TITLE: Looking for an atlas with 1 chart
QUESTION [8 upvotes]: Can we provide the set $\{(x,y,z)\in\mathbb{R^3}|x^2+y^2=1\}$ with a 2-dimensional manifold structure involving only 1 chart? I can see it with 2 charts with cylindrical coordinates, but not with only one...

REPLY [7 votes]: You can't do it if you want a chart that is homeomorphic to (e.g.)  the unit disk in $\mathbb{R}^2$. The latter is homotopy equivalent to a point, but your manifold is not. But you can do it with the unit disk with the origin removed.