Overview
- 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
Part of the book series: Springer Monographs in Mathematics (SMM)
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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.
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Keywords
- Finsler geometry
- Finsler manifolds
- Introduction to Finsler geometry
- Comparison geometry in Finsler context
- Functional inequalities in Finsler context
- Nonlinear Laplacian on Finsler manifolds
- Heat flow on Finsler manifolds
- Ricci curvature
- Weighted Ricci curvature
- Lower Ricci curvature bounds
- Gamma-calculus
- Curve-dimension condition
- Needle decomposition
- Finsler geometry for Riemannian geometers
Table of contents (19 chapters)
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Front Matter
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Part III
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Front Matter
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Reviews
“Finsler geometry is an active area of research in mathematics and has led to numerous real-world applications. This book is a comprehensive introduction to Finsler geometry and its applications. It covers the basic concepts of this geometry. More intuitively, this book provides an accessible introduction to recent developments in comparison geometry and geometric analysis on Finsler manifolds. … this book offers a valuable perspective for those familiar with comparison geometry and geometric analysis.” (Behroz Bidabad, Mathematical Reviews, May, 2023)
Authors and Affiliations
About the author
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.
Bibliographic Information
Book Title: Comparison Finsler Geometry
Authors: Shin-ichi Ohta
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-3-030-80650-7
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Hardcover ISBN: 978-3-030-80649-1Published: 10 October 2021
Softcover ISBN: 978-3-030-80652-1Published: 10 October 2022
eBook ISBN: 978-3-030-80650-7Published: 09 October 2021
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XXII, 316
Number of Illustrations: 8 b/w illustrations
Topics: Differential Geometry, Global Analysis and Analysis on Manifolds