Existence of partitions of unity on paracompact manifolds

Partitions of unity are the main technical tool that lets local constructions on charts be assembled into global geometric objects.

Theorem (Smooth partitions of unity)

Let $M$ be a paracompact smooth manifold and let $\{U_i\}_{i\in I}$ be an open cover of $M$. Then there exists a family of smooth functions $\{\rho_i\}_{i\in I}$ on $M$ such that:

1. $0\le \rho_i\le 1$ for all $i$,
2. the family is locally finite (every point has a neighborhood where all but finitely many $\rho_i$ vanish),
3. $\mathrm{supp}(\rho_i)\subset U_i$ for all $i$ (subordinate to the cover), and
4. $\sum_{i\in I}\rho_i = 1$ everywhere on $M$.

In particular, any collection of local data defined over the $U_i$ that is affine or convex (e.g. local connection 1-forms, local metrics, local differential forms) can be glued into a global object by weighting with the $\rho_i$ and summing.

Examples

1. Bump functions on Euclidean space. On $\mathbb{R}^n$, for a cover by balls one can choose smooth bump functions supported in slightly smaller balls and normalize their sum to obtain a partition of unity.
2. Gluing differential forms. If $\alpha_i$ are local differential forms on $U_i$ agreeing on overlaps, then $\sum_i \rho_i \alpha_i$ defines a global form; smoothness follows from local finiteness.
3. Building global connections. Local connection 1-forms on a trivializing cover can be combined using a partition of unity to produce a global connection on a vector bundle (or a principal connection after ensuring the correct transformation behavior).