1Manifolds
III Differential Geometry
1.1 Manifolds
As mentioned in the introduction, manifolds are spaces that look locally like
R
n
.
This local identification with R
n
is done via a chart.
Many sources start off with a topological space and then add extra structure
to it, but we will be different and start with a bare set.
Definition
(Chart)
.
A chart (
U, ϕ
) on a set
M
is a bijection
ϕ
:
U → ϕ
(
U
)
⊆ R
n
,
where U ⊆ M and ϕ(U ) is open.
A chart (U, ϕ) is centered at p for p ∈ U if ϕ(p) = 0.
Note that we do not require
U
to be open in
M
, or
ϕ
to be a homeomorphism,
because these concepts do not make sense!
M
is just a set, not a topological
space.
p
U
ϕ(p)
ϕ
With a chart, we can talk about things like continuity, differentiability by
identifying U with ϕ(U ):
Definition
(Smooth function)
.
Let (
U, ϕ
) be a chart on
M
and
f
:
M → R
.
We say
f
is smooth or
C
∞
at
p ∈ U
if
f ◦ ϕ
−1
:
ϕ
(
U
)
→ R
is smooth at
ϕ
(
p
) in
the usual sense.
R
n
⊇ ϕ(U) U R
ϕ
−1
f
p
U
ϕ(p)
f ◦ ϕ
−1
R
f
ϕ
We can define all other notions such as continuity, differentiability, twice differ-
entiability etc. similarly.
This definition has a problem that some points might not be in the chart, and
we don’t know how to determine if a function is, say, smooth at the point. The
solution is easy — we just take many charts that together cover
M
. However,
we have the problem that a function might be smooth at a point relative to some
chart, but not relative to some other chart. The solution is to require the charts
to be compatible in some sense.
Definition
(Atlas)
.
An atlas on a set
M
is a collection of charts
{
(
U
α
, ϕ
α
)
}
on
M such that
(i) M =
S
α
U
α
.
(ii)
For all
α, β
, we have
ϕ
α
(
U
α
∩U
β
) is open in
R
n
, and the transition function
ϕ
α
◦ ϕ
−1
β
: ϕ
β
(U
α
∩ U
β
) → ϕ
α
(U
α
∩ U
β
)
is smooth (in the usual sense).
U
β
U
α
ϕ
β
ϕ
α
ϕ
α
ϕ
−1
β
Lemma.
If (
U
α
, ϕ
α
) and (
U
β
, ϕ
β
) are charts in some atlas, and
f
:
M → R
,
then
f ◦ ϕ
−1
α
is smooth at
ϕ
α
(
p
) if and only if
f ◦ ϕ
−1
β
is smooth at
ϕ
β
(
p
) for all
p ∈ U
α
∩ U
β
.
Proof. We have
f ◦ ϕ
−1
β
= f ◦ ϕ
−1
α
◦ (ϕ
α
◦ ϕ
−1
β
).
So we know that if we have an atlas on a set, then the notion of smoothness
does not depend on the chart.
Example. Consider the sphere
S
2
= {(x
1
, x
2
, x
3
) :
X
x
2
i
= 1} ⊆ R
3
.
We let
U
+
1
= S
2
∩ {x
1
> 0}, U
−
1
= S
2
∩ {x
1
< 0}, · · ·
We then let
ϕ
+
1
: U
+
1
→ R
2
(x
1
, x
2
, x
3
) 7→ (x
2
, x
3
).
It is easy to show that this gives a bijection to the open disk in
R
2
. We similarly
define the other ϕ
±
i
. These then give us an atlas of S
2
.
Definition
(Equivalent atlases)
.
Two atlases
A
1
and
A
2
are equivalent if
A
1
∪A
2
is an atlas.
Then equivalent atlases determine the same smoothness, continuity etc.
information.
Definition
(Differentiable structure)
.
A differentiable structure on
M
is a choice
of equivalence class of atlases.
We want to define a manifold to be a set with a differentiable structure. How-
ever, it turns out we can find some really horrendous sets that have differential
structures.
Example.
Consider the line with two origins given by taking
R × {
0
} ∪ R × {
1
}
and then quotienting by
(x, 0) ∼ (x, 1) for x 6= 0.
Then the inclusions of the two copies of R gives us an atlas of the space.
The problem with this space is that it is not Hausdorff, which is bad. However,
that is not actually true, because
M
is not a topological space, so it doesn’t
make sense to ask if it is Hausdorff. So we want to define a topology on
M
, and
then impose some topological conditions on our manifolds.
It turns out the smooth structure already gives us a topology:
Exercise.
