Differential Geometry

1.3 Directional Derivatives

Definition 3.1

Let be a differentiable real-valued function on , and let be a tangent vector to .

Then the number

is called the derivative of f with respect to

is directional derivative.

Lemma 3.2

If is a tangent vector to, then

proof

Let then

we use the chain rule to compute the derivative at t= 0 of the function

Since

we obtain

Q.E.D

Lemma 1.4.6

Let be a curve in and let be a differentialble function on . Then

proof

Since

From lemma 1.3.2, that

By chain rule, the lemma is proved. ()

By definition, is the rate of change of along the line through in the direction.

1.5 1-forms

The differential of the

1 form : : Tangent space 를 스칼라로 보내는 함수

If , and then

is 1 form such that

: f : scalar function, 1 form : scalar function , so...

Definition 5.1 : Definition of 1-form

A 1-form on is a real-valued function on the set of all tangent vectors on such that is linear at each point, that is

for any number and tangent vectors at the same pont of

Definition 5.2

For , the differential df of f is the 1 form such that

는 의 모든 방향으로의 f의 변화율 즉,

Remind the following simple form

and

Example

(1)

(2)

when

Simply We regard it as

, so that

Thus,

Corollary 1.5.5

Lemma 1.5.7

proof)

Since

1.6 Differential Forms

Differential Forms는 Associative와 Distribution은 성립 Group은 된다.

Commutative는 성립하지 않으므로 Ring, Field는 안된다.

Example

Computation of Wedge Products : p29

Remind :

(1)

(2) Let

Lemma

If are 1-forms, then

proof

Let

Definition : The exterior derivatives

If is a 1-form on , the exterior derivative of is

the 2-form

Meaning :

Theorem

Let be functions, be 1-forms.

proof

For final function :

Let

2.5 Covariant Derivatives

Definition

Let be a vector field on , and let v be a tangent vector field to at the point .

Then the covariant derivative of with respect to is the tangent vector

measures the initial rate of changes of as p moves in the direction.

Lemma

is a vector field on and is a tengent vector at , then

proof of Lemma

By definition

By definition of , the lemma is proved.

(Q.E.D)

It means that

conserves the linearity and Leibnizian properties of covariant derivartives.

The covariant derivative of a vector field with respect to a vector field :

then

Basic Identity of :

The example of the covariant derivative of a vector field with respect to a vector field.

-------------------------------

2.6 Frame Fields

Definition

Vector fields on constitute a frame field on provided

where is the Kronecker delta

2.7 Connection Forms

Express the covariant derivatives of these vector fields in terms of the vector fields themselves.

then

The coefficient depends on the particular tangent vector , so that

Lemma : The connection forms (1-form) of the frame field

Let, be a frame field on . For each tangent vector to at the point . Let

Then each is a 1-form, and .

proof By the definition 5.1 of ch.1, we should a linearity of .

The 1-form is proved by linearities.

To prove , we remind that and , where is a constant value. Thus,

Q.E.D

Meaning of :

The initial rate at which rotates toward as moves in the direction.

Theorem

For any vector field on

---------

By , . Therefore,

The connection forms as the entries of a skew-symmetric matrix of 1-forms.

Relation to the Frenet Formulas

Frenet Formula는 임의 커브에 대한 것 결국 1-dimension

Connection forms 는 임의 Tangent vector or Vector field 3-dimension

이 차이로 약간 다르다.

Frenet Formula는 curve에 대한 정보 T, N, B 제공

Connection forms 는 frame field의 rate of rotation 정보 제공 -> 직접적인 해답을 준다.

More fundamental

Example

Assume that the natural frame field on and a frame field be connected as

Attitude Matrix of the frame field

where

The differential of is defined as

The entries of are 1-forms

Theorem

If is the attitude matrix and the matrix of connection forms of a frame field , then

or equivalently

Sketch of Proof

Example

Attitude matrix of the cylindrical frame field

It can be reduced as

2.8 Structual Equation

The connection form is the rate of rotation of a frame field.

Furthermore, the frame field itself can be described in terms of the 1-forms

Definition

If is a frame field on , then the dual 1-forms of the frame field are the 1-forms such that

for each tangent vector to at the point

... Tangent Vector 를 Frame Field 에 정사영 하는 것이다.

... Connection Form은 를 에 정사영 하는 것이다.

... 둘다 i-th component의 coefficient

.... The dual 1 form은 Frame Field에 대한 Tangent Vector 성분이며

... Connection Form은 Frame Field에 대한 다른 Frame field의 v 방향으로의 변화율이다.

