partition of unity

Definition

Let $M$ be a topological space with an open cover $(X_{\alpha }{)}_{\alpha \in A}$. A paritition of unity subordinate to $(X_{\alpha })$ is a family of continuous functions

$$({\psi }_{\alpha }{)}_{\alpha \in A}{\psi }_{\alpha }:M⟶\mathbb{R}$$

with the following properties:

1. $0\le {\psi }_{\alpha }\le 1$ for all $\alpha \in A$ and all $x\in M$.

2. $\{\text{supp}{\psi }_a\subset X_{\alpha }\}$ for each $\alpha \in A$. (Outside the open set $X_{\alpha }$, the function is 0).

3. $\{\text{supp }{\psi }_{\alpha }{\}}_{\alpha \in A}$ is locally finite. (Each point in $M$ has a neighborhood that intersects with finitely many non-zero ${\psi }_{\alpha }$ non-zero.)

4. ${\sum }_{\alpha \in A}{\psi }_{\alpha }(x)=1$ for all $x\in M$.

Smooth partition of unity

If $M$ is a smooth manifold, then each of the functions ${\psi }_{\alpha }$ must be smooth.

Why care?

Partitions of unity give a good way to “glue” structures together on smooth manifolds. Since $\sum {\psi }_{\alpha }=1$ then “multiplying” by a structure will allow us to (informally) spread it around open sets but ensure that it won’t be changed on the open set that it is defined on.

Partitions of unity exist

Given an open cover $\{V_{\alpha }\}$ of a smooth manifold $M$, there exists a partition of unity $\{{\psi }_{\alpha }\}$ subordinate to $\{V_{\alpha }\}$.

Proof

Since a smooth manifold $M$ is paracompact, we can find a locally finite countable refinement $\{B_i\}$ of coordinate balls.

Let $x\in X_{\alpha }\subset M$, then we can take the coordinate neighborhood $x\in U_x\subset X_{\alpha }$. Then chose a bump function ${\phi }_i$ which is $1$ on a ball $x\in B_i\subset U_x$ and 0 outside $U_x$. This can be extended to the entire manifold by making it 0 everywhere for every other point. Since $B_i$ is locally finite, then there are only finitely many non-zero terms coming from neighborhoods of $x$.

Therefore, we can know that the sum

$$0<\sum _i{\phi }_i(x)\le \infty$$

is finite and must be positive.

Then for each $X_{\alpha }$ we can take the function build from the above steps to be

$$\psi =\frac{{\phi }_i}{{\sum }_i{\phi }_i}$$

for $B_i\subset X_{\alpha }$.

Thus

$$\text{supp}{\psi }_i=\text{supp }{\phi }_i\subset U_i\subset X_{\alpha }$$

and

$$\sum {\psi }_i=\frac{{\sum }_i{\phi }_i}{{\sum }_i{\phi }_i}=1.$$

These functions make the partition of unity.