Q:

Application of the Poisson equation on a parametric surface

Now if I have the electric potential as a function of $x$, $y$ and $z$. I use the Poisson equation to calculate the charge distribution and I obtain a charge distribution as a function of $x$, $y$ and $z$. Now, if I know that the charge lies entirely on a surface in 3d Cartesian coordinates and the potential on the surface is described by a function of its parametric variables. Can I use the Poisson equation as follows? 
--- I take the laplacian of the potential function (which is a function of the parametric variables) with respect to the parametric variables and equate it to $-\rho/\epsilon$ not?
$$\frac{d^2v}{ds^2}+\frac{d^2v}{dt^2} =-\frac{\rho}{\epsilon}$$ (where $s$ and $t$ are the parametric variables.)
This will give the charge density as a function of $s$ and $t$ (the parametric variables) and is defined on the surface.
Is this true? Thanks

A:

In general, the Poisson equation in a Riemannian manifold $M$ is given by,
$$\nabla_i \nabla^i \phi = -\frac{\rho}{\epsilon}$$
where $\nabla_i \nabla^i$ is the Laplace-Beltrami operator, which acting on 
a scalar is simply,
$$\nabla_i \nabla^i \phi = \frac{1}{\sqrt{|g|}}\partial_i (\sqrt{|g|}g^{ij}\partial_j \phi)$$
for the metric $g_{ij}$ on $M$, and coordinates $\{x^i\}$ on $M$. Now, if we have a surface or sub-manifold $\Sigma$ embedded in $M$ with coordinates $\{\sigma^a\}$, then the charge density in terms of $\{\sigma^a\}$ satisfies,
$$\nabla_a \nabla^a \phi= -\frac{\rho(\sigma)}{\epsilon}$$
where $\nabla_a \nabla^a$ is now the Laplace-Beltrami operator, but for the induced metric on $\Sigma$, given by,
$$\gamma_{ab} = \frac{\partial x^i}{\partial \sigma^a}\frac{\partial x^j}{\partial \sigma^b} g_{ij}$$
which is the pullback of $g$ onto $\Sigma$ and $x^i(\sigma)$ are the embedding functions of $\Sigma$. 

