绪 论几何学是数学中一门古老的分支学科. 几何学产生于现实生产活动. “geometry ”就是“土地测量”.Pythagoras 定理和勾股定理(《周髀算经》). 数学：人类智慧的结晶，严密的逻辑系统. 以欧几里德(Euclid)的《几何原本》(Elements )为代表.《自然辩证法》和《反杜林论》：数学与哲学；数与形的统一：解析几何；坐标系：笛卡儿和费马引入.对微分几何做出突出贡献的数学家：欧拉(Euler)，蒙日(Monge)，高斯(Gauss)，黎曼(Riemann). 克莱因(Klein)关于变换群的观点. E. Cartan 的活动标架方法.微分几何：微积分，拓扑学，高等代数与解析几何知识的综合运用. 内容简介第一章：预备知识. 第二章：曲线论. 第三章至第五章：曲面论. 第六章：曲面上的曲线，非欧几何. 第七章*：活动标架和外微分.第一章 预备知识本章内容：向量代数知识复习；正交标架；刚体运动；等距变换；向量函数 计划学时：3学时难点：正交标架流形；刚体运动群；等距变换群引言为什么要研究向量函数？在数学分析中，我们知道一元函数()y f x =的图像是xy 平面上的一条曲线，二元函数(,)z f x y =的图像是空间中的一张曲面.采用参数方程，空间一条曲线可以表示成()()(),(),()r r t x t y t z t ==.这是一个向量函数，它的三个分量都是一元函数.所有这些例子中，都是先取定了一个坐标系. 所以标架与坐标是建立“形”与“数”之间联系的桥梁.§ 1.1 三维欧氏空间中的标架一、向量代数复习向量即有向线段：AB ，r ，r. 向量相等的定义：大小和方向. 零向量：0，0 . 反向量：a - . 向量的线性运算. 加法：三角形法则，多边形法则. 向量的长度. 三角不等式. 数乘.内积的定义：:||||cos (,)ab a b a b =∠外积的定义.二重外积公式：()()()a b c a c b b c a ⨯⨯=⋅-⋅ ；()()()a b c a c b a b c ⨯⨯=⋅-⋅内积的基本性质：对称性，双线性，正定性. 外积的基本性质：反对称性，双线性.二、标架仿射标架{};,,O OA OB OC. 定向标架.正交标架(即右手单位正交标架)：{};,,O i j k. 笛卡尔直角坐标系. 坐标.内积和外积在正交标架下的计算公式. 两点距离公式. 三维欧氏空间3E 和3.三、正交标架流形取定一个正交标架{};,,O i j k (绝对坐标系). 则任意一个正交标架{}123;,,P e e e被P 点的坐标和三个基向量{}123,,e e e的分量唯一确定：123111121322122233313233,,,.OP a i a j a k e a i a j a k e a i a j a k ea i a j a k ⎧=++⎪=++⎪⎨=++⎪⎪=++⎩(1.6) 其中123(,,)a a a a =可以随意取定，而(,1,2,3)ij a i j =应满足31ikjk ij k aa δ==∑， (1.7)即过渡矩阵()ij a A =是正交矩阵. 又因为123,,e e e是右手系，det 1A =，即矩阵111213212223313233(3)a a a A a a a SO a a a ⎛⎫ ⎪=∈ ⎪ ⎪⎝⎭(1.8, 1.9) 是行列式为1的正交矩阵. 我们有一一对应：{正交标架}←→3(3)E SO ⨯，{}123;,,(,)P e e e a A ←→.所以正交标架的集合是一个6维流形.四、正交坐标变换与刚体运动，等距变换空间任意一点Q 在两个正交标架{};,,O i j k 和{}123;,,P e e e中的坐标分别为(,,)x y z 和(,,)xy z ，则两个坐标之间有正交坐标变换关系式： 111213*********132333,,.x a xaya za y a xaya za z a xa ya za =+++⎧⎪=+++⎨⎪=+++⎩ (1.10) 如果一个物体在空间运动，不改变其形状和大小，仅改变其在空间中的位置，则该物体的这种运动称为刚体运动.QOPki1e j2e 3e QO()P O σ=ki1e j2e 3e ()QQ σ=在刚体运动33:E E σ→下，若σ将正交标架{};,,O i j k 变为{}123;,,P e e e，则空间任意一点(,,)Q x y z 和它的像点 (,,)Q xy z (均为在{};,,O i j k 中的坐标)之间的关系式为 111213121222323132333,,.x a xa ya za y a xa ya za za xa ya za =+++⎧⎪=+++⎨⎪=+++⎩ (1.11) 定理1.1 3E 中的刚体运动把一个正交标架变成一个正交标架；反过来，对于3E 中的任意两个正交标架，必有3E 的一个刚体运动把其中的一个正交标架变成另一个正交标架.空间3E 到它自身的、保持任意两点之间的距离不变的变换33:E E σ→称为等距变换. 刚体运动是等距变换，但等距变换不一定是刚体运动. 一般来说，等距变换是一个刚体运动，或一个刚体运动与一个关于某平面的反射的合成(复合映射).仿射坐标变换与仿射变换.§ 1.2 向量函数所谓的向量函数是指从它的定义域D 到3中的映射3::()r p r p →D .设有定义在区间[,]a b 上的向量函数()((),(),()),r t x t y t z t a t b =≤≤. 如果(),(),()x t y t z t 都是t 的连续函数，则称向量函数()r t是连续的；如果(),(),()x t y t z t 都是t 的连续可微函数，则称向量函数()r t是连续可微的. 向量函数()r t的导数和积分的定义与数值函数的导数和积分的定义是相同的，即0000()()lim t t t r t t r t drdt t∆→=+∆-=∆0000000()()()()()()lim ,,t x t t x t y t t y t z t t z t t t t ∆→+∆-+∆-+∆-⎛⎫= ⎪∆∆∆⎝⎭()000(),(),()x t y t z t '''=，0(,)t a b ∈， (2.6)()1()lim ()(),(),()nbbbbi i aaaai r t dt r t t x t dt y t dt z t dt λ→='=∆=∑⎰⎰⎰⎰， (2.7)其中01n a t t t b =<<<= 是区间[,]a b 的任意一个分割，1i i i t t t +∆=-，1[,]i i i t t t -'∈，并且{}max |1,2,,i t i n λ=∆= . (由向量加法和数乘的定义可以得到)向量函数的求导和积分归结为它的分量函数的求导和积分，向量函数的可微性和可积性归结为它的分量函数的可微性和可积性.由(1.6)可得()()()()()(),()()()()()()a t b t a t b t t at t a t t a t λλλ''''''+=+=+. 定理2.1 (Leibniz 法则) 假定(),(),()a t b t c t是三个可微的向量函数，则它们的内积、外积、混合积的导数有下面的公式：(1) ()()()()()()()a t b t a t b t a t b t '''⋅=⋅+⋅；(2) ()()()()()()()a t b t a t b t a t b t '''⨯=⨯+⨯；(3) ()()()()(),(),()(),(),()(),(),()(),(),()a t b t c t a t b t c t a t b t c t a t b t c t ''''=++.定理2.2 设()a t是一个处处非零的连续可微的向量函数，则 (1) 向量函数()a t 的长度是常数当且仅当()()0a t a t '⋅≡. (2) 向量函数()a t的方向不变当且仅当()()0a t a t '⨯≡.(3) 设()a t 是二阶连续可微的. 如果向量函数()a t与某个固定的方向垂直，那么 ()(),(),()0a t a t a t '''≡. 反过来，如果上式成立，并且处处有()()0a t a t '⨯≠，那么向量函数()a t必定与某个固定的方向垂直.证明 (1) 因为()()22()()()()|()|a t a t a t a t a t '''== ，所以|()|a t 是常数2|()|a t ⇔是常数()()0a t a t '⇔⋅≡.(2) 因为()a t 处处非零，取()a t方向的单位向量1()|()|()b t a t a t -= . 则()()()a t f t b t = ，其中()|()|f t a t =连续可微. 于是()()2()()()()()()()()()()(),.a t a t f t b t f t b t f t b t f t b t b t t ''''⨯=⨯+=⨯∀“⇒”由条件知()b t c = 是常向量，()0b t c ''== . 从而()()0a t a t '⨯≡.“⇐”由条件得()()0b t b t '⨯≡，所以()b t ，()b t ' 处处线性相关. 因为()b t 是单位向量，处处非零，所以()()()b t t b t λ'= . 用()b t 作内积，得()12()()()()()0t b t bt b t b t λ''=⋅=⋅≡ . 于是()0b t '≡ ，()b t c =是常向量.(3) 设向量函数()a t与某个固定的方向垂直，那么有单位常向量1e 使得1()0a t e ⋅≡ . 求导得到1()0a t e '⋅≡ ，1()0a t e ''⋅≡ . 从而(),(),()a t a t a t '''共面，()(),(),()0a t a t a t '''≡ .反之，设()(),(),()0a t a t a t '''≡ . 令()()()b t a t a t '=⨯. 由条件，()b t 处处非零. 且()b t '= ()()a t a t ''⨯连续. 根据二重外积公式，()()()()()()()()()()()(),(),()()(),(),()()(),(),()()0.b t b t a t a t a t a t a t a t a t a t a t a t a t a t a t a t a t a t ''''⨯=⨯⨯⨯''''''=-'''=≡根据已经证明的(2)，()b t 的方向不变. 设这个方向为1e . 则1()|()|b t b t e = . 用()a t作内积，得()1|()|()()()()()()0b t a t e a t b t a t a t a t '⋅=⋅=⋅⨯≡.由于()b t 处处非零，得到1()0a t e ⋅≡ ，即()a t与固定方向1e 垂直. □课外作业： 1. 证明定理2.1.2. 设33:E E σ→为等距变换. 在3E 中取定一个正交标架{};,,O i j k . 令3 为3E 中全体向量构成的向量空间. 定义映射33::()()AB A B σσ→ . 如果()O O σ=，证明 是线性映射.3. 设向量函数()r t 有任意阶导(函)数. 用()()k r t 表示()r t 的k 阶导数，并设()(1)()()k k r t r t +⨯处处非零. 试求()()(1)(2)(),(),()0k k k r t r t r t ++≡的充要条件.第二章 曲线论本章内容：弧长，曲率，挠率；Frenet 标架，Frenet 公式；曲线论基本定理 计划学时：14学时，含习题课3学时. 难点：曲线论基本定理的证明§ 2.1 参数曲线三维欧氏空间3E 中的一条曲线C 是一个连续映射3:[,]p a b E →，称为参数曲线. 几何上，参数曲线C 是映射p 的象.取定正交标架{};,,O i j k，则曲线上的点()([,])p t t a b ∈与它的位置向量()Op t 一一对应. 令()()r t Op t =. 则()()()()((),(),())r t x t i y t j z t k x t y t z t =++=，[,]t a b ∈， (1.3)其中t 为曲线的参数，(1.3)称为曲线的参数方程.由定义可知()()01()lim (),(),()()()t r t x t y t z t r t t r t t∆→''''==+∆-∆，(,)t a b ∈. (1.4)如果坐标函数(),(),()x t y t z t 是连续可微的，则称曲线()r t是连续可微的. 此概念与标架的取法无关.(为什么？)导数()r t '的几何意义：割线的极限位置就是曲线的切线.如果()0r t '≠ ，则()r t '是该曲线在()r t 处的切线的方向向量，称为该曲线的切向量. 这样的点称为曲线的正则点. 曲线在正则点的切线方程为()()()X u r t ur t '=+， (1.5) 其中t 是固定的，u 是切线上点的参数，()X u是切线上参数为u 的点的位置向量.定义. 如果()r t是至少三次以上的连续可微向量函数，并且处处是正则点，即对任意的t ，()0r t '≠ ，则称曲线()r t是正则参数曲线. 将参数增大的方向称为曲线的正向.上述定义与3E 中直角坐标系的选取无关. 正则曲线：正则参数曲线的等价类.曲线的参数方程中参数的选择不是唯一的. 在进行参数变换时，要求参数变换()t t u =满足：(1)()t u 是u 的三次连续可微函数；(2) ()t u '处处不为零. 这样的参数变换称为可允许的参数变换. 当()0t u '>时，称为保持定向的参数变换.根据复合函数的求导法则，[]()(())()()d d du dt t t u r t u r t t u ='=⋅ .这种可允许的参数变换在所有正则参数曲线之间建立了一种等价关系. 等价的正则参数曲线看作是同一条曲线，称为一条正则曲线. 以下总假定()r t是正则曲线.如果一条正则参数曲线只允许作保持定向的参数变换，则这样的正则参数曲线的等价类被称为是一条有向正则曲线. (返回Frenet 标架)例1.1 圆柱螺线()(cos ,sin ,),()r ta t a t bt t =∈ ，其中,ab 是常数，0a >.()()sin ,cos ,r t a t a t b '=- ，|()|0()0r t r t ''=>⇒≠所以圆柱螺线是正则曲线.例1.2 半三次曲线32()(,),()r t t t t =∈.2()(3,2)r t t t '= ，(0)0r '= .这条曲线不是正则曲线.连续可微性和曲线的正则性（光滑性）是不同的概念. （与数学分析中的结论比较） 平面曲线的一般方程()y f x =和隐式方程(,)0F x y =. 空间曲线的一般方程(),()y f x z g x == (1.6)和隐式方程(,,)0,(,,)0.F x y zG x y z =⎧⎨=⎩ (1.8) 这些方程可以化为参数方程. (习题4：正则曲线总可以用一般方程表示)曲线(1.8)的切线方向，正则性. 课外作业：习题2，5§ 2.2 曲线的弧长设3E 中一条正则曲线C 的方程为(),[,]r r t t a b =∈. 则|()|b as r t dt '=⎰(2.1)是该曲线的一个不变量，即它与正交标架的选取无关，也与曲线的可允许参数变换无关.不变量s 的几何意义是该曲线的弧长，因为1max||01|()|lim|()()|i nbi i at i s r t dt r t r t +∆→='==-∑⎰.其中01n a t t t b =<<<= 是区间[,]a b 的任意一个分割，1i i i t t t +∆=-，max λ={|1,i t i ∆=2,,n . (为什么？)令()|()|t as t r d ττ'=⎰. (2.4)则()s s t =是曲线C 的保持定向的可允许参数变换，称为弧长参数. 它是由曲线本身确定的，至多相差一个常数，与曲线的坐标表示和参数选择都是无关的. 因此任何正则曲线都可以采用弧长s 作为参数，当然，允许相差一个常数.注意|()|ds r t dt '=也是曲线的不变量，称为曲线的弧长元素(或称弧微分).虽然理论上任何正则曲线都可以采用弧长参数s ，但是具体的例子中，曲线都是用一般的参数t给出的. 由(2.4)，即使|()|r t '是初等函数，()s t 也不一定是初等函数. 下面的定理给出了判别一般参数是否是弧长参数的方法.定理 2.1 设(),[,]r r t t a b =∈是3E 中一条正则曲线，则t 是它的弧长参数的充分必要条件是|()|1r t '=. 即t 是弧长参数当且仅当(沿着曲线C )切向量场是单位切向量场.证明. “⇐”由(2.4)可知，s t a =-. “⇒”如果t 是弧长参数，则s t =，从而|()|1ds r t dt '==. □以下用“﹒”表示对弧长参数s 的导数，如()r s ，()r s 等等，或简记为,rr 等等. 而“'”则用来表示对一般参数t 的导数.课堂练习：4课外作业：习题1，2(1)，3.§ 2.3 曲线的曲率和Frenet 标架设曲线C 的方程为()r r s =，其中s 是曲线的弧长参数. 令()()s r s α=. (3.1) 对于给定的s ，令θ∆是()s α 与()s s α+∆之间的夹角，其中0s ∆≠是s 的增量.定理3.1 设()s α 是曲线()r r s = 的单位切向量场，s 是弧长参数. 用θ∆表示向量()s s α+∆与()s α之间的夹角，则lim|()|ss s θα∆∆∆→= . (3.2) 证明. ()001||lim lim ()()s s d s s s ds s s ααααα∆→∆→∆===+∆-∆∆()()2200022sin sin lim lim lim ||s s s s s s θθθθθ∆∆∆∆→∆→∆→∆∆===∆∆∆， ()r s 0s =图2-5O()s αs L=()s s α+∆()r s s +∆()s s α+∆()s α()()s s s αα+∆-θ∆因为θ∆=定义 )为该曲线的曲率向量.把曲线C . 其方程就是(3.3)当然，s (3.4) 所以(3.5) 即曲率κ由|()s α 如果在一点s 处()0s κ≠. 于是在该点有(3.6) 在()s κ (3.7)}),()s s γ ，称为曲线在该点的Frenet 标架(见图2-2). 它的确定不受曲线的保持定向的参数变换的影响.注意. 如果在一点0s 处0()0s κ=，则一般来说无法定义在该点的Frenet 标架. 1. 若()0s κ≡，则C 是直线，可以定义它的Frenet 标架.2. 若0s 是κ的孤立零点, 则在0s 的两侧都有Frenet 标架. 如果00()()s s ββ-+=，则可以将Frenet 标架延拓到0s 点.3. 在其他的情况下将曲线分成若干段来考察.切线、主法线和次法线，法平面、从切平面和密切平面，以及它们的方程.切线：()()()u r s u s ρα=+；主法线：()()()u r s u s ρβ=+ ；次法线：()()()u r s u s ργ=+法平面：[()]()0X r s s α-= ；从切平面：[()]()0X r s s β-= ；密切平面：[()]()0X r s s γ-=在一般参数t 下，曲率κ和Frenet 标架的计算方法.3|()()|()|()|r t r t t r t κκ'''⨯==' ，()|()|r t r t α'=' ，()()|()()|r t r t r t r t γ'''⨯='''⨯，βγα=⨯ . (3.13) 证明. 设()s s t =为弧长参数，()t t s =为其反函数. 则由(2.4)，()|()|ds s t r t dt''==. 故(())()()()|()|(())()(),():(())|()|dr s t ds t r t r t r t s t s t t s t ds dt r t αααα''''====='. (3.12) 由曲率κ的定义，||0κα=≥ ，可知主法向量||αβα= 满足ακβ= . 上式再对t 求导，得 2d d ds r s s s s s s dt ds dtααααακβ'''''''''''=+=+=+.于是2333()()||r r s s s s s r r s αακβκαβκγκ'''''''''''''⨯=⨯+=⨯=⇒⨯= .所以33|()()||()()||()|r t r t r t r t s r t κ''''''⨯⨯==''. 代入上式得()()|()()|r t r t r t r t γ'''⨯='''⨯. □ 例3.1 求圆柱螺线()(cos ,sin ,),()r t a t a t bt t =∈的曲率和Frenet 标架，其中0a >.解. ()r t 'r ' 所以例3.2 .解法1. 22t k ππ=+于是当/t π=r 所以在1)-，γ=解法2. 对应的参数为0s =. 则有 (0)((0),(0),(0))(0,0,1)r x y z ==， (1)以及22222222()()()1,(,).()()()0,()()()1,x s y s z s s x s y s x s xs y s z s εε⎧++=⎪∀∈-+-=⎨⎪++=⎩ (3.14) 求导得到()()()()()()0,2()()2()()()0,()()()()()()0.x s x s y s y s z s z s x s x s y s y s x s x s x s y s y s z s z s ++=⎧⎪+-=⎨⎪++=⎩(3.15) 令0s =，由(1)和上述方程组得到(0)(0)0xz == ，(0)1y =± . 通过改变曲线的正方向，可设(0)1y= ，于是 (0)((0),(0),(0))(0,1,0)xy z α==. (3.16) 对(3.15)前两式再求导，利用(3.14)得22()()()()()()1,2()()2()2()()2()()0.x s x s y s y s z s z s x s x s x s y s y s y s x s ++=-⎧⎨+++-=⎩(3.17) 令0s =，由(3.15)和(3.16)得(0)0y= ；由(1)和(3.17)第1式得(0)1z =- ；再由(3.17)第2式得(0)2x = . 所以(0)(0)((0),(0),(0))(2,0,1)r x y zα===-. 由此得(0)(0,0,1)r =处的曲率(0)|(0)|κα== ，Frenet 标架为：(0)(0,0,1)r = ；(0)(0,1,0)α=，1(0)(0)(0)1)κβα==-，(0)(0)(0)1,0,2)γαβ=⨯=-- . □课外作业：习题1(2,4)，4，7§ 2.4 曲线的挠率和Frenet 公式密切平面对弧长s 的变化率为||γ，它刻画了曲面偏离密切平面的程度，即曲线的扭曲程度. 定义4.1 函数τγβ=-⋅ ，即()()()s s s τγβ=-⋅ 称为曲线的挠率.注. 由0γγ⋅= ，()0γαγαγκβ⋅=-⋅=-⋅= 可知//γβ . 因此可设γτβ=- ， (4.1)从而||||τγ= ，即挠率的绝对值刻画了曲线的扭曲程度. 定理4.1 设曲线C 不是直线，则C 是平面曲线的充分必要条件是它的挠率0τ≡.证明. 设曲线C 的弧长参数方程为()r r s =，[0,]s L ∈. 因为C 不是直线，0κ≠(见定理3.2 )，存在Frenet 标架{};,,r αβγ.“⇒” 设C 是平面曲线，在平面:()0X a n ∏-= 上，其中a是平面上一个定点的位置向量，n 是平面的法向量，a 和n均为常向量. 则有(())0,[0,]r s a n s L -=∀∈.求导得()0,()()0()0,s n s s n s n s ακββ==⇒=∀.于是()//s n γ , 由于|()|||1s n γ== ，所以()s n γ=± 是常向量，从而0γ≡ ，||||0τγ=≡ . 即有0τ≡.“⇐”设0τ≡. 由(4.1)得0γτβ=-= . 所以()0s c γ=≠ 是常向量. 由(())()()()0d r s c r s c s s ds αγ=== 可知()r s c是一个常数，即0()()r s c r s c = ，其中0[0,]s L ∈是固定的. 于是曲线C 上的点满足平面方程0[()()]0r s r s c -= ，其中0()r s 是平面上一个定点的位置向量，c是平面的法向量. □设正则曲线C 上存在Frenet 标架. 对Frenet 标架进行求导，得到Frenet 公式,,,.r αακββκατγγτβ⎧=⎪=⎪⎨=-+⎪⎪=-⎩(4.8) 上式中的后三式可以写成矩阵的形式00000ακαβκτβγτγ⎛⎫⎛⎫⎛⎫⎪ ⎪ ⎪=- ⎪ ⎪ ⎪ ⎪⎪ ⎪- ⎪⎝⎭⎝⎭⎝⎭. (4.9) 作为Frenet 公式的一个应用，现在来证明定理4.2 设曲线()r r s =的曲率()s κ和挠率()s τ都不为零，s 是弧长参数. 如果该曲线落在一个球面上，则有222111d a ds κτκ⎡⎤⎛⎫⎛⎫+= ⎪ ⎪⎢⎥⎝⎭⎝⎭⎣⎦， (4.10) 其中a 为常数.证明. 由条件，设曲线所在的球面半径是a ，球心是0r，即有()220()rs r a -= . (4.11)求导得到()0()()0rs r s α-=. 这说明0()r s r - 垂直于()s α，可设 0()()()()()r s r s s s s λβμγ-=+. (4.12)再求导，利用Frenet 公式得()()()()[()()()()]()()()()()s s s s s s s s s s s s s αλβλκατγμγμτβ=+-++-. 比较两边,,αβγ的系数，得1λκ=-，λμτ= ，μλτ=- ， (4.13) 其中略去了自变量s . 所以1λκ=-，111d d ds ds λλμτττκ⎛⎫===- ⎪⎝⎭. (4.14)将(4.12)两边平方可得()22220r r a λμ+=-=，再将(4.14)代入其中，即得(4.10). □注记 由证明过程中的(4.13)第3式还可得110d d ds ds τκτκ⎡⎤⎛⎫+= ⎪⎢⎥⎝⎭⎣⎦. (4.16) 在一般参数下挠率的计算公式.2(,,)||r r r r r τ''''''='''⨯ . (4.18)证明. 因为()|()|ds s t r t dt''==，利用Frenet 公式，有 ()()(())ds dr r t s t s t dt dsα''==，2()()(())()(())(())r t s t s t s t s t s t ακβ'''''=+，23(())()()(())3()()(())(())()(())()(())[(())(())(())(())].d s t r t s t s t s t s t s t s t s t s t dts t s t s t s t s t s t κακββκκατγ''''''''''=++'+-+于是3()()()(())(())r t r t s t s t s t κγ''''⨯= ，从而()362()()()()(())(())()(),(),()()(())(()).r t r t r t s t s t s t r t rt r t r t s t s t s t κγκτ''''''''''''''''=⨯⋅=⋅'=由(3.13)可知622()(())|()()|s t s t r t r t κ''''=⨯ ，代入上式即得(4.18). □定理4.3 曲线()r r t = 是平面曲线的充要条件是(,,)0r r r ''''''=. □例 求圆柱螺线()(cos ,sin ,)r t a t a t bt =的挠率.解. ()(sin ,cos ,)r t a t a t b '=- ，()(cos ,sin ,0)r t a t a t ''=-，|()|r t '=2(sin ,cos ,)(sin ,cos ,)r r ab t ab t a a b t b t a '''⨯=-=-，||r r '''⨯=()(sin ,cos ,0)r t a t t '''=-所以2(,,)r r r a b ''''''= ，22b a b τ=+. □课外作业：习题1(2, 4)，4，10§ 2.5 曲线论基本定理已经知道正则参数曲线的弧长、曲率、挠率是曲线的不变量，与坐标系取法及保持定向的参数无关，都是曲线本身的内在不变量. 在空间的刚体运动下，弧长、曲率、挠率保持不变(为什么？). 反之，这三个量也是曲线的完备不变量系统，对确定空间曲线的形状已经足够了，即有定理 5.1 (唯一性定理) 设111222:(),:()C r r s C r r s ==是3E 中两条以弧长s 为参数的正则参数曲线，[0,]s l ∈. 如果它们的曲率处处不为零，且有相同的曲率函数和挠率函数，即12()()s s κκ=，12()()s s ττ=，则有3E 中的一个刚体运动σ将1C 变成2C .证明 选取3E 中的刚体运动σ将2C 在0s =处的Frenet 标架{}2222(0);(0),(0),(0)r αβγ变为1C 在0s =处的Frenet 标架{}1111(0);(0),(0),(0)r αβγ. 则这个刚体运动σ将2C 变为正则曲线3C .设3C 的弧长参数方程为33()r r s =. 由于在刚体运动下，弧长、曲率、挠率保持不变，1C 与3C 也有相同的曲率和挠率函数：13()()s s κκ=，13()()s s ττ=.且在0s =处它们有相同的Frenet 标架：13131313(0)(0),(0)(0),(0)(0),(0)(0).r r ααββγγ====令{}1111();(),(),()r s s s s αβγ 和{}3333();(),(),()r s s s s αβγ分别为1C 和3C 的Frenet 标架. 则它们都满足一阶线性常微分方程组初值问题,,,.r αακββκατγγτβ⎧=⎪=⎪⎨=-+⎪⎪=-⎩(5.6) 1111(0)(0),(0)(0),(0)(0),(0)(0).r r ααββγγ=⎧⎪=⎪⎨=⎪⎪=⎩(5.7)根据解的唯一性(见附录定理1.1)，有13()()r s r s =，即1C 与3C 重合. □注 常微分方程组(5.6)中，共有12个未知函数：()()(),(),()r s x s y s z s =，()123()(),(),()s s s s αααα= ， ()123()(),(),()s s s s ββββ= ，()123()(),(),()s s s s γγγγ=.初始条件为：()1123(0)(,,)(0),(0),(0)r a a a x y z ==，()123111213(0),(0),(0)(,,)a a a ααα=，()123212223(0),(0),(0)(,,)a a a βββ=，()123313233(0),(0),(0)(,,)a a a γγγ=.