Consider the manifold of constant negative curvature $G=SL(2, R)/\Gamma$ where $\Gamma$ is such that $G$ is compact (I have no special constraint on $\Gamma$). I know that by the Whitney embedding theorem it can embedded in $R^m$ for some $m$. However, I would like to know an actual explicit embedding. Any idea, pointers?
In order to clarify my request, maybe it is worth to explain why I need the embedding. I would like to numerically solve some differential equations on G (let me skip lots of details). In order to do that, you could represent in your code an element of $G$ like a triple of floating point numbers, but in this case you would need at each step to "normalize" your triple by doing a kind of "reduction modulo $\Gamma$." This reduction introduces a "discontinuity" that causes several numerical issues. Therefore, I would like to "represent" $G$ in my code in a smooth way and an embedding in $R^m$ looks like a nice solution.