Example

In case of the natural frame field , the dual forms are just such that

hence

Dual form 을 사용하여 the orthornormal expansion 를 이렇게 쓸 수 있다.

....(1)

..... Dual form dx 의 개념이므로 Frame field 에 대하여 Tangent Vector를 의 형태로 쓸 수 있음을 의미한다.

Lemma

Let be the dual 1-forms of a frame field .

Then any 1-form on has a unique expression

proof

2->3 항의 경우는 1-form 정의에서 간단한 Inner Product 이므로 당연하다.

는 Vedctor 이다. 를 entity로 한다. (1) 참조

.... 임의의 1 form은 Frame Field의 dual form()과 Frame field의 임의 1-form간() 1차 결합으로 표현 가능하다.

Example

a Frame field and the natural frame field

The dual formulation

has same coefficient.

** Frame field 와 Dual formulation (dual form의 결합)은 같은 coefficient.

Really? 증명하자.

바로 위의 Lemma 에서

즉, 는 the natural frame field 의 dual form

By definition of

By definition of E_i

그래서 놀라운 결과!!

Frame Field에 대한 Natural Frame Field의 계수는

Frame Field에 대한 dual 1-form ()에 대한 Natural Frame Field의 dual form ()의 계수와 같다.

즉, 와 는 같은 계수를 가진다.

다시말해, 와 는 같은 축상에 있으며 전자가 Frame Field (vector)이고 후자는 1-form인 것.

자연스럽게 1-form에 대한 미분 : the exterior derivatives가 유도된다.

Theorem (Cartan Structual eqation) : Exterior derivatives

Let be a frame field on with dual forms and connection forms ()

The exterior derivatives of these forms satisfy

(1) The first structual equations :

.... It looks like that a dual form of the connection forms as

(2) The second structual equations :

증명에 필요한 것들

Proof of the first structual equations

Proof of the second structual equations

For functions

Relation between dual forms and Frame fields

Chapter 3. Euclidean Geometry

3.1 Isometries of

Definition

An isometry (or rigid motion) of is a mapping such that

for all points in

Lemma

If is an orthogonal transformation, then is an isometry of

proof

Lemma

If is an isometry of such that , then is an orthogonal tramsformation.

3.2 The tangent map of an isometry

An arbitrary mapping has a tangent map that carries each tangent vector at to

a tangent vector at

Isometry에서 는 preserve됨

Example : 대표적인 는 Orthogonal or Orthonormal Matrix

3.3 Orientation

Thus,

Lemma

Let be a frame at a point of . If and , then

where

proof

위 식과 이 식을 비교하면 자명하다.

cross product는 Form 연산과 똑같이 보인다.

Theorem

Let be a tangent vectors to at . If is an isometry of , then

3.4 Euclidean Geometry

Both Dot product and Cross product are preserved by an isometries.

Velocity is preserved by arbitrary mappings ,

-> , is a tangent vector.

But acceleration os not preserved by arbitrary mappings.

However, Euclidean geometry 에서는 속도와 가속도가 모두 보존된다.

Theorem 4.2

Let be a unit-speed curve in with positive curvature, and let

be the image curve of under an isometry of . Then

3.5 Conguence of Curves

Theorerm 5.3

If are unit-speed curves such that and

k Curvature , t : Torsion

Remind : Frenet Formula

Basic Formula for the Frenet Formula

, , Binormal Vector ,

Chapter 4. Calculus on a Surface

4.1 Surface in

Definition 4.1.1 : coordinate patch

A coordinate patch is a one-to-one regular mapping of an open set of into

Proper patch

A coordinate patch that has an inverse function is continuous.

Definition 4.1.2 : surface in

A surface in is a subset of such that for eacg point of there exists a proper patch in

whose image contains a neighborhood of in

Example

The surface . Every differentiable real-valued function on determines a surface of

: The graph of , that is, the set of all points of whose coordinates satisfy the equation .

Evidently is an image of the Monge patch : (다음과 같은 형식의 patch)

hence by the remarks above, is a simple surface.

Theorem 4.1.4: surface by Implicit function theorem

Let be a differentiable real-valued function on , and a number.

The subset of is a surface if differential at any point of .

Sketch of proof

이러면 최소 z에 대하여 미분 방정식을 풀 수 있다. 이렇게 되면 최소한 이렇게 표시할 수 있다.

.... Monge patch 형태로 만들 수 있다.

Example : Sphere at

when -> Sphere의 중심일때만 0이 된다.