定理5.2设111222:(),:()C r r t C r r u ==是3E 中两条正则参数曲线，它们的曲率处处不为零.如果存在三次以上的连续可微函数()u t λ=([,]t a b ∈)，()0t λ'≠，使得这两条曲线的弧长函数、曲率函数和挠率函数之间满足121212()(()),()(()),()(())s t s t t t t t λκκλττλ===， (5.4) 则有3E 中的一个刚体运动σ将1C 变成2C .证明 不妨设()0t λ'>. 对2C 作可允许参数变换()u t λ=，可将2C 的参数方程写成32()(())r t r t λ=. 则1C 的弧长为11()|()|t a s t r d ξξ'=⎰ ，2C 的弧长为 ()23322()()|()||()|(())()t t t a a a dr s t r d d d s t r duλλξξλξξηλη'''====⎰⎰⎰.由条件，可取132()()()s s t s t s t λ=== 作为1C 和2C 的弧长参数. 因为13()()s t s t =有相同的反函数()t s μ=，即111111322()s s s s μλλ-----==== ，12s λμ-= . 于是 1111112222()()()()()()s s s s s s s s κκκμκλμκκ--≡===≡ .同理，21()()s s ττ= 根据定理5.1，有3E 中的一个刚体运动σ将1C 变成2C . □定理5.3 (存在性定理) 设(),()s s κτ是定义在区间[,]a b 上的任意二个给定的连续可微函数，并且()0s κ>. 则除了相差一个刚体运动之外，存在唯一的3E 中的正则曲线:()C r r s =，[,]s a b ∈，使得s 是C 的弧长参数，且分别以给定的函数()s κ和()s τ为它的曲率和挠率.证明 唯一性由定理5.1即得. 只要证明存在性.考虑含有12个未知函数的一阶线性常微分方程组初值问题(5.6)，(5.7).：,,,.dr ds d ds d ds d dsαακββκατγγτβ⎧=⎪⎪⎪=⎪⎨⎪=-+⎪⎪⎪=-⎩(5.6) 0000(0),(0),(0),(0).r r ααββγγ=⎧⎪=⎪⎨=⎪⎪=⎩ (5.7)根据解的唯一存在定理(见附录定理1.1)，对任意给定的初始条件(5.7)，(5.6)都有定义在区间[,]a b 上[,]a b 11[,]a b [0,]l λ1s 2s1κ2κμ的解. 取(5.6)的满足初始条件(0)0,(0),(0),(0)r i j k αβγ====(5.7)’的解，其中{};,,O i j k是一个正交标架(即右手单位直角标架). 为了使用求和号，记123,,,ij i j e e e g e e αβγ====, (5.9)11121321222331323300000a a a a a a a a a κκττ⎛⎫⎛⎫⎪ ⎪=- ⎪ ⎪-⎝⎭⎝⎭. (5.5) 因为123,,,r e e e 是(5.6)的解，所以()r r s = 是三阶连续可微的. 下面来证明()r r s =就是所要求的曲线. 由(5.6)可得311,,1,2,3i ij j j de dr e a e i dsds ====∑(5.6)’ 首先来证明(),,1,2,3ij ij g s i j δ==. (5.10)由(5.6)得333111()()iji j j i j i ik k j jk i k ik kj jk ki k k k dg d e e de de e e a e e a e e a g a g ds ds ds ds =====+=+=+∑∑∑， 由初始条件(5.7)’可知有(0)(0)(0)ij i j ij g e e δ==，,1,2,3i j =. 这说明9个函数()ij g s 满足一阶线性常微分方程组初值问题31()ij ik kj jk ki k dF a F a F ds==+∑，(0)ij ij F δ=，,1,2,3i j =.另一方面由(5.5)可知ij ji a a =-，,1,2,3i j =. 于是9个函数()ij ij F s δ=也满足上面的一阶线性常微分方程组初值问题. 由解的唯一性，必有()()ij ij ij g s F s δ==.因此123(),(),()e s e s e s 是两两正交的单位向量. 从而混合积()123(),(),()1e s e s e s =±. 但是函数()123()(),(),()f s e s e s e s = 是连续的，并且由初始条件得()123(0)(0),(0),(0)1f e e e ==. 所以123(),(),()e s e s e s构成右手系.现在，由(5.6)’可知11dr e ds==. 所以()r r s = 是正则曲线，并且s 是:()C r r s = 的弧长参数，1()()s e s α=是C 的单位切向量场. 由(5.6)第2式及()0s κ>可知C 的曲率为()s κ，主法向量场为2()()s e s β=. 最后，因为123(),(),()e s e s e s 是右手单位正交基，所以3()()s e s γ= 是次法向量场. 再由(5.6)第3式可知C 的挠率为()()()s s s γβτ-= . □例 求曲率和挠率分别是常数00κ>，0τ的曲线C 的参数方程.解 我们已经知道圆柱螺线()(cos ,sin ,)r t a t a t bt =的曲率和挠率都是常数，分别为22aa b +和22b a b +. 根据定理 5.1，曲线C 一定是圆柱螺线. 由022a a b κ=+和022ba bτ=+解出02200a κκτ=+，02200b τκτ=+. 因此所求曲线C 的参数方程为()00022001()cos ,sin ,r t t t t κκτκτ=+ .因为C的弧长参数s ==t就可得到C 的弧长参数方程：))()00022001()cos ,sin,r s κκτκτ=+ . □课外作业：习题1，4，6§ 2.6 曲线参数方程在一点的标准展开对于定义在区间[,]a b 上的n 次连续可微的函数()f x ，可以在区间(,)a b 内任意一点0x 邻近展开为Taylor 展式：2()11000000002!!()()()()()()()()()n n n n f x f x f x x x f x x x fx x x o x x '''=+-+-++-+- . 同样，对于一条三次连续可微的弧长参数曲线(),(,)r r s s εε=∈-，可在0s =处展开为 233112!3!()(0)(0)(0)(0)()r s r sr s r s r o s =++++ ， (6.1) 其中3()o s是一个向量函数，满足330()lim 0s o s s→=. (6.2) 由Frenet 公式可得 2(0)(0),(0)(0)(0),(0)(0)(0)(0)(0)(0)(0)(0)r r r ακβκακβκτγ===-++ (6.3)代入(6.1)得23233300000()(0)(0)(0)(0)()6266r s r s s s s s o s κκκκταβγ⎛⎫⎛⎫=+-++++ ⎪ ⎪⎝⎭⎝⎭ ，其中000(0),(0),(0)κκκκττ=== . 以0s =处的Frenet 标架{}(0);(0),(0),(0)r αβγ 建立右手直角坐标系，则曲线C 在0s =附近的参数方程为2330123300233003(),6(),26().6x s s o s y s s o s z s o s κκκκτ⎧=-+⎪⎪⎪=++⎨⎪⎪=+⎪⎩(6.4) 上式称为曲线:()C r r s =在0s =处的标准展开式.在标架{}(0);(0),(0),(0)r αβγ下，考虑C 的近似曲线232300000011:(),,(0)(0)(0)(0)2626C r s s s s r s s s κκτκκταβγ⎛⎫=≡+++ ⎪⎝⎭. (6.5)近似曲线1C 与原曲线C 在0s =处有相同的Frenet 标架{}(0);(0),(0),(0)r αβγ，有相同的曲率0κ和相同的挠率0τ. 这是因为s 是1C 的一般参数，并且1(0)(0,0,0)(0)r r ==，1(0)(1,0,0)(0)r α'== ，100(0)(0,,0)(0)r κκβ''==，10000(0)(0,0,)(0)rκτκτγ'''== ， 从而1(0)1r '= ，111(0)(0)(0)(0)r r αα'==' ，()1100(0)(0)(0)(0)(0)r r ακβκγ'''⨯=⨯=，110(0)(0)r r κ'''⨯=，111031(0)(0)(0)(0)r r r κκ'''⨯==' ，11111(0)(0)(0)(0)(0)(0)r r r r γγ'''⨯=='''⨯ ， 111(0)(0)(0)(0)(0)(0)βγαγαβ=⨯=⨯=，2111001022011(0)(0)(0)(0)(0)(0)r r r r r κτττκ''''''⨯⋅==='''⨯ . 在0s =邻近，近似曲线1C 的性状近似地反映了原曲线C 的性状. 近似曲线1C 的图形见下图，其在各坐标平面上的投影见书上图2-6.在密切平面上的投影是抛物线：20,,02x s y s z κ===，在从切平面上的投影是三次曲线：300,0,6x s y z s κτ===，在法平面上的投影是半三次曲线：230000,,26x y s z s κκτ===.定义 设两条弧长参数曲线111222:(),:()C r r s C r r s ==相交于0p ，012(0)(0)Op r r == . 取1122,p C p C ∈∈，使得 0102p p p p s ==∆. 若有正整数n 使得121200|||()()|lim lim 0n n s s p p r s r s s s ∆→∆→∆-∆==∆∆ ，1210|()()|lim 0n s r s r s s +∆→∆-∆≠∆， (6.9) 则称1C 与2C 在0p 处有n 阶切触.定理6.1 设两条弧长参数曲线111222:(),:()C r r s C r r s ==在0s =处相交. 则它们在0s =处有n 阶切触的充分必要条件是()()12(0)(0)k k r r =，1,2,,k n = ，(1)(1)12(0)(0)n n r r ++≠ . (6.10)证明 在0s =处，有0s s s ∆=-=. 因为12,C C 在0s =处相交，所以12(0)(0)r r =. 根据Taylor 公式，12()()12121()()()(0)(0)!kn n k k k s r s r s o s r r k ++=-=+⎡⎤-⎣⎦∑ . 充分性. 由(6.10)，12(1)(1)1212()()()(0)(0)(1)!n n n n s r s r s o s r r n ++++-=+⎡⎤-⎣⎦+ ，所以 2(1)(1)12121210001()||()()lim lim lim ||0(0)(0)(1)!n n n n n n s s s o s p p r s r s s r r n s s s++++∆→→→-===+⎡⎤-⎣⎦+∆， 2(1)(1)1212121110001()||()()lim lim lim 0(0)(0)(1)!n n n n n n s s s o s p p r s r s r r n s s s++++++∆→→→-==≠+⎡⎤-⎣⎦+∆. 即12,C C 在0s =处有n 阶切触.必要性. 由条件，12,C C 在0s =处有n 阶切触，则1n ≥. 如果12(0)(0)r r ''≠ ，则12121200||()()lim lim 0(0)(0)s s p p r s r s r r s s∆→→-''==>-∆， 从而120||lim0ns p p s ∆→≠∆，矛盾. 设1m ≥是满足()()12(0)(0)k k r r = ，1,2,,k m = ，(1)(1)12(0)(0)m m r r ++≠的正整数. 由充分性，12,C C 在0s =处有m 阶切触. 由条件得m n =，故(6.10)成立. □ 推论 (1) 一条曲线与它在一点的Taylor 展开式中的前1n +项之和(即略去()ns ∆的高阶无穷小)至少有n 阶切触；与它在一点的切线至少有1阶切触；与它在一点的近似曲线至少有2阶切触. (2) 两条相交曲线在交点处有二阶以上切触的充分必要条件是这两条曲线在该点处相切，且有相同的有向密切平面和相同的曲率.曲率圆(密切圆)：在弧长参数曲线:()C r r s = 上一点()r s处的密切平面上，以曲率中心1()()()r s s s βκ+ 为圆心，以曲率半径1()R s κ=为半径的圆. 它的方程是：()11()()()cos ()sin ()()()X t r s s t s t s s s βαβκκ=+++ . 曲线与曲面的切触阶，密切球面，曲率轴. (略) 课外作业：习题2，3§2.7 存在对应关系的曲线偶设两条正则参数曲线111222:(),:()C r r t C r r u ==之间存在一个一一对应关系()t u t ↔=，()0u t '≠. 对曲线2C 作参数变换，可设222:()C r r t =，从而12,C C 之间的一一对应就是参数相同的点之间的一一对应.定义7.1 如果两条互不重合的曲线12,C C 之间存在一个一一对应，使得它们在对应点有公共的主法线，则称这两条曲线为Bertrand 曲线偶，其中每一条曲线称为另一条曲线的侣线，或共轭曲线.事实上，因为，所以，. 另一方面由可知. 因此//n α . 设rn κα=. 于是C 的曲率 ()()|()||||()|||(),()rs s n s x s y s κακ=====. 当常数λ充分小时，1()[1()]()0r r s s s λκα'=+≠ ，所以1C 是正则参数曲线. 因为0λ≠，所以曲线C 和1C 不重合.现在来证明在对应点C 和1C 有相同的主法线. 在相同的参数s 点处，C 的主法线l 是过()r s(的终)点且垂直于()s α 的直线，所以l 的方程为()()()X u r s un s =+，u ∈ .同理，在相同的参数s 点处，1C 的主法线1l 是过1()r s 点且垂直于1()//()r s s α' 的直线. 所以1//l l (因为它们都垂直于()s α ). 由定义可知1()r s在直线l 上，所以l 与1l 重合. □下面考虑空间挠曲线，即挠率0τ≠的曲线.定理7.1 设1C 和2C 是Bertrand 曲线偶. 则1C 和2C 在对应点的距离是常数，并且1C 和2C 在对应点的切线成定角.证明 设曲线1C 的弧长参数方程为11()r r s = ，Frenet 标架为{}1111();(),(),()r s s s s αβγ，曲率和挠率分别为1()s κ和1()s τ. 因为1C 和2C 之间存在一一对应，设2C 上与1()r s 对应的点是22()r r s = ，s 是2C 的一般参数，2C 的Frenet 标架为{}2222();(),(),()r s s s s αβγ，曲率和挠率分别为2()s κ和2()s τ. 再设2C 的弧长参数为()ss s = . 