How to specify a diffeomorphism between real and complex manifolds?

Created by: tobiasdiez

How can I specify a diffeomorphism between a real and a complex manifold, say for example S^2 and the Riemann sphere. I tried

Phi = S2.diffeomorphism(Cinfty,
                    {(stereoN, Zorigin): [Xn + Yn*I],
                     (stereoS, Zinfty): [Xs - Ys*I]},
                     name='Iso', latex_name=r'\Phi')

where

Cinfty = Manifold(1, r'C_\infty', field='complex')
S2 = Manifold(2, 'S^2', latex_name=r'\mathbb{S}^2', start_index=1)

But this results in the following error

ValueError: for an isomorphism, the source manifold and target manifold must have the same dimension

So I suspect that SageManifold doesn't take dim_C = 2 dim_R into account. But even if I replace diffeomorphism by diff_map above, it doesn't really work: Phi.display() prints no coordinate expression. However, the pullback of a function is ok (except that the pullback of a complex-valued function is printed as real-valued).

On a related note: is the concept of embedding somewhere defined? For example, I would like to invert the natural embedding of S^2 into R^3 on the image (to display the coordinate expression of the inverse).