a partition of unity

Manifolds are built locally, chart by chart, but most things you want — a metric, a connection, an integral — must be defined globally. A partition of unity is the gluing device that lets you build a global object by patching together local pieces, each defined only on one chart, with no seams. It is a family of smooth bump functions that add up to 1 everywhere, weighting each local contribution so they blend smoothly.

Given an open cover {U_alpha} of M, a partition of unity subordinate to it is a collection of smooth functions psi_alpha: M -> [0, 1] such that the support of each psi_alpha lies inside U_alpha, the supports form a locally finite family (every point has a neighborhood meeting only finitely many), and at every point the sum of all psi_alpha equals 1. The key existence theorem: on any smooth manifold (Hausdorff, second-countable) a smooth partition of unity subordinate to any open cover exists. The construction rests on smooth bump functions — the existence of a C^infinity function that is 1 on a ball and 0 outside a larger ball, the hallmark of the smooth (not analytic) category.

Now the magic: to define a Riemannian metric, choose any metric g_alpha on each chart U_alpha (just pull back the Euclidean one) and set g = sum psi_alpha g_alpha; the sum is a smooth, positive-definite metric because a convex combination of inner products is an inner product. The same trick globalizes connections, builds the function for Whitney's embedding, and shows manifolds carry plenty of smooth functions. The honest limit: partitions of unity are why the smooth category is so flexible, but the trick fails badly in the holomorphic or real-analytic categories — there are no analytic bump functions (an analytic function vanishing on an open set is identically zero), which is precisely why complex geometry is rigid where smooth geometry is soft.