漫谈微分几何、多复变函数与代数几何

微分几何科普(1):浅谈度规和曲率Shanqin(萍踪浪迹)前言：从现在开始，写一些大学理科生可以轻松看得懂的科普帖子,作出的牺牲就是让其他更高学历的人看起来很平庸.从现在开始,要把看起来要写比较长的文章分开写，不在一个帖子里搞连载。

这样主要是为了避免没有时间续写自己的主题而让自己的帖子变成TJ帖(啥叫TJ呢？就是和DJ有一定联系的)。

正文：初步的微分几何，必须掌握基本的曲线论，必须适应以弧长为参数的方程.Frenet公式是曲线论基本公式,Frenet标架是活动标架在曲线时的特殊情形.两条曲率和挠率都一样的曲线可以通过刚体运动重合在一起,这是曲线论基本定理.曲线的内蕴曲率为零。

所以所有曲线都可以拉直而不改变其上任意两点间弧长.我们知道,曲面论中这一点通常不能成立,除非此曲面可以等距映射为平面,我们称这种可以和平面进行等距映射的曲面为平坦曲面,如柱面.因此,我们必须深入研究曲面的曲率问题,首先要熟悉曲线坐标,在切平面上讨论问题,这个是整个微分几何的基础.因为即使到高维情形，我们仍要讨论切空间及其上的Levi-Civita联络.在切平面上任意点引入切矢量^du,dv),切向量在这个基下的分量则为「—1「—▽定义切向量内积系数：E=<r_u.r_u>=g_11,F=<r_u.r_v>=<r_v.r_u>=g_12,G=<r_v.r_v>=g_22,这三个量就是极其重要的度量(度规)系数.曲面的第一基本形式于是可以写成:I=<dr.dr>=Edudu+2Fdudv+Gdvdv=g_ijdu_idu_j最后一式我们将du,dv写成du_1,du_2,i,j取值为1,2,这里采用了Einstein求和约定:重复指标自动求和.这样的符号约定和求和约定可以让我们轻松将2维情形推广到n维流形的n维切空间，其上切向量内积系数(度量系数)就是g_ij(i,j=1,2,用)若n等于4,就是广义相对论中的度规张量情形.我们开始讨论曲面的第二基本形式.引入曲面上任意点的法向量n,定义两点间法向量的变化：dn=n_udu+n_vdv.其中n_u,n_v为dn在基(du,dv)下的展开系数.则我们可以定义内积：L=-<r_u.n_u>=h_11M=-<r_u.n_v>=<r_v.n_u>h_12N=-<r_v.n_v>=h_22L,M,N(h_11,h_12,h_22)称为第二形式基本量，于是第二基本形式可以写成：II=-<dr.dn>=Ldudu+2Mdudv+Ndvdv=h_ijdu_idu_j.最后一个等式采用的符号和求和约定同上.第一基本形式决定了曲面的内蕴结构，以后我们会发现，联络系数(Christoffel符号)由度规张量和度规张量的一次导数决定，而曲面的Gauss曲率(广而言之,流形的Riemann截面曲率)由联络系数及其一阶导数决定.什么是Gauss曲率和Riemann截面曲率我们可以从曲面的法曲率出发，定义主曲率.我们想象拿着一把刀，贴着曲面上某点(u,v)的法线往下切，在曲面上切出一条曲线，这条曲线的曲率就是曲面在该点(u,v)沿(du,dv)方向的法曲率.如果想象我们切一个椭球面,在同一点贴着法线,沿不同方向切下去,切出的所有曲线(称为法截线,相应的这一刀所在的平面称为法截面)的曲率不一定一样.我们把这些曲线的曲率进行比较，最大和最小的法曲率称为主曲率，记为k_1,k_2.这两个法曲率对应的法截线必定垂直.定义Gauss曲率为k_1,k_2的乘积:K=k_1.k_2.若K=0,则曲面必然平坦.定义平均曲率为k_1,k_2的算术平均：H=(k_1+_2)/2.若H=0,则该曲面就是极小曲面.Gauss绝妙定理指出，Gauss曲率K在曲面的等距变换下保持不变.即曲面的内蕴性质由第一基本形式决定决定,与它在外围空间中的形状无关.而曲面的第二基本形式则决定了曲面在外围空间中的形状.这些结论可以可以推广到高维空间中的超曲面（维数比外围空间低一的曲面称为超曲面）.1854年Riemann推广了Gauss的想法，将抽象曲面研究推广到高维抽象弯曲空间（流形）进行研究.在高维情形，我们将面对切空间.与前面类似，我们定义度规系数g_ij（i,j=1,2,,n）,此时我们可以让其他方向都退化,留下两个方向，用曲面论观点看问题.这样就可以将Gauss曲率搬到这里，由于方向很多,我们将面对不止一个的Gauss曲率,我们将这些曲率称为Riemann截面曲率.显然，当弯曲空间为2维曲面时，Riemann截面曲率就是Gauss曲率.Riemann截面曲率为常数的空间称为常曲率空间，如果这个常曲率空间是单连通的，我们就称为空间形式，最重要的三种空间形式分别是正曲率的球空间,零曲率的欧空间,负曲率的双曲空间.Riemann在世时，并未将这个想法进行详细发展，后世的Christoffel进行了很大的扩充，这个曲率由Christoffel 符号的导数和乘积表示，所以Riemann截面曲率也称为Riemann-Christoffel曲率.将Riemann截面曲率缩并（取迹，即让R_ijkl中的两个字母相同而求和），就得到了Ricci曲率R_ij,W Ricci 曲率缩并，就得到标量曲率（数量曲率，纯量曲率）R.这些概念在后来Einstein创立的引力论（GR）之中都成为核心概念.GR确定了时空曲率和物质分布的关系.其基本方程就是Einstein方程：R_ij-1/2Rg_ij+g_ij=8T_ij其中R_ij为时空的Ricci曲率,R为时空的标量曲率，g_ij为时空的度规张量.为宇宙学常数，T_ij为物质的物质的能-动张量.我们可以记G_ij=R_ij-1/2Rg_ij,G_ij就是通常所说的Einstein张量.因此我们研究四维时空时,只要知道它的度规张量（第一基本形式系数）,就可以直接以这个四维时空为研究对象，而不用考虑将这个时空嵌入更高维数的空间进行研究所以不管是Minkowski空间,deSitter空间还是反deSitter 空间，都是写成度规后进行研究.但是在很多时候，我们要研究时空中的超曲面.即使是deSitter空间和反deSitter空间，我们也可以将它们分别嵌入五维欧氏空间R八5里面的双曲面.而在广义相对论中我们以Lorentz流形作为基本研究框架（尽管我们可以赋予时空其他形式的度规结构，但是我们最经常使用的还是Lorentz度规.）我们通常要研究Lorentz流形中的类空超曲面M八3,为了研究其上的内蕴特征和外在特征在时间演化下的变化，就必须引入初始数据集（M八3,g_ij,h_ij）,此处g_ij,h_ij分别为M 八3上的度规张量和第二基本形式量.g_ij和h_ij必须满足的相容性条件是著名的Gauss-Codazzi方程.因为Gauss-Codazzi方程是（超）曲面存在的充分必要条件.因此可见看似初等的微分几何曲面论中的一些概念在广义相对论的现代研究中实际上是非常重要的.但是在很多时候，我们要研究时空中的超曲面.即使是deSitter空间和反deSitter空间，我们也可以将它们分别嵌入五维欧氏空间R八5里面的双曲面.昌海兄，请将上面这一段替换成下面这一段，然后删除此回帖：但是在很多时候，我们要研究时空中的超曲面.即使是deSitter空间和反deSitter空间，我们也可以将它们分别嵌入五维伪欧氏空间（pseudo-Euclideanspaces）R人5里面的双曲面。

