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The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold?

I've found that a similar question has been asked at Mathematics.SE but that question was asking for a FT on a Lorentzian Manifold. The answer given is actually very good but then it gets a little too technical and quite cryptic to me...

The first comment on that question actually points at the right direction but could someone give me an example on a specific Riemannian Manifold (aside Euclidean), for example on a 2-cylinder with metric $g=dz^2+r^2d\theta^2$?

PS: Maybe I should have posted the question on Mathematics.SE but I was hopping for the physicists more direct approach to these kind of problems...

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1 Answer 1

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By the Pontryagin duality you find Fourier transforms on locally compact groups and you can translate the Haar measure over them so that (I think) in the Lie group case you obtain a Riemannian metric which evaluates on tangent vectors as the inner product of the Lie-algebra. E.g. as a smooth group $SU(2)$ has a Fourier transform by the duality and certainly a metric as $SU(2)∼\mathbb{S}^3⊂\mathbb{R}^4$.

In the Pontryagin case the key is to consider the group homomorphisms $\phi:G\rightarrow U(1)$ from your the group $G$ into the circle group. The maps $\phi$ translate the group operation in $G$ to the group operation in $U(1)$, which can be thought of the multiplication of phases. In the euclidean case $G=\mathbb{R}$, you have vectors $\vec x,\vec y\in \mathbb{R}$ and the group operation is the addition $\vec x+\vec y$. Then the functions $x\mapsto e^{i \vec p\vec x}$ (there is one such function for each $\vec p$) are the only homomorphisms: $e^{i \vec p\vec x}e^{i \vec p\vec y}=e^{i \vec p(\vec x+\vec y)}$ ($U(1)$ group operation on the left side, $G$ group operation on the right). This then relates to the the suitable function space $G\rightarrow \mathbb{C}$. The duality implies a representations of the functions $f(x)$ on $G$ involving an integral with the Haar measure and the homomorphism. In the case $G=\mathbb{R}$ you have $f(x)=\int g(p)e^{ipx}\text dp$ for some $g$. $x$- and $p$-space (translation- and phase-space) turn out to be $\mathbb{R}$.

The key is that on a Lie group you can translate stuff around and hence construct the $\phi$-maps. For any manifold without proper flow, this might be more difficult.

(Wait for other answers and comment though.)

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