Atlas of torus and sphere. Atlas of Calabi–Yau manifold.

[Spinnor]

Is it true that the atlas for a torus can consist of a single map while the atlas for a sphere requires at least two maps?

Can we ever get by with a single map for some Calabi–Yau manifolds assuming that question makes sense? If not is there some maximum number required?

Thanks for any help!

 

[pasmith]

The torus is compact. (An open subset of) $\mathbb{R}^2$ is not. Hence the atlas for a torus also requires at least two charts.

 

[mathwonk]

If by "atlas" you mean a covering family of homeomorphisms from open sets of the manifold to open sets of Euclidean space, as is usual, then this cannot happen for a torus using only one map. If you mean instead a single map from Euclidean space to the torus, such that one can obtain an atlas of the usual sort by taking more than one restriction of that one map, then this does happen for the torus using the universal covering map of the torus by Euclidean space.

 

[Spinnor]

Thanks for the quick replys! Will study.

Thought there was a difference between the sphere and torus regarding "maps".

Thanks for the help!