由条件，2()r s 在曲线1C 上的点1()r s 处的主法线11()()()X u r s u s β=+上，所以()121//()()()s r s r s β-，并且12()()s s ββ=± . 因此可设211()()()()r s r s s s λβ=+，21()()s s βεβ= ， (7.3)其中1ε=±是常数，()121()()()()s s r s r s λβ=-是可微函数.将(7.3)两边对s 求导，利用Frenet 公式，得21111()()()()()()[()()()()]ss s s s s s s s s s ααλβλκατγ''=++-+111[1()()]()()()()()()s s s s s s s s λκαλβλτγ'=-++. (7.4)以21βεβ=分别与上式两边作内积，可得()0s λ'=，()s c λ=是常数. 再由(7.3)得211|()()||()()|||r s r s s s c λβ-==，即1C 和2C 在对应点的距离是常数||(0c >，因为1C 和2C 不重合).设12()((),())s s s θαα=∠ ，则()12()()cos ()s s s ααθ=. 因为()112212122211120d ss dsκβακαβεκβαεκαβαα''=+=+=， 所以()cos ()s θ是常数，从而()s θ是常数. □定理7.2 设正则曲线C 的曲率κ和挠率τ都不为零. 则C 是Bertrand 曲线的充分必要条件是：存在常数,λμ，且0λ≠，使得1λκμτ+=.证明 必要性. 设曲线C 有侣线1C ，它们的参数方程分别是()r s 和1()r s，其中s 是C 的弧长参数. 如同定理7.1的证明过程一样，设{}();(),(),()r s s s s αβγ和{}1111();(),(),()r s s s s αβγ分别是C和1C 的Frenet 标架，11,κτ分别是1C 的曲率和挠率，s是1C 的弧长参数. 现在(7.3)和(7.4)分别成为 1()()()r s r s s λβ=+，1()()s s βεβ= ， (7.3) 1()()[1()]()()()ss s s s s s αλκαλτγ'=-+. (7.5) 其中0λ≠是常数. 因此由0τ≠得|()|0ss '=≠，()s s ε'= 其中11ε=±也是一个常数.由定理7.1，1()()s s c αα= 是常数. 用()s α与(7.5)两边作内积，得22221()(1)[1()][()]c s c s c s ελκλκλτ=-⇒--=.由()0s λτ≠可知2(1)0c -≠，从而1()()s s λκμτ-==是常数. 这就是说，存在常数0,λμ≠，使得.充分性. 设正则弧长参数曲线:()C r r s =的曲率κ和挠率τ满足1λκμτ+=，其中,λμ是常数，且0λ≠. 令1()()()r s r s s λβ=+，则1()[1()]()()()()[()()]0r s s s s s s s s λκαλτγτμαλγ'=-+=+≠. 所以由参数方程11()r r s =定义的曲线1C 是正则曲线，并且与曲线C 不重合(因为0λ≠).由于1|r τ'= 1C 的单位切向量场1()[sin ()cos ()]s s s αθαθγ=±+，其中arctan(/)θμλ=是常数，满足sin θ=，cos θ=.设s是1C 的弧长参数，利用Frenet 公式，有111(sin cos )d ds ds ds ακβθκθτβ==±- .如果sin cos 0θκθτ-≠，则有1ββ=±，从而曲线1C 是C 的侣线，1C 和C 是Bertrand 曲线偶(在参数s 相同的点，1C 和C 得主法线有相同方向，并且1()r s 在()r s处的主法线上). 如果sin cos 0θκθτ-=，则μκλτ=. 结合1λκμτ+=可知κ和τ都是非零常数，C 是圆柱螺线，从而是Bertrand 曲线. □定义7.2 如果两条曲线12,C C 之间存在一个一一对应，使得曲线1C 在任意一点的切线正好是2C 在对应点的法线(即垂直于2C 在该点的切线)，则称曲线2C 是1C 的渐伸线. 同时称曲线1C 是2C 的渐缩线.定理7.3 设:()C r r s =是正则弧长参数曲线. 则C 的渐伸线的参数方程为1()()()()r s r s c s s α=+-. (7.7) 证明 设渐伸线1C 上与()r s 对应的点为1()r s . 则1()r s 在曲线C 上()r s点处的切线上，故有函数()s λλ=使得1()()()()r s r s s s λα=+. (7.8) 由渐伸线的定义，1()()r s s α'⊥，所以10()()[()()()()()()]()1()r s s s s s s s s s s ααλαλκβαλ'''==++=+. 由此得()1s λ'=-，()s c s λ=-. 代入(7.8)即得(7.7). □曲线C 的渐伸线可以看作是该曲线的切线族的一条正交轨线，位于C 的切线曲面∑上. 定理7.4设:()C r r s =是正则弧长参数曲线. 则C 的渐缩线的参数方程为()111()()()tan ()()()()r s r s s s ds s s s βτγκκ=+-⎰. (7.10) 证明 设渐缩线1C 上与()r s 对应的点为1()r s . 由定义，1[()()]()()rs r s r s s α-⊥=，可设 1()()()()()()r s r s s s s s λβμγ=++. (7.11) 求导得1()()()()()[()()()()]()()()()()r s s s s s s s s s s s s s s αλβλκατγμγμτβ'''=++-++-[1()()]()[()()()]()[()()()]()s s s s s s s s s s s λκαλμτβμλτγ''=-+-++.因为11()//[()()]()()()()r s r s r s s s s s λβμγ'-=+，所以1()[()()()()]0r s s s s s λβμγ'⨯+=，即有()()1s s λκ=，()[()()()]()[()()()]s s s s s s s s μλμτλμλτ''-=+. (7.12)所以()1/()s s λκ=，且由(7.12)第2式得22()μλλμμλτ''-=+，arctan μτλ'⎛⎫⇒=- ⎪⎝⎭，()()()tan ()s s s ds μλτ⇒=-⎰.所以有(7.10). □课外作业：习题4，8§2.8 平面曲线本节研究平面曲线的特殊性质.一、平面曲线的Frenet 标架在平面2E 上取定一个正交标架(右手直角标架){};,O i j. 则平面曲线C 的弧长参数方程为()((),())r s x s y s =， [,]s a b ∈. (8.1)它的单位切向量为()()()(),()cos(()),sin(())s xs y s s s αθθ==， (8.2) 其中()(,())s i s θα=是由i到()s α的有向角(允许相差2π的整数倍)，逆时针方向为正. 当区间[,]a b 是闭区间时，函数()s θ可以成为定义在整个[,]a b 上的连续可微函数.将()s α 右旋/2π，得到与()s α正交的单位向量()s β ，()()()22()cos(()),sin(())sin(()),cos(())(),()s s s s s y s x s ππβθθθθ=++=-=- . (8.3)这样，得到沿曲线C 的(平面)Frenet 标架{}();(),()r s s s αβ.二、平面曲线的Frenet 公式由于()s α 是单位切向量场，有0αα⋅= ，故//αβ ，可设 ()()()rs s s ακβ= ， (7.4) 其中()()()()()()()(),()(),()()()r x s y s s s s x s y s y s x s x s y s καβ=⋅=⋅-= (7.5)称为曲线C 的相对曲率. 曲线C 的曲率为()|()|r s s κκ=. ()r s κ的符号的几何意义见图2-8.利用(7.4)得到平面曲线的Frenet 公式Cyxs =s l=O()s α ()s β(),()x f x i。
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辛拓扑与整体微分几何介绍辛拓扑与整体微分几何是数学领域中两个重要的研究方向。
辛拓扑主要研究辛流形上的辛结构与辛同调，而整体微分几何则关注微分流形上的几何性质与微分形式。
本文将深入探讨辛拓扑与整体微分几何的关系，并介绍两个领域的共同基础以及发展方向。
二级标题一：辛拓扑（Symplectic Topology）三级标题一：什么是辛流形（Symplectic Manifold）辛流形是一种特殊的微分流形，具有辛结构。
辛结构是一个非退化闭二次型，它与切空间上的微分形式相关联。
辛流形的一个重要性质就是它可以定义哈密顿力学系统，使得系统的运动方程可以通过辛形式和哈密顿函数来描述。
三级标题二：什么是辛同调（Symplectic Cohomology）辛同调是辛拓扑的主要研究对象之一。
它可以看作是辛流形上的拓扑不变量，描述了辛流形中的闭辛形式的性质。
辛同调理论的发展使得我们可以通过辛结构来研究辛流形的拓扑性质，进一步深入了解辛流形的几何结构。
三级标题三：辛拓扑的应用辛拓扑在数学与物理领域中有广泛的应用。
在几何学中，可以通过研究辛拓扑来研究曲面的分类和拓扑不变量；在物理学中，辛结构被用于描述经典与量子力学系统的运动方程。
二级标题二：整体微分几何（Global Differential Geometry）三级标题一：什么是微分流形（Differential Manifold）微分流形是拓扑空间与微分结构的结合体，是现代微分几何的基础概念之一。
微分流形可以通过局部欧氏空间的粘合来构造，使得局部上的欧氏几何可以推广到整个流形上。
三级标题二：什么是微分形式（Differential Form）微分形式是微分流形上的一种重要的几何工具。
它可以看作是切向量的推广，可以被积分，用于描述流形上的几何性质。
微分形式的外微分运算与微分流形的拓扑性质密切相关，提供了一种深入理解流形结构的方法。
三级标题三：整体微分几何的应用整体微分几何在数学和物理领域都有广泛的应用。
Kernels and Regularization on GraphsAlexander J.Smola1and Risi Kondor21Machine Learning Group,RSISEAustralian National UniversityCanberra,ACT0200,AustraliaAlex.Smola@.au2Department of Computer ScienceColumbia University1214Amsterdam Avenue,M.C.0401New York,NY10027,USArisi@Abstract.We introduce a family of kernels on graphs based on thenotion of regularization operators.This generalizes in a natural way thenotion of regularization and Greens functions,as commonly used forreal valued functions,to graphs.It turns out that diffusion kernels canbe found as a special case of our reasoning.We show that the class ofpositive,monotonically decreasing functions on the unit interval leads tokernels and corresponding regularization operators.1IntroductionThere has recently been a surge of interest in learning algorithms that operate on input spaces X other than R n,specifically,discrete input spaces,such as strings, graphs,trees,automata etc..Since kernel-based algorithms,such as Support Vector Machines,Gaussian Processes,Kernel PCA,etc.capture the structure of X via the kernel K:X×X→R,as long as we can define an appropriate kernel on our discrete input space,these algorithms can be imported wholesale, together with their error analysis,theoretical guarantees and empirical success.One of the most general representations of discrete metric spaces are graphs. Even if all we know about our input space are local pairwise similarities between points x i,x j∈X,distances(e.g shortest path length)on the graph induced by these similarities can give a useful,more global,sense of similarity between objects.In their work on Diffusion Kernels,Kondor and Lafferty[2002]gave a specific construction for a kernel capturing this structure.Belkin and Niyogi [2002]proposed an essentially equivalent construction in the context of approx-imating data lying on surfaces in a high dimensional embedding space,and in the context of leveraging information from unlabeled data.In this paper we put these earlier results into the more principled framework of Regularization Theory.We propose a family of regularization operators(equiv-alently,kernels)on graphs that include Diffusion Kernels as a special case,and show that this family encompasses all possible regularization operators invariant under permutations of the vertices in a particular sense.2Alexander Smola and Risi KondorOutline of the Paper:Section2introduces the concept of the graph Laplacian and relates it to the Laplace operator on real valued functions.Next we define an extended class of regularization operators and show why they have to be es-sentially a function of the Laplacian.An analogy to real valued Greens functions is established in Section3.3,and efficient methods for computing such functions are presented in Section4.We conclude with a discussion.2Laplace OperatorsAn undirected unweighted graph G consists of a set of vertices V numbered1to n,and a set of edges E(i.e.,pairs(i,j)where i,j∈V and(i,j)∈E⇔(j,i)∈E). We will sometimes write i∼j to denote that i and j are neighbors,i.e.