4.2 Patch Computation

Definition

If is a patch, for each point in :

(1) The velocity vector at of the u-parameter curve, , is denoted by

(2) The velocity vector at of the v-parameter curve, , is denoted by

The vectors and are called the partial velocities of at .

If an Euclidean coordinate function is defined by a formula

Then the partial velocity functions are given by

무엇으로 편미분 하느냐로 결정됨 :

Definition

A regular mapping whose image lies in a surface is called a parameterization of the region in

The regularity of : is never zero / the partial velocity vectors of are linearly independent.

Definition

A ruled surface is a surface swept out by a straight line L moving along a curve .

The various positions of the generating line L are called the rulings of the surface.

Such a surface always has a ruled parameterization

: Base curve, : Director curve

Notion 1

same to Lemma 2.1.8 at p47

4.3 Differential Functions and Tangent Vectors

Suppose that

: a real valued function defined on a surface .

: a coordinate patch in

then

A Coordinate Expression :

For a function , each patch in gives

a coordinate expression for

즉, x 도 F도 모두 M에 있다.

Lemma 4.3.1

Overlapped Patch

Definition

Let be a point of a surface in .

A tangent vector to at is tangent to at provided is a velocity of some curve in .

Tangent plane of at :

is the linear approximation of the surface near

Definition

A Euclidean vector field on a surface in is a function

that assigns to each point of a tangent vector to at .

A Euclidean vector at a point point of is normal

A Euclidean vector field on is a normal vector field provided each vector is normal to .

** 은 기본적으로 . 따라서 하나의 Dimension이 남는다.

그러므로 normal vector 나머지 하나의 Dimension을 만든다.

Lemma 4.3.8 (Gradient vector is perpendicular to Tangent plane)

If is a surface in , then the gradient vector field is a nonvanishing

normal vector field on the entire surface .

sketch of proof

을 보이면 된다.

인 것은 (by theorem 4.1.4 에서)

Definition 4.3.10

Let be a tangent vector to at , and let be a differentiable real-valued function on .

The derivative of with respect to is the common value of for all curves in with initial velocity .

Tips from the exercise 4.3

1. Let be a patch in . If is the tangent map of ,

2. If is a differentiable function on , then

4.4 Differential Forms on a Surface

Surface 이므로 2-form 을 사용하게 된다.

2-form 은 2개의 tangent vector에 대응하여 만들어진다.

2개의 tangent vector는 하나의 surface를 정의하기 때문

Definition 4.4.1

A 2-form on a surface is a real-valued function on all ordered pairs of tangent vectors to such that

(1) is linear in

(2)

all p-forms with p>2 is zero, since the dimension of surface is 2.

Lemma 4.4.2

Let be a 2-form on a surface , and let be (linearly independent) tangent vectors at some point of . Then

proof

Using the linearity of (by the definition 4.1 (1))

Wedge product에 의한 2-form 생성, surface 이므로 2개의 1-form Wedge product로 만들어진다.

Definition 4.4.3

If and are 1-forms on a surface , the wedge product is the 2-form on such that

for all pairs of tangent vectors to

: 매우 중요한 정의 : (ad -bc : 에서 유도 : determinent 개념)

If is a p-form and is a q-form, then

The differential calculus of forms is based on the exterior derivative .

0-form : function

1-form : such that

Definition 4.4 :Exterior Derivative

Let be a 1-form on a surface . Then the exterior derivative of is the 2-form such that for any patch in .

.... Look like Lie Derivative or (ad-bc .. determinent a : u에 대한 편미분, b : v에 대한 편미분, c : u에 대한 1 form, d : v 에 대한 1-form)

....Regard as . It is easy to understand.

.... 우측 첫째항에서 x는 v에 대해 편미분되어 u항 살아있는 벡터가 된다 (). 이것이 에 의해 1 form이 된다.

.... 다음 1 form을 u에 대해 미분한다.

Example (my)

.... , ,

Definition 4.4.4는 불완전한 정의

What we have actually defined is a form on the image of each patch in .

따라서, 와 가 on the overlap of 에서 같을 때만이 를 정의할 수 있다.

Lemma 4.4.5 : How to use the definitions and calculus for the 2-forms.

Let be a 1-form on . If are patches on , then on the overlap of .

proof

It means that

Set on the overlap of , then

then by lemma 4.4.2

증명 되려면

임을 증명하면 된다.

즉 By definition 4.4.4

에서

임을 증명하면 된다.

이므로...

. . .

