Not sure how this would be received here. This question is about smooth partitions of unity.

Let $M$ be a manifold. Consider an open cover $\{U_\alpha\}_{\alpha\in \Lambda}$ of $M$. A collection of smooth functions$\{p_\alpha:U_\alpha\rightarrow \mathbb{R}\}_{\alpha\in \Lambda}$ is called a smooth partition of unity subordinate to the cover $\{U_\alpha\}$ if

- $\text{supp}(p_\alpha)\subseteq U_\alpha$ for each $\alpha\in \Lambda$,
- the collection of supports $\{\text{supp}(p_\alpha)\}_{\alpha\in \Lambda}$ is a locally finite set,
- $\sum_{\alpha\in \Lambda}p_\alpha=1$.

It is known that given any open cover $\{U_\alpha\}$, we can produce a partition of unity on $M$ subordiante to this cover. Next question is, what is the use of partitions?

- Suppose I am given a $n$-form $\omega$ on a (oriented) $n$-manifold $M$. I have to make sense of (give a reasonable definition for) $\int_M\omega$. Suppose that $\omega$ is (compactly supported) zero outside a chart $(U,\phi:U\rightarrow \mathbb{R}^n)$ (let $\psi:\phi(U)\subseteq \mathbb{R}^n\rightarrow U$ be its inverse). Then, pullback $\omega$ along $\psi$ to get $n$-form $\psi^*\omega$ on $\phi(U)\subseteq \mathbb{R}^n$ and we know how to integrate an $n$-form on an open subset of $\mathbb{R}^n$. So, we know what is $\int_{\phi(U)}\psi^*\omega$ is and define $\int_M\omega:=\int_{\phi(U)}\psi^*\omega$. Suppose $\omega$ is arbitrary (compactly supported), then, ask for partition of unity $\{p_\alpha\}$ and consider $p_\alpha\omega$. This is compactly supported in $U_\alpha$. So, $\int_{U_\alpha}p_\alpha\omega$ make sense and define $\int_{M}\omega=\sum \int_{U_\alpha}p_\alpha\omega$.
- Given a manifold $M$, how do I know that there exists a Riemannian metric on $M$? Partition of Unity.
- Given a manifold $M$, how do I know there exists a connection on the tangent bundle $TM\rightarrow M$? Partitions of unity.
- Given a principal bundle over manifold $M$, how do I know connection exists on the principal bundle $P\rightarrow M$? Partitions of unity (along with trivialization of course).

My question is the following:

Is partition of unity used for anything serious than making sure some structures can be glued to give a global structure? Do you, as a research mathematician, come across the necessity of using the partition of unity for any reason other than similar to what I mentioned above?