A partition of unity is a collection of smooth non-negative functions {ρα} on a manifold that sum to 1 everywhere, each supported in a single coordinate chart. Partitions of unity allow you to patch together local constructions — metrics, forms, connections — into global objects. Their existence on smooth manifolds (guaranteed by paracompactness and the existence of smooth bump functions) is what makes the passage from local to global possible throughout differential geometry.
Many constructions in differential geometry start locally — in a single coordinate chart, it is easy to define a metric, a connection, or a volume form using the coordinate structure of ℝⁿ. The challenge is patching these local constructions into a coherent global object. Partitions of unity are the glue that makes this possible. They are collections of smooth functions that decompose the manifold into weighted pieces, each living inside a single chart.
A smooth bump function is the fundamental building block: a smooth non-negative function that equals 1 on a compact set K and vanishes outside a slightly larger open set. Such functions exist in the smooth category because C∞ functions can be "flat" (all derivatives zero) at a point without being identically zero. The standard construction uses the function e^{-1/x} for x > 0 and 0 for x ≤ 0, which is smooth but not analytic. Starting from bump functions, you construct a partition of unity subordinate to any open cover by taking bump functions for each chart and normalizing by dividing by their sum.
The construction of a Riemannian metric on any smooth manifold illustrates the power of the technique. Each chart provides a local metric (the pullback of the Euclidean metric on ℝⁿ). These local metrics may disagree on overlaps, but the partition-of-unity average g = Σ ρα gα produces a globally defined metric. This works because the set of positive-definite inner products is convex: any positive combination of inner products is again an inner product. The same technique constructs connections, embeddings into Euclidean space (the Nash embedding theorem uses partitions of unity), and many other global geometric objects.
Not every structure can be patched with partitions of unity. The technique works for structures defined by convex conditions (metrics, connections) but fails for structures defined by non-convex conditions. Symplectic forms, for instance, cannot be averaged — a convex combination of symplectic forms need not be symplectic. Complex structures similarly resist partition-of-unity arguments. This is why some geometric structures (Riemannian metrics) always exist on smooth manifolds while others (symplectic forms, complex structures) impose genuine topological constraints. Understanding which constructions survive the local-to-global passage — and which do not — is one of the organizing themes of differential geometry.