课程编号：34102574

课程名称（中文）：微分几何                 

课程名称（英文）： Differential Geometry

学分数/学时数：4/80

开课单位/开课学期：数学系/三年级上学期

课程类别：必修课

面向专业：数学与应用数学

课程负责人：朱熹平

课程内容简介（中文）：

微分几何是数学的一个重要分支，它起源于微积分在几何上的应用，并与微分方程，复分析，代数，拓扑以及理论物理等相互渗透成为推动这些理论发展的一项重要工具。此外，微分几何在机械工程，力学等领域有广泛应用。

大学本科的微分几何内容包括两部分：局部理论和整体理论。微分几何的局部理论研究三维欧氏空间中的曲线和曲面在一点附近的性质，其中的一个主要问题是寻求几何不变量并确定这些不变量能在什么程度刻划曲线和曲面。这就是所谓曲线论和曲面论基本定理的内容。局部微分几何的一个里程碑是Gauss关于曲面的理论，他建立了基于曲面第一基本形式的几何，并把欧几里得几何推广到曲面上“弯曲”的几何。由此开创了曲面的内蕴几何学的研究，使微分几何成为一门真正独立的学科。Riemann将Gauss的理论推广到高维空间而创立了Riemann几何并为Einstein的广义相对论奠定了基础。

二十世纪三四十年代发展起来的整体微分几何，其中的一个重要部分是讨论流形的曲率是如何影响流形的拓扑乃至度量性质。这方面最早的结果是Gauss-Bonnet定理，它表明曲面的Euler示性数能用Gauss曲率的积分表示。而微分几何中的刚性问题讨论曲率等如何确定流形的度量性质。

学习微分几何可了解与掌握几何概念与方法，培养几何直观和图形想像的能力以及从具体到抽象的能力。

课程内容简介（英文）：

    Differential Geometry is one of the important branches in mathematics. It originated from the application of calculus to geometry and develops through interaction with other subjects of mathematics such as Differential Equation, Complex Analysis, Algebra, Topology and Theoretic Physics. Differential Geometry has brought powerful tools to the study of these theories and has been applied extensively to the field of Engineering and Mechanics, among others.

   The undergraduate differential geometry deals with the local and global theory of curves and surfaces lying in three-dimensional Euclidean spaces. The local differential geometry concerns the study of properties of curves and surfaces in a neighborhood of a given point. The main problem is to find geometric invariants and to determine to what extent they can locally characterize the curves and surfaces. The answer consist in the fundamental theorems of curves and surfaces.

   One of the landmarks in the development of Differential geometry is C.F. Gauss's work on surfaces. In the definitive paper "General investigations of curved surfaces", Gauss advanced the totally new concept that a surface is a space in itself. More specifically, Gauss studied the geometry of surfaces based on the first fundamental form (also called "line element") of surfaces and generalized Euclidean geometry to "curved geometry" on surfaces. The idea of Gauss initiated the study of "intrinsic geometry" and made differential geometry an independent subject of mathematics. Later, Riemann generalized Gauss's theory on surfaces to higher dimensional spaces (called "manifolds") and established Riemannian geometry which constitutes the foundation of Einstein's General Relativity theory.

    The global differential geometry progressed rapidly in the twentieth century. One of its main goals is to classify topologically, or even metrically, manifolds with certain geometry conditions on, say, curvature, volume, diameter. The earliest such theorem is the Gauss-Bonnet theorem, which gives the Euler characteristic in terms of the integral of the Gaussian curvature. Also the rigidity problems in differential geometry discuss how curvature influences the metric properties of manifolds.

    Through studying geometry, you will learn the geometry concepts and methods, improve your geometric intuition and imagination. Also you will learn to use the method of figuring out the abstract thing by way of investigating the corresponding concrete one.