Here are some things I don't understand, I would be very grateful for any help!
I am trying to find a chart on the unit sphere that preserves area. The most natural map that springs to my mind is taking the inverse of $\phi(x,y)=(\sin x\cos y, \sin x\sin y, \cos x)$. But the first fundamental form of this map is $EG-F^2=\sin ^2 x$, so presumably the inverse map has E'G'-F'^2={1\over \sin^2 x}. So it doesn't preserve area? What is a chart that does?
Since the unit sphere cannot be fully represented by one chart, I will need another one, which I believe we can just take the same map but change the domain so that we exclude a disjoint half great circle? I want the transition functions relating the charts to have derivative s with determinant 1 but I don't understand what that means. Presumably it implies area- and orientation- preservation.
I am very sorry about the original version of my question.