代数几何学代数几何学是数学中一个重要的分支，它采用代数方法来研究几何图形的性质。

代数几何学主要研究几何图形构成的方程组，从而确定图形的形状和特征，给出关于这些图形的性质和结论。

它主要涉及几何学中的几何图形，其包括点、直线、圆、椭圆、抛物线、双曲线、曲面等形状。

代数几何学研究的技术和方法主要有一元几何、多元几何、曲线几何、曲面几何等。

其中，一元几何主要研究直线、圆形和椭圆的性质，例如用一元方程分析直线、圆形和椭圆的性质；多元几何主要研究多维空间的图形，例如用多元方程来分析几何图形；曲线几何主要研究双曲线、抛物线、螺线等曲线，其中双曲线和抛物线常用于推导相关的重要定理；曲面几何主要研究平面曲面、柱面曲面等曲面，用于描述立体几何。

代数几何学的发展为其他数学领域提供了很多技术和思想。

其中，代数几何学被应用于解析几何、概率论、复变函数、拓扑学、函数解析学等数学分支中，形成了一个结构完整的数学系统。

此外，代数几何学最重要的作用是为信息科学和计算机科学提供重要的理论和技术。

有许多应用代数几何学的计算机科学研究，比如几何图形学、图论、几何处理、计算机视觉和虚拟现实等，都用代数几何学来分析复杂的几何图形，并使用它的概念、方法和理论来分析空间结构和处理复杂的数学问题。

