For the sake of this articles, one at least need to understand the
following concept in that article.
Topological Manifold
Smooth Manifold
Manifold with Boundary
Partition of Unity
Submersion, Immersion and Embedding
etc.
Lie Algebra and Lie Groups
Smooth Differential
Operators
A smooth differential operator of order on is a linear map that
can be expressed locally as a finite sum of the form on each coordinate
chart : where are smooth
functions on , , and is a multi-index
with .
And by definition, a smooth vector field on is a smooth differential operator of
order .
Using partition of unity, we can prove that any smooth differential
operator can be expressed as composition of at most smooth vector fields.
And for any smooth function .
Peetre proves that any linear differential operator that satisfies
the above condition is a smooth differential operator.
Lie Bracket on Smooth
Vector Fields
Given two smooth vector fields on a smooth manifold , the Lie bracket of
and , denoted as , on a local coordinate it is defined as: note the second order derivative is diminished, it is a the
smooth vector field on .
Generally speaking, if is a
differential operator of order ,
is of order . shall be a differential operator of order .
The Lie bracket
satisfy the following property for any
Antisymmetry:
Jacobi Identity: .
-Linearity:
Note that it is not linear with respect to the second argument.
The Lie bracket gives a Lie Algebra structure on
.
Integral Curves
Let , if a
smooth curve is
such that
for all , then
is called an
integral curve of the vector field .
if , then is called the initial
point of the integral curve.
Example:
For the vector field , the integral
curve is: where
are the initial point of the integral curve.
Assume is in a coordinate
chart , we write .
We denote as the -th coordinate of on .
So the equation can be written as:
So we have the system of ordinary differential equations: for .
Conversely, every system of ordinary differential equations of the
form above defines a unique integral curve on of the vector field .
By the existence and uniqueness theorem of ordinary differential
equations on , we have
the following theorem:
Assume is a
smooth vector field on a smooth manifold , then for every , there exists a neighborhood of , and a smooth mapping such that: 1. for all . 2. For fixed , is the integral
curve of with initial point for all . 3. The
integral curve in is unique in
the sense that if
is another smooth curve with , then for all .
Reparametrization
Generally, the reparametrization of an integral curve is not an integral curve of the
vector field . But if the
reparametrization is linear, the reparametrization is still an integral
curve of the vector field .
If is an integral
curve of , then: 1. Let , be the curve defined by
for all
. Then is also an integral curve of
. 2. Let , then
is an integral curve of
For any , the integral
curve starting at has a maximal
interval of existence . It is
easy to prove that must be
open.
Denote the maximal curve as
We have the following property:
For , if
.
And is
called a complete vector field if
Flow
The flow is a mapping on set such that .
In certain situations one might also consider local flows, which are
defined only in some subset is an
open subset of .
We can see that a vector field
on induced a flow on .
We have the following theorem stating the smoothness of such flow:
> For any , the flow is smooth on its
domain .
Proof
By the property of flow, it suffices to prove that is smooth around any point . And then we can transfer the
smoothness to any point by
the property of flow.
By the fundamental theorem above, there exists a neighborhood of and such that
is smooth. So is smooth around
.
If the vector field is
complete, then the flow is
defined on the whole .
We call the flow of a
complete vector field the
global flow of .
Otherwise, we call it a local flow.
Completeness and
the induced diffeomorphism
We shall naturally ask, when is a vector field complete?
A sufficient condition is that the vector field has compact
support.
We define the support of a vector field as:
We state the proof idea here:
For , the integral curve
is constant, so it is defined on .
For , then the
integral curve shall be
always in .
And there exists a such
that is defined on
.
Now since is compact, we can
cover with finite many , and let . Then for
any , is defined on . And by proper
reparametrization, we can extend the integral curve to the whole .
As a corollary, a vector field on a compact manifold is always
complete.
Induced Diffeomorphism
Let be a
complete vector field on , then
the flow induces a one-parameter group of diffeomorphisms defined
as: And satisfies the
following properties: - - for all . - Each is a
diffeomorphism and .
That is to say, a complete vector field on induces a family of diffeomorphisms
from to itself.
Dynamical Systems
Induced by Vector Fields
Mathematically, a dynamical system is a triple , where is a set called the state
space, is a semi-group
of transformation parameters called the time set, and
is a mapping from to that satisfies the properties of
flow.
