LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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A locally finite positive smooth family normalizes to a partition of unity

Statement

Let $(U_{i})_{i \in I}$ be an open cover of a smooth manifold $M$ , and let $(g_{i})_{i \in I}$ be nonnegative smooth functions whose supports form a locally finite family with $supp (g_{i}) \subseteq U_{i}$ for every $i$ . Suppose that for every $p \in M$ at least one $g_{i} (p)$ is strictly positive. Put $G : = \sum_{i} g_{i}$ . Then $G$ is a positive smooth function, each $ϕ_{i} : = g_{i} / G$ is smooth, and $(ϕ_{i})_{i \in I}$ is a smooth partition of unity subordinate to $(U_{i})_{i \in I}$ .

Facts & Assumptions

Given: An indexed open cover $(U_{i})$ and nonnegative smooth functions $(g_{i})$ with locally finite supports contained in the corresponding $U_{i}$ , pointwise positive as a family.

[L1]

A locally finite sum of smooth functions is smooth (A locally finite sum of smooth functions is smooth).

[L2]

If the supports are locally finite, then the cozero sets are locally finite (Locally finite supports have locally finite cozero sets).

[L3]

The definition of a smooth partition subordinate to an open cover requires locally finite supports, support containment in the corresponding open sets, and pointwise sum one (Smooth partitions of unity subordinate to an open cover).

Proof

technique · direct

1.1L1given

By [L1], the sum $G : = \sum_{i} g_{i}$ is smooth; because the $g_{i}$ are nonnegative and some $g_{i} (p)$ is positive at each point, one has $G (p) > 0$ for all $p \in M$ .

2.1step 1.1algebra

Fix a chart near $p$ . Since $G (p) > 0$ , continuity gives a smaller chart neighbourhood on which $G > 0$ . The real function $r (t) = 1 / t$ is smooth on $(0, + \infty)$ , with $r^{(k)} (t) = (- 1)^{k} k! t^{- k - 1}$ ; the chartwise chain rule therefore makes $1 / G = r \circ G$ smooth there. As $p$ was arbitrary, $1 / G$ is smooth on $M$ , and the product rule makes every $ϕ_{i} = g_{i} / G$ smooth and nonnegative.

3.1L2L3step 1.1step 2.1∎

Since $G > 0$ , $ϕ_{i} (p) > 0$ exactly when $g_{i} (p) > 0$ . Their cozero sets, and hence their supports, are equal. Thus the supports of $(ϕ_{i})$ are locally finite and satisfy $supp (ϕ_{i}) = supp (g_{i}) \subseteq U_{i}$ ; their cozero sets are locally finite by [L2]. Finally $\sum_{i} ϕ_{i} = (1 / G) \sum_{i} g_{i} = 1$ pointwise. These are exactly the conditions of [L3].

Depends on

Used by

Smooth partitions of unity exist on manifolds Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)