陈省身微分几何讲义：数学与物理学的交汇

s.s.chern

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"Lectures on Differential Geometry" 是一本由S.S. Chern、W.H. Chen和K.S. Lam合著的数学教材，属于"Series on University Mathematics"系列。该书致力于保留陈省身教授讲义的精髓，包括简洁、直接和经济的方法，以及对基础主题的深入探讨，适合于数学领域的高级本科生、初阶研究生，以及对物理学中应用微分几何感兴趣的物理学家作为教材或参考书。书中的内容旨在为更高级和专门的研究提供坚实的基础。微分几何是数学的一个分支，主要研究流形（如曲线和曲面）上的几何结构。陈省身教授是这一领域的大师，他的工作对现代数学和理论物理学有着深远的影响。本书可能涵盖了以下几个核心知识点： 1. **微分流形**：微分几何的基础是微分流形，它是一类局部类似于欧几里得空间的拓扑空间。书中可能会介绍流形的基本概念，如维度、子流形、切空间和切向量。 2. **联络与曲率**：联络是定义在流形上的结构，允许我们在流形上进行平行移动。曲率则描述了这个结构的局部性质，是理解和研究流形的重要工具。陈省身教授的工作之一就是发展了联络的几何理论。 3. **黎曼几何**：黎曼几何是微分几何的核心部分，它引入了黎曼度量，通过度量可以定义距离和角度，从而量化流形的几何性质。书中可能详细讨论了黎曼度量的概念、黎曼曲率张量及其与局部几何的关系。 4. **李群与李代数**：微分几何与李群和李代数有紧密联系，特别是在描述对称性和守恒定律时。书中可能包含李群和李代数的基础知识，以及它们如何与微分几何相融合。 5. **几何积分**：陈省身引入了陈示性类，这是一种特殊的几何积分，对于理解流形的拓扑和几何性质至关重要。这部分内容可能涉及外微分形式、Stokes定理和特征类理论。 6. **几何在物理学中的应用**：微分几何在广义相对论、量子场论和弦理论等物理学分支中扮演着关键角色。书中可能会探讨这些应用，解释如何用微分几何的语言来描述物理空间的时间和空间结构。 7. **实例与习题**：为了帮助读者理解抽象概念，书中很可能包含许多具体的几何例子，如球面、双曲平面、克莱因瓶等，并配有丰富的练习题以加深理解。通过阅读这本教材，读者将能够建立起对微分几何的深入理解，为进一步研究高级主题如规范场论、弦理论或几何分析打下坚实基础。此外，书中可能还包括了数学家们对如何处理复杂问题的思考方式，这对于培养独立研究和解决问题的能力大有裨益。

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Chapter

Differentiable Manifolds

The numbers of the (m+l)-tuple

(zl,

. . .

zm+')

are called the

homogeneous

coordinates

[z].

They are determined by

[z]

up to

nonzero factor.

is thus the space of all straight lines in

Rm+l

which pass through the origin.

Let

uz=

{[XI,

...

,Z"+']I5i#O),

(1.11)

{

([XI)

(

it1

iti-1,

iti+l7..

ilm+l)

where

iJh

zh/zz

i).

Obviously,

{Ui,

forms an open covering of

P".

Uj,

the change of coordinates is

given by

(1.12)

Hence

{(Ui,pi)}15z5m+l

suffices to generate

smooth structure on

Pm.

Remark.

In each of the above examples, the respective coordinate charts

given are in fact Cw-compatible also, and

provide the structures for

IR",

S", and

as analytic manifolds.

Example

(Milnor's Exotic Sphere).

There may exist distinct differen-

tiable structures on a single topological manifold.

Milnor gave a famous

example (Milnor 1956), which shows that there exist nonisomorphic smooth

structures on homeomorphic topological manifolds (see the discussion follow-

ing the remark to definition 1.3 below). Hence

differentiable structure is

more than

topological structure.

complete understanding of the Milnor

sphere is outside the scope of this text. Here we will give only a brief descrip-

tion of the main ideas.

more recent example is the existence of distinct

smooth structures on

IR4

discovered by

K. Donaldson (see Donaldson and

Kronheimer 1991)].

Choose two antipodal points

and

S4.

Let

{A},

{B}.

(1.13)

Then

and

form an open covering of

S4.

We wish to paste the trivial

sphere bundles

and

together to get the 3-sphere bundle

over

S4.

Under the stereographic projection,

and

are both homeomorphic

IR4,

and

is homeomorphic to

IR4

(0).

Identify the elements of

IR4

(0)

as quaternions, and choose an odd number

K,,

where

K,'

mod

51-1:

Definition

Differentiable Manifolds

Consider the map

(R4

(0))

--+

(IR4

(0))

S3,

such that for every

(u,v)

(R4

(0))

S3,

we have

where

1-6

J=-

6+1

h=-

(1.14)

(1.15)

and in (1.14) the multiplication and the norm

are in the sense of quaternions.

Obviously

smooth map. We can thus paste

and

together

using

can be proved that the

constructed in this way is homeomorphic

to the 7-dimensional unit sphere

S',

but its differentiable structure is different

from the standard differentiable structure of

(Example

2).

On a smooth manifold, the concept of a smooth function is well-defined.