(i,j)∈E. The adjacency matrix of G is an n×n real matrix W,with W ij=1if i∼j,and 0otherwise(by construction,W is symmetric and its diagonal entries are zero). These definitions and most of the following theory can trivially be extended toweighted graphs by allowing W ij∈[0,∞).Let D be an n×n diagonal matrix with D ii=jW ij.The Laplacian of Gis defined as L:=D−W and the Normalized Laplacian is˜L:=D−12LD−12= I−D−12W D−12.The following two theorems are well known results from spectral graph theory[Chung-Graham,1997]:Theorem1(Spectrum of˜L).˜L is a symmetric,positive semidefinite matrix, and its eigenvaluesλ1,λ2,...,λn satisfy0≤λi≤2.Furthermore,the number of eigenvalues equal to zero equals to the number of disjoint components in G.The bound on the spectrum follows directly from Gerschgorin’s Theorem.Theorem2(L and˜L for Regular Graphs).Now let G be a regular graph of degree d,that is,a graph in which every vertex has exactly d neighbors.ThenL=d I−W and˜L=I−1d W=1dL.Finally,W,L,˜L share the same eigenvectors{v i},where v i=λ−1iW v i=(d−λi)−1L v i=(1−d−1λi)−1˜L v i for all i.L and˜L can be regarded as linear operators on functions f:V→R,or,equiv-alently,on vectors f=(f1,f2,...,f n) .We could equally well have defined Lbyf,L f =f L f=−12i∼j(f i−f j)2for all f∈R n,(1)which readily generalizes to graphs with a countably infinite number of vertices.The Laplacian derives its name from its analogy with the familiar Laplacianoperator∆=∂2∂x21+∂2∂x22+...+∂2∂x2mon continuous spaces.Regarding(1)asinducing a semi-norm f L= f,L f on R n,the analogous expression for∆defined on a compact spaceΩisf ∆= f,∆f =Ωf(∆f)dω=Ω(∇f)·(∇f)dω.(2)Both(1)and(2)quantify how much f and f vary locally,or how“smooth”they are over their respective domains.Kernels and Regularization on Graphs3 More explicitly,whenΩ=R m,up to a constant,−L is exactly thefinite difference discretization of∆on a regular lattice:∆f(x)=mi=1∂2∂x2if≈mi=1∂∂x if(x+12e i)−∂∂x if(x−12e i)δ≈mi=1f(x+e i)+f(x−e i)−2f(x)δ2=1δ2mi=1(f x1,...,x i+1,...,x m+f x1,...,x i−1,...,x m−2f x1,...,x m)=−1δ2[L f]x1,...,x m,where e1,e2,...,e m is an orthogonal basis for R m normalized to e i =δ, the vertices of the lattice are at x=x1e1+...+x m e m with integer valuedcoordinates x i∈N,and f x1,x2,...,x m=f(x).Moreover,both the continuous and the dis-crete Laplacians are canonical operators on their respective domains,in the sense that they are invariant under certain natural transformations of the underlying space,and in this they are essentially unique.Regular grid in two dimensionsThe Laplace operator∆is the unique self-adjoint linear second order differ-ential operator invariant under transformations of the coordinate system under the action of the special orthogonal group SO m,i.e.invariant under rotations. This well known result can be seen by using Schur’s lemma and the fact that SO m is irreducible on R m.We now show a similar result for L.Here the permutation group plays a similar role to SO m.We need some additional definitions:denote by S n the group of permutations on{1,2,...,n}withπ∈S n being a specific permutation taking i∈{1,2,...n}toπ(i).The so-called defining representation of S n consists of n×n matricesΠπ,such that[Ππ]i,π(i)=1and all other entries ofΠπare zero. Theorem3(Permutation Invariant Linear Functions on Graphs).Let L be an n×n symmetric real matrix,linearly related to the n×n adjacency matrix W,i.e.L=T[W]for some linear operator L in a way invariant to permutations of vertices in the sense thatΠ πT[W]Ππ=TΠ πWΠπ(3)for anyπ∈S n.Then L is related to W by a linear combination of the follow-ing three operations:identity;row/column sums;overall sum;row/column sum restricted to the diagonal of L;overall sum restricted to the diagonal of W. Proof LetL i1i2=T[W]i1i2:=ni3=1ni4=1T i1i2i3i4W i3i4(4)with T∈R n4.Eq.(3)then implies Tπ(i1)π(i2)π(i3)π(i4)=T i1i2i3i4for anyπ∈S n.4Alexander Smola and Risi KondorThe indices of T can be partitioned by the equality relation on their values,e.g.(2,5,2,7)is of the partition type [13|2|4],since i 1=i 3,but i 2=i 1,i 4=i 1and i 2=i 4.The key observation is that under the action of the permutation group,elements of T with a given index partition structure are taken to elements with the same index partition structure,e.g.if i 1=i 3then π(i 1)=π(i 3)and if i 1=i 3,then π(i 1)=π(i 3).Furthermore,an element with a given index index partition structure can be mapped to any other element of T with the same index partition structure by a suitable choice of π.Hence,a necessary and sufficient condition for (4)is that all elements of T of a given index partition structure be equal.Therefore,T must be a linear combination of the following tensors (i.e.multilinear forms):A i 1i 2i 3i 4=1B [1,2]i 1i 2i 3i 4=δi 1i 2B [1,3]i 1i 2i 3i 4=δi 1i 3B [1,4]i 1i 2i 3i 4=δi 1i 4B [2,3]i 1i 2i 3i 4=δi 2i 3B [2,4]i 1i 2i 3i 4=δi 2i 4B [3,4]i 1i 2i 3i 4=δi 3i 4C [1,2,3]i 1i 2i 3i 4=δi 1i 2δi 2i 3C [2,3,4]i 1i 2i 3i 4=δi 2i 3δi 3i 4C [3,4,1]i 1i 2i 3i 4=δi 3i 4δi 4i 1C [4,1,2]i 1i 2i 3i 4=δi 4i 1δi 1i 2D [1,2][3,4]i 1i 2i 3i 4=δi 1i 2δi 3i 4D [1,3][2,4]i 1i 2i 3i 4=δi 1i 3δi 2i 4D [1,4][2,3]i 1i 2i 3i 4=δi 1i 4δi 2i 3E [1,2,3,4]i 1i 2i 3i 4=δi 1i 2δi 1i 3δi 1i 4.The tensor A puts the overall sum in each element of L ,while B [1,2]returns the the same restricted to the diagonal of L .Since W has vanishing diagonal,B [3,4],C [2,3,4],C [3,4,1],D [1,2][3,4]and E [1,2,3,4]produce zero.Without loss of generality we can therefore ignore them.By symmetry of W ,the pairs (B [1,3],B [1,4]),(B [2,3],B [2,4]),(C [1,2,3],C [4,1,2])have the same effect on W ,hence we can set the coefficient of the second member of each to zero.Furthermore,to