Partition of unity subordinate to an open cover

Let $M$ be a smooth manifold and let $\{U_i\}_{i\in I}$ be an open cover of $M$ .

A smooth partition of unity subordinate to $\{U_i\}$ is a family of smooth functions $\{\varphi_i:M\to[0,1]\}_{i\in I}$ such that:

(Support condition) For each $i$ , the support $\mathrm{supp}(\varphi_i)$ is contained in $U_i$ .
(Local finiteness) The family $\{\mathrm{supp}(\varphi_i)\}$ is locally finite: every point of $M$ has a neighborhood meeting only finitely many supports.
(Sum to one) For all $x\in M$ , $\sum_{i\in I}\varphi_i(x)=1,$ where the sum is well-defined because of local finiteness.

Existence

A fundamental theorem states that if $M$ is a paracompact manifold, then every open cover admits such a partition of unity.

Examples

Three-arc cover of the circle. Cover $S^1$ by three open arcs with pairwise overlaps. One can build smooth bump functions supported in each arc and normalize their sum to obtain a partition of unity.

Cover of $\mathbb R^n$ by balls. For an open cover of $\mathbb R^n$ by (possibly overlapping) balls, choose a locally finite refinement and bump functions supported in the refined sets; normalizing yields a subordinate partition of unity.

Gluing local data. If $\alpha_i$ are differential forms defined on $U_i$ , then $\sum_i \varphi_i\,\alpha_i$ defines a global form after extending each product by zero outside $U_i$ ; no agreement of the local forms on overlaps is required for this weighted construction, and local finiteness ensures the sum is pointwise finite.

Quadratic normalization

When a decomposition requires the squares of its coefficients to sum to one, use a squared partition of unity. Normalizing a smooth family by the square root of its positive sum of squares preserves smoothness.