几何图形的理解和研究对现代社会的发展至关重要，这也是代数几何学最重要的意义之一。

尤其是现代社会的核心工程，如机器人技术、运动控制技术等，都需要研究几何图形的基本性质，而这些性质正是代数几何学可以提供的。

因此，在现代社会中，发展代数几何学是非常必要的。

总之，代数几何学是数学中一个重要的分支，它主要研究几何图形构成的方程组，分析图形的性质，给出性质的结论，并探究它的应用。

由于它具有重要的意义，因此在现代社会中，发展代数几何学是非常必要的。

数学王国的凯撒大帝丘成桐发布时间：2022-09-29T06:05:54.232Z 来源：《教学与研究》2022年第11期作者：巴英[导读] 本文从多个维度比较全面地介绍了华裔数学家丘成桐，巴英（江汉大学人工智能学院，湖北武汉 430056）摘要：本文从多个维度比较全面地介绍了华裔数学家丘成桐，包括他的求学经历、在数学和物理上的卓越贡献、对中国数学教育事业的付出以及他的文学素养，再现了他几十年来对数学事业的炽热追求，道出了他坚韧顽强和锲而不舍的成功基石。

关键词：卡拉比猜想、偏微分方程、微分几何、正质量猜想前言丘成桐（1949-）是当代公认的最具影响力的数学家之一，他的工作极大地拓展了偏微分方程在微分几何中的应用，其影响遍及拓朴学、代数几何、表示理论、广义相对论等数学、物理领域，《纽约时报》曾经称他为数学王国的凯撒大帝，对待科学具有不屈不挠、勇往直前的拼搏精神。

1丘成桐的经历丘成桐1949年生于广东汕头，后随父母全家定居香港，他14岁时，在大学工作的父亲去世，生活陷入困窘，但是丘成桐不改对数学的痴迷，无论外界环境如何，从不间断学习和研究数学。

1966年，半工半读的丘成桐以优异成绩考入香港中文大学崇基学院数学系，并以三年时间提前修完全部课程。

1969年，丘成桐被推荐到美国加利福尼亚大学伯克利分校攻读博士学位，在这个云集了许多几何学家和年轻学者的世界微分几何中心，他师从著名微分几何学家陈省身教授，潜心研究微分几何。

