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The question is basically in the title, but I‘ll explain a bit more what I mean.

Let‘s say I want to construct a function with a certain property in a neighborhood of an embedded submanifold $S\subset M,$ where $M$ is a smooth manifold (the same question would apply for constructing other things such as e.g. vector fields). Assume now that for every $p \in S$ I can locally construct a function $f_p:U_p\to \mathbb{R}$ with the desired property (where $U_p$ is a neighborhood of $p$ in $M$).

What I would like to do now is take something like a partition of unity subordinate to the collection $\{U_p\}_{p\in S},$ so that I could define a function $f$ in $U=\cup_{p \in S} U_p$ as usual by \begin{equation} f=\sum_{p\in S}\psi_p f_p. \end{equation}

My thoughts I had so far are the following:

1) If $S$ is properly embedded, i.e. it is also a closed subset of $M$, then I would just take the cover $\{U_p\}_{p\in S} \cup \{M\setminus S\}$ and a partition of unity subordinate to this cover.

2) In the general case I could maybe do the following: For each $p \in S$ I choose a neighborhood $V_p$ such that $\bar{V_p}\subset U_p$ and then take a locally finite open refinement $\{V_\alpha\}_{\alpha \in A}$ ( I hope this is possible when the $V_p$‘s do not form an open cover) and them look at the cover $\{U_p\}_{p\in S} \cup \{M\setminus \cup_{\alpha \in A}\bar{V_\alpha}\}$ and a partition of unity subordinate to it.

My question now is: Is there something like a partition of unity for things which are not a cover? Are my thoughts correct and if not what are the mistakes?

Thanks in advance

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I dont know precisely what object you want to construct, but seems like you want to extend smooth function defined on an embedded submanifold $S \subseteq M$. In this case, the partition of unity like you mention is that the partition of unity subordinate to the open cover $\{ U_p : p\in S\}$ by regard $U = \bigcup_{p\in S} U_p$ as an open submanifold of $M$ (or embedded submanifold of codimension-$0$). To extend $f : S \to \mathbb{R}$ to $U$, we choose each $U_p$ to be domain of a slice chart for $S$, so the subset $S \cap U_p$ is closed in $U_p$, since it has representation $(x^1,\dots,x^k,0,\dots,0)$. Since $f|_{U_p\cap S}$ is a smooth map on the closed subset $U_p\cap S \subseteq U_p$, there is a smooth function $f_p : U_p \to \mathbb{R}$ such that $f_p|_{U_p\cap S} = f|_{U_p\cap S}$. Then use partition of unity above to construct the extension of $f$ to $U$.

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  • $\begingroup$ Thanks. The only trick I needed is the (now obvious) observation that $U$ is a manifold in its own right. How to construct the $f_p$‘s was clear to me. $\endgroup$ – Frieder Jäckel Apr 2 '18 at 16:37

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