An atlas determines a topology on
M
by saying
V ⊆ M
is open iff
ϕ
(
U ∩ V
) is open in
R
n
for all charts (
U, ϕ
) in the atlas. Equivalent atlases give
the same topology.
We now get to the definition of a manifold.
Definition
(Manifold)
.
A manifold is a set
M
with a choice of differentiable
structure whose topology is
(i)
Hausdorff, i.e. for all
x, y ∈ M
, there are open neighbourhoods
U
x
, U
y
⊆ M
with x ∈ U
x
, y ∈ U
y
and U
x
∩ U
y
= ∅.
(ii)
Second countable, i.e. there exists a countable collection (
U
n
)
n∈N
of open
sets in
M
such that for all
V ⊆ M
open, and
p ∈ V
, there is some
n
such
that p ∈ U
n
⊆ V .
The second countability condition is a rather technical condition that we
wouldn’t really use much. This, for example, excludes the long line.
Note that we will often refer to a manifold simply as
M
, where the differen-
tiable structure is understood from context. By a chart on
M
, we mean one in
some atlas in the equivalence class of atlases.
Definition
(Local coordinates)
.
Let
M
be a manifold, and
ϕ
:
U → ϕ
(
U
) a
chart of M . We can write
ϕ = (x
1
, · · · , x
n
)
where each x
i
: U → R. We call these the local coordinates.
So a point p ∈ U can be represented by local coordinates
(x
1
(p), · · · , x
n
(p)) ∈ R
n
.
By abuse of notation, if
f
:
M → R
, we confuse
f|
U
and
f ◦ ϕ
−1
:
ϕ
(
U
)
→ R
.
So we write f (x
1
, · · · , x
n
) to mean f (p), where ϕ(p) = (x
1
, · · · , x
n
) ∈ ϕ(U).
U M R
ϕ(U)
ι
ϕ
f
f|
U
Of course, we can similarly define
C
0
, C
1
, C
2
, · · ·
manifolds, or analytic manifolds.
We can also model manifolds on other spaces, e.g.
C
n
, where we get complex
manifolds, or on infinite-dimensional spaces.
Example.
(i)
Generalizing the example of the sphere, the
n
-dimensional sphere
S
n
=
{(x
0
, · · · , x
n
) ∈ R
n+1
:
P
x
2
i
= 1} is a manifold.
(ii)
If
M
is open in
R
n
, then the inclusion map
ϕ
:
M → R
n
given by
ϕ
(
p
) =
p
is a chart forming an atlas. So
M
is a manifold. In particular,
R
n
is
a manifold, with its “standard” differentiable structure. We will always
assume R
n
is given this structure, unless otherwise specified.
(iii) M
(
n, n
), the set of all
n × n
matrices is also a manifold, by the usual
bijection with
R
n
2
. Then
GL
n
⊆ M
(
n, n
) is open, and thus also a manifold.
(iv)
The set
RP
n
, the set of one-dimensional subspaces of
R
n+1
is a manifold.
We can define charts as follows: we let
U
i
to be the lines spanned by a
vector of the form (v
0
, v
1
, · · · , v
i−1
, 1, v
i+1
, · · · , v
n
) ∈ R
n+1
.
We define the map
ϕ
i
:
U
i
→ R
n
∼
=
{x ∈ R
n+1
:
x
i
= 1
}
that sends
ϕ
(
L
) = (
v
0
, · · · ,
1
, · · · , v
n
), where
L
is spanned by (
v
0
, · · · ,
1
, · · · , v
n
). It
is an easy exercise to show that this defines a chart.
Note that when we defined a chart, we talked about charts as maps
U → R
n
.
We did not mention whether
n
is fixed, or whether it is allowed to vary. It turns
out it cannot vary, as long as the space is connected.
Lemma.
Let
M
be a manifold, and
ϕ
1
:
U
1
→ R
n
and
ϕ
2
:
U
2
→ R
m
be charts.
If U
1
∩ U
2
6= ∅, then n = m.
Proof. We know
ϕ
1
ϕ
−1
2
: ϕ
2
(U
1
∩ U
2
) → ϕ
1
(U
1
∩ U
2
)
is a smooth map with inverse ϕ
2
ϕ
−1
1
. So the derivative
D(ϕ
1
ϕ
−1
2
)(ϕ
2
(p)) : R
m
→ R
n
is a linear isomorphism, whenever p ∈ U
1
∩ U
2
. So n = m.
Definition
(Dimension)
.
If
p ∈ M
, we say
M
has dimension
n
at
p
if for one
(thus all) charts
ϕ
:
U → R
m
with
p ∈ U
, we have
m
=
n
. We say
M
has
dimension n if it has dimension n at all points.