Use of partitions of unity

So, I'm working through introduction to smooth manifolds, and I've seen the fact that partitions of unity exist. They seem like extremely useful theoretical tools, but I haven't seen them used yet.

I'd like some nice examples as to where they can be used, and why they're "special" in some sense (I don't have a good intuition as to why you need a smooth manifold to exhibit this partition).

What's the intuition with partitions of unity? asks the intuition of this object. I am after uses.

Non-Theoretical Applications of Partitions of Unity asks about, well, applied uses.

I want something along the lines of "what is the most dazzling thing you can do with this tool" - Something like the awesome uses of orthogonalization, or the spectral theorem you would see in linear algebra.

• The purpose of a partition of unity is to allow you to define the global integral of a form using local representations in neighborhoods and not double count "overlaps" for a lack of a better term. However, I feel like you already know this and might be asking for something more in depth. – rubikscube09 Apr 29 '18 at 17:42

The existence of continuous partitions of the unity is guaranteed on any topological space (which is countable at infinity), the smooth manifold assumption is here only to make sense of the smoothness of the maps in the family.

Usually, partitions of the unity are used to construct objects on a smooth manifold by patching them together, here is a general recipe:

• The existence of the object you are interested in is locally guaranteed by the existence of charts.

• The set of all these objects is convex.

Example. Let $M$ be a smooth manifold, then there exists $g$ a Riemannian metric on $M$, that is a smooth family of inner products on the tangent spaces of $M$.

• I believe you need paracompactness to ensure the existence of a (continuous) partition of unity. – Aweygan Apr 29 '18 at 17:45
• @Aweygan I believe countable at infinity would suffice, I am going to add this. Thank you for pointing that out! – C. Falcon Apr 29 '18 at 17:47

On Riemannian manifold, if there is an locally finite open cover $U_i$ and $X_i$ is a vector field on $U_i$, then by partition of unity $f_i$, we have a global vector field $X =\sum_i\ f_iX_i$.