Partitions of Unity and Orientation

Partitions of unity are the essential technical tool that allows global constructions on manifolds by piecing together local data. Orientation provides the structure needed for consistent integration.

Partitions of Unity

Definition4.5Partition of unity

Let $M$ be a smooth manifold and $\{U_\alpha\}$ an open cover. A partition of unity subordinate to $\{U_\alpha\}$ is a collection of smooth functions $\{\rho_\alpha: M \to [0,1]\}$ such that:

$\operatorname{supp}(\rho_\alpha) \subset U_\alpha$ for each $\alpha$ .
The collection $\{\operatorname{supp}(\rho_\alpha)\}$ is locally finite (every point has a neighborhood meeting only finitely many supports).
$\sum_\alpha \rho_\alpha(p) = 1$ for all $p \in M$ .

Theorem4.4Existence of partitions of unity

For any open cover $\{U_\alpha\}$ of a smooth manifold $M$ , there exists a smooth partition of unity subordinate to $\{U_\alpha\}$ .

The proof relies on the paracompactness of smooth manifolds (guaranteed by second-countability and the Hausdorff property) and the existence of smooth bump functions.

ExampleApplications of partitions of unity

Partitions of unity enable:

Extending local objects globally: Any tensor field defined on a chart neighborhood can be multiplied by a bump function and extended to all of $M$ .
Defining integrals: $\int_M \omega = \sum_\alpha \int_{U_\alpha} \rho_\alpha \omega$ .
Constructing Riemannian metrics: If $g_\alpha$ are inner products on each chart domain, $g = \sum_\alpha \rho_\alpha g_\alpha$ is a global Riemannian metric.
Embedding theorems: Whitney's embedding theorem uses partitions of unity to construct embeddings $M \hookrightarrow \mathbb{R}^N$ .

Orientation

Definition4.6Orientation

An orientation on a smooth $n$ -manifold $M$ is a choice of nowhere-vanishing $n$ -form $\mu$ on $M$ , up to multiplication by a positive smooth function. Equivalently, it is a consistent choice of orientation for each tangent space $T_p M$ that varies continuously (smoothly) with $p$ . A manifold admitting an orientation is called orientable.

Theorem4.5Orientation and volume forms

A smooth manifold $M$ is orientable if and only if it admits a nowhere-vanishing top-degree form (a volume form). Equivalently, $M$ is orientable if and only if the top exterior power $\Lambda^n(T^*M)$ is a trivial line bundle.

ExampleOrientable and non-orientable manifolds

$\mathbb{R}^n$ , $S^n$ , $T^n$ (tori), and all Lie groups are orientable.
The Mobius band and the Klein bottle are non-orientable.
The real projective space $\mathbb{R}P^n$ is orientable if and only if $n$ is odd.
Every simply connected manifold is orientable.

RemarkIntegration on non-orientable manifolds

On non-orientable manifolds, one cannot integrate differential forms globally (the sign of the integral depends on the chart). However, one can integrate densities -- sections of the density bundle $|\Lambda|^n(T^*M)$ -- which transform by $|\det J|$ rather than $\det J$ under coordinate changes. This gives a well-defined notion of volume for any manifold.