1971年，22岁的丘成桐完成了博士论文《非正曲率紧流行的基本群》，文中巧妙地解决了沃尔夫猜想。

1974年到1987年，丘成桐成为斯坦福大学、普林斯顿高等研究院、加州大学圣地亚哥分校的数学教授。

1987年起任哈佛大学讲座教授。

1993年被选为美国国家科学院院士。

1994年成为台湾中央研究院院士和中国科学院外籍院士，并出任香港中文大学数学科学研究所所长。

2003年出任香港中文大学博文讲座教授。

复变函数与解析几何的关系复变函数和解析几何是数学中两个重要的分支，它们之间存在着紧密的联系和相互依存的关系。

复变函数是研究复数域上的函数，而解析几何则是研究几何图形和代数方程之间的关系。

本文将探讨复变函数与解析几何之间的关系，并探讨它们在数学和其他领域中的应用。

一、复变函数的基本概念复变函数是定义在复数域上的函数，它由实部和虚部组成。

复变函数的基本运算规则与实数函数类似，但复变函数的特殊性在于它具有解析性。

解析性是指函数在其定义域内处处可导，并且导函数也是解析函数。

复变函数的解析性使得它在数学和物理学中有着广泛的应用，特别是在解析几何中。

二、解析几何的基本概念解析几何是研究几何图形和代数方程之间的关系。

它将几何问题转化为代数问题，通过代数方程的解析方法来研究几何图形的性质和特征。

解析几何的基本概念包括坐标系、曲线方程和曲线的性质等。

通过解析几何的方法，我们可以用代数方程来描述和分析几何图形，从而深入研究它们的特征和性质。

三、复变函数与解析几何的联系复变函数与解析几何之间存在着紧密的联系和相互依存的关系。

一方面，复变函数可以通过解析几何的方法来研究和描述。

例如，通过将复变函数表示为实部和虚部的形式，我们可以将其与坐标系中的点相对应，从而将复变函数与几何图形联系起来。

这样，我们可以通过解析几何的方法来研究复变函数的性质和特征。

另一方面，解析几何也可以通过复变函数的方法来研究和描述。

复变函数的解析性使得它在解析几何中有着广泛的应用。

例如，我们可以通过复变函数的导数和积分来研究曲线的切线和曲率等几何性质。

复变函数的解析性还可以用来研究曲线的拓扑结构和变形等问题。

因此，复变函数和解析几何之间的联系不仅体现在它们的相互应用上，还体现在它们的理论基础和方法论上。

四、复变函数与解析几何的应用复变函数和解析几何在数学和其他领域中有着广泛的应用。

在数学中，复变函数和解析几何是研究复数域和几何图形的重要工具。

它们在数学分析、代数几何和微分几何等领域中有着广泛的应用。

微分几何的理论与应用微分几何（Differential geometry）是研究曲线、曲面以及流形等对象的性质和应用的数学分支学科。

在现代物理学和工程学等领域中，微分几何是一门极为重要的工具性学科。

在该领域已经有许多伟大的学者付出了艰辛的探索和研究，在高维空间和广义相对论等领域中得到了广泛应用。

一、微分几何的概念及发展历程微分几何是研究曲线、曲面以及流形等对象的性质和应用的数学分支学科，起源于高斯等学者的研究。

它发展的主要难点是高维度空间的研究，由于其复杂性很大，所以在目前仍是有待深入研究的领域。

由于其应用价值极高，所以引起了许多研究者的关注和研究。

二、微分几何的重要性微分几何作为一个重要的数学分支，在现代物理学和工程学等领域中发挥着重要作用。

近年来，随着计算能力和计算机技术的提高，微分几何正在得到越来越广泛的应用。

由于其性质复杂且运算高度抽象，所以具有很强的工具性。

在大规模计算、机器学习、自然语言处理、人工智能等领域均得到了应用，尤其是在机器学习和人工智能中的神经网络的架构设计、优化方法中，微分几何学理论成为实现机器学习算法的根本基础。

三、微分几何在物理学中的应用在物理学中，微分几何扮演着非常重要的角色，特别是在空间和时间的相对性理论中。

广义相对论是利用微分几何所建立的一种描述太阳系和宇宙的理论。

在相对论框架中，重力场是动力学和几何的交互作用，可以通过几何工具来描述其性质、演化、变形等，成为广义相对论领域研究的核心。

微分几何的工具在测量、空间定位、物体运动的模拟等方面，均有着广泛的应用。

四、微分几何在工程学中的应用微分几何在工程学中的应用也非常广泛，如在许多科研领域中要求对形态进行描述和分析，用于形状识别、图像处理等领域，并且可以在地质勘查、机械制造、飞行器设计、建筑、船舶设计等领域中得到应用。

