Determine the expression of the Riemannnian metric of $S^2$ induced by $\mathbb{R}^3$ 
Consider usual local coordinates  $(\theta, \varphi)$ em $S^2 \subset \mathbb{R}^3$ defined by the parametrization $\phi:(0, \pi) \times (0, 2 \pi) \rightarrow \mathbb{R}^3$ given by
$$\phi(\theta, \varphi) = (\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta)$$
Using those coordinates determine the expression of the riemannian metric induced over $S^2$ by the euclidian metric of $\mathbb{R}^3$.

Conceptually the riemannian metric will associate each point $p \in S^2$ to a product $\langle \cdot,\cdot \rangle_p$ in its tangent space $T_pM$. I think that what the question means by metric induced by $\mathbb{R}^3$ is taking $\langle \cdot,\cdot \rangle_p$ to be the usual internal product in $\mathbb{R}^3$. I'm having trouble with getting an explicit expression for the metric.
 A: Essentially, the metric will have the form $${\rm d}s^2 = E(\theta,\varphi){\rm d}\theta^2 +2F(\theta,\varphi){\rm d}\theta\,{\rm d}\varphi + G(\theta,\varphi){\rm d}\varphi^2,$$where $$E(\theta,\phi)=\left\langle\frac{\partial\phi}{\partial \theta}(\theta,\varphi),\frac{\partial\phi}{\partial \theta}(\theta,\varphi)\right\rangle,\quad F(\theta,\varphi)=\left\langle\frac{\partial\phi}{\partial \theta}(\theta,\varphi),\frac{\partial\phi}{\partial \varphi}(\theta,\varphi)\right\rangle,\quad\mbox{and}\quad G(\theta,\varphi)=\left\langle\frac{\partial\phi}{\partial \varphi}(\theta,\varphi),\frac{\partial\phi}{\partial \varphi}(\theta,\varphi)\right\rangle,$$where these inner products are computed with the ambient metric. This is a general mechanism: if $(M^n,g)$ is a Riemannian manifold, $\iota:S \to M$ is a submanifold, and $(u^1,...,u^k)$ are coordinates for $S$, then $\iota^*g$ is described in these coordinates as $$\sum_{i,j=1}^k a_{ij}\,{\rm d}u^i\,{\rm d}u^j,$$where $$a_{ij} = g\left({\rm d}\iota\left(\frac{\partial}{\partial u^i}\right),{\rm d}\iota\left(\frac{\partial}{\partial u^j}\right)\right).$$
A: Let $(\theta,\varphi)$ be the coordinates defined by $\phi^{-1}$, and $(x,y,z)$ be the standard coordinates in $\mathbb{R}^{3}$, so that the euclidean metric is
$$g_{0}=dx\otimes dx+dy\otimes dy+dz\otimes dz$$
We identify, for $p\in \mathbb{S}^{2}$, $T_{p}\mathbb{S}^{2}$ as a subspace of $T_{p}\mathbb{R}^{3}$ (via the differential $dj_{p}$ of the inclusion). Now if $U$ is the domain of $(\theta,\varphi)$, and $p\in U$, we have
$$ \dfrac{\partial}{\partial \theta}=\dfrac{\partial x}{\partial \theta}\dfrac{\partial}{\partial x}+\dfrac{\partial y}{\partial \theta}\dfrac{\partial}{\partial y}+\dfrac{\partial z}{\partial \theta}\dfrac{\partial}{\partial z} \\
\dfrac{\partial}{\partial \varphi}=\dfrac{\partial x}{\partial \varphi}\dfrac{\partial}{\partial x}+\dfrac{\partial y}{\partial \varphi}\dfrac{\partial}{\partial y}+\dfrac{\partial z}{\partial \varphi}\dfrac{\partial}{\partial z}
 $$
Since $x=\sin \theta \cos \varphi$, $y=\sin \theta \sin \varphi$, $z=\cos \theta$, we get:
$$ \dfrac{\partial}{\partial \theta}=\cos \theta \cos \varphi\dfrac{\partial}{\partial x}+\cos \theta \sin \varphi\dfrac{\partial}{\partial y}-\sin \theta\dfrac{\partial}{\partial z} \\
\dfrac{\partial}{\partial \varphi}=-\sin \theta \sin \varphi\dfrac{\partial}{\partial x}+\sin \theta \cos \varphi\dfrac{\partial}{\partial y}
 $$
Finally, $g(v,w)=g_{0}(v,w)$ for $v,w\in T_{p}\mathbb{S}^{2}$. Since the basis vectors of $T_{p}\mathbb{R}^{3}$ are orthonormal:
$$ g_{\theta \theta}=\cos^{2}\theta \cos^{2}\varphi+\cos^{2}\theta \sin^{2}\varphi+\sin^{2}\theta=1 \\
g_{\theta \varphi}=g_{\varphi \theta}=-\cos \theta \cos \varphi \sin \theta \sin \varphi+\cos \theta \cos \varphi \sin \theta \sin \varphi=0 \\
g_{\varphi \varphi}=\sin^{2}\theta\sin^{2}\varphi+\sin^{2}\theta\cos^{2}\varphi=\sin^{2}\theta. $$
Hope this helps!
