Partition of Unity

Given a smooth manifold M with an open cover U_i, a partition of unity subject to the cover U_i is a collection of smooth, nonnegative functions psi_i, such that the support of psi_i is contained in U_i and sum_(i)psi_i=1 everywhere. Often one requires that the U_i have compact closure, which can be interpreted as finite, or bounded, open sets. In the case that the U_i is a locally finite cover, any point x in M has only finitely many i with psi_i(x)!=0.

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 M with coordinate charts U_i such that only finitely many overlap at any point. On each coordinate chart U_i, there are the local vector fields partial/partialx_j. Label these v_(i,j) and, for each chart, pick the vector field v_(i,1)=partial/partialx_1. Then sum_(i)psi_iv_(i,1) is a global vector field. The sum converges because at any x, only finitely many psi_i(x)!=0.

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 sum_(i)psi_i 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.

See also

Convex Set, Open Cover, Riemannian Metric, Section, Smooth Manifold, Vector Field

This entry contributed by Todd Rowland

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Rowland, Todd. "Partition of Unity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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