Springer Monographs in Mathematics

Comparison Finsler Geometry

Authors: Ohta, Shin-ichi

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  • Generalizes the weighted Ricci curvature and develops comparison geometry and geometric analysis in the Finsler context
  • Offers an accessible entry point to studying Finsler geometry for those familiar with differentiable manifolds
  • Illustrates and compares three methods for studying lower Ricci curvature bounds: Gamma-calculus, curvature-dimension condition, needle decomposition
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  • ISBN 978-3-030-80650-7
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Hardcover $159.99
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About this book

This monograph presents recent developments in comparison geometry and geometric analysis on Finsler manifolds. Generalizing the weighted Ricci curvature into the Finsler setting, the author systematically derives the fundamental geometric and analytic inequalities in the Finsler context. Relying only upon knowledge of differentiable manifolds, this treatment offers an accessible entry point to Finsler geometry for readers new to the area.

Divided into three parts, the book begins by establishing the fundamentals of Finsler geometry, including Jacobi fields and curvature tensors, variation formulas for arc length, and some classical comparison theorems. Part II goes on to introduce the weighted Ricci curvature, nonlinear Laplacian, and nonlinear heat flow on Finsler manifolds. These tools allow the derivation of the Bochner–Weitzenböck formula and the corresponding Bochner inequality, gradient estimates, Bakry–Ledoux’s Gaussian isoperimetric inequality, and functional inequalities in the Finsler setting. Part III comprises advanced topics: a generalization of the classical Cheeger–Gromoll splitting theorem, the curvature-dimension condition, and the needle decomposition. Throughout, geometric descriptions illuminate the intuition behind the results, while exercises provide opportunities for active engagement.

Comparison Finsler Geometry offers an ideal gateway to the study of Finsler manifolds for graduate students and researchers. Knowledge of differentiable manifold theory is assumed, along with the fundamentals of functional analysis. Familiarity with Riemannian geometry is not required, though readers with a background in the area will find their insights are readily transferrable.

About the authors

Shin-ichi Ohta is Distinguished Professor of Mathematics at Osaka University, Japan. His research interests lie in comparison geometry and its applications. He is a leading expert in the geometry and analysis of weighted Ricci curvature.

Table of contents (19 chapters)

Table of contents (19 chapters)

Buy this book

eBook $119.00
price for USA in USD
  • ISBN 978-3-030-80650-7
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $159.99
price for USA in USD
  • ISBN 978-3-030-80649-1
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
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Bibliographic Information

Bibliographic Information
Book Title
Comparison Finsler Geometry
Authors
Series Title
Springer Monographs in Mathematics
Copyright
2021
Publisher
Springer International Publishing
Copyright Holder
The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
eBook ISBN
978-3-030-80650-7
DOI
10.1007/978-3-030-80650-7
Hardcover ISBN
978-3-030-80649-1
Series ISSN
1439-7382
Edition Number
1
Number of Pages
XXII, 316
Number of Illustrations
8 b/w illustrations
Topics