So a complete vector field on
a smooth manifold induces a
dynamical system , where is the
flow of .
Application in Morse Theory
The following theorem is a fundamental result in Morse theory
indicating the topological structure of the manifold is determined by the properties of the
vector field around the critical
points.
Let be a smooth manifold, and
be a smooth
function on . For any , we define the
sublevel set
For , assuming that is compact and is a regular value of
, then there exists a
diffeomorphism such
that (a
deformation retract, more precisely).
Proof
First we embed into a Eulidean
space, so a inner product is given in every tangent space .
We define a gradient vector field as following: We take a compact support bump function such that and is a open set not containing any
critical point.
And is a compact support vector field on .
Let be the global flow generated by , then
So we have for any .
Thus, the diffeomorphism maps to .
Lie Derivative
The Lie derivative of a function with respect to a
vector field
is defined as: where is the
flow of .
The Lie derivative of a vector field with respect to a
vector field
is defined as: where is the
flow of .
The second equality is because for any defined around :
And
The Lie derivative is a derivation on the Lie algebra . That
is to say, for any and , we have the Leibniz rule:
Proof
By the Jacobi identity, we have:
So we can see the Jacobi identity as the Leibniz rule of the Lie
derivative.
Lie Group
Let be a group, we say is a Lie group if it
is a smooth manifold and the group operation is smooth mappings.
Examples of Lie groups include: - with addition as the group
operation. - with
multiplication as the group operation. - The general linear group , which
consists of all invertible matrices with real entries, with matrix multiplication as the
group operation. - The special orthogonal group , which consists of
all orthogonal matrices
with determinant 1, with matrix multiplication as the group operation. -
, the circle group,
which can be identified with the unit complex numbers under
multiplication. - , the
3-sphere, which can be identified with unit quaternions under
multiplication.
A Lie group must satisfy the following properties: - The base space
(i.e. the underlying topological
space) is orientable. - The fundamental group is a abelian group for a
connected Lie group . - Every Lie
group is parallelizable, i.e., its tangent bundle is trivial , where
.
Left and Right
Multiplication
Assume is a Lie group, for any
, the left multiplication and
right multiplication induce two mappings on :
Note we can write them as the composition of the group operation and
the inclusion map: where is
the inclusion map from the identity element to .
And .
Thus, and are diffeomorphisms on .
To show the usefulness of left and right multiplication, we leveraged
them to prove that every Lie group is parallelizable.
Theorem Every Lie group is parallelizable, i.e., its tangent
bundle is trivial , where .
Proof Consider where is the identity
element of . It is easy to see
that is a bijection. And since
is a diffeomorphism, is a linear isomorphism from
to . So is a diffeomorphism.
Use the same technique, we can also show that the inverse mapping
is a smooth
diffeomorphism on .
And the differential of the inverse mapping is given by:
And .
Lie Algebra
From the proof above, we see that the tangent space determines the structure around any
point .
So we focus on the structure around the identity element of the Lie group .
Definition
Let be a vector space over
(or any field ) with a map
satisfying the following properties: - Antisymmetry: for all . - Linearity: . - Jacobi Identity: for all .
Then is called
a Lie algebra and the map is called the Lie
bracket.
Left Invariant Vector Fields
Assume is a Lie group, is a vector in the tangent
space at the identity element . We
can define a vector field as follows: This vector field is called a left invariant vector
field because it is invariant under left multiplication, i.e.,
for all .
It is obvious that any left invariant vector field is uniquely
determined by its value at the identity element .
Denote the set of all left invariant vector fields on as .
The set is a vector
space over and , .
For any , we can verify that: Thus is also a left
invariant vector field.
So is a Lie algebra
with the Lie bracket induced by the Lie bracket of , which is called
the Lie Algebra on Lie group .
Example
is a
open subset of
So
as the tangent space of on is isomorphic to .
And for all
.
Homomorphism of Lie
Groups and Lie Algebras
There are two types of homomorphism, one is the Lie group
homomorphism and the other is the Lie algebra
homomorphism, which are morphism between Lie group categories
and Lie algebra categories respectively.
Let be Lie groups, if smooth
mapping is a group
homomorphism, we call a Lie
group homomorphism. If is a
diffeomorphism, we call a Lie
group isomorphism.
Let be Lie
algebras, if a linear mapping satisfies
for all
, we call a Lie algebra homomorphism. If is invertible, we call a Lie algebra isomorphism. Note that if
is invertible, then is also a Lie algebra
homomorphism.