enforce symmetry on L ,the coefficient of B [1,3]and B [2,3]must be the same (without loss of generality 1)and this will give the row/column sum matrix ( k W ik )+( k W kl ).Similarly,C [1,2,3]and C [4,1,2]must have the same coefficient and this will give the row/column sum restricted to the diagonal:δij [( k W ik )+( k W kl )].Finally,by symmetry of W ,D [1,3][2,4]and D [1,4][2,3]are both equivalent to the identity map.The various row/column sum and overall sum operations are uninteresting from a graph theory point of view,since they do not heed to the topology of the graph.Imposing the conditions that each row and column in L must sum to zero,we recover the graph Laplacian.Hence,up to a constant factor and trivial additive components,the graph Laplacian (or the normalized graph Laplacian if we wish to rescale by the number of edges per vertex)is the only “invariant”differential operator for given W (or its normalized counterpart ˜W ).Unless stated otherwise,all results below hold for both L and ˜L (albeit with a different spectrum)and we will,in the following,focus on ˜Ldue to the fact that its spectrum is contained in [0,2].Kernels and Regularization on Graphs5 3RegularizationThe fact that L induces a semi-norm on f which penalizes the changes between adjacent vertices,as described in(1),indicates that it may serve as a tool to design regularization operators.3.1Regularization via the Laplace OperatorWe begin with a brief overview of translation invariant regularization operators on continuous spaces and show how they can be interpreted as powers of∆.This will allow us to repeat the development almost verbatim with˜L(or L)instead.Some of the most successful regularization functionals on R n,leading to kernels such as the Gaussian RBF,can be written as[Smola et al.,1998]f,P f :=|˜f(ω)|2r( ω 2)dω= f,r(∆)f .(5)Here f∈L2(R n),˜f(ω)denotes the Fourier transform of f,r( ω 2)is a function penalizing frequency components|˜f(ω)|of f,typically increasing in ω 2,and finally,r(∆)is the extension of r to operators simply by applying r to the spectrum of∆[Dunford and Schwartz,1958]f,r(∆)f =if,ψi r(λi) ψi,fwhere{(ψi,λi)}is the eigensystem of∆.The last equality in(5)holds because applications of∆become multiplications by ω 2in Fourier space.Kernels are obtained by solving the self-consistency condition[Smola et al.,1998]k(x,·),P k(x ,·) =k(x,x ).(6) One can show that k(x,x )=κ(x−x ),whereκis equal to the inverse Fourier transform of r−1( ω 2).Several r functions have been known to yield good results.The two most popular are given below:r( ω 2)k(x,x )r(∆)Gaussian RBF expσ22ω 2exp−12σ2x−x 2∞i=0σ2ii!∆iLaplacian RBF1+σ2 ω 2exp−1σx−x1+σ2∆In summary,regularization according to(5)is carried out by penalizing˜f(ω) by a function of the Laplace operator.For many results in regularization theory one requires r( ω 2)→∞for ω 2→∞.3.2Regularization via the Graph LaplacianIn complete analogy to(5),we define a class of regularization functionals on graphs asf,P f := f,r(˜L)f .(7)6Alexander Smola and Risi KondorFig.1.Regularization function r (λ).From left to right:regularized Laplacian (σ2=1),diffusion process (σ2=1),one-step random walk (a =2),4-step random walk (a =2),inverse cosine.Here r (˜L )is understood as applying the scalar valued function r (λ)to the eigen-values of ˜L ,that is,r (˜L ):=m i =1r (λi )v i v i ,(8)where {(λi ,v i )}constitute the eigensystem of ˜L .The normalized graph Lapla-cian ˜Lis preferable to L ,since ˜L ’s spectrum is contained in [0,2].The obvious goal is to gain insight into what functions are appropriate choices for r .–From (1)we infer that v i with large λi correspond to rather uneven functions on the graph G .Consequently,they should be penalized more strongly than v i with small λi .Hence r (λ)should be monotonically increasing in λ.–Requiring that r (˜L) 0imposes the constraint r (λ)≥0for all λ∈[0,2].–Finally,we can limit ourselves to r (λ)expressible as power series,since the latter are dense in the space of C 0functions on bounded domains.In Section 3.5we will present additional motivation for the choice of r (λ)in the context of spectral graph theory and segmentation.As we shall see,the following functions are of particular interest:r (λ)=1+σ2λ(Regularized Laplacian)(9)r (λ)=exp σ2/2λ(Diffusion Process)(10)r (λ)=(aI −λ)−1with a ≥2(One-Step Random Walk)(11)r (λ)=(aI −λ)−p with a ≥2(p -Step Random Walk)(12)r (λ)=(cos λπ/4)−1(Inverse Cosine)(13)Figure 1shows the regularization behavior for the functions (9)-(13).3.3KernelsThe introduction of a regularization matrix P =r (˜L)allows us to define a Hilbert space H on R m via f,f H := f ,P f .We now show that H is a reproducing kernel Hilbert space.Kernels and Regularization on Graphs 7Theorem 4.Denote by P ∈R m ×m a (positive semidefinite)regularization ma-trix and denote by H the image of R m under P .Then H with dot product f,f H := f ,P f is a Reproducing Kernel Hilbert Space and its kernel is k (i,j )= P −1ij ,where P −1denotes the pseudo-inverse if P is not invertible.Proof Since P is a positive semidefinite matrix,we clearly have a Hilbert space on P R m .To show the reproducing property we need to prove thatf (i )= f,k (i,·) H .(14)Note that k (i,j )can take on at most m 2different values (since i,j ∈[1:m ]).In matrix notation (14)means that for all f ∈Hf (i )=f P K i,:for all i ⇐⇒f =f P K.(15)The latter holds if K =P −1and f ∈P R m ,which proves the claim.In other words,K is the Greens function of P ,just as in the continuous case.The notion of Greens functions on graphs was only recently introduced by Chung-Graham and Yau [2000]for L .The above theorem extended this idea to arbitrary regularization operators ˆr (˜L).Corollary 1.Denote by P =r (˜L )a regularization matrix,then the correspond-ing kernel is given by K =r −1(˜L ),where we take the pseudo-inverse wherever necessary.More specifically,if {(v i ,λi )}constitute the eigensystem of ˜L,we have K =mi =1r −1(λi )v i v i where we define 0−1≡0.