Let

be a real-valued function defined on an m-dimensional smooth manifold

and

(U,

cp,)

is a compatible coordinate chart containing

then

'pi'

is a real-valued function defined on the open subset

cp,(U)

of the

Euclidean space

R".

cp;'

the point

'p,

(p)

R",

we say that

the function

The differentiability of the function

at the point

is independent of the

choice of the compatible coordinate chart containing

In fact, for another

compatible coordinate chart

(V,pv)

containing

such that

have

'p;'

.;l)

(Yu

'pi1)

Since

cp,

pi'

is smooth, we see that

cp;'

and

cp;'

are differentiable

at the same point

If the real-valued function

every point in

then we call

C",

smooth,

function on

We shall denote the set of all smooth

functions on

Cm(M).

Smooth real-valued functions are just important special cases

smooth

maps between smooth manifolds.

Definition

1.3.

Suppose

continuous map from one smooth

manifold

to another,

where dim

m and dim

there exist

compatibleb coordinate charts

(U,cp,)

at the point

and

(V,$v)

f(p)

such that the map

'pi'

cp,(W

--+

$v(V)

bThat is, contained in the smooth structures

the respective manifolds.

Chapter

Differentiable Manifolds

C""

at the point

cpv(p),

then the map

is called

C""

the map

at every point

then we say that

smooth

map

from

Remark.

Since

cp-'

continuous map from an open set

cp,(U)

IR"

to another open set

$v(V)

IR",

its differentiability

the point

cp,(p)

is defined. Obviously the differentiability off at

is independent of the choice

of compatible coordinate charts

(U,

cp,)

and

(V,

cp,).

In the case dim

dim

homeomorphism and

f-l

are both smooth maps, then we call

--+

diffeomorphism.

the smooth manifolds

and

are diffeomorphic, then we say that the

corresponding smooth structures of the manifolds are

isomorphic.

In the

above example, the Milnor sphere

is homeomorphic but not diffeomorphic

S7.

Hence their smooth structures are not isomorphic.

Another important special case of smooth maps between smooth manifolds

is that of

parametrized curves

on manifolds, in which

is an open interval

(a,

IR'.

A smooth map

(a,

from

to the manifold

is a

parametrized curve in the manifold

Now

suppose

and

are m-dimensional and n-dimensional manifolds

with differentiable structures

{

(U,,

'pa)}aEA

and

{(Vp,

q!~p)}~~%,

respectively.

We can construct

new (m

n)-dimensional smooth manifold

by the

following method. First, we see that

{U,

Vp},.EA,PEIB

forms an open covering

of the topological product space

Then we define maps

cp,

IRm+n

such that

(1.16)

(Pa

dJO(P,Q)

(cp,(P)>h(4))

(P,

v,.

Thus

(U,

Vp,

'p,

$0)

coordinate chart of

It is easy

prove

that all the coordinate charts obtained in this way are Cco-compatible, and

hence they determine

smooth differentiable structure on

Definition

1.4.

The smooth differentiable structure determined by the

C"-

compatible coordinate covering

{(U,

Vp,

cp,

$p)},~~,p~%

of the topological

product space

makes

(rn

n)-dimensional smooth manifold,

called the

product manifold

and

The natural projections of the product manifold

onto its factors are

denoted by

+M,

~2:

where,

for

any

(z,

m(s,y)

7Tz(S,Y)

Obviously these are both smooth maps.

51-2:

Tangent Spaces

$1-2

Tangent

Spaces

every point on a regular curve

(or

surface), we have the notion of the

tangent line (or tangent plane). Similarly] given

differentiable structure on

topological manifold, we can approximate

neighborhood of any point by

linear space. More precisely] the concepts of the tangent space and the

cotangent space can be introduced. We begin with the cotangent space.

Suppose

is an m-dimensional smooth manifold. Fix

point

and

let

be a

function' defined in

neighborhood of

Denote the set of all

these functions

CF.

Naturally, the domains of two different functions in

may be different, but addition and multiplication in the function space

are still well-defined. Suppose

with domains

and

respectively.

Then

is also

neighborhood containing

Thus

and

can be

defined

functions on

that is,

and

CF.

Define a relation

follows. Suppose

f,g

CF.

Then

if and only if there exists an open neighborhood

of the point

such that

fl,

91,.

Obviously

is an equivalence relation in

Cr.

We will denote the

equivalence class of

If],

which is called a

C"-germ

Let

c,"/

{[fl

c;}.

Then, by defining addition and scalar multiplication,

becomes

linear space

over

IR:

for

[

f],

[g]

3p1

lR,

define

In this definition] the right hand sides of (2.2) are independent of the choices of

[f]

and

[g].

The reader should verify that

is an infinite-dimensional

real linear space.

Suppose

is a parametrized curve in

through

point

Then there

exists a positive number

such that

(-6,s)

map and

y(0)

Denote the set of all these parametrized curves by rp.

For

rp,

[f]

3p,

let (see Figure 3)

(2.3)

'Suppose

is a function defined on an open set

If the function

pi1

on the open set

pU(U

for any admissible coordinate chart

(U,p,),

where

then we say

function defined on

In fact,

has

differentiable

structure induced from

(see section

51-3).

Thus

Coo

function on the differentiable

manifold

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