Examples
For any Lie group , the
conjugation mapping
defined as: is a Lie group isomorphism for any . The inverse mapping is given by
.
For any ,
the adjoint mapping
defined as: is a Lie group isomorphism. The inverse mapping is given by
.
Induced Lie Algebra
Homomorphism
Let be a Lie group
homomorphism, then the differential of at the identity element , induced a mapping , is a Lie algebra homomorphism
from the Lie algebra
of to the Lie algebra of .
To write it more explicitly, for any ,
is the value of at .
To see that is a Lie
algebra homomorphism, we need to show that for
any .
For any , we
have:
To move on, we need:
Lemma:
Proof: For any , we have:
For the third step, is a
group homomorphism, we know that , so , differentiating both sides at
gives us .
That is to say, the definition of the push forward by the differential of the
diffeomorphism is equivalent
to the corresponding vector field induced by the left action on .
So we have:
The above proof can be summerized as the following:
Step 1: is -related to .
For , is
left-invariant, and for , since , we have .
Differentiating at : Thus:
So, , meaning is -related to .
Step 2: Lie Bracket Preservation
If are
-related to , then
is -related to .
At : Since both are left-invariant, .
Linearity of follows from
.
Thus, is a Lie algebra
homomorphism.
Adjoint Map
Since every element gives
a isomorphism on Lie group: Such map induces a isomorphism on Lie algebra:
So this yields a new mapping as a group homomorphism between and , which is called
the adjoint representation of .
Again, take the induced mapping of , we get
We shall see that .
Exponential Map
Recall that Lie algebra can be viewed as the tangent space of of some Lie group . For now, we put whether such can always be found aside. And a
tangent vector induced a
left-invariant vector field by
the left multiplication on
. Such vector fields could induce
a flow on , i.e. we can build a connection between
the Lie algebra and the induced flow on the group.
We shall ask the following question:
Is complete?
How does change
regarding different choices of ?
The answer to the first question is yes.
Although is not necessarily
compact, but if we locally
have as the fundamental theorem states, then we can apply
to transform the local maps to
anywhere , thus the integral curve
can be extended to the whole manifold.
Note where are the
diffeomorphism induced by the flow at time .
Exercise: Prove it.
Hint: First assume , and
show that any can be
written as finite product of
So And for all
Thus
So the mapping is a smooth group homomorphism.
Exercise: > Prove it is smooth.
So is called the
one-parameter subgroup of .
And every gives
rise to a one-parameter subgroup of .
Conversely, let be a
one-parameter subgroup of . Then
is a left-invariant vector field on .
Proof:
So has a bijection with
, which is the
third interpretation of the Lie algebra—the infinitesimal generator of
all the one-parameter subgroup of .
Exercise > Now, prove .
To study the second problem, we define the exponential map Note .
The exponential map has the following properties: 1.
Differentiable: The exponential map is a smooth
(infinitely differentiable) map.
Local Diffeomorphism: The exponential map is a
local diffeomorphism around the zero vector in the Lie algebra . This means that for small
enough , the map
is invertible in a
neighborhood of .
Examples
(1). , .
To calculate , write .
And Note yields .
(2). For , (3). For where .
is a
smooth mapping, and its differential at is the identity map
Proof Consider a vector field manifold ,
The integral curve just be
The flow is smooth, thus is smooth.
The identity property is given by the following:
Particularly, is a
diffeomorphism around . That is to
say, locally around , the Lie
group looks like the corresponding Lie algebra.
Naturality
Given any Lie group homomorphism , the diagram
commute.
Exercise Proof it > Hint: For any , represents a
one-parameter groups in , what is
the infinitesimal generator?
Particularly, for mapping and , we have: 1.
2.
Baker-Campbell-Hausdorff
Formula
TODO.
Exterior Algebra,
Differential Forms
Note: we discuss multilinear functions over -vector spaces, but the theory
can be generalized to -vector spaces or other vector
spaces over a field.
Dual Space
Let be a -vector space, we denote all of
the mapping from to as .
The dual space of , denoted as or , is the vector space of all linear
maps from to , i.e., .
The elements of the dual space are called covectors
or dual vectors.
The basis of the dual space
is defined as the set of linear functionals that map each basis vector
of to , while mapping all other basis
vectors to zero.
