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I am trying to solve a geometry problem:

Let $T$ be the torus given by rotating the circle $\{ (x,0,z) \in \mathbb{R}^3 | \ (x-2)^2 + z^2 = 1 \}$ around the $z$-axis and let $G : T \to S^2$ be the Gauss map. Compute the integral $$\int_T G^* \ \eta_{S^2}$$ where $\eta_{S^2}$ is the area form on $S^2$.

My first step is to write \begin{equation*} \int_T G^* \ \eta_{S^2} = \int_{G(T)} \eta_{S^2} . \end{equation*} I am not really sure where to go from here. Not sure if Stokes' theorem will be of any use since we don't have a boundary or a "$d$" appearing already. Any hints or pointers would be greatly appreciated!

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    I think you just need to know how many times $G(T)$ covers $S^2$. As far as I can see that is $2$ so the integral is just twice the area of $S^2$ – 2017-01-04
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    I see that, but shouldn't this involve a little more than that? – 2017-01-04
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    I wrote it below explicitly – 2017-01-04
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    I see that now, thank you! – 2017-01-04

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So as a explained above the answer is $2\times {\rm Area}_{S^2}=8\pi$

More explicitly: we can parametrize the above torus by $\phi$ and $\theta$ as $(x,y,z)=(\cos (\theta ) \sin (\phi ),\sin (\theta ) \sin (\phi ),\cos (\phi ))$ where both angles go from $0$ to $2\pi$.

The Gauss map explicitly is $$ \{\cos (\theta ) \sin (\phi ),\sin (\theta ) \sin (\phi ),\cos (\phi )\}\in S^2 $$ as these are usual spherical coordinates on $S^2$ (with the only difference that both angles allowed to go from $0$ to $2\pi$) we get twise the usual answer $$ \int_0^{2\pi}d\phi\int_0^{2\pi}d\theta |\sin\phi|=8\pi $$