Fundamentals of Differential Geometry (Graduate Texts in Mathematics)

出版者:Springer

作者:Serge Lang

页数:557

出版时间:1998-12-30

价格:USD 69.95

装帧:Hardcover

isbn号码:9780387985930

丛书系列:Graduate Texts in Mathematics

具体描述

This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo-Riemannian manifolds; applications to the exponential map, including the Cartan-Hadamard theorem and the first basic theorem of calculus of variations. Although the book grew out of the author's earlier book "Differential and Riemannian Manifolds", the focus has now changed from the general theory of manifolds to general differential geometry, and includes new chapters on Jacobi lifts, tensorial splitting of the double tangent bundle, curvature and the variation formula, a generalization of the Cartan-Hadamard theorem, the semiparallelogram law of Bruhat-Tits and its equivalence with seminegative curvature and the exponential map distance increasing property, a major example of seminegative curvature (the space of positive definite symmetric real matrices), automorphisms and symmetries, and immersions and submersions. These are all covered for infinite-dimensional manifolds, modeled on Banach and Hilbert Spaces, at no cost in complications, and some gain in the elegance of the proofs. In the finite-dimensional case, differential forms of top degree are discussed, leading to Stokes' theorem (even for manifolds with singular boundary), and several of its applications to the differential or Riemannian case. Basic formulas concerning the Laplacian are given, exhibiting several of its features in immersions and submersions.

作者简介

Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He was a member of the Bourbaki group.

Lang was born in Paris in 1927, and moved with his family to California as a teenager, where he graduated in 1943 from Beverly Hills High School. He subsequently graduated from the California Institute of Technology in 1946, and received a doctorate from Princeton University in 1951. He held faculty positions at the University of Chicago and Columbia University (from 1955, leaving in 1971 in a dispute). At the time of his death he was professor emeritus of mathematics at Yale University.

用户评价

我发现这本书的参考价值，远超出了它作为一本“教科书”的范畴。它在某些核心主题上的论述深度，已经达到了次级文献的水平。特别是关于特定拓扑空间与几何结构的联系那一章，作者的处理方式非常独特且具有启发性，很多经典教材处理得比较敷衍的地方，在这里得到了详尽的展开和清晰的论证。这使得它不仅适合课堂教学，更成为了我个人研究中随时会翻阅的参考手册。当我在进行深入探索，需要回溯某个基本性质的最初推导时，这本书总能提供一个清晰、可靠的源头。它不像一些流行的、更注重“应用”的教材那样强调工具箱，而是更偏向于对“原理”本身的探究，这对于致力于理论研究的人来说，是无价之宝。

作为一本研究生级别的教材，它在习题设计上的深度和广度，着实考验了读者的功力。这些习题绝非简单的概念验证，而是对所学理论的深度挖掘和创造性应用。很多题目需要读者跳出书本的框架，将不同章节的工具融会贯通，甚至需要一些“旁征博引”的技巧才能攻克。我特别喜欢那些开放式或者需要构造性证明的难题，它们迫使你真正地“与数学对话”，而不是被动地接受知识。当然，对于那些基础薄弱的章节，习题的难度可能会带来一定的挫败感，但正是这种挑战，确保了知识的内化，而不是浅尝辄止地翻阅。做完这些习题，你会有一种豁然开朗的感觉，像是真正掌握了一项强大的分析工具。

这本书的排版和装帧设计简直是一场视觉盛宴。厚重的封面，那种沉甸甸的质感，拿在手里就有一种被知识重量包裹的感觉。纸张的选用也十分考究，光洁而不反光，使得阅读时的眼睛负担很小。更值得称赞的是，内页的字体和行距处理得极其到位，几何图形的绘制清晰锐利，无论是曲线的描摹还是向量场的表示，都精准得像是艺术品。对于像我这样需要长时间沉浸在数学公式和图示中的读者来说，这种对细节的极致追求，极大地提升了阅读体验。很多数学书虽然内容深刻，但排版粗糙，让人望而却生畏，但这本书却反其道而行之，用优雅的呈现方式，邀请你进入深奥的理论殿堂。每一页的边距都留有足够的空间，方便读者在旁边做笔记和批注，这种体贴入微的设计，体现了出版者对学习者的尊重和理解。

这本书的章节逻辑编排，简直可以用“行云流水”来形容，它不是简单地堆砌定理和定义，而是在构建一个完整的、自洽的理论框架。初学者可能会觉得开篇的预备知识部分略显精简，但对于有一定基础的人来说，这种直奔主题的叙述方式反而高效得多。作者非常擅长在引入复杂概念时，先提供一个直观的几何洞察，而不是直接抛出冰冷的代数表达式。例如，在介绍黎曼曲率张量时，它不是上来就给一长串坐标分量，而是先通过一个思想实验，让你“感受”到空间弯曲的本质。这种“先知后术”的叙述策略，使得高深的微分几何概念变得不再那么高不可攀，真正做到了循序渐进，层层递进，为后续更复杂的流形理论打下了坚实的基础。

这本书的语言风格保持了一种极其严谨的学术口吻，措辞精准到每一个介词和副词都不能有丝毫马虎。这对于需要精确理解数学定义的读者来说，是极大的福音，避免了因模糊的表述而产生的歧义。然而，这种严谨性也带来了一个副作用：对于初次接触这方面内容的读者，初读时的确会感到有些“晦涩难懂”。它很少使用非正式的类比或幽默来调剂，完全专注于数学本身的纯粹性。可以说，这本书更像是一位技艺精湛的工匠，为你提供了最优质的原材料和最精密的工具，但最终的雕刻和打磨，则完全依赖于阅读者自身的努力和背景知识储备。它要求读者带着一份严肃的敬意去对待每一个符号和每一个论断。

build up your concepts

build up your concepts

本书包含了《微分流形和黎曼流形》的全部内容，是那本书的“再版”。

科大学子亲切地称Serge Lang为“色狼”（笑），不过书还是蛮好的啦

build up your concepts