To be specific, let be -dimensional vector space, if is a basis of
, then the dual basis
of is defined by: where is the Kronecker delta.
To show that is a basis of , we need to show that it is linearly
independent and spans .
Let be a linear
functional. We can express in
terms of the dual basis as:
where are the
coefficients of the linear functional in the dual basis. This shows that
spans .
If
for some coefficients , then for
any basis vector , we
have: for all . This implies that
all coefficients , showing
that the dual basis is linearly independent.
Multilinear function
Let be the -fold Cartesian product of a vector
space , i.e., . A multilinear function
or -linear
function is a function that is linear in each argument separately. It is a
type tensor.
One can prove that all multilinear functions on forms a vector space, denoted as or .
Examples
Dot Product: For a -vector space with a standard basis , the dot
product is a bilinear function defined as: where and are vectors in .
Determinant: The determinant is a multilinear
function
defined on the space of
matrices. It is linear in each row (or column) of the matrix.
Let be a
permutation of the indices . The left action of on a multilinear function is defined as:
A symmetric multilinear function is a multilinear
function that is invariant under any permutation of its arguments.
Formally, for any
permutation .
An alternating multilinear function is a multilinear
function that changes sign when two of its arguments are swapped, or
equivalently, changes by a factor of when permuted by .
Formally, for any permutation , where is the sign of the
permutation.
All symmetric multilinear functions on forms a vector space, denoted as or . All
alternating multilinear functions on forms a vector space, denoted as or .
Symmetrizing and
Alternating Operators
Let be a vector space and
be a
multilinear function. We can define the symmetrizing
operator and alternating operator on .
The symmetrizing operator is defined as:
The symmetrizing operator takes a multilinear function and produces a
symmetric multilinear function by averaging over all permutations of its
arguments.
The alternating operator is defined as:
The alternating operator takes a multilinear function and produces an
alternating multilinear function by summing over all permutations of its
arguments, weighted by the sign of the permutation.
We shall only prove the alternating operator does produce an
alternating multilinear function. The proof for the symmetrizing
operator is trivial.
Proof: Let be
a multilinear function and be
the result of the alternating operator. For any permutation , we have:
The alternating operator
satisfies the following properties: 1. Linearity: For
and , we have:
2. 3.
Tensor Product
Let and be two multilinear functions
on . The tensor
product of and , denoted as , is a multilinear function on
defined as:
The tensor product satisfies associativity and bilinearity. Formally,
we have: 1. Associativity: for any multilinear function . 2. Bilinearity: For and ; , we have:
Example: Let be a basis of , be the dual
basis of . Then the inner
product is a bilinear function defined as: where and are vectors in
.
Now, let , .
So we can write the inner product as a tensor product of the dual
basis:
Wedge Product
The wedge product or exterior
product is an operation on alternating multilinear functions
that produces a new alternating multilinear function. It is denoted by
and is defined as
follows:
For all and , the wedge product is defined by:
or equivalently,
To avoid division by factorials, we can define the wedge product as
sum over sorted permutations:
Let and be alternating multilinear
functions. The wedge product satisfies the following properties: 1.
Bilinearity: It is bilinear on and . 2. Associativity: It
is associative, i.e., for any alternating multilinear
function . 3.
Anticommutativity: It is anticommutative, i.e., .
Particularly, when
is odd. 4. , .
Let , for , the wedge product
is , where is
the multinomial coefficient.
or equivalently,
This shows the associativity directly.
Relation to determinant
Let , ,
then:
This is the determinant of the matrix formed by evaluating the dual
basis vectors on the vectors .
Graded Algebra and Exterior
Algebra
Graded Algebra
Let be a field, the
graded algebra is a vector space over equipped with a direct sum
decomposition , where each is a
vector space of degree . The
multiplication operation satisfies:
is called
anticommutative if for all and , .
A homomorphism of graded algebra is an algebra homomorphism that
preserves the degree.
Exterior Algebra
The exterior algebra or Grassmann
algebra is a anticommutative graded algebra
defined on a vector space with
the wedge product , where
. When is
finite-dimensional, we can write or , where .
This is because for
, which we shall prove
later.
is called the
-th exterior
power of .
Basis of
Let be a
-indices, and are the basis of
the vector space and its dual
respectively. Denote as
, and . , where is the generalized Kronecker
delta, which is if and otherwise.
Let , we claim that is a basis of .
The proof is similar to the proof in the dual space section.
