Partition of Unity for defining Riemannian metric.

Question

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

Answer 1

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.

Answer 2

Once you absorb the idea of covering a manifold with "patches", your first idea might be to piece-wise define a metric: at each point in the manifold, the metric tensor is one determined by a chart domain in which the point lies. But since "patches"/chart domains overlap in general, if the point lies in two chart domains, then you've got two metric tensors to choose from. This does not lead to a well-defined metric...

The special properties of a partition of unity subordinate to the collection of chart domains provide a solution.

Subordination to the collection ensures that the supports of the partition functions are contained in just one open set each. At any one point in the manifold, only one of the partition functions has nonzero value, so the sum $$\sum_{\alpha}g_{\alpha}\phi_{\alpha}$$ where the $g_{\alpha}$ are the metrics on the open patches $U_{\alpha}$ and the $\phi_{\alpha}$ are the partition functions, reduces to just one $g_{\beta}\phi_{\beta}$. Because $\sum_{\alpha}\phi_{\alpha}=1$ at every point, the "nonzero value" is 1. The smoothness of the $\phi_{\alpha}$ ensures that the sum $\sum_{\alpha}g_{\alpha}\phi_{\alpha}$ is smooth on the manifold. Bilinearity, symmetry, and positive-definiteness at each point clearly still hold for this sum, and one can easily see that it provides a Riemannian metric on the manifold.