An application of partition of unity

Partition of unity builds a bridge between local properties and global properties. A nice example is the Stokes' theorem on manifolds.

Suppose \(\omega\) is a \((n-1)\)-form with compact support on a oriented manifold \(M\) of dimension \(n\) and if \(\partial{M}\) is given the induced orientation, then \[\int_M d\omega=\int_{\partial{M}}\omega\]

This theorem can be proved in two steps. First, by Fubini's theorem, one proves the identity on \(\mathbb{R}^n\) and \(\mathbb{H}^n\). Second, for the general case, let \((U_\alpha)\) be an oriented atlas for \(M\) and \((\rho_\alpha)\) a partition of unity to \((U_\alpha)\), one naturally writes \(\omega=\sum_{\alpha}\rho_\alpha\omega\). Since \(\int_M d\omega=\int_{\partial M}\omega\) is linear with respect to \(\omega\), it suffices to prove it only for \(\rho_\alpha\omega\). Note that the support of \(\rho_\alpha\omega\) is contained in the intersection of supports of \(\rho_\alpha\) and \(\omega\), hence a compact set.

On the other hand, \(U_\alpha\) is diffeomorphic to either \(\mathbb{R}^n\) or \(\mathbb{H}^n\), it is immediate that \[\int_M d\rho_\alpha\omega=\int_{U_\alpha}d\rho_\alpha\omega=\int_{\partial U_\alpha}\rho_\alpha\omega=\int_{\partial{M}}\rho_\alpha\omega.\] Which furnishes the proof for the general case.

As is seen, to prove a global thing, we do it locally. If you have trouble with these terminologies, never mind. We will go through this right now (in a more abstract way however). If you are familiar with them however, fell free to skip.

Prerequisites

Manifold (of finite or infinite dimension)

Throughout, we use bold letters like \(\mathbf{E}\), \(\mathbf{F}\) to denote Banach spaces. We will treat Euclidean spaces as a case instead of our restriction. Indeed since Banach spaces are not necessarily of finite dimension, our approach can be troublesome. But the benefit is a better view of abstraction.

Let \(X\) be a set. An atlas of class \(C^p\) (\(p \geq 0\)) on \(X\) is a collection of pairs \((U_i,\varphi_i)\) where \(i\) ranges through some indexing set, satisfying the following conditions:
AT 1. Each \(U_i\) is a subset of \(X\) and \(\bigcup_{i}U_i=X\).
AT 2. Each \(\varphi_i\) is a bijection of \(U_i\) onto an open subset \(\varphi_iU_i\) of some Banach space \(\mathbf{E}_i\) and for any \(i\) and \(j\), \(\phi_i(U_i \cap U_j)\) is open in \(E_i\).
AT 3. The map \[\varphi_j\circ\varphi_i^{-1}:\varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j)\] is a \(C^p\)-isomorphism for all \(i\) and \(j\).

One should be advised that isomorphism here does not come from group theory, but category theory. Precisely speaking, it's the isomorphism in the category \(\mathfrak{O}\) whose objects are the continuous maps of Banach spaces and whose morphisms are the continuous maps of class \(C^p\).

Also, by setting \(\tau_X=(U_i)_i\), we see \(\tau_X\) is a topology, and \(\varphi_i\) are topological isomorphisms. Also, we see no need to assume that \(X\) is Hausdorff unless we start with Hausdorff spaces. Lifting this restriction gives us more freedom (also sometimes more difficulty to some extent though).

For condition AT 2, we did not require that the vector spaces be the same for all indexes \(i\), or even that they be toplinearly isomorphic. If they are all equal to the same space \(\mathbf{E}\), then we say that the atlas is an \(\mathbf{E}\)-atlas.

Suppose that we are given an open subset \(U\) of \(X\) and a topological isomorphism \(\phi:U \to U'\) onto an open subset of some Banach space \(\mathbb{E}\). We shall say that \((U,\varphi)\) is compatible with the atlas \((U_i,\varphi_i)_i\) if each map \(\varphi\circ\varphi^{-1}\) is a \(C^p\)-isomorphism. Two atlas are said to be compatible if each chart of one is compatible with other atlas. It can be verified that this is a equivalence relation. An equivalence relation of atlases of class \(C^p\) on \(X\) is said to define a structure of \(C^p\)-manifold on \(X\). If all the vector spaces \(\mathbf{E}_i\) in some atlas are toplinearly isomorphic, we can find some universal \(\mathbf{E}\) that is equal to all of them. In this case, we say \(X\) is a \(\mathbf{E}\)-manifold or that \(X\) is modeled on \(\mathbf{E}\).

As we know, \(\mathbb{R}^n\) is a Banach space. If \(\mathbf{E}=\mathbb{R}^n\) for some fixed \(n\), then we say that the manifold is \(n\)-dimensional. Also we have the local coordinates. A chart \[\varphi:U \to \mathbb{R}^n\] is given by \(n\) coordinate functions \(\varphi_1,\cdots,\varphi_n\). If \(P\) denotes a point of \(U\), these functions are often written \[x_1(P),\cdots,x_n(P),\] or simply \(x_1,\cdots,x_n\).

Topological prerequisites

Let \(X\) be a topological space. A covering \(\mathfrak{U}\) of \(X\) is locally finite if every point \(x\) has a neighborhood \(U\) such that all but a finite number of members of \(\mathfrak{U}\) do not intersect with \(U\) (as you will see, this prevents some nonsense summation). A refinement of a covering \(\mathfrak{U}\) is a covering \(\mathfrak{U}'\) such that for any \(U' \in \mathfrak{U}'\), there exists some \(U \in \mathfrak{U}\) such that \(U' \subset U\). If we write \(\mathfrak{U} \leq \mathfrak{U}'\) in this case, we see that the set of open covers on a topological space forms a direct set.

A topological space is paracompact if it is Hausdorff, and every open covering has a locally finite open refinement. Here follows some examples of paracompact spaces.

Any compact Hausdorff space.
Any CW complex.
Any metric space (hence \(\mathbb{R}^n\)).
Any Hausdorff Lindelöf space.
Any Hausdorff \(\sigma\)-compact space

These are not too difficult to prove, and one can easily find proofs on the Internet. Below are several key properties of paracompact spaces.

If \(X\) is paracompact, then \(X\) is normal. (Proof here)

Let \(X\) be a paracompact (hence normal) space and \(\mathfrak{U}=(U_i)\) a locally finite open cover, then there exists a locally finite open covering \(\mathfrak{V}=(V_i)\) such that \(\overline{V_i} \subset U_i\). (Proof here. Note the axiom of choice is assumed.

One can find proofs of the following propositions on Elements of Mathematics, General Topology, Chapter 1-4 by N. Bourbaki. It's interesting to compare them to the corresponding ones of compact spaces.

Every closed subspace \(F\) of a paracompact space \(X\) is paracompact.

The product of a paracompact space and a compact space is paracompact.

Let \(X\) be a locally compact paracompact space. Then every open covering \(\mathfrak{R}\) of \(X\) has a locally finite open refinement \(\mathfrak{R}'\) formed of relatively compact sets. If \(X\) is \(\sigma\)-compact then \(\mathfrak{R}'\) can be taken to be countable.

Partition of unity

A partition of unity (of class \(C^p\)) on a manifold \(X\) consists of an open covering \((U_i)\) of \(X\) and a family of functions \[\psi_i:X \to \mathbb{R}\] satisfying the following conditions:
PU 1. For all \(x \in X\) we have \(\phi_i(x) \geq 0\).
PU 2. The support of \(\psi_i\) is contained in \(U_i\).
PU 3. The covering is locally finite
PU 4. For each point \(x \in X\) we have \[\sum_{i}\psi_i(x)=1\]

The sum in PU 4 makes sense because for given point \(x\), there are only finite many \(i\) such that \(\psi_i(x) >0\), according to PU 3.

