d
Textonly Preview
0.1
Smooth manifolds
Definition 1. Given a topological space M , we say M is a topological nmanifold if
it has the following properties:
* M is Hausdorff
* M is second countable, meaning it has a countable basis for its topology
* M is locally Euclidean of dimension n.
The last condition is equivalent to saying every point of M has a neighborhorhood
homeomorphic to an open set in
n
R . We'll commonly call M an nmanifold if it satisfies
these properties. We can think of manifolds as spaces that look like Euclidean space on
a small enough scale. The following definition captures this idea.
Definition 2. Let M be a topological nmanifold. A coordinate chart on M is a pair
(U, ), where U is an open subset of M and : U U is a homeomorphism from U
to an open subset U = (U )
n
R . U and are called the coordinate domain and
coordinate map respectively of the coordinate chart.
Example 1. Let U
n
k
R be an open set, and F : U R be a continuous function. The
graph of F is the subset of
n
k
R x R defined by
(F ) = {(x, y)
n
k
R x R : x U and y = F (x)},
with the subspace topology. This is a topological manifold of dimension n.
Example 2. The unit nsphere n
n+1
S R
with the subspace topology is an nmanifold.
Example 3. The ndimensional real projective space, denoted by
n
RP
is defined as
the set of 1dimensional linear subspaces of
n+1
R
. In another words, this is the set of all
lines that run through the origin. It has the quotient topology given by the natural map
:
n+1
n
R
\{0} RP that sends a point in the domain to its equivalence class (the line
through the origin that this point intersects). This is an nmanifold.
Example 4. If M1, . . . , Mk are manifolds of dimensions n1, . . . , nk, then the product
space M1 x . . . x Mk is a topological manifold of dimension n1 + . . . + nk.
Before we can do calculus on manifolds, we need to equip the manifold with some
additional structure so that we can perform derivatives in a way that makes sense. Recall
that if U and V are open subsets of
n
m
R and R
respectively, then a function F : U V
is called smooth if each of its component functions has partial derivatives of all orders.
If F is a bijection and has a smooth inverse, then F is called a diffeomorphism. We'll
now define a way to give a topological manifold a notion of "smoothness".
Definition 3. Let M be a topological nmanifold. If (U, ) and (V, ) are two charts
such that U V = , the composite map 1 : (U V ) (U V ) is called the
transition map from to . Two charts are said to be smoothly compatible if either
U V = or the transition map 1 is a diffeomorphism.
Definition 4. We define an atlas for M to be a collection of charts whose domains cover
M. An atlas A is called a smooth atlas if any two charts in A are smoothly compatible
with each other.
1
Note that every topological manifold M has an atlas, since every point in M is con
tained in the domain of some chart by the property of M being locally Euclidean. The
above definitions set us up to be able to talk about a "smooth structure" on a manifold.
We could define a smooth structure by giving a smooth atlas on a manifold and say a
function f : M R is smooth iff f is smooth (in the ordinary sense) for each chart
in the given atlas.
A small problem right now though is that we could have two different smooth atlases
that describe the same smooth structure, that is, the same family of functions f are
smooth on a manifold with respect to different atlases. A way around this is to define an
equivalence relation on the set of all smooth atlases on M , where A1 A2 iff the charts
in A1 in A2 are smoothly compatible with each other. A smooth structure on M could
then just be an equivalence class, and it's clear that any representative would determine
the same smooth structure. In this way we avoid the problem of having different smooth
structures that are essentially identical (that is, they describe the same family of smooth
functions on the manifold). However, it is easier to just define smooth structure in the
following way:
Definition 5. A smooth atlas A is called maximal if it is not contained in any strictly
larger smooth atlas. In another words, any chart that is smoothly compatible with a chart
in A is already in A. A smooth structure on M is a maximal smooth atlas, and a
smooth manifold is a pair (M, A), where M is a manifold and A is a smooth structure
on M.
As a remark, not every topological manifold can be equipped with a smooth structure.
