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Smooth locally defined functions can be glued by a partition of unity

Statement

Let $(U_{i})_{i \in I}$ be an open cover of a smooth manifold $M$ , let $f_{i} : U_{i} \to R$ be smooth for each $i$ , and let $(ϕ_{i})_{i \in I}$ be a smooth partition of unity subordinate to $(U_{i})$ . Then the pointwise formula $F (p) : = \sum_{i} ϕ_{i} (p) f_{i} (p)$ defines a smooth function $F : M \to R$ .

Facts & Assumptions

Given: An open cover $(U_{i})_{i \in I}$ of $M$ , smooth functions $f_{i} : U_{i} \to R$ , and a smooth partition of unity $(ϕ_{i})_{i \in I}$ subordinate to $(U_{i})$ .

[F1]

In a smooth partition of unity subordinate to $(U_{i})$ , the support family $(supp (ϕ_{i}))$ is locally finite and each support lies in $U_{i}$ (Smooth partitions of unity subordinate to an open cover).

[L1]

Smooth maps paste over an open cover (Smooth maps paste over an open cover).

[L2]

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

[A1]

For each $i$ , the product $ϕ_{i} f_{i}$ is smooth on $U_{i}$ .

Proof

technique · direct

1.1F1L1A1givenconstruct

Fix $i \in I$ . On the open set $U_{i}$ let $G_{i} : = ϕ_{i} f_{i}$ , and on the open set $M ∖ supp (ϕ_{i})$ let $H_{i} : = 0$ . By [A1], the map $G_{i}$ is smooth on $U_{i}$ ; by [F1], the two open sets cover $M$ and on the overlap $U_{i} ∖ supp (ϕ_{i})$ one has $ϕ_{i} = 0$ , so $G_{i} = H_{i}$ . Therefore [L1] pastes them to a smooth global function $F_{i} : M \to R$ with $F_{i} = ϕ_{i} f_{i}$ on $U_{i}$ and $supp (F_{i}) \subseteq supp (ϕ_{i})$ .

2.1F1L2step 1.1

By [F1] and step 1.1, the family $(supp (F_{i}))_{i \in I}$ is locally finite. Hence the sum $F : = \sum_{i} F_{i}$ is well defined and smooth by [L2].

3.1F1step 1.1step 2.1∎

Let $p \in M$ . If $p \in U_{i}$ , step 1.1 gives $F_{i} (p) = ϕ_{i} (p) f_{i} (p)$ , including when $ϕ_{i} (p) = 0$ . If $p \notin U_{i}$ , then [F1] implies $p \notin supp (ϕ_{i})$ , so the zero extension in step 1.1 gives $F_{i} (p) = 0$ ; the summand $ϕ_{i} (p) f_{i} (p)$ is interpreted as zero there. Local finiteness in [F1] ensures that only finitely many supports contain $p$ , and all other summands vanish. Thus the stated pointwise sum equals $\sum_{i} F_{i} (p) = F (p)$ and is smooth by step 2.1.

Depends on

Used by

Dependency tree · two levels

9 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)