# Introduction

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.

# Limbering Up

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.

First define

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 ).

Next define

Finally we can define by . This has the properties

- if
- if

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

by

Then is smooth, and . Thus is the required partition of unity.

# Paracompactness

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.

Now define

where, since is compact, is a finite subcover.

And further define

where again, since is compact, is a finite subcover.

Now define

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.

# Bibliography

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.

Interesting. Every time I saw the definition of manifold at undergrad it was always paracompact by definition, but paracompact and second-countable were the kind of terms that weren’t examined too closely and lecturers often stated it wasn’t quite a precise definition. Admittedly I never took the actual “manifolds” course. Thanks for the detail!