I have asked a similar question on the math stackexchange website, but since this type of question might have an answer that is known to physicists better than mathematicians I'm posting the question here as well.

What is the right way of making sense of generalized functions over manifolds? For concreteness, let me restrict my question to the dirac delta function. The article on Wikipedia on Dirac delta function gives a definition of the composition of $\delta$ with continuously differentiable functions $g\colon \mathbb{R}\to \mathbb{R}$ as follows: $$\delta(g(x)):=\sum_i\frac{\delta(x-x_i)}{|g'(x_i)|}$$ where the sum extends over all roots of $g$ which are assumed to be simple. A similar definition is also given for the case where $g$ is a multivariable function.

My question is whether such definitions can be extended to functions $g$ defined over manifolds, say $S^1$ or $S^2$. Over the circle, it seems to me that the only reasonable way of making sense of something like $$\int_{S^1}\delta(|x-i|)f(x)\, dx$$ is by assigning the value $f(i)$ to it which agrees with the naive approach of parameterizing the circle and reducing the problem to the real case: $$\int_{S^1}\delta(|x-i|)f(x)\, dx"="\int_0^{2\pi}\delta(\cos\theta, \sin\theta-1)f(\cos\theta, \sin\theta)\, d\theta.$$

However, for $S^2$ I think a similar computation with, say the north pole $n$ in place of $i$, is ambiguous: $$\int_{S^2}\delta(|x-n|)f(x)\, dx"="\int_{0}^{2\pi}\int_{0}^{\pi}\delta(\text{blah})f(\text{blah})\sin\phi\, d\phi\, d\theta.$$

It seems to me that the values of the right-hand integral depends on the parameterization and is affected by the value of the sine function.

I think the first step in answering such a question is perhaps to make sense of a composition like $\delta(|x|)$ for $x\in \mathbb{R}$.

Question Is there anywhere in math/physics literature that I can find the definition of $\delta(|x|)$ (or a similar composition)?

Any thoughts, comments, insights, or references are highly appreciated.


First of all, the concept of $\delta(\lvert x - x_0\rvert)$ seems not to be well defined neither as a usual distribution. Nevertheless, if you would like to interpret it coherently with the interpretation given for $\delta(f(x))$, i.e. that is the delta distribution giving the value of the test function at the zeros of $f$ (and forget about the derivative multiplicative factor) then the $\delta(\lvert x\rvert)=\delta(x)$.

Anyways, returning to your question of defining distributions on manifolds, that should be perfectly possible using the correct definition of distributions: the objects belonging to the topological dual of the given topological space (the manifold has usually a topology), i.e. the continuous linear functionals of the manifold.

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