Partitions of Unity

Smooth weights that sum to one, gluing local pieces into a global whole.

The idea

A partition of unity is the standard device for assembling objects defined locally, chart by chart, into one object defined on a whole manifold; it is how the integral of a differential form over a manifold is defined at all.

Let $M$ be a smooth manifold and let $\{U_{i}\}_{i \in I}$ be an open cover of $M$. A partition of unity subordinate to $\{U_{i}\}$ is a family of smooth functions $\varphi_{i} \colon M \to [0,1]$ such that each $\varphi_{i}$ vanishes outside its own $U_{i}$; the family is locally finite, meaning every point has a neighbourhood on which all but finitely many $\varphi_{i}$ are identically $0$; and $\sum_{i \in I} \varphi_{i}(x) = 1 \qquad \text{for every } x \in M.$ Local finiteness makes the sum at each point a finite sum, so no question of convergence arises.

Suppose now that an object $\omega_{i}$ is defined on each $U_{i}$ and nowhere else. The combination $\sum_{i} \varphi_{i}\,\omega_{i}$ is defined on all of $M$, since each term is smooth and vanishes outside its own $U_{i}$; and because the weights sum to $1$ at every point, the overlaps are averaged rather than counted twice. Every smooth manifold whose topology has a countable basis admits a partition of unity subordinate to any open cover, built out of smooth bump functions.

Ways to work on it

• Walkthrough. The defining property, sum-to-one, and smooth bump functions.
• Practice. Recover the missing weight so the partition sums to one.
• Hardest. What partitions of unity are for, and why smoothness matters.

Not sure where to start? Take the ten-question placement test.