The maximal smooth atlas on M described above will, in general, contain an extremely
large amount of charts. The following lemma shows that fortunately we only need to
specify some smooth atlas on M to talk about its smooth structure. It also shows that
the previously proposed definition of smooth structures via equivalence classes is in fact
an equivalent way to define the smooth structure.
Lemma 6. Let M be a topological manifold.
(a) Every smooth atlas for M is contain in a unique maximal smooth atlas.
(b) Two smooth atlases for M determine the same maximal smooth atlas if and only
if their union is a smooth atlas.
If an nmanifold has a global coordinate chart, that is, a chart (M, ) where is a
homeomorphism from M to some open subset of
n
R , then this chart determines a smooth
structure on M . This is because the smooth compatibility condition is trivially satisfied
because there are no other charts in the atlas consisting of just the chart (M, ). Note
that this does not mean (M, ) is the only chart in the smooth structure determined by
this chart, but rather this smooth structure is the set of all charts that are smoothly
compatible with (M, ). In this way, this single chart determines a smooth structure on
M .
Example 5. A 0manifold M is just a countable discrete space. This is because each
p M has a neighborhood U homeomorphic to
0
R , but this forces U = {p}. If each one
point set is open in M , then M is necessarily a discrete space, and it must be countable
2
since it it has a countable basis (the one point sets). Since there is only map : {p}
0
R
for each p, the set of all charts on M trivially satisfies the smooth compatibility condition
(the intersection of the domains is trivial), thus M has a unique smooth structure.
Example 6.
n
R is a smooth nmanifold with smooth structure determined by the global
coordinate chart ( n
n
R , Id n ). This is called the standard smooth structure on
.
R
R
Other smooth structures exist though. For example, on R, consider the homeomorphism
: R R given by (x) = x3. This global chart determines a smooth structure on
R; however, it is not the same as the structure determined by (R, Id ). This is because
R
the transition map Id 1 :
1(x) = x1/3. This map is not
R
R R is given by IdR
smooth at the origin, so by the previous lemma, these two charts cannot determine the
same smooth structure.
We'll want to talk about finitedimensional vector spaces as manifolds now, but first
we need to show that any norm on a vector space generates the same topology. This is
the essence of the following lemma.
Lemma 7. Any two norms  * 1 and  * 2 on a finitedimensional vector space over R are
equivalent. That is, there exist some c, C > 0 such that for any x V ,
cx1 x2 Cx1.
Proof. Fix a basis {E
n
i}n
for V , and let T :
V be the basis isomorphism defined
i=1
R
T (x) = T (x1, . . . , xn) =
n
x
i=1
iEi.
Let  * 1 be any norm on V , and define a second
norm  * 2 on V by y2 = T 1(y), where T 1(y) is the Euclidean norm of T 1(y) in
n
R . Note that this also means x = T (x)2. It will suffice to show  * 1 and  * 2 are
equivalent norms in V . Denote by (V,  * 2) the space V with norm topology induced by
 * 2.
First, verify T : (V,  * 
n
n
2) R
is also a homeomorphism, where R has the standard
topology. Continuity of T follows easily from the linearity of T and the definition of  * 2:
Given some
> 0, for any x  y2 < where = , then T (x)  T (y) = T (x  y) =
x  y
n
2 <
. Similarly T 1 is continuous, and it follows V
R . It follows now as a
consequence of the HeineBorel Theorem, any subset of V that is closed and bounded
(with respect to the topology induced by  * 2) is compact. We now observe that if we let
C =
(E
i
i1)2, then for any x V :
x1 = 
x
x
x
x
(E
i
iEi1
i
iEi1 =
i
i * Ei1
i
i2 *
i
i1)2 = C x2
We'll show the map  * 1 : (V,  * 2) R is continuous. Given any > 0, let = /C,
then x1  y1 x  y1 < Cx  y2, and the claim follows. Consider now the
set S = {x V : x2 = 1}, which is closed and bounded in (V,  * 2), and hence
compact. Thus,  * 1 attains a minimum on S, that is, there exists some x0 S such that
x
0 < c := x01 x1 for any x S. We then have that for x V ,
S. It follows
x2
x
x
that
=
1 c, and so x

1 cx2.
x2 1
x2
Example 7. Let V be a finitedimensional vector space with the norm topology. Fix a
basis {E
n
i}n
for V and let T :
V be the basis isomorphism defined by the map that
i=1
R
sends the standard basis in
n
R to {Ei}n . This map is also a homeomorphism, and the
i=1
chart (V, T 1) defines a smooth structure on V . This is the standard smooth structure
on V , and doesn't depend on the choice of basis for V .
