This should be a simple problem, yet I'm having trouble seeing the answer. Given two Riemannian metrics related by a conformal transformation $e^{2\sigma(x^{i})}$ . The relationship between them is simply:
$$\bar{g}_{\mu\nu}=e^{-2\sigma(x)}g_{\mu\nu}$$
Yet when it comes to the sphere $S^{3}$ and the flat space $R^{3}$, I know they are conformally related, and yet I can't find a way to write them in the above form. I'm supposing it may have to do with the fact that $R^{3}$ has had a point removed (ie. one point compactification). Is there actually a way to write this? As an aside I study physics, but would like to understand the deeper mathematical relationships that are often “swept under” the proverbial rug.
A possible conceptual issue here is the distinction between the following (very closely related) concepts:
The relationship between them is straightforward: $g_1,g_2$ are conformally related if the identity map $\mathrm{id}_M:(M,g_1) \to (M,g_2)$ is conformal, and a map $\phi: (M,g) \to (N,h)$ is conformal if $g, \phi^* h$ are conformally related.
Since $\mathbb S^3$ and $\mathbb R^3$ are distinct manifolds, what you really want to show is that there is a conformal map $\mathbb R^3 \to \mathbb S^3$, the most likely candidate for which is (as the question linked in the comments suggests) just the inverse of stereographic projection.
Alternatively, you can use the stereographic projection to identify $\mathbb S^3 \setminus \{ p \}$ with $\mathbb R^3$, so that you have two metrics on $\mathbb R^3$: the Euclidean metric $\delta$ , and the metric $g$ obtained from $\mathbb S^3$ via this identification. The problem is then in the form you originally posed it: show that $ g = e^{2 \sigma} \delta$ for some $\sigma$.
The relationship between the two concepts means that the calculation you need to do will be the same either way.