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!

 

[blue_raver22]

Yes, it is true that the atlas for a torus can consist of a single map, while the atlas for a sphere requires at least two maps. This is because the torus is a two-dimensional surface with no singularities, so it can be covered by a single coordinate chart. On the other hand, the sphere is a three-dimensional surface with a singularity at the north and south poles, so it requires at least two coordinate charts to cover the entire surface.

As for Calabi-Yau manifolds, it is not always possible to have a single map covering the entire surface. This is because Calabi-Yau manifolds are complex, six-dimensional spaces with special geometric properties, and they can have more complicated topologies than tori or spheres. In general, the number of maps required in an atlas for a Calabi-Yau manifold will depend on the specific manifold and its topological features. There is no maximum number of maps required, as it can vary depending on the manifold.