書名 Lectures on Finsler geometry
著作者等 Shen, Zhongmin
出版元 World Scientific
刊行年月 c2001
ページ数 xiv, 307 p.
大きさ 23 cm
ISBN 9810245319
9810245300
NCID BA52622426
※クリックでCiNii Booksを表示
言語 英語
出版国 シンガポール
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