例如，在机器人定位和导航、工业机器人中的路径规划和运动装置的控制等方面，都需要用到微分几何的理论。

五、未来发展方向当前，微分几何的研究仍有很大发展空间，随着计算机技术的飞速发展、大数据技术的出现，微分几何将更好地结合实际应用场景，发挥出更大的应用价值。

漫谈微分几何、多复变函数与代数几何（Differential geometry, functions of complex variable and algebraic geometry）Differential geometry and tensor analysis, developed with the development of differential geometry, are the basic tools for mastering general relativity. Because general relativity's success, to always obscure differential geometry has become one of the central discipline of mathematics.Since the invention of differential calculus, the birth of differential geometry was born. But the work of Euler, Clairaut and Monge really made differential geometry an independent discipline. In the work of geodesy, Euler has gradually obtained important research, and obtained the famous Euler formula for the calculation of normal curvature. The Clairaut curve of the curvature and torsion, Monge published "analysis is applied to the geometry of the loose leaf paper", the important properties of curves and surfaces are represented by differential equations, which makes the development of classical differential geometry to reach a peak. Gauss in the study of geodesic, through complicated calculation, in 1827 found two main curvature surfaces and its product in the periphery of the Euclidean shape of the space not only depends on its first fundamental form, the result is Gauss proudly called the wonderful theorem, created from the intrinsic geometry. The free surface of space from the periphery, the surface itself as a space to study. In 1854, Riemann made the hypothesis about geometric foundation, and extended the intrinsic geometry of Gauss in 2 dimensional curved surface, thus developing n-dimensional Riemann geometry, with the development of complex functions. A group of excellentmathematicians extended the research objects of differential geometry to complex manifolds and extended them to the complex analytic space theory including singularities. Each step of differential geometry faces not only the deepening of knowledge, but also the continuous expansion of the field of knowledge. Here, differential geometry and complex functions, Lie group theory, algebraic geometry, and PDE all interact profoundly with one another. Mathematics is constantly dividing and blending with each other.By shining the charming glory and the differential geometric function theory of several complex variables, unit circle and the upper half plane (the two conformal mapping establishment) defined on Poincare metric, complex function theory and the differential geometric relationships can be seen distinctly. Poincare metric is conformal invariant. The famous Schwarz theorem can be explained as follows: the Poincare metric on the unit circle does not increase under analytic mapping; if and only if the mapping is a fractional linear transformation, the Poincare metric does not change Poincare. Applying the hyperbolic geometry of Poincare metric, we can easily prove the famous Picard theorem. The proof of Picard theorem to modular function theory is hard to use, if using the differential geometric point of view, can also be in a very simple way to prove. Differential geometry permeates deep into the theory of complex functions. In the theory of multiple complex functions, the curvature of the real differential geometry and other series of calculations are followed by the analysis of the region definition metric of the complex affine space. In complex situations, all of the singular discrete distribution, and in more complex situations, because of the famous Hartogsdevelopment phenomenon, all isolated singularities are engulfed by a continuous region even in singularity formation is often destroyed, only the formation of real codimension 1 manifold can avoid the bad luck. But even this situation requires other restrictions to ensure safety". The singular properties of singularities in the theory of functions of complex functions make them destined to be manifolds. In 1922, Bergman introduced the famous Bergman kernel function, the more complex function or Weyl said its era, in addition to the famous Hartogs, Poincare, Levi of Cousin and several predecessors almost no substantive progress, injected a dynamic Bergman work will undoubtedly give this dead area. In many complex function domains in the Bergman metric metric in the one-dimensional case is the unit circle and Poincare on the upper half plane of the Poincare, which doomed the importance of the work of Bergman.The basic object of algebraic geometry is the properties of the common zeros (algebraic families) of any dimension, affine space, or algebraic equations of a projective space (defined equations),The definitions of algebraic clusters, the coefficients of equations, and the domains in which the points of an algebraic cluster are located are called base domains. An irreducible algebraic variety is a finite sub extension of its base domain. In our numerical domain, the linear space is the extension of the base field in the number field, and the dimension of the linear space is the number of the expansion. From this point of view, algebraic geometry can be viewed as a study of finite extension fields. The properties of algebraic clusters areclosely related to their base domains. The algebraic domain of complex affine space or complex projective space, the research process is not only a large number of concepts and differential geometry and complex function theory and applied to a large number of coincidence, the similar tools in the process of research. Every step of the complex manifold and the complex analytic space has the same influence on these subjects. Many masters in related fields, although they seem to study only one field, have consequences for other areas. For example: the Lerey study of algebraic topology that it has little effect on layer, in algebraic topology, but because of Serre, Weil and H? Cartan (E? Cartan, eldest son) introduction, has a profound impact on algebraic geometry and complex function theory. Chern studies the categories of Hermite spaces, but it also affects algebraic geometry, differential geometry and complex functions. Hironaka studies the singular point resolution in algebraic geometry, but the modification of complex manifold to complex analytic space and blow up affect the theory of complex analytic space. Yau proves that the Calabi conjecture not only affects algebraic geometry and differential geometry, but also affects classical general relativity. At the same time, we can see the important position of nonlinear ordinary differential equations and partial differential equations in differential geometry. Cartan study of symmetric Riemann space, the classification theorem is important, given 1, 2 and 3 dimensional space of a Homogeneous Bounded Domain complete classification, prove that they are all homogeneous symmetric domains at the same time, he guessed: This is also true in the n-dimensional equivalent relation. In 1959, Piatetski-Shapiro has two counterexample and find the domain theory of automorphic function study in symmetry, in the 4 and 5dimensional cases each find a homogeneous bounded domain, which is not a homogeneous symmetric domain, the domain he named Siegel domain, to commemorate the profound work on Siegel in 1943 of automorphic function. The results of