Comparison Finsler Geometry
暫譯: 比較芬斯勒幾何學
Ohta, Shin-Ichi
- 出版商: Springer
- 出版日期: 2022-10-10
- 售價: $6,400
- 貴賓價: 9.5 折 $6,080
- 語言: 英文
- 頁數: 316
- 裝訂: Quality Paper - also called trade paper
- ISBN: 3030806529
- ISBN-13: 9783030806521
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相關分類:
物理學 Physics
海外代購書籍(需單獨結帳)
商品描述
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.
商品描述(中文翻譯)
這本專著介紹了在 Finsler 流形上的比較幾何和幾何分析的最新發展。作者將加權 Ricci 曲率推廣到 Finsler 設定中,系統性地推導出 Finsler 背景下的基本幾何和分析不等式。這本書僅依賴於可微流形的知識,為新手讀者提供了一個進入 Finsler 幾何的可接觸點。
本書分為三個部分,首先建立 Finsler 幾何的基本概念,包括 Jacobi 場和曲率張量、弧長的變分公式以及一些經典的比較定理。第二部分介紹了加權 Ricci 曲率、非線性拉普拉斯算子和 Finsler 流形上的非線性熱流。這些工具使得能夠推導出 Bochner-Weitzenböck 公式及相應的 Bochner 不等式、梯度估計、Bakry-Ledoux 的高斯等周不等式以及 Finsler 背景下的函數不等式。第三部分包含進階主題:經典 Cheeger-Gromoll 分裂定理的推廣、曲率-維度條件和針狀分解。整體而言,幾何描述闡明了結果背後的直覺,而練習題則提供了主動參與的機會。
比較 Finsler 幾何 為研究 Finsler 流形的研究生和研究人員提供了一個理想的入門途徑。假設讀者具備可微流形理論的知識,以及函數分析的基本概念。雖然不要求熟悉黎曼幾何,但具備該領域背景的讀者會發現他們的見解可以輕易轉移。