Partition of Unity for defining Riemannian metric. Why do we need Partition of Unity for defining a Riemannian metric on a manifold ?
What role does it play ?

Manifolds are built from coordinate patches, if you do it the old-fashioned way. This means that in general functions are only defined locally on open sets (there is, for example, no global coordinate system on a sphere, from the hairy ball theorem). The partition of unity is a way to patch compatible definitions of functions on different open subsets together to form a global definition of an object.
Fundamentally, I would say in the case of the metric it is that you cannot expect to be able to define global coordinates: you take some local coordinate chart in each open set of an atlas, construct a metric on the tangent bundle over this set, then join the bundles together so that the metric function (and other things like its derivatives) agree on overlaps.