3
Example 8. Given a smooth nmanifold M and an open subset U M , we can define
an atlas on U by AU = {smooth charts (V, ) for M such that V U }. U is then a
smooth nmanifold, called an open open submanifold of M .
As an application of the previous examples, we can show GLn(R) Mn(R) is a n
manifold. Since Mn(R) is a (finite) n2dimensional vector space and det : Mn(R) R
is continuous, it follows det1(0) = Mn(R)\GLn(R) is closed. Thus GLn(R) is an open
submanifold of Mn(R).
Lemma 8 (Smooth Manifold Construction). Let M be a set, and suppose we are
given a collection U
n
of subsets of M, together with an injective map : U R
for
each , such that the following properties hold:
* For each ,
n
(U) is an open subset of R .
* For each and ,
n
(U U ) and (U U ) are open in R .
* Whenever U U = , 1 :
(U U ) (U U ) is a diffeomorphism.
* Countably many of the sets U cover M .
* Whenever p,q are distinct points in M, either there exists some U containing both
p and q or there exist disjoint sets U, U with p U and q U.
This lemma allows us to essentially create a smooth manifold from a set by gluing
together charts on the set in a "nice" way. We'll now want to account for certain spaces
that would be manifolds, except they have a "boundary" that prevents them from being
a regular smooth manifold. Let
n
n
H = {x R xn 0}.
Definition 9. An nmanifold with boundary is a secondcountable Hausdorff space
M in which every point has a neighborhood homeomorphic to a (relatively) open subset
of
n
H . M is called a smooth manifold with boundary if the transition maps can be
extended to smooth maps on open subsets of
n
R .
4
Smooth manifolds
Definition 1. Given a topological space M , we say M is a topological nmanifold if
it has the following properties:
* M is Hausdorff
* M is second countable, meaning it has a countable basis for its topology
* M is locally Euclidean of dimension n.
The last condition is equivalent to saying every point of M has a neighborhorhood
homeomorphic to an open set in
n
R . We'll commonly call M an nmanifold if it satisfies
these properties. We can think of manifolds as spaces that look like Euclidean space on
a small enough scale. The following definition captures this idea.
Definition 2. Let M be a topological nmanifold. A coordinate chart on M is a pair
(U, ), where U is an open subset of M and : U U is a homeomorphism from U
to an open subset U = (U )
n
R . U and are called the coordinate domain and
coordinate map respectively of the coordinate chart.
Example 1. Let U
n
k
R be an open set, and F : U R be a continuous function. The
graph of F is the subset of
n
k
R x R defined by
(F ) = {(x, y)
n
k
R x R : x U and y = F (x)},
with the subspace topology. This is a topological manifold of dimension n.
Example 2. The unit nsphere n
n+1
S R
with the subspace topology is an nmanifold.
Example 3. The ndimensional real projective space, denoted by
n
RP
is defined as
the set of 1dimensional linear subspaces of
n+1
R
. In another words, this is the set of all
lines that run through the origin. It has the quotient topology given by the natural map
:
n+1
n
R
\{0} RP that sends a point in the domain to its equivalence class (the line
through the origin that this point intersects). This is an nmanifold.
Example 4. If M1, . . . , Mk are manifolds of dimensions n1, . . . , nk, then the product
space M1 x . . . x Mk is a topological manifold of dimension n1 + . . . + nk.
Before we can do calculus on manifolds, we need to equip the manifold with some
additional structure so that we can perform derivatives in a way that makes sense. Recall
that if U and V are open subsets of
n
m
R and R
respectively, then a function F : U V
is called smooth if each of its component functions has partial derivatives of all orders.