Piatetski-Shapiro has profound impact on the theory of complex variable functions and automorphic function theory, and have a profound impact on the symmetry space theory and a series of topics. As we know, Cartan transforms the study of symmetric spaces into the study of Lie groups and Lie algebras, which is directly influenced by Klein and greatly develops the initial idea of Klein. Then it is Cartan developed the concept of Levi-Civita connection, the development of differential geometry in general contact theory, isomorphic mapping through tangent space at each point on the manifold, realize the dream of Klein and greatly promote the development of differential geometry. Cartan is the same, and concluded that the importance of the research in the holonomy manifold twists and turns, finally after his death in thirty years has proved to be correct. Here, we see the vast beauty of differential geometry.As we know, geodesic ties are associated with ODE (ordinary differential equations), minimal surfaces and high dimensional submanifolds are associated with PDE (partial differential equations). These equations are nonlinear equations, so they have high requirements for analysis. Complex PDE and complex analysis the relationship between Cauchy-Riemann equations coupling the famous function theory, in the complex case, the Cauchy- Riemann equations not only deepen the unprecedented contact and the qualitative super Cauchy-Riemann equations (the number of variables is greater than the number of equations) led to a strange phenomenon. This makes PDE and the theory ofmultiple complex functions closely integrated with differential geometry.Most of the scholars have been studying the differential geometry of the intrinsic geometry of the Gauss and Riemann extremely deep stun, by Cartan's method of moving frames is beautiful and concise dumping, by Chern's theory of characteristic classes of the broad and profound admiration, Yau deep exquisite geometric analysis skills to deter.When the young Chern faced the whole differentiation, he said he was like a mountain facing the shining golden light, but he couldn't reach the summit at one time. But then he was cast as a master in this field before Hopf and Weil.If the differential geometry Cartan development to gradually change the general relativistic geometric model, then the differential geometry of Chern et al not only affect the continuation of Cartan and to promote the development of fiber bundle in the form of gauge field theory. Differential geometry is still closely bound up with physics as in the age of Einstein and continues to acquire research topics from physicsWhy does the three-dimensional sphere not give flatness gauge, but can give conformal flatness gauge? Because 3D balls and other dimension as the ball to establish flat space isometric mapping, so it is impossible to establish a flatness gauge; and n-dimensional balls are usually single curvature space, thus can establish a conformal flat metric. In differential geometry, isometry means that the distance between the points on the manifold before and after the mapping remains the same. Whena manifold is equidistant from a flat space, the curvature of its Riemann cross section is always zero. Since the curvature of all spheres is positive constant, the n-dimensional sphere and other manifolds whose sectional curvature is nonzero can not be assigned to local flatness gauge.But there are locally conformally flat manifolds for this concept, two gauge G and G, if G=exp{is called G, P}? G between a and G transform is a conformal transformation. Weyl conformal curvature tensor remains unchanged under conformal transformation. It is a tensor field of (1,3) type on a manifold. When the Weyl conformal curvature tensor is zero, the curvature tensor of the manifold can be represented by the Ricci curvature tensor and the scalar curvature, so Penrose always emphasizes the curvature =Ricci+Weyl.The metric tensor g of an n-dimensional Riemann manifold is conformally equivalent to the flatness gauge locally, and is called conformally flat manifold. All Manifolds (constant curvature manifolds) whose curvature is constant are conformally flat, so they can be given conformal conformal metric. And all dimensions of the sphere (including thethree-dimensional sphere) are manifold of constant curvature, so they must be given conformal conformal metric. Conversely, conformally flat manifolds are not necessarily manifolds of constant curvature. But a wonderful result related to Einstein manifolds can make up for this regret: conformally conformally Einstein manifolds over 3 dimensions must be manifolds of constant curvature. That is to say, if we want conformally conformally flat manifolds to be manifolds of constant curvature, we must call Ric= lambda g, and this is thedefinition of Einstein manifolds. In the formula, Ric is the Ricci curvature tensor, G is the metric tensor, and lambda is constant. The scalar curvature S=m of Einstein manifolds is constant. Moreover, if S is nonzero, there is no nonzero parallel tangent vector field over it. Einstein introduction of the cosmological constant. So he missed the great achievements that the expansion of the universe, so Hubble is successful in the official career; but the vacuum gravitational field equation of cosmological term with had a Einstein manifold, which provides a new stage for mathematicians wit.For the 3 dimensional connected Einstein manifold, even if does not require the conformal flat, it is also the automatic constant curvature manifolds, other dimensions do not set up this wonderful nature, I only know that this is the tensor analysis summer learning, the feeling is a kind of enjoyment. The sectional curvature in the real manifold is different from the curvature of the Holomorphic cross section in the Kahler manifold, and thus produces different results. If the curvature of holomorphic section is constant, the Ricci curvature of the manifold must be constant, so it must be Einstein manifold, called Kahler- Einstein manifold, Kahler. Kahler manifolds are Kahler- Einstein manifolds, if and only if they are Riemann manifolds, Einstein manifolds. N dimensional complex vector space, complex projective space, complex torus and complex hyperbolic space are Kahler- and Einstein manifolds. The study of Kahler-Einstein manifolds becomes the intellectual enjoyment of geometer.Let's go back to an important result of isometric mapping.In this paper, we consider the isometric mapping between M and N and the mapping of the cut space between the two Riemann manifolds, take P at any point on M, and select two non tangent tangent vectors in its tangent space, and obtain its sectional curvature. In the mapping, the two tangent vectors on the P point and its tangent space are transformed into two other tangent vectors under the mapping, and the sectional curvature of the vector is also obtained. If the mapping is isometric mapping, then the curvature of the two cross sections is equal. Or, to be vague, isometric mapping does not change the curvature of the section.Conversely, if the arbitrary points are set, the curvature of the section does not change in nature, then the mapping is not isometric mapping The answer was No. Even in thethree-dimensional Euclidean space on the surface can not set up this property. In some cases, the