Partitions of Unity

In this section, we develop bump functions and partitions of unity, which are tools for patching together local smooth objects into global ones.

Partition of Unity

Let be a manifold, and an open cover of . A (smooth) partition of unity subordinate to the cover is a collection of functions satisfying:

1. For every and every , .
2. The support is contained in for each .
3. The set of supports is locally finite. That is, for every point , there exists an open neighborhood of in such that is finite.
4. For each , .

Theorem

Let be a manifold and an open cover. Then there exists a smooth partition of unity subordinate to .

First, we will develop the basic building blocks for the construction, which are smooth compactly supported functions on Euclidean space:

Lemma

If , then there exists such that

• for ,
• for , and
• for .

Proof Let satisfy Then the function such that will do the job.

Applications of Partitions of Unity

Existence of Bump Functions

Let be a smooth manifold. For any closed subset and any open subset containing , there exists a smooth bump function for supported in . That is, a smooth function such that for and .

Proof Let , , and . Then we can find a partition of unity subordinate to . Since for , we have that for . Moreover, the support of is contained in .

Proposition

Smooth functions on a closed submanifold are restrictions. That is, if is a closed -dimensional submanifold of , then for any smooth function , there exists such that .

It is an important question whether an abstract manifold can always be realized as a submanifold of some Euclidean space. A simple case is when the manifold is compact. In this case, we can use the following theorem:

Embedding Compact Manifolds in High Dimension

Let be a compact manifold. Then for sufficiently large there exists a smooth embedding .