In their paper Betancourt et al. (2014), the authors give a corollary which starts with the phrase “Because the manifold is paracompact”. It wasn’t immediately clear why the manifold was paracompact or indeed what paracompactness meant although it was clearly something like compactness which means that every cover has a finite sub-cover.
It turns out that every manifold is paracompact and that this is intimately related to partitions of unity.
Most of what I have written below is taken from some hand-written anonymous lecture notes I found by chance in the DPMMS library in Cambridge University. To whomever wrote them: thank you very much.
Let be an open cover of a smooth manifold . A partition of unity on M, subordinate to the cover is a finite collection of smooth functions
where for some such that
and for each there exists such that
We don’t yet know partitions of unity exist.
Techniques of classical analysis easily show that is smooth ( is the only point that might be in doubt and it can be checked from first principles that for all ).
Finally we can define by . This has the properties
Now take a point centred in a chart so that, without loss of generality, (we can always choose so that the open ball and then define another chart with ).
Define the images of the open and closed balls of radius and respectively
and further define bump functions
Then is smooth and its support lies in .
By compactness, the open cover has a finite subcover . Now define
Then is smooth, and . Thus is the required partition of unity.
Because is a manifold, it has a countable basis and for any point , there must exist with . Choose one of these and call it . This gives a countable cover of by such sets.
where, since is compact, is a finite subcover.
And further define
where again, since is compact, is a finite subcover.
Then is compact, is open and . Furthermore, and only intersects with and .
Given any open cover of , each can be covered by a finite number of open sets in contained in some member of . Thus every point in can be covered by at most a finite number of sets from and and which are contained in some member of . This is a locally finite refinement of and which is precisely the definition of paracompactness.
To produce a partition of unity we define bump functions as above on this locally finite cover and note that locally finite implies that is well defined. Again, as above, define
to get the required result.
Betancourt, M. J., Simon Byrne, Samuel Livingstone, and Mark Girolami. 2014. “The Geometric Foundations of Hamiltonian Monte Carlo,” October, 45. http://arxiv.org/abs/1410.5110.