The Unapologetic Mathematician

Mathematics for the interested outsider

Partitions of Unity Subordinate to a Cover

We know what a partition of unity is, but not all partitions of unity are very useful. For instance, the single function defined by \phi(p)=1 for all points p\in M is a partition of unity all on its own — its support is M itself, which is clearly a locally finite cover of M, and it adds up to the constant unit function. But we can’t really do anything with it.

What we need is a partition of unity subordinate to an open cover. That is, given a collection \{U_\alpha\} of open sets that cover M, we want a partition of unity \{\phi_\beta\} such that for every \beta there is some \alpha so that \phi_\beta is supported in U_\alpha. In particular, we can let \{U_\alpha\} be the collection of coordinate patches in a smooth atlas, so each of the functions \phi_\beta “lives in” a single local coordinate system.

But do any such things exist? Remember, except for the trivial example above I haven’t actually given any examples of a smooth partition of unity at all. The example last time was differentiable, and even twice-differentiable, but not smooth. So this is a nice concept, but it might well be vacuous.

Still, all is not lost: I say that given any open cover of a smooth manifold, there is a countable smooth partition of unity subordinate to that cover. In particular, given any smooth structure on a manifold we can always find a partition of unity with each function supported completely within a single coordinate patch. The proof of this fact, however, is one of the few really annoying, fiddly, technical bits in differential geometry. It will take a few days of doing, and I fully understand if you’d rather just skip it. All you really need to know is: whenever we need a partition of unity to break global things defined over our entire manifold up into nice chunks that fit into coordinate patches, we can do it.

However, I should point this out: analytic manifolds are not nearly so forgiving. The basic (but sketchy) idea is that in order to construct our partitions of unity we’ll need to create “bump” functions sort of like the one we did last time, but ones that are smooth instead of just twice-differentiable. This means using a piecewise definition, just like last time, and at the edge of a piece we’ll have points such that in any neighborhood of that point we need two different definitions of the function. But if the function is supposed to be analytic, then the definition that works on one side should keep working on the other side, and so we can’t make the bump functions we need.

This is a big reason why people stop at smooth manifolds rather than working with analytic ones, despite the fact that analytic functions are arguably “nicer”. Unfortunately, this also means that not everything we do carries over quite so easily to complex manifolds — based on complex vector spaces — which must always be analytic.

About these ads

March 8, 2011 - Posted by | Differential Topology, Topology

5 Comments »

  1. [...] first step in finding partitions of unity subordinate to a given cover is actually to set up a nice [...]

    Pingback by Atlases Refining Covers, part 1 « The Unapologetic Mathematician | March 10, 2011 | Reply

  2. [...] we can prove what we’ve asserted: given any open cover of a smooth manifold we can find a countable smooth partition of unity [...]

    Pingback by Partitions of Unity (proof) « The Unapologetic Mathematician | March 14, 2011 | Reply

  3. [...] an immediate application of our partitions of unity, let’s show that we can always get whatever bump functions we [...]

    Pingback by Bump Functions, part 2 « The Unapologetic Mathematician | March 16, 2011 | Reply

  4. [...] native orientations to cover the whole manifold. And as usual for this sort of thing, we pick a partition of unity subordinate to our atlas. That is, we have a countable, locally finite collection of functions so [...]

    Pingback by Orientable Atlases « The Unapologetic Mathematician | August 31, 2011 | Reply

  5. [...] can then find a partition of unity subordinate to this cover of . We can decompose into a (finite) [...]

    Pingback by Integrals over Manifolds (part 2) « The Unapologetic Mathematician | September 7, 2011 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 393 other followers

%d bloggers like this: