I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold (curved space). This transformation is quite simple in Euclidean space. One can consider it as a Fourier transformation of a Guassian function: $$ \exp\left(-\frac{a}{2}\frac{d^2}{dx^2}\right) f(x)~=~\int_{\mathbb{R}} \!\frac{dy}{\sqrt{2\pi a}}\exp\left(-\frac{y^2}{2a}\right) f(x+y), \qquad a~>~0. $$ However, my question is can one apply HS transformation to a scalar function on the compact Riemannian manifold $(M,g)$? If so, how?
I couldn't understand how to define the Fourier transform on a compact Riemannian manifold. It seems to me the Fourier transform is defined by the Pontryagin duality.. I've found that a question about the Fourier transform has been asked at this post.. The answer given is actually nice but it goes little technical and I'm not familiar with that.. Nevertheless, I guess one can write the Hubbard-Stratonovich transform: $$ \exp\left(-\frac{a}{2}\Delta\right) f({x})~=~ (2\pi a)^{-\frac{n}{2}}\int d^n{y}\sqrt {g(y)}\exp\left(-\frac{{y}^2}{2a}\right) f({x+y})~, $$ where $\Delta$ is the Laplace–Beltrami operator. In addition, I read that the Weierstrass transform can be defined on any Riemannian manifold, but it doesn't make sense to me, I couldn't find any proper references and don't know how to write this transform.
Context:
My real problem involves the free energy of a harmonic oscillator on a Riemannian manifold which leads to an trasformation similar to the HS transformation. Indeed, potential energy of harmonic oscillator is $U=\frac{1}{2}kd(y,y_0)^2$ which $k$ is a constant, $d(y,y_0)$ is distance. I simply assumed that $d(y, .)^2=y^2$. Under what conditions on the Riemannian manifold $M$ can I write a transformation similar to to the one mentioned above? As far as I understand, one can define the FT on a Riemannian manifold when it is either a Lie group or a symmetric space.
PS: I have originally posted this question at physics.SE but perhaps math.SE is a better place for this kind of integral-transforms.
I would appreciate answers not demanding an all profound background...