SageManifolds metaticket

[egourgoulhon]

This is the implementation of manifolds resulting from the ​SageManifolds project.

Algebraic part

• tensors on free modules of finite rank: #15916 (merged in Sage 6.6)
• parallelization of tensor computations on free modules of finite rank: #18100 (merged in Sage 6.10)

Topological and differential part

• Topological manifolds (over R, C or a topological field K):
  • basics (charts, subsets): #18529 (merged in Sage 7.1.beta1)
  • scalar fields (continuous functions to the base field): #18640 (merged in Sage 7.3.beta0)
  • morphisms (continuous maps between manifolds): #18725 (merged in Sage 7.3.beta0)
• Differentiable manifolds (over R, C or a non-discrete topological field K):
  • basics (charts, transition maps, scalar fields, morphisms): #18783
  • vector fields, tensor fields and p-forms: #18843
  • tangent spaces: #19092
  • curves: #19124
  • affine connections: #19147
• Complex and almost complex manifolds:
  • almost complex structures through Hodge structures: #18786
• Pseudo-riemannian manifolds:
  • pseudo-riemannian metrics: #19209
  • Schouten, Cotton, and Cotton-York tensors: #19823
  • Euclidean spaces: #19978

[egourgoulhon]

All the tickets, except for #18786, are now ready for review.

[egourgoulhon]

All the tickets are now based on the category ticket #18175, so that the manifold categories are

• Manifolds(K) for topological manifolds over a topological field K
• Manifolds(K).Differentiable() for differentiable manifolds
• Manifolds(K).Smooth() for smooth manifolds

[tscrim]

Something I would like to see once the basics are done is a catalog of examples and common interesting manifolds:

• n-sphere
• n-torus
• real/complex projective n-space
• surfaces
• (Affine) Grassmannians
• Classical Lie groups (more for my info, a description of charts is on page 5 of ​https://www.dpmms.cam.ac.uk/~agk22/mfds.pdf, but this probably isn't a good atlas for doing computations)

I understand that some of these could be considered more wishlist than others. Some other wishlist items:

• ​Morse theory to compute homology of manifolds.
• Manifolds with boundary
• Cartesian products of manifolds (or more generally, fiber bundles)
• DeRham? cohomology (see, e.g., lecture notes above)

[egourgoulhon]

Replying to tscrim:

Something I would like to see once the basics are done is a catalog of examples and common interesting manifolds:

Thanks for these suggestions. For sure, one should have a catalog of standard manifolds. For the time being, there are only examples available as worksheets at ​http://sagemanifolds.obspm.fr/examples.html, for instance

• the 2-sphere at ​http://sagemanifolds.obspm.fr/examples/html/SM_sphere_S2.html
• the real projective plane at ​http://sagemanifolds.obspm.fr/examples/html/SM_projective_plane_RP2.html
• the hyperbolic plane at ​http://nbviewer.ipython.org/github/sagemanifolds/SageManifolds/blob/master/Worksheets/v0.8/SM_hyperbolic_plane.ipynb

I understand that some of these could be considered more wishlist than others. Some other wishlist items:

• ​Morse theory to compute homology of manifolds.
• Manifolds with boundary
• Cartesian products of manifolds (or more generally, fiber bundles)
• DeRham? cohomology (see, e.g., lecture notes above)

All the above seem indeed desirable extensions. Even if they are not implemented yet, we should have them in mind when setting the basics.

[egourgoulhon]

PS: could you point to some existing catalog in Sage, in order to have some example?

[tscrim]

• algebras.<tab> in sage/algebras/catalog.py
• crystals.<tab> in sage/combinat/crystals.catalog.py
• designs.<tab> in sage/combinat/designs.designs_catalog.py
• groups.<tab> in sage/groups/groups_catalog.py

[egourgoulhon]

Thanks!