If F is a bijection and has a smooth inverse, then F is called a diffeomorphism. We'll
now define a way to give a topological manifold a notion of "smoothness".
Definition 3. Let M be a topological nmanifold. If (U, ) and (V, ) are two charts
such that U V = , the composite map 1 : (U V ) (U V ) is called the
transition map from to . Two charts are said to be smoothly compatible if either
U V = or the transition map 1 is a diffeomorphism.
Definition 4. We define an atlas for M to be a collection of charts whose domains cover
M. An atlas A is called a smooth atlas if any two charts in A are smoothly compatible
with each other.
1
Note that every topological manifold M has an atlas, since every point in M is con
tained in the domain of some chart by the property of M being locally Euclidean. The
above definitions set us up to be able to talk about a "smooth structure" on a manifold.
We could define a smooth structure by giving a smooth atlas on a manifold and say a
function f : M R is smooth iff f is smooth (in the ordinary sense) for each chart
in the given atlas.
A small problem right now though is that we could have two different smooth atlases
that describe the same smooth structure, that is, the same family of functions f are
smooth on a manifold with respect to different atlases. A way around this is to define an
equivalence relation on the set of all smooth atlases on M , where A1 A2 iff the charts
in A1 in A2 are smoothly compatible with each other. A smooth structure on M could
then just be an equivalence class, and it's clear that any representative would determine
the same smooth structure. In this way we avoid the problem of having different smooth
structures that are essentially identical (that is, they describe the same family of smooth
functions on the manifold). However, it is easier to just define smooth structure in the
following way:
Definition 5. A smooth atlas A is called maximal if it is not contained in any strictly
larger smooth atlas. In another words, any chart that is smoothly compatible with a chart
in A is already in A. A smooth structure on M is a maximal smooth atlas, and a
smooth manifold is a pair (M, A), where M is a manifold and A is a smooth structure
on M.
As a remark, not every topological manifold can be equipped with a smooth structure.
The maximal smooth atlas on M described above will, in general, contain an extremely
large amount of charts. The following lemma shows that fortunately we only need to
specify some smooth atlas on M to talk about its smooth structure. It also shows that
the previously proposed definition of smooth structures via equivalence classes is in fact
an equivalent way to define the smooth structure.
Lemma 6. Let M be a topological manifold.
(a) Every smooth atlas for M is contain in a unique maximal smooth atlas.
(b) Two smooth atlases for M determine the same maximal smooth atlas if and only
if their union is a smooth atlas.
If an nmanifold has a global coordinate chart, that is, a chart (M, ) where is a
homeomorphism from M to some open subset of
n
R , then this chart determines a smooth
structure on M . This is because the smooth compatibility condition is trivially satisfied
because there are no other charts in the atlas consisting of just the chart (M, ). Note
that this does not mean (M, ) is the only chart in the smooth structure determined by
this chart, but rather this smooth structure is the set of all charts that are smoothly
compatible with (M, ). In this way, this single chart determines a smooth structure on
M .
Example 5. A 0manifold M is just a countable discrete space. This is because each
p M has a neighborhood U homeomorphic to
0
R , but this forces U = {p}. If each one
point set is open in M , then M is necessarily a discrete space, and it must be countable
2
since it it has a countable basis (the one point sets). Since there is only map : {p}
0
R
for each p, the set of all charts on M trivially satisfies the smooth compatibility condition
(the intersection of the domains is trivial), thus M has a unique smooth structure.
Example 6.
n
R is a smooth nmanifold with smooth structure determined by the global
coordinate chart ( n
n
R , Id n ). This is called the standard smooth structure on
.
R
R
Other smooth structures exist though. For example, on R, consider the homeomorphism
: R R given by (x) = x3. This global chart determines a smooth structure on
R; however, it is not the same as the structure determined by (R, Id ). This is because
R
the transition map Id 1 :
1(x) = x1/3. This map is not
R
R R is given by IdR
smooth at the origin, so by the previous lemma, these two charts cannot determine the
same smooth structure.
We'll want to talk about finitedimensional vector spaces as manifolds now, but first
we need to show that any norm on a vector space generates the same topology. This is
the essence of the following lemma.
