现代几何结构和场论 影印版
出版时间： 2018年版
内容简介
本书将黎曼几何现代形式的基础表示为微分流形的几何及其上重要的结构。作者的处理方法是：黎曼几何的所有构造都源于一个可以让我们计算切向量之标量积的流形。按此方式，作者展示了黎曼几何对于现代数学几个基础领域及其应用的巨大影响。 \n● 几何是纯数学与自然科学首先是物理学之间的一个桥梁。自然界基本规律严格表示为描述各种物理量的几何场之间的关系。 \n● 对几何对象整体性质的研究导致了拓扑学深远的发展，这包括了纤维丛的拓扑与几何。 \n● 描述许多物理现象的哈密顿系统的几何理论导致了辛几何和泊松几何的发展。本书讲述的场论和高维变分学将数学与理论物理统一了起来。 \n● 复几何和代数流形将黎曼几何和现代复分析、代数和数论统一了起来。 \n本书的预备知识包括几门基础的本科课程，如高等微积分、线性代数、常微分方程以及拓扑要义。
目录
Preface to the English Edition
Preface
Chapter 1． Cartesian Spaces and Euclidean Geometry
1．1． Coordinates． Space-time
1．1．1． Cartesian coordinates
1．1．2． Change of coordinates
1．2． Euclidean geometry and linear algebra
1．2．1． Vector spaces and scalar products
1．2．2． The length of a curve
1．3． Affine transformations
1．3．1． Matrix formalism． Orientation
1．3．2． Affine group
1．3．3． Motions of Euclidean spaces
1．4． Curves in Euclidean space
1．4．1． The natural parameter and curvature
1．4．2． Curves on the plane
1．4．3． Curvature and torsion of curves in R Exercises to Chapter
Chapter 2． Symplectic and Pseudo-Euclidean Spaces
2．1． Geometric structures in linear spaces
2．1．1． Pseudo-Euclidean and symplectic spaces
2．1．2． Symplectic transformations
2．2． The Minkowski space
2．2．1． The event space of the special relativity theory
2．2．2． The Poincare group
2．2．3． Lorentz transformations
Exercises to Chapter
Chapter 3． Geometry of Two-Dimensional Manifolds
3．1． Surfaces in three-dimensional space
3．1．1． Regular surfaces
3．1．2． Local coordinates
3．1．3． Tangent space
3．1．4． Surfaces as two-dimensional manifolds
3．2． Riemannian metric on a surface
3．2．1． The length of a curve on a surface
3．2．2． Surface area
3．3． Curvature of a surface
3．3．1． On the notion of the surface curvature
3．3．2．Curvature of lines on a surface
3．3．3． Eigenvalues of a pair of scalar products
3．3．4． Principal curvatures and the Gaussian curvature
3．4． Basic equations of the theory of surfaces
3．4．1． Derivational equations as the “zero curvature”
condition． Gauge fields
3．4．2． The Codazzi and sine-Gordon equations
3．4．3． The Gauss theorem
Exercises to Chapter
Chapter 4． Complex Analysis in the Theory of Surfaces
4．1． Complex spaces and analytic functions
4．1．1． Complex vector spaces
4．1．2． The Hermitian scalar product
4．1．3． Unitary and linear-fractional transformations
4．1．4．． Holomorphic functions and the Cauchy
Riemann equations
4．1．5． Complex-analytic coordinate changes
4．2． Geometry of the sphere
4．2．1． The metric of the sphere
4．2．2． The group of motions of a sphere
4．3． Geometry of the pseudosphere
4．3．1． Space-like surfaces in pseudo-Euclidean spaces
4．3．2． The metric and the group of motions of the pseudosphere
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Chapter 5． Smooth Manifolds
Chapter 6． Groups of Motions
Chapter 7． Tensor Algebra
Chapter 8． Tensor Fields in Analysis
Chapter 9． Analysis of Differential Forms
Chapter 10． Connections and Curvature
Chapter 11． Conformal and Complex Geometries
Chapter 12． Morse Theory and Hamiltonian Formalism
Chapter 13． Poisson and Lagrange Manifolds
Chapter 14． Multidimensional Variational Problems
Chapter 15． Geometric Fields in Physics
Bibliography
Index
