Find an atlas of charts on $S^2$ with certain properties Find an atlas of charts on $S^2$ for which each chart preserves area, and the transition functions relating charts have derivatives with determinant 1.
I have been thinking that I should consider the projection from the sphere to the cylinder around it, as by Archimedes this map is area preserving. I am however having difficulty explicitly expressing this map as $\sigma(u,v)$. Also, I am not sure how I make the transition functions have determinant of derivative 1.
Many thanks in advance
 A: One solutions is to take your favorite area preserving map of the Earth, such as the Gall-Peters projection.  Restricting the domain, appropriately, this becomes a chart.
For a sphere or radius $1$, the Gall-Peters projection has the simple formula which maps a point $(\theta, \phi)$ is spherical coordinates to the point $$(x,y) = f(\theta, \phi) = (\theta, \cos(\phi)) \in \mathbb{R}^2.$$
To verify that this preserves area, note the the area form on $\mathbb{R}^2$ is $dx\wedge dy$.  Pulling this back by $f$, we get \begin{align*} f^\ast(dx\wedge dy) &= (f^\ast dx)\wedge (f^\ast dy) \\ &= d(f^\ast x) \wedge d(f^\ast y)\\ &= d(x\circ f)\wedge d(y\circ f)\\ &= d\theta \wedge d\cos\phi\\ &= -\sin\phi d\theta \wedge d\phi\end{align*} which is, up to sign, the usual volume form on $S^2$.
Suppose one restricts to, say, $0<\theta <2\pi$ and $0<\phi<\pi$ to get a chart.  To complete this to an atlas, first rotate the sphere about the $z$-axis half way around, and then rotate in the $xz$ plane a quarter of the way around, and then apply the same $f$.  
As Henning noted in the comments, because the second chart is also area preserving, on the overlap between the two sets, the determinant is $\pm 1$.  Since the overlap is connected, verifying at one point will determine whether or not everything is orientation preserving or orientation reversing.  If reversing, modify the second chart by postcomposing with a reflection about the $x$ axis in $\mathbb{R}^2$.
A: The projection on the cylinder is not area preserving but angle preserving !
