Let's suppose we have some space M that is topologically a three-sphere (but arbitrary in specifics) with metric $g_{\mu\nu}$. The affirmative solution to the Yamabe problem leads me to believe that the metric here should be conformal to the metric of a unit three-sphere centered at the origin (which we will denote $\mathbb{S}_{ab}$). In this case, we expect a relation of the form:
$$g_{\mu\nu}=|\phi|^{2}\mathbb{S}_{\mu\nu}$$
Suppose we now write this three-sphere as an embedding in $\mathbb{R}^{4}$, we might then write then write the metric as induced:
$$g_{\mu\nu}=\partial_{\mu}\phi^{A}\eta_{AB}\partial_{\nu}\phi^{B}$$
Where $\phi^{A}$ are coordinates and $\eta_{AB}$ is the flat space metric (both in $\mathbb{R}^{4}$). We might now also write the unit three-sphere metric as induced:
$$\mathbb{S}_{\mu\nu}=\partial_{\mu}\Phi^{A}\eta_{AB}\partial_{\nu}\Phi^{B}$$
Where it becomes obvious that:
$$\Phi^{A}=\frac{\phi^{A}}{2|\phi^{A}|}$$
In order that $\Phi^{A}\Phi_{A}=1$. Let us try a simple substitution then:
$$g_{\mu\nu}=|\phi|^{2}\mathbb{S}_{\mu\nu}\rightarrow\phi^{A}\phi_{A}\frac{1}{4}\partial_{\mu}\left(\frac{\phi^{A}}{|\phi^{A}|}\right)\eta_{AB}\partial_{\nu}\left(\frac{\phi^{B}}{|\phi^{B}|}\right)$$
$$=\left(\frac{1}{4}\partial_{\mu}\left(\phi^{A}\right)\eta_{AB}\partial_{\nu}\left(\phi^{B}\right)+\frac{1}{4}\left(\phi^{A}\phi_{A}\right)^{2}\partial_{\mu}\left(\frac{1}{|\phi^{A}|}\right)\partial_{\nu}\left(\frac{1}{|\phi^{B}|}\right)\right)$$
The first term is indeed the induced metric we started with, but the second term???? It simplifies a bit more:
$$=\left(\partial_{\mu}\left(\phi^{A}\right)\eta_{AB}\partial_{\nu}\left(\phi^{B}\right)+\partial_{\mu}\left(\phi^{A}\right)\eta_{AB}\partial_{\nu}\left(\phi^{B}\right)\right)$$
This looks right except for the factor of two issue. I'm guessing I just made a mistake somewhere? so we might also say that we can choose coordinates such that:
$$g_{\mu\nu}=|\phi^{A}\phi_{A}|\frac{g_{\mu\nu}}{|\phi^{A}\phi_{A}|}=|\phi^{A}|^{2}\mathbb{S}_{\mu\nu}$$
Is this correct?