Questions: Partition of Unity

Question 1

A partition of unity subordinate to an open cover {Uα} of a manifold M is a collection {ρα} of smooth functions satisfying three conditions. Which of the following is NOT one of them?

• A. Each ρα ≥ 0 everywhere on M
• B. The support of each ρα is contained in the corresponding Uα
• C. Σα ρα(p) = 1 for every p ∈ M
• D. Each ρα is strictly positive on all of Uα

Answer: (D)

Option D is not required and is generally false. Each ρα must be non-negative and supported inside Uα, but it can be zero on much of Uα — it just needs to be nonzero somewhere inside Uα. The three actual conditions are: (1) ρα ≥ 0, (2) supp(ρα) ⊂ Uα, and (3) Σα ρα = 1 everywhere. The sum is locally finite (only finitely many ρα are nonzero near any point), which makes the sum well-defined. The functions need not cover each Uα uniformly.

Question 2

Partitions of unity exist on every smooth manifold.

• T. True
• F. False

Answer: True

This is a theorem, not obvious. The proof requires two ingredients: (1) smooth manifolds are paracompact (every open cover has a locally finite refinement), which follows from second-countability; and (2) smooth bump functions exist — compactly supported smooth functions that are identically 1 on a given compact set. Paracompactness provides a locally finite cover, and bump functions provide the raw material. The bump functions are then normalized (divided by their sum) to produce the partition. This fails for analytic manifolds — analytic bump functions do not exist.

Question 3

How are partitions of unity used to construct a Riemannian metric on any smooth manifold?

Model answer: In each coordinate chart (Uα, φα), the standard Euclidean inner product on ℝⁿ pulls back to a Riemannian metric gα on Uα. Using a partition of unity {ρα} subordinate to {Uα}, define g = Σα ρα · gα. This sum is well-defined (locally finite), smooth, and at each point is a convex combination of inner products — hence positive definite. The result is a globally defined Riemannian metric. This construction shows that every smooth manifold admits a Riemannian metric.

The key insight is that the set of inner products on a vector space is convex — a positive combination of inner products is again an inner product. Since partition-of-unity functions are non-negative and sum to 1, the combination g = Σ ρα gα is a convex combination at each point, hence positive definite. This convexity argument works for metrics but fails for other structures (like symplectic forms) where the 'space of structures' is not convex.

Question 4

Partitions of unity do not exist in the analytic (Cω) category because analytic bump functions do not exist.

• T. True
• F. False

Answer: True

A real-analytic function that vanishes on an open set is identically zero (by the identity theorem). Therefore, a compactly supported analytic function that is not identically zero cannot exist — it would need to vanish outside a compact set but be nonzero inside. Since bump functions are the building blocks of partitions of unity, the partition-of-unity technique is fundamentally a smooth (C∞) phenomenon. This is one reason smooth manifolds are more flexible than analytic manifolds, and why many constructions in differential geometry require C∞ smoothness.