Lemma 7. Any two norms  * 1 and  * 2 on a finitedimensional vector space over R are
equivalent. That is, there exist some c, C > 0 such that for any x V ,
cx1 x2 Cx1.
Proof. Fix a basis {E
n
i}n
for V , and let T :
V be the basis isomorphism defined
i=1
R
T (x) = T (x1, . . . , xn) =
n
x
i=1
iEi.
Let  * 1 be any norm on V , and define a second
norm  * 2 on V by y2 = T 1(y), where T 1(y) is the Euclidean norm of T 1(y) in
n
R . Note that this also means x = T (x)2. It will suffice to show  * 1 and  * 2 are
equivalent norms in V . Denote by (V,  * 2) the space V with norm topology induced by
 * 2.
First, verify T : (V,  * 
n
n
2) R
is also a homeomorphism, where R has the standard
topology. Continuity of T follows easily from the linearity of T and the definition of  * 2:
Given some
> 0, for any x  y2 < where = , then T (x)  T (y) = T (x  y) =
x  y
n
2 <
. Similarly T 1 is continuous, and it follows V
R . It follows now as a
consequence of the HeineBorel Theorem, any subset of V that is closed and bounded
(with respect to the topology induced by  * 2) is compact. We now observe that if we let
C =
(E
i
i1)2, then for any x V :
x1 = 
x
x
x
x
(E
i
iEi1
i
iEi1 =
i
i * Ei1
i
i2 *
i
i1)2 = C x2
We'll show the map  * 1 : (V,  * 2) R is continuous. Given any > 0, let = /C,
then x1  y1 x  y1 < Cx  y2, and the claim follows. Consider now the
set S = {x V : x2 = 1}, which is closed and bounded in (V,  * 2), and hence
compact. Thus,  * 1 attains a minimum on S, that is, there exists some x0 S such that
x
0 < c := x01 x1 for any x S. We then have that for x V ,
S. It follows
x2
x
x
that
=
1 c, and so x

1 cx2.
x2 1
x2
Example 7. Let V be a finitedimensional vector space with the norm topology. Fix a
basis {E
n
i}n
for V and let T :
V be the basis isomorphism defined by the map that
i=1
R
sends the standard basis in
n
R to {Ei}n . This map is also a homeomorphism, and the
i=1
chart (V, T 1) defines a smooth structure on V . This is the standard smooth structure
on V , and doesn't depend on the choice of basis for V .
3
Example 8. Given a smooth nmanifold M and an open subset U M , we can define
an atlas on U by AU = {smooth charts (V, ) for M such that V U }. U is then a
smooth nmanifold, called an open open submanifold of M .
As an application of the previous examples, we can show GLn(R) Mn(R) is a n
manifold. Since Mn(R) is a (finite) n2dimensional vector space and det : Mn(R) R
is continuous, it follows det1(0) = Mn(R)\GLn(R) is closed. Thus GLn(R) is an open
submanifold of Mn(R).
Lemma 8 (Smooth Manifold Construction). Let M be a set, and suppose we are
given a collection U
n
of subsets of M, together with an injective map : U R
for
each , such that the following properties hold:
* For each ,
n
(U) is an open subset of R .
* For each and ,
n
(U U ) and (U U ) are open in R .
* Whenever U U = , 1 :
(U U ) (U U ) is a diffeomorphism.
* Countably many of the sets U cover M .
* Whenever p,q are distinct points in M, either there exists some U containing both
p and q or there exist disjoint sets U, U with p U and q U.
This lemma allows us to essentially create a smooth manifold from a set by gluing
together charts on the set in a "nice" way. We'll now want to account for certain spaces
that would be manifolds, except they have a "boundary" that prevents them from being
a regular smooth manifold. Let
n
n
H = {x R xn 0}.
Definition 9. An nmanifold with boundary is a secondcountable Hausdorff space
M in which every point has a neighborhood homeomorphic to a (relatively) open subset
of
n
H . M is called a smooth manifold with boundary if the transition maps can be
extended to smooth maps on open subsets of
n
R .
4