고로 이들을 잘 정리하여 은근과 끈기로 전개한 후 빼주면 증명 된다.

Theorem 4.4.6

If is a real-valued function on , then .

proof

Let

(는 patch 이므로 벡터 형태다. 는 와 동일한 차원을 가진다. 고로 를 로 보면 차원이 맞지 않는다.

단, 는 스칼라 이므로 이렇게 정의 되면 쉽다.

(1) Tangent 벡터 가 형태로 픽스 되면 Tangent의 정의 : 에 의해

(2) Tangent 벡터에 미분 정의가 들어가 있으면 i.e. Lemma 1.4.6 에 의해

즉, 에서 (f 가 x의 함수가 아니므로) 가 된다.

의 두 가지를 모두 기억해야 한다.)

Example 4.4.7 : Differential forms on the plane

Let be the natural coordinate functions.

the natural frame field on .

The differential calculus of forms on is expressed in terms of as follows:

If is a function, is an 1-form, and is a 2-form, then

(1) where .

(2) , where .

(3) for and as above,

(4)

(5)

Definition 4.4.8 : closed, exact

A differential form is closed if its exterior derivative is zero, ; and is exact if it is the exterior derivative of some form .

Every exact form is closed since for an exact .

Some important calculus

are functions on , and is a 1-form.

(1) ,

(2) ->

(3)

(a)

(b)

(c)

since

(ad -bc 로 기억하자..)

Set , then

* Differential form 에 의한 함수 Derivative의 Wedge product는 Determinant 꼴

* 단순 Differential form의 Wedge product는 Differential form의 단순 곱

4.5 Mapping of surfaces

Definition 4.5.1

A function from one surface to another is differentiable provided that each patch in and in the composite function is Euclidean differentiable (and defined on an open set of ). is then called a mapping of surfaces.

Definition 4.5.2

Let be a mapping of surfaces.

The tangent map of assigns to each tangent vector to the tangent vector to

such that if is the initial velocity of a curve in , then is the initial velocity vector of the image curve in .

Tangent map is a linear transformation, so that preserves velocities of curves.

is regular : all of its derivative maps are one to one.

Linera algebra에서 one to one 이면 isomorphism

Theorem 4.5.2

Let be a mapping of surfaces, and suppose that is a linear isomorphism at some one point p of M

Then there exist a neighborhood of p in such that the restriction of to is a diffeomorphism onto a neighborhood of in .

* 0-form을 가정하면 0-form은 a real-valued function

1. Let be a mapping of surfaces.

1-1 is a functon on , it means that , then there is no reasonable general way to move over to a function on ,

since is a function on , not a function on

1-2 s a functon on , then we pull back to the composite function on . ( f(y in N) on N... 고로 f(F(x in M)) on M 이건 정의 된다. Pull-back 개념)

즉, N 에 정의된 form을 pull-back 해서 M위에 정의한다.

Definition 4.5.6

Let be a mapping of surfaces,

(1) If is a 1-form on , let be the 1-form on such that

for all tangent vectors to

(2) If is a 2-form on , let be a 2-form on such that

for all tangent vectors on

* 0-form의 경우 instead of

Theorem 4.5.7

Let be a mapping of surfaces, and let and be forms on . Then

(1)

(2)

(3)

proof

(1) Let then

(2) Let then

(3) Let be a 1-form then is a 2-form such that .

In addition, let such that and , then

Therefore,

* Full-back to an Exterior Derivative of a form is equal to an exterior derivative of a full-back to a form.

즉, Full-Back 연산은 미분과 관계 없는 무슨 스칼라 상수 처럼 작용하는 연산이다.

4.6 Integration Forms

Let be a curve segment on a surface M.

Set

Definition 4.6.1

Let is a 1-form on . and let be a curve segment in . Then the integral of over is

Problem 2.

(a) If , compute

sol)

에서

, 고로

(기억할 것, 같은 Form 연산은 form과 tangent vector의 inner product를 의미 그래야 scalar가 나온다. ..기본 가끔, dt 연산도 있다. 2개 기억)

Theorem 4.6.2

Let be a function on . and let be a curve segment in from , then

proof)

By definition,

whereas, (form과 tangent vector의 inner product가 아닌 두번째 정의, 증명에 자주 이용)

If is a 2-form on , then the pullback, then is..

... 원래 U1 U2 함수이다. pull-back 되었으므로.., 정의에 의해 M위의 xu xv의 함수로 표현하는 것이다.

