Question

The usual way to model a vibrating membrane is by using the wave equation. Is it possible to do that from "within"? Probably the answer is yes, but where can I see it done explicitly. What I mean is can we model the membrane as a two dimensional Riemannian manifold without any specific embedding in R3 plus equations involving the metric (or form, tensors etc.) such that at any time it is isometric to the surface given by the graph of the function satisfying the wave equation at that time. Higher dimensional generalizations are also interesting.   

Answer 1

A part of your question sounds like the covariant treatment of membranes (and higher-dimensional branes) in string/M-theory. Of course, to do actual quantitative calculation, one has to choose a particular embedding and world volume coordinates along the membrane, and impose the coordinate redefinition symmetry (the same thing is done for strings).

However, the embedding in some actual spacetime still exists. If it is guaranteed or required to exist, it makes no sense to pretend that it doesn't exist: the system will effectively be all about wave equations for the transverse coordinates (to the brane).

If you wanted to encode the curvature of the brane in its metric tensor only, assuming that the metric tensor behaves just like an induced metric from a higher-dimensional space, you would get different equations of motion. The simplest equations for the metric are Einstein-like equations which are second-order in the metric; however, the induced metric is proportional to the derivatives of spatial coordinates, so the Einstein-like equations would be third-order in the spacetime coordinates, and moreover nonlinear to contract the odd number of indices.

But I probably don't understand what exactly you have in mind - and apologies, it's probably because what you have in mind is impossible mathematically.