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

The projection on the cylinder is not area preserving but angle preserving !