Corollary 1: The dimension of is , where .
Corollary 2: The dimension of is , where .
Corollary 3: The dimension of is for .
Proof: For , the
set of indices cannot be chosen
such that , hence .
Interior Product
Let be a vector space, , and . The
interior product of and , denoted as , is a linear map defined by:
where means
that the -th term is omitted from
the wedge product.
Let , the interior product satisfies the following
properties: 1. Nilpotency: for any . 2. . 3.
Linearity: for any
and .
The interior product is
called the interior product or
contraction with respect to the vector . It reduces the degree of the form by
1.
Observe the definition, we see that it just the evaluation of the
first argument of the alternating multilinear function at the vector , and then take the wedge product of the
remaining arguments.
Pullback
Let be a linear map
between vector spaces and . The pullback of
exterior forms under , denoted as
, is defined by: for all .
The pullback satisfies the following properties: 1. 2. .
Differential Forms
We have defined the exterior algebra and the wedge product on an abstract vector space . Now we can define differential
forms on a manifold ,
where is the tangent space at a point .
Cotangent Space and
Cotangent Bundle
The cotangent space at a point is the dual space of the tangent
space . It consists of all
linear functionals on .
Elements of the cotangent space are called covectors or
differential 1-forms.
The cotangent space is a
vector space of dimension , where
is the dimension of the manifold
.
The cotangent bundle is the disjoint union of all
cotangent spaces at each point in :
The cotangent bundle is a
manifold of dimension .
Local coordinates on the cotangent bundle can be defined as , where are local coordinates on and
are the components of a covector in the cotangent space with respect to
the dual basis .
Differential -Forms
Let be a smooth manifold of
dimension . A differential
-form on is a smooth section of the -th exterior power of the cotangent
bundle, denoted as .
In other words, a differential -form on a open subset is a mapping that assigns to each point an alternating multilinear function and
for all .
Let be a basis of
. The corresponding dual basis
of is , where .
A basis of is then
given by , where is a -tuple of indices with , and denotes the
wedge product .
Therefore, a differential -form
can be expressed in local
coordinates as: where are functions on .
A differential -form is called smooth
if all the functions are smooth
functions on .
Let be the space of all smooth differential -forms on a manifold . It is a vector space over .
The wedge product of two differential forms and is pointwise
defined as: where and are the coefficients of and in local coordinates,
respectively. Note that if and
are such that , then due to the
anticommutativity of the wedge product. The wedge product of
differential forms is bilinear and associative, and it satisfies the
anticommutativity property:
Let be the space of all smooth differential forms on
. It is a graded algebra with
respect to the wedge product .
Coordinate
Functions and Their Differentials
Let be a coordinate chart on a smooth manifold of dimension . The coordinate functions are smooth
functions that assign to each point its -th coordinate
.
The coordinate 1-forms are smooth sections of the cotangent
bundle, each
satisfies where is
the bundle projection.
We have defined by for any tangent vector . The coordinate 1-forms satisfy the dual basis
property: making the dual basis of corresponding to the coordinate
basis of .
Any differential -form on can be uniquely expressed as , where
with ,
, and are smooth functions.
Pullback of Differential
Forms
Let be a smooth map
between manifolds, and let be a differential -form on . The pullback of by , denoted as , is a differential -form on defined by: where
is the
differential of at the point
.
Local Coordinate Expression
Let and
be coordinate
charts on and respectively, with .
Let have local
coordinate representation , where are coordinates on
and are
coordinates on .
If has the local
expression: where are smooth functions.
Then the pullback has
the expression:
Note:
Therefore:
Properties of Pullback
The pullback operation satisfies the following important
properties:
Linearity: for
Preservation of wedge product:
Functoriality: If and are
smooth maps, then:
Identity:
Exterior Derivative
The Exterior Derivative is an operator that
generalizes the concept of differentiation to differential forms. It is
defined as follows: where is a differential -form on , and for .
It can be naturally extended to the exterior algebra as follows:
where and .
The exterior derivative satisfies the following properties: 1.
Linearity: for . 2.
Nilpotency: , i.e.,
for any differential form .
3. Antiderivation Property: for and . 4.
Pullback Compatibility: For any smooth map , we have . One can
verify this property using local coordinates and chain rules. 5.
Vanishing of Top Forms: if , where .
We only prove the nilpotency here:
Proof:
Only to prove , $$ $$ since mixed partial derivatives commute for
smooth functions.