(16)3.4Examples of KernelsBy virtue of Corollary 1we only need to take (9)-(13)and plug the definition of r (λ)into (16)to obtain formulae for computing K .This yields the following kernel matrices:K =(I +σ2˜L)−1(Regularized Laplacian)(17)K =exp(−σ2/2˜L)(Diffusion Process)(18)K =(aI −˜L)p with a ≥2(p -Step Random Walk)(19)K =cos ˜Lπ/4(Inverse Cosine)(20)Equation (18)corresponds to the diffusion kernel proposed by Kondor and Laf-ferty [2002],for which K (x,x )can be visualized as the quantity of some sub-stance that would accumulate at vertex x after a given amount of time if we injected the substance at vertex x and let it diffuse through the graph along the edges.Note that this involves matrix exponentiation defined via the limit K =exp(B )=lim n →∞(I +B/n )n as opposed to component-wise exponentiation K i,j =exp(B i,j ).8Alexander Smola and Risi KondorFig.2.Thefirst8eigenvectors of the normalized graph Laplacian corresponding to the graph drawn above.Each line attached to a vertex is proportional to the value of the corresponding eigenvector at the vertex.Positive values(red)point up and negative values(blue)point down.Note that the assignment of values becomes less and less uniform with increasing eigenvalue(i.e.from left to right).For(17)it is typically more efficient to deal with the inverse of K,as it avoids the costly inversion of the sparse matrix˜L.Such situations arise,e.g.,in Gaussian Process estimation,where K is the covariance matrix of a stochastic process[Williams,1999].Regarding(19),recall that(aI−˜L)p=((a−1)I+˜W)p is up to scaling terms equiv-alent to a p-step random walk on the graphwith random restarts(see Section A for de-tails).In this sense it is similar to the dif-fusion kernel.However,the fact that K in-volves only afinite number of products ofmatrices makes it much more attractive forpractical purposes.In particular,entries inK ij can be computed cheaply using the factthat˜L is a sparse matrix.A nearest neighbor graph.Finally,the inverse cosine kernel treats lower complexity functions almost equally,with a significant reduction in the upper end of the spectrum.Figure2 shows the leading eigenvectors of the graph drawn above and Figure3provide examples of some of the kernels discussed above.3.5Clustering and Spectral Graph TheoryWe could also have derived r(˜L)directly from spectral graph theory:the eigen-vectors of the graph Laplacian correspond to functions partitioning the graph into clusters,see e.g.,[Chung-Graham,1997,Shi and Malik,1997]and the ref-erences therein.In general,small eigenvalues have associated eigenvectors which vary little between adjacent vertices.Finding the smallest eigenvectors of˜L can be seen as a real-valued relaxation of the min-cut problem.3For instance,the smallest eigenvalue of˜L is0,its corresponding eigenvector is D121n with1n:=(1,...,1)∈R n.The second smallest eigenvalue/eigenvector pair,also often referred to as the Fiedler-vector,can be used to split the graph 3Only recently,algorithms based on the celebrated semidefinite relaxation of the min-cut problem by Goemans and Williamson[1995]have seen wider use[Torr,2003]in segmentation and clustering by use of spectral bundle methods.Kernels and Regularization on Graphs9Fig.3.Top:regularized graph Laplacian;Middle:diffusion kernel with σ=5,Bottom:4-step random walk kernel.Each figure displays K ij for fixed i .The value K ij at vertex i is denoted by a bold line.Note that only adjacent vertices to i bear significant value.into two distinct parts [Weiss,1999,Shi and Malik,1997],and further eigenvec-tors with larger eigenvalues have been used for more finely-grained partitions of the graph.See Figure 2for an example.Such a decomposition into functions of increasing complexity has very de-sirable properties:if we want to perform estimation on the graph,we will wish to bias the estimate towards functions which vary little over large homogeneous portions 4.Consequently,we have the following interpretation of f,f H .As-sume that f = i βi v i ,where {(v i ,λi )}is the eigensystem of ˜L.Then we can rewrite f,f H to yield f ,r (˜L )f = i βi v i , j r (λj )v j v j l βl v l = iβ2i r (λi ).(21)This means that the components of f which vary a lot over coherent clusters in the graph are penalized more strongly,whereas the portions of f ,which are essentially constant over clusters,are preferred.This is exactly what we want.3.6Approximate ComputationOften it is not necessary to know all values of the kernel (e.g.,if we only observe instances from a subset of all positions on the graph).There it would be wasteful to compute the full matrix r (L )−1explicitly,since such operations typically scale with O (n 3).Furthermore,for large n it is not desirable to compute K via (16),that is,by computing the eigensystem of ˜Land assembling K directly.4If we cannot assume a connection between the structure of the graph and the values of the function to be estimated on it,the entire concept of designing kernels on graphs obviously becomes meaningless.10Alexander Smola and Risi KondorInstead,we would like to take advantage of the fact that ˜L is sparse,and con-sequently any operation ˜Lαhas cost at most linear in the number of nonzero ele-ments of ˜L ,hence the cost is bounded by O (|E |+n ).Moreover,if d is the largest degree of the graph,then computing L p e i costs at most |E | p −1i =1(min(d +1,n ))ioperations:at each step the number of non-zeros in the rhs decreases by at most a factor of d +1.This means that as long as we can approximate K =r −1(˜L )by a low order polynomial,say ρ(˜L ):= N i =0βi ˜L i ,significant savings are possible.Note that we need not necessarily require a uniformly good approximation and put the main emphasis on the approximation for small λ.However,we need to ensure that ρ(˜L)is positive semidefinite.Diffusion Kernel:The fact that the series r −1(x )=exp(−βx )= ∞m =0(−β)m x m m !has alternating signs shows that the approximation error at r −1(x )is boundedby (2β)N +1(N +1)!,if we use N terms in the expansion (from Theorem 1we know that ˜L≤2).For instance,for β=1,10terms are sufficient to obtain an error of the order of 10−4.Variational Approximation:In general,if we want to approximate r −1(λ)on[0,2],we need to solve the L ∞([0,2])approximation problemminimize β, subject to N i =0βi λi −r −1(λ) ≤ ∀λ∈[0,2](22)Clearly,(22)is equivalent to minimizing sup ˜L ρ(˜L )−r−1(˜L ) ,since the matrix norm is determined by the largest eigenvalues,and we can find ˜Lsuch that the discrepancy between ρ(λ)and r −1(λ)is attained.Variational problems of this form have been studied in the literature,and their solution may provide much better approximations to r −1(λ)than a truncated power series expansion.4Products of GraphsAs we have already pointed out,it is very expensive to compute K for arbitrary ˆr and ˜L.For special types of graphs and regularization,however,significant computational savings can be made.4.1Factor GraphsThe work of this section is a direct extension of results by Ellis [2002]and Chung-Graham and Yau [2000],who study factor graphs to compute inverses of the graph Laplacian.Definition 1(Factor Graphs).Denote by (V,E )and (V ,E )the vertices V and edges E of two graphs,then the factor graph (V f ,E f ):=(V,E )⊗(V ,E )is defined as the graph where (i,i )∈V f if i ∈V and i ∈V ;and ((i,i ),(j,j ))∈E f if and only if either (i,j )∈E and i =j or (i ,j )∈E and i =j .Kernels and Regularization on Graphs 11For instance,the factor graph of two rings is a torus.The nice property of factor graphs is that we can compute the eigenvalues of the Laplacian on products very easily (see e.g.,Chung-Graham and Yau [2000]):Theorem 5(Eigenvalues of Factor Graphs).The eigenvalues and eigen-vectors of the normalized Laplacian for the factor graph between a regular graph of degree d with eigenvalues {λj }and a regular graph of degree d with eigenvalues {λ l }are of the form:λfact j,l =d d +d λj +d d +d λ l(23)and the eigenvectors satisfy e j,l(i,i )=e j i e l i ,where e j is an eigenvector of ˜L and e l is an eigenvector of ˜L.This allows us to apply Corollary 1to obtain an expansion of K asK =(r (L ))−1=j,l r −1(λjl )e j,l e j,l .