1Manifolds
III Differential Geometry
1.3 Bump functions and partitions of unity
Recall that there is one thing we swept under the carpet — to define the tangent
space, we needed to pick an open set
U
. Ways to deal with this can be found in
the example sheet, but there are two general approaches — one is to talk about
germs of functions, where we consider all open neighbourhoods, and identify two
functions if they agree on some open neighbourhood of the point. The other way
is to realize that we can “extend” any function on
U ⊆ M
to a function on the
whole of M , using bump functions.
In general, we want a function that looks like this:
Lemma.
Suppose
W ⊆ M
is a coordinate chart with
p ∈ W
. Then there is an
open neighbourhood
V
of
p
such that
¯
V ⊆ W
and an
X ∈ C
∞
(
M, R
) such that
X = 1 on V and X = 0 on M \ W .
Proof.
Suppose we have coordinates
x
1
, · · · , x
n
on
W
. We wlog suppose these
are defined for all |x| < 3.
We define α, β, γ : R → R by
α(t) =
(
e
−t
−2
t > 0
0 t ≤ 0
.
We now let
β(t) =
α(t)
α(t) + α(1 − t)
.
Then we let
γ(t) = β(t + 2)β(2 − t).
Finally, we let
X(x
1
, · · · , x
n
) = γ(x
1
) · · · γ(x
n
).
on W . We let
V = {x : |x
i
| < 1}.
Extending
X
to be identically 0 on
M \ W
to get the desired smooth function
(up to some constant).
Lemma.
Let
p ∈ W ⊆ U
and
W, U
open. Let
f
1
, f
2
∈ C
∞
(
U
) be such that
f
1
= f
2
on W . If X ∈ Der
p
(C
∞
(U)), then we have X(f
1
) = X(f
2
)
Proof.
Set
h
=
f
1
− f
2
. We can wlog assume that
W
is a coordinate chart. We
pick a bump function
χ ∈ C
∞
(
U
) that vanishes outside
W
. Then
χh
= 0. Then
we have
0 = X(χh) = χ(p)X(h) + h(p)X(χ) = X(h) + 0 = X(f
1
) − X(f
2
).
While we’re doing boring technical work, we might as well do the other one,
known as a partition of unity. The idea is as follows — suppose we want to
construct a global structure on our manifold, say a (smoothly varying) inner
product for each tangent space
T
p
M
. We know how to do this if
M
=
R
n
,
because there is a canonical choice of inner product at each point in
R
n
. We
somehow want to patch all of these together.
In general, there are two ways we can do the patching. The easy case is that
not only is there a choice on
R
n
, but there is a unique choice. In this case, just
doing it on each chart suffices, because they must agree on the intersection by
uniqueness.
However, this is obviously not the case for us, because a vector space can
have many distinct inner products. So we need some way to add them up.
Definition
(Partition of unity)
.
Let
{U
α
}
be an open cover of a manifold
M
. A
partition of unity subordinate to
{U
α
}
is a collection
ϕ
α
∈ C
∞
(
M, R
) such that
(i) 0 ≤ ϕ
α
≤ 1
(ii) supp(ϕ
α
) ⊆ U
α
(iii) For all p ∈ M, all but finitely many ϕ
α
(p) are zero.
(iv)
P
α
ϕ
α
= 1.
Note that by (iii), the final sum is actually a finite sum, so we don’t have to
worry about convergence issues.
Now if we have such a partition of unity, we can pick an inner product on
each
U
α
, say
q
α
(
· , ·
), and then we can define an inner product on the whole
space by
q(v
p
, w
p
) =
X
α
ϕ
α
(p)q
α
(v
p
, w
p
),
where
v
p
, w
p
∈ T
p
M
are tangent vectors. Note that this makes sense. While
each
q
α
is not defined everywhere, we know
ϕ
α
(
p
) is non-zero only when
q
α
is
defined at p, and we are also only taking a finite sum.
The important result is the following:
Theorem.
Given any
{U
α
}
open cover, there exists a partition of unity subor-
dinate to {U
α
}.
Proof.
We will only do the case where
M
is compact. Given
p ∈ M
, there exists
a coordinate chart
p ∈ V
p
and
α
(
p
) such that
V
p
⊆ U
α(p)
. We pick a bump
function
χ
p
∈ C
∞
(
M, R
) such that
χ
p
= 1 on a neighbourhood
W
p
⊆ V
p
of
p
.
Then supp(χ
p
) ⊆ U
α(p)
.
Now by compactness, there are some
p
1
, · · · , p
N
such that
M
is covered by
W
p
1
∪ · · · ∪ W
p
N
. Now let
˜ϕ
α
=
X
i:α(p
i
)=α
χ
p
i
.
Then by construction, we have
supp( ˜ϕ
α
) ⊆ U
α
.
Also, by construction, we know
P
α
˜ϕ
α
> 0. Finally, we let
ϕ
α
=
˜ϕ
α
P
β
˜ϕ
β
.
The general proof will need the fact that the space is second-countable.
We will actually not need this until quite later on in the course, but we might
as well do all the boring technical bits all together.