Definition 4.6.3

Let is a 2-form on , and let be a 2-segment in . The integral of over is

Definition 4.6.4

The edge curves of are the curve segment such that

The boundary of the 2-segment x is the formal expression

The integral of over the boundary of x is

Theorem 4.6.5 : Stokes' theorem

If is a 1-form on , and is a 2-segment, then

proof

.. 우항의 적분은 x가 아닌 R range 이다.

Let then

Set the Rectangle R is given as

By definition,

따라서,

마찬가지로

Q.E.D

적분 경로는 밑이 출발점이고 위점이 도착점이다. 즉,

에서 a 에서 출발하여 b 점에 도착하는 방향이다.

Problem

4.6.1

4.6.2

, ,

(1) by defintion 4.6.1

...답은 2

(2)

4.6.4 중요함 풀어봐야 함.

4.6.5

, ,

(b) prove

(c)

4.6.7

(a)

4.6.8

4.6.10

by theorem 4,5,7

적분 경로는 밑이 출발점이고 위점이 도착점이다. 즉,

에서 a 에서 출발하여 b 점에 도착하는 방향이다.

답은 -1/2 이다.

4.7 Topological Properties of surface

Definition 4.7.1 : Connected

A surface connected provided that for any two points and of there is a curve segment in from to .

Lemma 4.7.2 : Compact

A surface is compact if and only if it can be covered by the images of a finite number of 2-segments in .

Lemma 4.7.3 : A Properties of compact

A continuous function on a compact region ina surface takes on a maximum (or minimum) at some point of .

Definition 4.7.4 : Orientable

A surface is orientable if there exists a differential (or merely continuous) 2-form on that is nonzero at every point of .

.... 즉 미분가능하고 0이 아닌 2-form이 존재한다는 것. Good... 그래서 면적에 대한 정보를 알 수 있고 이는 Cross-Product 기반이므로

Proposition 4.7.5

A surface is orientable if and only if there exists a unit normal vector field on .

If is connected as well as orientable, there are exactly two unit normals, .

Definition 4.7.6 : homotopic

A closed curve in is homotopic to a constant provided there is a 2-segment (called a homotopy) defined on

such that is the base curve of x and the other three edge curves are constant at .

... 즉 어떤 p 점에서 어떤 loop 가 존재한다는 것

Definition 4.7.7 : Simply Connected

A surface is simply connected provided it is connected and every loop in is homotopic to a constant.

Simply connected 와 differential form 가 closed : 을 생각해보자.

Lemma 4.7.8

Let be a closed 1-form on a surface . If a loop in is homotopic to a constant, then

proof:

(by the definition of the closed differential)

(by Stokes theorem) therefore,

Q.E.D)

Differential 의 의미

Let and . Assume that , Let a cross product of the normalized vector of and

OK.

Lemma 4.7.9 (Poincare)

On a simply connected surface, every closed 1-form is exact.

Note.

1. Closed : , exact : 가 어떤 form의 exterior derivative 일때.

Thus,

2. Path independent : if are curve segment from p to q, then

proof

Simply connected 이므로 위에 정의되는 임의의 curve segment 에 대하여

정의 가능, 여기에 대해 p에서 출발하며 되는 curve 를 정의하면

양변 미분하면

Q.E.D

Theorem 4.7.10

A compact surface in is orientable.

Theorem 4.7.11

A simply connected surface is orientable.

4.8 Manifolds

Definition 4.8.1

A surface is a set furnished with a collection of abstract patches in satisfying

(1) The covering axiom

The images of the patches in the collection cover .

(2) The smooth overlap axiom

For any patches in , the composite functions and are Euclidean differentiable,

and defined on opens sets of .

(3) The Haussdorff axiom

For any points in , there exist disjoint patches with in and in .

* Abstract surface 에서 should be defined the velocity of a curve in .

... Tangent vectors, vector fields, differentiable forms...

Definition 4.8.2 : General Definition of velocity on an abstract surface

Let be a curve in an abstract surface .

For each , the velocity vector is the function such that

for every differentiable real-valued function on .

Definition 4.8.1을 사용하여 Manifold를 정의한다. (4.8.1의 가 이 됨)

Definition 4.8.2 : Manifold

An n-dimensional manifold is a set furnished with a collection of abstract patches

(one-to-one functions , is an open set in ) satisfying

(1) The covering property

The images of the patches in the collection cover

(2)The smooth overlap property

For any patches in the composite functions and are Euclidean differentiable,

and defined on opens sets of

(3) The Haussdorff axiom

For any points in , there exist disjoint patches with in and in .