Chart vs ambient coördinates for smooth affine varieties

In SageManifolds, coördinates are defined by means of charts and transition functions. For instance, I understand one can define a sphere by giving two charts and the usual transition arising from stereographic projections. But a sphere can also come with its own coordinates by being defined as a variety. Is there a way (or even a nice way!) to reconcile these two views when working with locally defined objects? For instance, I might have a function on a manifold that vanishes in some way, which when defined on charts is obvious, but when defined on the subvariety one needs to apply the defining equation(s) to see this. Other properties defined via equations in function rings (or other rings of sections of sheaves or bundles) are similar.

edit retag close merge delete