Source For Partitions of Unity Problems

Question

I’ve realized that I need to work more with partitions of unity, but unfortunately there are not a lot of problems in the partitions of unity section in Lee which enable you to practice using them, and also get used to some of the tricks one might need to know in their applications. Are there any good sources for partitions of unity problems?

Answer 1

One good way to understand partitions of unity and why they are so useful is to study the comparison with the complex case. Take a compact Riemann surface $X$ (maybe $X=\mathbb{CP}^1$). A useful object we can define on $X$ is the sheaf of holomorphic functions $\mathcal{O}_X$, which assigns to each $U\subseteq X$ $$\mathcal{O}_X(U)=\{f:U\to \mathbb{C}\:\text{holomorphic}\}.$$ By the open mapping theorem, a globally defined function $f$ on $X$ is constant because its image would be open and compact in $\mathbb{C}$ were it not constant. So, this tells us that global sections $f\in \mathcal{O}_X(X)$ are just constant functions. Thus, in the compact complex case there are in general very few global objects. In particular, it is hard to take a local object and extend it to a global one in a nontrivial manner.

Over $\mathbb{R}$, the situation is vastly different, due essentially to the existence of bump functions. A bump function is a compactly supported $\mathcal{C}^\infty$ function on $\mathbb{R}^n$. Because holomorphicity is very restrictive, such functions do not exist in $\mathcal{H}(\mathbb{C})$. Bump functions (and by extension partitions of unity) allow us to extend local objects defined in coordinates to global objects, namely global sections. So by contrast, given a compact manifold $M$, $\mathcal{C}^\infty(M)$ has unimaginably many elements. The fact that we can extend local objects by zero using bump functions lets us construct many global objects that are very important.

As for resources, try Tu's Introduction to Manifolds, Bott and Tu, or maybe Guillemin and Pollack. If you can't find enough exercises on this, why not study the proof of the existence of Riemannian metrics on smooth manifolds, and see what other kinds of objects you can come up with.