limit of the geodesic line must be added, and the properties of the Jacobi field can be used to do so. This is the famous Cartan isometry theorem. This theorem is a wonderful application of the Jacobi field. Its wide range of promotion is made by Ambrose and Hicks, known as the Cartan-Ambrose-Hicks theorem.Differential geometry is full of infinite charm. We classify pseudo-Riemannian spaces by using Weyl conformal curvature tensor, which can be classified by Ricci curvature tensor, or classified into 9 types by Bianchi. And these things are all can be attributed to the study of differential geometry, this distant view Riemann and slightly closer to the Klein point of the perfect combination, it can be seen that the great wisdom Cartan, here you can see the profound influence of Einstein.From the Hermite symmetry space to the Kahler-Hodge manifold, differential geometry is not only closely linked with the Lie group, but also connected with algebra, geometry and topologyThink of the great 1895 Poicare wrote the great "position analysis" was founded combination topology unabashedly said differential geometry in high dimensional space is of little importance to this subject, he said: "the home has beautiful scenery, where Xuyuan for." (Chern) topology is the beauty of the home. Why do you have to work hard to compute the curvature of surfaces or even manifolds of high dimensions? But this versatile mathematician is wrong, but we can not say that the mathematical genius no major contribution to differential geometry? Can not. Let's see today's close relation between differential geometry and topology, we'll see. When is a closed form the proper form? The inverse of the Poicare lemma in the region of the homotopy point (the single connected region) tells us that it is automatically established. In the non simply connected region is de famous Rham theorem tells us how to set up, that is the integral differential form in all closed on zero.Even in the field of differential geometry ignored by Poicare, he is still in a casual way deeply affected by the subject, or rather is affecting the whole mathematics.The nature of any discipline that seeks to be generalized after its creation, as is differential geometry. From the curvature, Euclidean curvature of space straight to zero, geometry extended to normal curvature number (narrow Riemann space) andnegative constant space (Lobachevskii space), we know that the greatness of non Euclidean geometry is that it not only independent of the fifth postulate and other alternative to the new geometry. It can be the founder of triangle analysis on it. But the famous mathematician Milnor said that before differential geometry went into non Euclidean geometry, non Euclidean geometry was only the torso with no hands and no feet. The non Euclidean geometry is born only when the curvature is computed uniformly after the metric is defined. In his speech in 1854, Riemann wrote only one formula: that is, this formula unifies the positive curvature, negative curvature and zero curvature geometry. Most people think that the formula for "Riemann" is based on intuition. In fact, later people found the draft paper that he used to calculate the formula. Only then did he realize that talent should be diligent. Riemann has explored the curvature of manifolds of arbitrary curvature of any dimension, but the quantitative calculations go beyond the mathematical tools of that time, and he can only write the unified formula for manifolds of constant curvature. But we know,Even today, this result is still important, differential geometry "comparison theorem" a multitude of names are in constant curvature manifolds for comparison model.When Riemann had considered two differential forms the root of two, this is what we are familiar with the Riemann metric Riemannnian, derived from geometry, he specifically mentioned another case, is the root of four four differential forms (equivalent to four yuan product and four times square). This is the contact and the difference between the two. But he saidthat for this situation and the previous case, the study does not require substantially different methods. It also says that such studies are time consuming and that new insights cannot be added to space, and the results of calculations lack geometric meaning. So Riemann studied only what is now called Riemann metric. Why are future generations of Finsler interested in promoting the Riemann's not wanting to study? It may be that mathematicians are so good that they become a hobby. Cartan in Finsler geometry made efforts, but the effect was little, Chern on the geometric really high hopes also developed some achievements. But I still and general view on the international consensus, that is the Finsler geometry bleak. This is also the essential reason of Finsler geometry has been unable to enter the mainstream of differential geometry, it no beautiful properties really worth geometers to struggle, also do not have what big application value. Later K- exhibition space, Cartan space will not become mainstream, although they are the extension of Riemannnian geometry, but did not get what the big development.In fact, sometimes the promotion of things to get new content is not much, differential geometry is the same, not the object of study, the more ordinary the better, but should be appropriate to the special good. For example, in Riemann manifold, homogeneous Riemann manifold is more special, beautiful nature, homogeneous Riemann manifolds, symmetric Riemann manifold is more special, so nature more beautiful. This is from the analysis of manifold Lie group action angle.From the point of view of metric, the complex structure is given on the even dimensional Riemann manifold, and the complexmanifold is very elegant. Near complex manifolds are complex manifolds only when the near complex structure is integrable. The complex manifold must be orientable, because it is easy to find that its Jacobian must be nonnegative, whereas the real manifold does not have this property in general. To narrow the scope of the Kahler manifold has more good properties, all complex Submanifolds of Kahler manifolds are Kahler manifolds, and minimal submanifolds (Wirtinger theorem), the beautiful results captured the hearts of many differential geometry and algebraic geometry, because other more general manifolds do not set up this beautiful results. If the first Chern number of a three-dimensional Kahler manifold is zero, the Calabi-Yau manifold can be obtained, which is a very interesting manifold for theoretical physicists. The manifold of mirrors of Calabi-Yau manifolds is also a common subject of differential geometry in algebraic geometry. The popular Hodge structure is a subject of endless appeal.Differential geometry, an endless topic. Just as algebraic geometry requires double - rational equivalence as a luxury, differential geometry requires isometric transformations to be difficult. Taxonomy is an eternal subject of mathematics. In group theory, there are single group classification, multi complex function theory, regional classification, algebraic geometry in the classification of algebraic clusters, differential geometry is also classified.The hard question has led to a dash of young geometry and old scholars, and the prospect of differential geometry is very bright.。