Lie Derivative on
Differential Forms
The Lie derivative of a differential form with respect to a
vector field is defined as: where is the flow of the vector field
at time .
The Lie derivative satisfy the following property: 1.
2. 3.
4. 5. Cartan’s Magic Formula
Proof
Use (5).
Do induction as (5).
For the -form and -form, it is trivial proved to be true.
Then inductively prove by decomposing it on local coordinates. Let be a -form. Locally, we have:
Integration of Differential
Forms
Highest Degree Forms
Let be a smooth manifold of
dimension . The highest
degree differential form on is a differential -form, denoted as . It can be
expressed in local coordinates as: where
is a smooth
function and is the basis of the cotangent space at each point in
.
If we change the local coordinates to by the pullback
map , , then the highest
degree form transforms as:
We can define the integral of a highest degree form over as: where the integral is taken
over the open set .
To make integral is well-defined and does not depend on the choice of
local coordinates, we need to ensure that the integral is invariant
under coordinate changes.
But according to the change of variables formula on integral over
, we only have:
So to make the integral well-defined, we need to restrict the
transformation to be orientation-preserving, i.e., .
We call the two charts and
orientation-compactible if the Jacobian determinant
is positive for all points in .
For a manifold, if all of the transition maps between charts are
orientation-preserving, we say that the manifold is
oriented. And if such a choice of orientation can be
made, we call the manifold orientable. And we call such
atlas an
oriented atlas or orientation of the
manifold .
Integration on Manifolds
Let be a smooth oriented
manifold of dimension . And let
be a smooth partition of
unity subordinate to an open cover of , where are smooth functions
such that
on and . The
integral of a differential -form
over the
manifold is defined as: where is a
highest degree form on and the
integral is taken over the open set .
Note is compactly
supported in and is a smooth
differential -form on . So the integral is well-defined
and finite. The integral is independent of the choice of partition of unity and
the open cover .
Change of Variables
Let be a smooth
map between oriented manifolds
and of dimension , with orientation respectively. We say that
is
orientation-preserving if for every chart , the chart is
orientation-compactible with . We say that is
orientation-reversing if for every chart , the chart is not
orientation-compactible with .
Actually, mapping is either
orientation-preserving or orientation-reversing if and are connected. Otherwise, it either
preserves or reverses the orientation on each connected component of
and .
The change of variables formula for the integral of
a differential -form under the map
is given by:
if is an
orientation-preserving map, then: if is an
orientation-reversing map, then:
The relation of the orientability and the differential forms is given
by the following theorem:
Theorem: A smooth manifold is orientable if and only if there
exists a nowhere vanishing differential -form .
And we call such -form a volume form on
.
Stokes Theorem
Induced Orientation on the
Boundary
Let be an oriented smooth
manifold with boundary .
The boundary is a smooth
manifold of dimension . And
is a volume form on that defines the orientation of .
We can define an orientation on induced by the orientation of as follows: At each point , we can choose a local
chart around such that
on and on . The orientation on is defined by the volume form:
where is
the outward-pointing normal vector field on .
Or explicitly, if on .
And gives a nowhere
vanishing form on .
To integrate a -form on , we locally write Then
The Stokes Theorem state that, for any
Proof
If is compactly support
in coordinate chart
If ,
is on , then:
If , then:
The last step is because for and .
For general , we can use
partition of unity to prove it.
de Rham Theory
Let be a smooth manifold,
be a smooth
differential -form. We call a closed form if
, and we call an exact form if
there exists a differential -form such that .
We denote the space of closed -forms on as and the space of exact
-forms as .
Let , we have the
following result:
if
.
, , where is
the number of connected components of .
By definition,
Since , we have the
following property:
We define the de Rham cohomology of as: and is the equivalence class of
in , called the de Rham
cohomology class of .
is a vector space
over and we shall see
that it is linear with respect to the cup product.
Example:
For , , , hence , and for
.
For , we know
and for
. So we only need to
compute .
Now, consider , we have for some . Since is a smooth function on , it can be expressed as
for some periodic
function with period . Thus, we have: where is a periodic
function with period and .
Thus, , which is the first de Rham cohomology group of the
circle since the cohomology class is represented by the integral value
of the 1-form over the circle.
Let , we have the
following results: - for . -
,
where is the number of connected
components of .
