This book is an English translation of the famous "Green Book" by Lafontaine and Pansu (1979). It has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices, by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures, as well as an extensive bibliography and index round out this unique and beautiful book.

These Lecture Notes are an expanded version of the Fermi Lectures Atiyah gave at Scuola Normale Superiore in Pisa, the Loeb Lectures at Harvard and the Whittemore Lectures at Yale, in 1978. In all cases he was addressing a mixed audience of mathematicians and physicists and the presentation had to be tailored accordingly. Throughout, Atiyah presented the mathematical material in a somewhat unorthodox order, following a pattern which he felt would relate the new techniques to familiar ground for physicists.
The main new results presented in the lectures, namely the construction of all multi-istanton solutions of Yang-Mills fields, is the culmination of several years of fruitful interaction between many physicists and mathematicians. The major breakthrough came with the observation by Ward that the complex methods developed by Penrose in his “twistor programme” were ideally suited to the study of the Yang-Mills equations. The instanton problem was then seen to be equivalent to a problem in complex analysis and to one in algebraic geometry. Using the powerful methods of modern algebraic geometry it was not long before the problem was finally solved.

《黎曼几何和几何分析(第4版)》是一部值得一读的研究生教材（全英文版），内容主要涉及黎曼几何基本定理的研究，如霍奇定理、Rauch比较定理、Lyusternik和Fet定理调和映射的存在性等，书中还有当代数学研究领域中的最热门论题，有些内容则是首次出现在教科书中。《黎曼几何和几何分析(第4版)》各章均附有习题。

This book consists mainly of lecture notes of some talks and courses at Mathematical Sciences Center (MSC) of Tsinghua University together with several other papers.
Since it was founded in December 2009, one of the missions of MSC has been to teach both undergraduate and graduate students important ideas, theories and results of contemporary mathematics. These lecture notes reflect this philosophy.They are expository and accessible to both students and nonexperts. On the other hand, they also contain novel ideas or presentations of important topics in mathematics. Therefore, this book is also useful to experts. Especially we would like to point out that the last paper in this book Open Problems in Differential Geometry by the third editor of this book is only the first three of many lectures given by him in both Beijing and Taipei, which can be considered as a reviewing and updating of the very influential open problem lists by him.
Besides these lecture notes from MSC, this book also contains four other papers. The first is a paper by James Milne based on his talk at the seminar "What is ..." at University of Michigan. The concept of motives is important and difficult, and the talk and this paper are attempts by an expert to explain it in concrete terms. The second is a master thesis in 2002 by Joris van Hoboken who gives a coherent and accessible exposition of the ubiquity of the important ADE classification in mathematics, which originally occurred in the classification of simple complex Lie algebras. Joris van Hoboken switched to study law right after obtaining his Master degree and is now a senior researcher at a law school. The ADE classification occurs at many different situations, and it is still a mystery whether there are some deep, intrinsic connections between them. This master thesis was never published and has been highly cited and circulated on the web.We are grateful that Dr. van Hoboken has given us permission to include it in the current book. We hope that this will make the ADE classification better known to the reader and also give a permanent record of this beautiful master thesis. The other two are reprints of papers of the third editor. The short paper A note on the distribution of critical points of eigenfunctions considered a novel question. As it is well-known, the location and distribution of the zero sets (i.e., nodal sets) of eigenfunctions of Riemannian manifolds have been extensively and intensively studied. Critical points of eigenfunctions are also special and deserve to be understood better. Analysis on nonsmooth spaces has been becoming quite important and applied to several subjects in mathematics. The paper is one of the early papers in this subject.1 Due to inaccessibility and no review of it in MathSciNet, this paper has been largely unknown. We hope that its inclusion in this book will be valuable to the reader as well.
It has been a lot of work for the speakers at MSC to write up their lecture notes. We would like to thank them, especially the four note-takers and co-authors (Hui Ma, Chun-Jun Tsai, Mu-Tao Wang, En-Tao Zhao) of the last paper in this book, for their efforts and contributions. We would also like to thank reviewers of the papers in this book for their help.
This book marks the beginning of publication from MSC and we hope and expect that future volumes will appear regularly.
Editors: Lizhen Ji, Yat-Sun Poon, Shing-Tung Yau
May 30, 2013

本书主要讨论紧黎曼曲面，中心是Riemann-Roch定理的证明及其应用，因为黎曼曲面是近代数学不少分支的最简单的模型．本书在讨论中采用一些必要的近代数学的概念与方法作为工具，以期使本书能成为近代数学很多方面的入门书．本书可供数学专业高年级学生、研究生、数学教师及其它数学工作者参今

Long regarded as a masterpiece in content and form, this work defines the concept of surface curvature and presents the important theorem stating that the "Gauss curvature" is invariant under arbitrary isometric deformation of a curved surface. This edition of Gauss's classic features a new introduction, bibliography, and notes by science historian Peter Pesic. 1902 edition.