复变函数知识点总结复变函数是数学中重要的概念，它在分析学、微分几何、数学物理等领域都有着广泛的应用。

本文将对复变函数的基本概念、性质和常见定理进行总结，希望能够帮助读者更好地理解和掌握复变函数的相关知识。

1. 复数与复变函数。

复数是由实部和虚部组成的数，通常表示为z=x+iy，其中x为实部，y为虚部，i为虚数单位，满足i^2=-1。

复数可以用平面上的点来表示，称为复平面，实部x对应横坐标，虚部y对应纵坐标。

复变函数是定义在复平面上的函数，通常表示为f(z)，其中z为复数变量。

2. 复变函数的导数与解析函数。

与实变函数类似，复变函数也有导数的概念，称为复导数。

如果一个函数在某点处可导，并且在该点的邻域内处处可导，那么称该函数在该邻域内解析。

解析函数具有很多良好的性质，比如在其定义域内可以展开成幂级数。

3. 共轭与调和函数。

对于复数z=x+iy，其共轭复数定义为z的实部不变，虚部取相反数，记为z=x-iy。

对于复变函数f(z)，如果它满足柯西-黎曼方程，即满足一阶偏导数存在且连续，并且满足偏导数的连续性条件，那么称f(z)为调和函数。

4. 柯西-黎曼方程与全纯函数。

柯西-黎曼方程是复变函数理论中的重要定理，它建立了解析函数与调和函数之间的联系。

柯西-黎曼方程指出，如果复变函数f(z)=u(x,y)+iv(x,y)在某点处可导，那么它满足柯西-黎曼方程，即u和v满足一阶偏导数的连续性条件。

满足柯西-黎曼方程的函数称为全纯函数，也称为解析函数。

5. 柯西积分定理与留数定理。

柯西积分定理是复变函数理论中的重要定理之一，它指出如果f(z)在闭合区域内解析，并且沿着闭合区域的边界进行积分，那么积分结果为0。

留数定理是计算闭合曲线积分的重要方法，它将积分结果与函数在奇点处的留数联系起来，从而简化了积分的计算。

6. 应用领域。

复变函数在物理学、工程学、经济学等领域都有着重要的应用，比如在电路分析中的传输线理论、振动理论中的阻尼比计算、流体力学中的势流与涡流等方面都需要用到复变函数的知识。

为什么要余切空间以三维空间为例。

对于平直空间，可以用一个坐标系（X,Y,Z）描述，X,Y,Z 代表三根坐标轴。

对于一般流形（一句话描述之：在每一点只存在一个唯一的切线或者切平面的玩意儿。

因此折线之类的东东，即使连续，也不是流形），只能采用局域坐标系来描述，此时通常在平直空间中讨论的量A，需要换成它的微分dA来代替。

我们既要跟矢量打交道，还要跟数量或者更一般地，跟标量函数打交道。

此时我们要考虑一个从矢量到函数的一个映射，这就是对矢量取内积。

一般情形下，是两个相互对偶的矢量之间才能定义内积（除非对偶矢量等同于矢量自身）。

由一组正交完备的矢量张成一个空间，则与之对偶的一组正交完备的对偶矢量张成的空间，称为对偶空间。

矢量概念可以推广到一般的抽象空间，比如平方可积的函数空间，此时矢量x,y之间的内积如果记做y(x)=<y,x>，此时y(x)又称为关于x的泛函，因此量子力学中通常所说的波函数ψ(p)=<p|ψ>，其实是Hilbert空间spaned by {|ψ>}中的泛函，也可以看作|ψ>在p表象中的具体表示。

）因此，你不难理解，为什么四维时空矢量有协变和逆变之分，二者是对偶的，它们之间的指标收缩（协变指标和逆变指标之间的收缩）对应时空矢量之间的内积。

下面用d/dx表示对x的偏微分（如此类推），而dx是x的微分（and so on）。

我们知道，对于三维空间中的坐标函数A(x,y,z)，我们有：dA=(dA/dx)dx+(dA/dy)dy+(dA/dz)dz=矢量（dA/dx，dA/dy，dA/dz）和矢量（dx，dy，dz）之间的内积。

其中（dA/dx，dA/dy，dA/dz）是A的梯度矢量，抽掉A，是梯度矢量（算子）:（d/dx，d/dy，d/dz）它与矢量（dx，dy，dz）构成一对对偶矢量。

由于在微分几何中，在局域坐标系中讨论问题，总是跟某个量的微分打交道，因此在现代微分几何中，干脆直接用坐标的偏微分（d/dx，d/dy，d/dz）或微分（dx，dy，dz）作为基矢量，来对某个矢量进行展开。