Metric on a $2$-sphere in different coordinates? I have a question about the metric on a $2$-sphere.
I know the line element in $\mathbb{R}^3$ is given by $$ds^2 = dr^2 + r^2 d \theta^2 + r^2 \sin^2 \theta d \phi^2. $$ If $r = R$ constant, as a for a sphere, we get $$ds^2 = R^2 d \theta^2 + R^2 \sin^2 \theta d \phi^2. $$ And so the standard metric on a $2$-sphere with coordinates $(\theta, \phi)$ is $$ g_{ij} = \begin{pmatrix} R^2 & 0 \\ 0 & R^2 \sin^2 \theta \end{pmatrix}. $$
I was wondering if it is possible to put certain coordinates $(t, \theta)$ on the $2$-sphere, as an embedded submanifold in $\mathbb{R}^3$, such that the metric on the $2$-sphere becomes \begin{align*} g_{ij} = \begin{pmatrix} 1 & 0 \\ 0 & \sin^2 (t) \cos^2 (t) \end{pmatrix}. \end{align*} If so, how does one find them?
 A: You seem to want a new coordinate $t(\theta,\phi)$ such that
\begin{align}
R^2d\theta^2 + R^2\mathrm{sin}^2(\theta)d\phi^2 &\stackrel{!}{=} dt^2 + \mathrm{sin}^2(t)\mathrm{cos}^2(t)d\theta^2
\end{align}
Here you can simply plug in the chain-rule
\begin{align}
dt = \frac{\partial t(\theta,\phi)}{\partial \theta} d\theta + \frac{\partial t(\theta,\phi)}{\partial \phi} d\phi
\end{align}
to get
\begin{align}
R^2d\theta^2 + R^2\mathrm{sin}^2(\theta)d\phi^2 &\stackrel{!}{=} \left(\frac{\partial t(\theta,\phi)}{\partial \theta} d\theta + \frac{\partial t(\theta,\phi)}{\partial \phi} d\phi\right)^2 + \mathrm{sin}^2(t)\mathrm{cos}^2(t)d\theta^2 \\\\
&= \left(\frac{\partial t}{\partial \theta}\right)^2 d\theta^2 + \left(\frac{\partial t}{\partial \phi}\right)^2 d\phi^2 + \frac{\partial t}{\partial \theta}\frac{\partial t}{\partial \phi}d\theta d\phi  + \mathrm{sin}^2(t)\mathrm{cos}^2(t)d\theta^2
\end{align}
So this gives a system of equations:
\begin{align}
R^2 &= \left(\frac{\partial t}{\partial \theta}\right)^2 + \mathrm{sin}^2(t)\mathrm{cos}^2(t) \\\\
0 &= \frac{\partial t}{\partial \theta} \frac{\partial t}{\partial \phi} \\\\
R^2\mathrm{sin}^2(\theta) &= \left(\frac{\partial t}{\partial \phi}\right)^2
\end{align}
To fulfil the second equation you would have to have either $\frac{\partial t}{\partial \theta}=0$ or $\frac{\partial t}{\partial \phi}=0$. In the first case you cant fulfil the first equation (at least not for all $t$). And in the second case you cant fulfil the third equation. So this is impossible.
If you choose a different ansatz of a coordinate transformation, these equations should be understood as (a system of partial) differential equations for the unknown coordinate-transformation-function $t(\theta,\phi)$. Solving that would give a the desired coordinate transformation.
