We say a compactly supported function $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}$ is a mollifier if
- $\int_{\mathbb{R}^n} \varphi(x) dx = 1$
- $\lim_{\epsilon \rightarrow 0} \varphi_\epsilon = \lim_{\epsilon \rightarrow 0}\epsilon^{-n}\varphi(x/\epsilon) = \delta(x)$
where $\delta$ is the delta function. Furthermore we may use $\varphi$ to smoothen any function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ using convolution: $$f_\epsilon= \int_{\mathbb{R}^n} \varphi_\epsilon(x-y) f(y) dy$$
How much of this carries over to a general smooth manifold $M$? I assume $M$ must be a Riemannian manifold so we have a canonical volume form to be able to integrate functions. The first criteria above can be satisfied on a Riemannian manifold, but how would one satisfy the second? There is no way to scale points on a manifold. Furthermore, if one wanted to use such a mollifier to smoothen a function $f: M \rightarrow \mathbb{R}$, then how would the difference $x-y$ be defined when carrying out the convolution? Taylor briefly mentions this in his three volume series on PDEs but he does not go into detail.