(24)While providing an explicit recipe for the computation of K ij without the need to compute the full matrix K ,this still requires O (n 2)operations per entry,which may be more costly than what we want (here n is the number of vertices of the factor graph).Two methods for computing (24)become evident at this point:if r has a special structure,we may exploit this to decompose K into the products and sums of terms depending on one of the two graphs alone and pre-compute these expressions beforehand.Secondly,if one of the two terms in the expansion can be computed for a rather general class of values of r (x ),we can pre-compute this expansion and only carry out the remainder corresponding to (24)explicitly.4.2Product Decomposition of r (x )Central to our reasoning is the observation that for certain r (x ),the term 1r (a +b )can be expressed in terms of a product and sum of terms depending on a and b only.We assume that 1r (a +b )=M m =1ρn (a )˜ρn (b ).(25)In the following we will show that in such situations the kernels on factor graphs can be computed as an analogous combination of products and sums of kernel functions on the terms constituting the ingredients of the factor graph.Before we do so,we briefly check that many r (x )indeed satisfy this property.exp(−β(a +b ))=exp(−βa )exp(−βb )(26)(A −(a +b ))= A 2−a + A 2−b (27)(A −(a +b ))p =p n =0p n A 2−a n A 2−b p −n (28)cos (a +b )π4=cos aπ4cos bπ4−sin aπ4sin bπ4(29)12Alexander Smola and Risi KondorIn a nutshell,we will exploit the fact that for products of graphs the eigenvalues of the joint graph Laplacian can be written as the sum of the eigenvalues of the Laplacians of the constituent graphs.This way we can perform computations on ρn and˜ρn separately without the need to take the other part of the the product of graphs into account.Definek m(i,j):=l ρldλld+de l i e l j and˜k m(i ,j ):=l˜ρldλld+d˜e l i ˜e l j .(30)Then we have the following composition theorem:Theorem6.Denote by(V,E)and(V ,E )connected regular graphs of degrees d with m vertices(and d ,m respectively)and normalized graph Laplacians ˜L,˜L .Furthermore denote by r(x)a rational function with matrix-valued exten-sionˆr(X).In this case the kernel K corresponding to the regularization operator ˆr(L)on the product graph of(V,E)and(V ,E )is given byk((i,i ),(j,j ))=Mm=1k m(i,j)˜k m(i ,j )(31)Proof Plug the expansion of1r(a+b)as given by(25)into(24)and collect terms.From(26)we immediately obtain the corollary(see Kondor and Lafferty[2002]) that for diffusion processes on factor graphs the kernel on the factor graph is given by the product of kernels on the constituents,that is k((i,i ),(j,j ))= k(i,j)k (i ,j ).The kernels k m and˜k m can be computed either by using an analytic solution of the underlying factors of the graph or alternatively they can be computed numerically.If the total number of kernels k n is small in comparison to the number of possible coordinates this is still computationally beneficial.4.3Composition TheoremsIf no expansion as in(31)can be found,we may still be able to compute ker-nels by extending a reasoning from[Ellis,2002].More specifically,the following composition theorem allows us to accelerate the computation in many cases, whenever we can parameterize(ˆr(L+αI))−1in an efficient way.For this pur-pose we introduce two auxiliary functionsKα(i,j):=ˆrdd+dL+αdd+dI−1=lrdλl+αdd+d−1e l(i)e l(j)G α(i,j):=(L +αI)−1=l1λl+αe l(i)e l(j).(32)In some cases Kα(i,j)may be computed in closed form,thus obviating the need to perform expensive matrix inversion,e.g.,in the case where the underlying graph is a chain[Ellis,2002]and Kα=Gα.Kernels and Regularization on Graphs 13Theorem 7.Under the assumptions of Theorem 6we haveK ((j,j ),(l,l ))=12πi C K α(j,l )G −α(j ,l )dα= v K λv (j,l )e v j e v l (33)where C ⊂C is a contour of the C containing the poles of (V ,E )including 0.For practical purposes,the third term of (33)is more amenable to computation.Proof From (24)we haveK ((j,j ),(l,l ))= u,v r dλu +d λv d +d −1e u j e u l e v j e v l (34)=12πi C u r dλu +d αd +d −1e u j e u l v 1λv −αe v j e v l dαHere the second equalityfollows from the fact that the contour integral over a pole p yields C f (α)p −αdα=2πif (p ),and the claim is verified by checking thedefinitions of K αand G α.The last equality can be seen from (34)by splitting up the summation over u and v .5ConclusionsWe have shown that the canonical family of kernels on graphs are of the form of power series in the graph Laplacian.Equivalently,such kernels can be char-acterized by a real valued function of the eigenvalues of the Laplacian.Special cases include diffusion kernels,the regularized Laplacian kernel and p -step ran-dom walk kernels.We have developed the regularization theory of learning on graphs using such kernels and explored methods for efficiently computing and approximating the kernel matrix.Acknowledgments This work was supported by a grant of the ARC.The authors thank Eleazar Eskin,Patrick Haffner,Andrew Ng,Bob Williamson and S.V.N.Vishwanathan for helpful comments and suggestions.A Link AnalysisRather surprisingly,our approach to regularizing functions on graphs bears re-semblance to algorithms for scoring web pages such as PageRank [Page et al.,1998],HITS [Kleinberg,1999],and randomized HITS [Zheng et al.,2001].More specifically,the random walks on graphs used in all three algorithms and the stationary distributions arising from them are closely connected with the eigen-system of L and ˜Lrespectively.We begin with an analysis of PageRank.Given a set of web pages and links between them we construct a directed graph in such a way that pages correspond。
