Use of partitions of unity

The existence of continuous partitions of the unity is guaranteed on any topological space (which is countable at infinity), the smooth manifold assumption is here only to make sense of the smoothness of the maps in the family.

Usually, partitions of the unity are used to construct objects on a smooth manifold by patching them together, here is a general recipe:

• The existence of the object you are interested in is locally guaranteed by the existence of charts.

• The set of all these objects is convex.

Example. Let $M$ be a smooth manifold, then there exists $g$ a Riemannian metric on $M$, that is a smooth family of inner products on the tangent spaces of $M$.

On Riemannian manifold, if there is an locally finite open cover $U_i$ and $X_i$ is a vector field on $U_i$, then by partition of unity $f_i$, we have a global vector field $X =\sum_i\ f_iX_i$.