发表机构
Universidad Nacional Autónoma de México（墨西哥国立自治大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本书系统介绍黎曼流形上的全局分析，融合几何、分析与PDE，涵盖指标定理、几何流及变分方法，强调显式证明与几何假设。
AI 中文摘要
本专著发展了一部以微分几何、泛函分析、偏微分方程与黎曼流形上变分方法之间相互作用为核心的全局分析导论。从光滑几何与黎曼几何开始，它发展了Sobolev空间、分布、插值与分数阶正则性、向量丛上的微分算子与伪微分算子、椭圆理论、热方法、有界几何与迹定理。随后，它处理了Fredholm理论与指标理论，最终达到Atiyah–Singer指标定理，接着是几何演化方程与Ricci流、Banach与Hilbert流形上的无穷维几何，以及变分方法，包括直接方法、Palais–Smale理论、形变论证、山路定理与Nehari方法。特别强调显式证明、从局部欧几里得估计到内蕴全局陈述的过渡，以及在紧致、非紧致与带边界情形下所需的精确几何假设。本书面向高年级本科生与研究生，以及从几何或微分方程角度接近全局分析的读者。
英文摘要
This monograph develops an introduction to global analysis centered on the interaction between differential geometry, functional analysis, partial differential equations, and variational methods on Riemannian manifolds. Beginning with smooth and Riemannian geometry, it develops Sobolev spaces, distributions, interpolation and fractional regularity, differential and pseudodifferential operators on vector bundles, elliptic theory, heat methods, bounded geometry, and trace theorems. It then treats Fredholm and index theory, culminating in the Atiyah--Singer index theorem, followed by geometric evolution equations and Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and variational methods including the direct method, Palais--Smale theory, deformation arguments, the mountain pass theorem, and the Nehari method. Particular emphasis is placed on explicit proofs, the passage from local Euclidean estimates to intrinsic global statements, and the precise geometric hypotheses required in compact, noncompact, and boundary settings. The text is intended for advanced undergraduate and graduate students, as well as readers approaching global analysis from geometry or differential equations.
Comments1845 pages. Preliminary version; comments and corrections are very welcome