Betti Numbers and Euler
Characteristic
If
for any , we define the
Betti number which is called the -th Betti number of the manifold .
The Euler characteristic of is defined as:
From the above result, we know , and .
Pullback
Let be a smooth map
between manifolds, the pullback of the de Rham cohomology is defined as:
by for
.
Since , maps closed forms to closed forms and
exact forms to exact forms.
Hence induces a well-defined
map on cohomology.
Note that the pullback of de Rham cohomology satisfies the
contravariant functoriality: 1. for smooth maps and . 2. .
Thus, the de Rham cohomology is a contravariant functor from the
category of smooth manifolds to the category of graded algebras over a
field. So a diffeomorphism induces an isomorphism .
Particularly, and are diffeomorphism
invariants.
Cup Product
Note that the wedge product on differential forms induces a
operation on de Rham cohomology, called the cup
product. defined by for and .
Assume and
, are closed forms, to prove
cup product is well-defined, we calculate Therefore, we have so the cup product is well-defined.
The cup product is bilinear, associative, and commutative up to a
sign:
The cup product induces a graded algebra structure on the de Rham
cohomology: with the cup product
as the multiplication
operation.
And
becomes a graded commutative ring with identity, called the de
Rham cohomology ring of .
Homotopy Invariance
We say and are homotopy equivalent if there exist
continuous maps and
such that is homotopic to the identity
map on and is homotopic to the identity
map on .
If and are homotopy equivalent, then for all
. This is stronger than the fact
that is a
diffeomorphism invariant, as homotopy equivalence does not even require
the manifolds to have the same dimension, e.g. and .
Note: This shows that the de Rham cohomology is determined by the
topological structure of the manifold, not relevant to the smooth
structure.
Poincaré Lemma
The Poincaré Lemma states that if is a star-shape area in , then for . Particularly, .
This is trivial since the star shape area can always contract to a
point.
Moreover, since for any point in a manifold has a neighborhood that
is diffeomorphic to a star shape area in , we have the following
collary:
For any -th closed form and any , there is a neighborhood and -form such that on .
Proof of Homotopy Invariance
It suffices to prove that functor is homotopic
invariant:
If two smooth maps
are homotopic, then .
Because assume this holds, if
and are homotopy equivalent, we
can use smooth approximations of the continuous maps to get smooth maps
and such that and are homotopic to the identity
maps on and , respectively. Then we have which implies
that and are isomorphisms.
Now we prove the claim that if are homotopic, then .
We define the cochain homotopy as follows: > if
are
homotopic. If there is a sequence of mapping
satisfying: we say the sequence is a cochain homotopy between
and .
If there exists such a cochain homotopy , for any
So .
Now we prove the existence of the cochain homotopy:
First, we prove a lemma
Let be a complete vector field
on , be the flow generated by . Then there exists a linear operator
s.t. : Proof Therefore, denote
Now we can construct the cochain homotopy between and as follows: Let , then is a
complete vector field on . Let
be the flow
generated by , then is a smooth map.
By the Lemma, we have a linear operator such
that:
By the Whitney Approximation theorem, we can find a smooth homotopy
such that
and for all .
Let be
the inclusion map, , then Then we have:
So and
is a
cochain homotopy between and
.
de Rham Theorem
The famous de Rham theorem states that the de Rham
cohomology is isomorphic to the singular cohomology with real
coefficients, i.e., for any smooth manifold ,
which we shall not prove here.
This theorem reveals the duality between the topological structure
and the algebraic structure (differential forms) on a manifold.
Chain Complex
The chain complex is a sequence of abelian groups
(or modules) connected by homomorphisms , such that the image
of one homomorphism is contained in the kernel of the next. The
composition of any two consecutive maps shall be the zero maps, or for short.
The cochain complex is the dual notion to the chain
complex.
where .
The elements in the kernel of
are called (co)cycles (or closed
elements), and the elements in the image of are called
(co)boundaries (or exact elements).
Right from the definition of the differential, all boundaries are
cycles. The -th
(co)homology group is the group of (co)cycles
modulo (co)boundaries in degree ,
that is,
A exact sequence is a (co)chain complex whose
(co)homology groups are all zero, which means all closed elements are
exact. A short exact sequence is a bounded exact sequence in which only
the groups , , may be nonzero. For example, the
following chain complex is a short exact sequence.
de Rham Complex
The de Rham complex is the sequence of differential
forms: