Partitions of Unity

A manifold is assembled from charts, and most constructions on it are first made chart by chart, where everything is Euclidean. The problem is to glue the local pieces into one global object without creating seams. The tool is a partition of unity: a family of smooth functions, each living inside one chart, that add up to $1$ everywhere. Multiply each local construction by its function and add: the result is smooth, global, and agrees with the local constructions where only one of them is active.

The headline application is the first step of the proof of the Poincaré conjecture: every smooth manifold carries a Riemannian metric. Take the Euclidean metric in each chart, weight it by a partition of unity, and add. A weighted average of inner products is an inner product, so the result is a metric. The Ricci flow then starts from this metric, any metric at all, and the rest of the guide is about what happens next.

The functions that make this possible are the bump functions of 2B.6 Power Series, Exponentials and Bump Functions: smooth, but zero outside a compact set. They exist because smooth functions need not be analytic. An analytic function that vanishes on an open set vanishes everywhere (5A.2 Cauchy’s Theorem and Its Consequences), so no analytic partition of unity can exist: the smooth category's flexibility is exactly what makes gluing possible.

By the end of this chapter you will be able to:

• construct bump functions and cut-off functions on $\mathbb{R}^n$ and on manifolds;
• define partitions of unity subordinate to an open cover, and prove they exist on compact manifolds;
• use partitions of unity to extend functions, to build exhaustion functions, and to put a Riemannian metric on any manifold;
• state Whitney's embedding theorem and prove an easy version for compact manifolds.

2.1Blending

2.2Bump functions

Recall from 2B.6 Power Series, Exponentials and Bump Functions that

is $C^\infty$ on $\mathbb{R}$, with all derivatives zero at $0$. From it:

• $\psi(t) = \frac{h(t)}{h(t) + h(1 - t)}$ is smooth, equal to $0$ for $t \leq 0$ and to $1$ for $t \geq 1$, and increasing in between: a smooth step;
• for $0 < r_1 < r_2$, $\rho(x) = 1 - \psi\big(\frac{|x| - r_1}{r_2 - r_1}\big)$ is a smooth function on $\mathbb{R}^n$ equal to $1$ on the closed ball $\overline B_{r_1}$, zero outside $B_{r_2}$, and between $0$ and $1$: a cut-off function (Exercise 2.5).

The support of a function, $\operatorname{supp}f$, is the closure of the set where it is non-zero. A bump function is a smooth function with compact support. On a manifold, composing a bump function on $\mathbb{R}^n$ with a chart and extending by zero gives a bump function on $M$ supported in the chart's domain (Exercise 2.6); this is where the Hausdorff condition is used, so that the support, compact in the chart, is closed in $M$.

2.3Partitions of unity

Local finiteness makes the sum a finite sum near each point, so it is smooth. The theorem is that partitions of unity always exist.

First applications.

• Extension. If $A \subseteq M$ is closed and $f$ is smooth on a neighbourhood $U$ of $A$, there is a smooth function on all of $M$ that equals $f$ on $A$: take a partition of unity $\{\psi_0, \psi_1\}$ subordinate to the cover $\{U, M\setminus A\}$, and use $\psi_0f$ (extended by $0$).
• Smooth Urysohn. For disjoint closed sets $A$, $B \subseteq M$ there is a smooth function equal to $0$ on $A$ and $1$ on $B$.
• Exhaustion functions. Every manifold has a smooth proper function $f : M \to [0, \infty)$ (7A.2 Compactness and Compactification): a function that goes to infinity at infinity. On a non-compact manifold it is the substitute for $|x|$ on $\mathbb{R}^n$, used to cut off functions near infinity, as in the analysis of complete non-compact Ricci flows (11B.3 Compactness of Ricci Flows).

2.4Every manifold has a Riemannian metric

A Riemannian metric on $M$ is a choice, for each $p \in M$, of an inner product $g_p$ on the tangent space $T_pM$ (8A.3 Tangent Vectors and Bundles), varying smoothly with $p$: in each chart it is given by a symmetric positive definite matrix $(g_{ij}(x))$ of smooth functions (9A.1 Riemannian Metrics and Model Spaces).

The proof uses only that a convex combination of inner products is an inner product: the set of inner products on a vector space is a convex cone. The analogous statement fails for Lorentzian metrics, of signature $(-, +, +, +)$, whose combinations can degenerate; whether a manifold admits one is a topological question (a closed manifold has a Lorentzian metric if and only if its Euler characteristic is $0$, because a Lorentzian metric gives a nowhere-vanishing direction field, 7A.7 Smooth Topology).

2.5Embedding manifolds in Euclidean space

Every smooth manifold, defined abstractly by charts, can in fact be realised as a submanifold of some $\mathbb{R}^N$ (8A.4 Submanifolds):

The easy version, for compact $M$ and a larger $N$, is a direct application of bump functions (Exercise 2.9): with finitely many charts $\varphi_1, \dots, \varphi_m$ and bump functions $\rho_j$ equal to $1$ on sets $V_j$ that still cover $M$, the map

is an injective immersion, hence an embedding of the compact $M$. The sharp dimension $2n$ is Hassler Whitney's theorem of 1944. So abstract manifolds are no more general than the submanifolds of 7A.7 Smooth Topology. They are used anyway, because an embedding adds structure that has nothing to do with the manifold: an abstract Riemannian manifold, such as hyperbolic space or a Ricci flow at a given time, has no preferred embedding, and its curvature must be computed intrinsically (8A.9 The Curvature of Surfaces, 9A.4 Curvature and What It Means).

2.6History

Partitions of unity were introduced by Jean Dieudonné in 1937 and became the standard gluing tool through the work of Hassler Whitney and others; the existence of Riemannian metrics on all paracompact manifolds is a consequence. Whitney proved that every $n$-manifold embeds in $\mathbb{R}^{2n+1}$ in 1936 and in $\mathbb{R}^{2n}$ in 1944. The partition of unity finite element method was introduced by Melenk and Babuška in 1996.

2.7Exercises

Solution

For every $t$, at least one of $t > 0$ and $1 - t > 0$ holds, so the denominator is positive. For $t \leq 0$, $h(t) = 0$; for $t \geq 1$, $h(1 - t) = 0$. Near $x = 0$, $|x| < r_1$, so the argument of $\psi$ is negative and $\rho \equiv 1$ there: the non-smoothness of $|x|$ at $0$ is never seen.

Solution

We need $\psi_2 = 0$ near $(-1, 0)$, where $\cos\theta = -1$, and $\psi_1 = 1 - \psi_2 = 0$ near $(1, 0)$, where $\cos\theta = 1$. Take $\psi_2 = \psi(\cos\theta + \frac12)$: it vanishes where $\cos\theta \leq -\frac12$, and equals $1$ where $\cos\theta \geq \frac12$. Then $\psi_1 = 1 - \psi_2$ vanishes where $\cos\theta \geq \frac12$. Both are smooth, with supports in $U_2$ and $U_1$ respectively.

Solution

$\sum_\alpha\int\psi_\alpha\omega = \sum_\alpha\sum_\beta\int\psi_\alpha\chi_\beta\omega$, since $\sum_\beta\chi_\beta = 1$ and the sums are finite on the compact support. Each $\psi_\alpha\chi_\beta\omega$ is supported in $U_\alpha\cap U_\beta$, where its integral can be computed in either chart with the same result. Exchanging the order of summation gives $\sum_\beta\int\chi_\beta\omega$.