A manifold \(X\) will be said to admit partition of unity if it is paracompact, and if, given a locally finite open covering \((U_i)\), there exists a partition of unity \((\psi_i)\) such that the support of \(\psi_i\) is contained in \(U_i\).

Bump function

This function will be useful when dealing with finite dimensional case.

For every integer \(n\) and every real number \(\delta>0\) there exist maps \(\psi_n \in C^{\infty}(\mathbb{R}^n;\mathbb{R})\) which equal \(1\) on \(B(0,1)\) and vanish in \(\mathbb{R}^n\setminus B(1,1+\delta)\).

Proof. It suffices to prove it for \(\mathbb{R}\) since once we proved the existence of \(\psi_1\), then we may write \[\psi_n(x_1,x_2,\cdots,x_n)=\psi_1(\sqrt{x_1^2+x_2^2+\cdots+x_n^2}).\] Consider the function \(\phi: \mathbb{R} \to \mathbb{R}\) defined by \[\phi(t)=\begin{cases}\exp\left(\frac{1}{(t-a)(t-b)}\right)&\quad\text{if } a<t<b,\\0 &\quad \text{otherwise}.\end{cases}\] The reader may have seen it in some analysis course and should be able to check that \(\phi \in C^{\infty}(\mathbb{R};\mathbb{R})\). Integrating \(\phi\) from \(-\infty\) to \(x\) and divide it by \(\lVert \phi \rVert_1\) (you may have done it in probability theory) to obtain \[\theta(x)=\frac{\int_{-\infty}^{x}\phi(t)dt}{\int_{-\infty}^{+\infty}\phi(t)dt};\] it is immediate that \(\theta(x)=0\) for \(x \leq a\) and \(\theta(x)=1\) for \(x \geq b\). By taking \(a=1\) and \(b=(1+\delta)^2\), our job is done by letting \(\psi_1(x)=1-\theta(x^2)\). Considering \(x^2=|x|^2\), one sees that the identity about \(\psi_n\) and \(\psi_1\) is redundant. \(\square\)

In the following blog posts, we will generalize this to Hilbert spaces.

Is partition of unity ALWAYS available?

Of course this is desirable. But we will give an example that sometimes we cannot find a satisfying partition of unity.

Let \(D\) be a connected bounded open set in \(\ell^p\) where \(p\) is not an even integer. Assume \(f\) is a real-valued function, continuous on \(\overline{D}\) and \(n\)-times differentiable in \(D\) with \(n \geq p\). Then \(f(\overline{D}) \subset \overline{f(\partial D)}\).

(Corollary) Let \(f\) be an \(n\)-times differentiable function on \(\ell^p\) space, where \(n \geq p\), and \(p\) is not an even integer. If \(f\) has its support in a bounded set, then \(f\) is identically zero.

It follows that for \(n \geq p\), \(C^n\) partitions of unity do not exists whenever \(p\) is not an even integer. For example,e \(\ell^1[0,1]\) does not have a \(C^2\) partition of unity. It is then our duty to find that under what condition does the desired partition of unity available.

Existence of partition of unity

Below are two theorems about the existence of partitions of unity. We are not proving them here but in the future blog post since that would be rather long. The restrictions on \(X\) are acceptable. For example \(\mathbb{R}^n\) is locally compact and hence the manifold modeled on \(\mathbb{R}^n\).

Let \(X\) be a manifold which is locally compact Hausdorff and whose topology has a countable base. Then \(X\) admits partitions of unity

Let \(X\) be a paracompact manifold of class \(C^p\), modeled on a separable Hilbert space \(E\), then \(X\) admits partitions of unity (of class \(C^p\))

References

N. Bourbaki, Elements of Mathematics
S. Lang, Fundamentals of Differential Geometry
M. Berger, Differential Geometry: Manifolds, Curves, and Surfaces
R. Bonic and J. Frampton, Differentiable Functions on Certain Banach Spaces