Add Examples to Manifolds Catalog

[gh-mjungmath]

The current catalog is nice to have but still very limited or incomplete. Constructing some examples in the current catalog also takes a lot of time.

While improving and extend this, it is also a good opportunity to expand the tutorials a little bit. Furthermore, most of our docstring examples are based on the 2-sphere. Hence we can perhaps shorten or simplify some parts of the documentation.

Step 1: rearrange folder structure (#30799)

Step 2: unification/improvement/extension of preexisting models

• Add Orientation to EuclideanSpace
• Sphere (#30804)
• Kerr
• Minkowski
• Torus

Step 3: add new examples

• Real Projective Space
• Complex Projective Space

Step 4: improve the tutorial/doctests

• simplify preexisting 2-sphere examples
• add thorough tutorial with old/new models

[egourgoulhon]

This sounds like a very good idea!

Note that in order not to clutter the global namespace, one can let access to the sphere and the hyperbolic space via the ​manifold catalog. The current version of the catalog is very primitive and the function Sphere() in it should call the constructor the sphere class, as soon as the latter is ready.

[gh-mjungmath]

Replying to egourgoulhon:

This sounds like a very good idea!

Note that in order not to clutter the global namespace, one can let access to the sphere and the hyperbolic space via the ​manifold catalog. The current version of the catalog is very primitive and the function Sphere() in it should call the constructor the sphere class, as soon as the latter is ready.

Ah, that is very nice. Actually, I was not aware that something like that exists. This should be populated somewhere more obvious. Probably in a tutorial?

So, the only thing that should be added now is the hyperbolic space, right?

[gh-mjungmath]

By the way, I noticed that the result coming from sage.manifolds.catalog.Sphere is not even correct:

sage: S.<th, ph> = manifolds.Sphere()
sage: S
2-dimensional Riemannian submanifold S embedded in the Euclidean
 space E^3
sage: S.atlas()
[Chart (S, (th, ph))]

This is not a correct atlas.

[egourgoulhon]

Replying to gh-mjungmath:

Ah, that is very nice. Actually, I was not aware that something like that exists. This should be populated somewhere more obvious. Probably in a tutorial?

Indeed, when it is more developed.

So, the only thing that should be added now is the hyperbolic space, right?

No, there remains to introduce a proper sphere class, on the model of EuclideanSpace, with methods like stereographic_coordinates.

[egourgoulhon]

Replying to gh-mjungmath:

By the way, I noticed that the result coming from sage.manifolds.catalog.Sphere is not even correct:

sage: S.<th, ph> = manifolds.Sphere()
sage: S
2-dimensional Riemannian submanifold S embedded in the Euclidean
 space E^3
sage: S.atlas()
[Chart (S, (th, ph))]

This is not a correct atlas.

Indeed. As I say in comment:1, the catalog is in a primitive stage.

[gh-mjungmath]

Now I see what you mean. I first thought that "primitive" was just referring to the amount of examples.

That's good: there's still work to do. :)