Manifolds and Differential Geometry
Manifolds and Differential Geometry
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DOI：
10.1090/gsm/107
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发表时间：
2009-11
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影响因子：
--
通讯作者：
John M. Lee
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文献类型：
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作者：
John M. Lee
差异几何形状始于使用微积分方法研究曲线和表面的研究。随着时间的流逝，将曲线和表面的概念与相关的概念（例如长度，体积和曲率）进行了概括。同时，该主题与拓扑的发展变得紧密相关。基本对象是平滑的歧管，已经连接了一些额外的结构，例如riemannian度量，形式，杰出的对称性组或切线束上的连接。这本书是对现代微分几何学工具和结构的研究生级介绍。其中包括通常在有关可区分流形的课程中发现的主题，例如向量束，张量，差异形式，de rham的共同体，Frobenius Theorem和基本的谎言组理论。这本书还包含有关向量束上的一般连接理论的材料，以及关于半帝国几何学的深入章节，涵盖了有关Riemannian歧管和Lorentz歧管的基本材料。这本书的一个不寻常的特征是将早期的章节包括在欧几里得空间中超曲面的差异几何形状。还有一个部分衍生了Maxwell方程的外部演算版本。这本书的第一章适用于一学学课程。有足够的材料可以容纳一年的流形和几何学。
Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature. At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle. This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory. The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hyper-surfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations. The first chapters of the book are suitable for a one-semester course on manifolds. There is more than enough material for a year-long course on manifolds and geometry.