Partition of Unity

Given a smooth manifold  with an open cover , a partition of unity subject to the cover  is a collection of smooth, nonnegative functions , such that the support of  is contained in  and  everywhere. Often one requires that the  have compact closure, which can be interpreted as finite, or bounded, open sets. In the case that the  is a locally finite cover, any point  has only finitely many  with .

A partition of unity can be used to patch together objects defined locally. For instance, there always exist smooth global vector fields, possibly vanishing somewhere, but not identically zero. Cover  with coordinate charts  such that only finitely many overlap at any point. On each coordinate chart , there are the local vector fields . Label these  and, for each chart, pick the vector field . Then  is a global vector field. The sum converges because at any , only finitely many .

Other applications require the objects to be interpreted as functions, or a generalization of functions called bundle sections, such as a Riemannian metric. By viewing such a metric as a section of a bundle, it is easy to show the existence of a smooth metric on any smooth manifold. The proof uses a partition of unity and is similar to the one used above.

Strictly speaking, the sum  doesn't have to be identically unity for the arguments to work. It goes with the name, because at every point the functions partition the value 1. Also, it is convenient when considered from the point of view of convexity.