This ticket implements affine connections on infinitely differentiable manifolds (i.e. smooth manifolds) . This is a follow-up of #19092 within the SageManifolds project (see the metaticket #18528 for an overview). As in #19092, the non-discrete topological field K over which the smooth manifold is defined is generic, although in most applications, K=R or K=C.

Affine connections are implemented via the Python class AffineConnection, the user interface being the method DifferentiableManifold.affine_connection(). At the user choice, CPU-demanding computations (like that of the curvature tensor) can be parallelized, thanks to #18100.

Various methods of the class AffineConnection allow the computation of

• the connection coefficients with respect to a given vector frame (from those w.r.t. another frame)
• the connection 1-forms with respect to a given vector frame
• the torsion tensor
• the torsion 2-forms with respect to a given vector frame
• the (Riemann) curvature tensor
• the curvature 2-forms with respect to a given vector frame
• the Ricci tensor
• the action of the affine connection on any tensor field

Documentation:
The reference manual is produced by
sage -docbuild reference/manifolds html
It can also be accessed online at http://sagemanifolds.obspm.fr/doc/19147/reference/manifolds/
More documentation (e.g. example worksheets) can be found here.

Depends on #18100
Depends on #19092

CC: @sagetrac-mbejger @man74cio

Component: geometry

Keywords: differentiable manifold, affine connection, curvature, torsion

Author: Eric Gourgoulhon, Michal Bejger, Marco Mancini

Branch/Commit: 906c030

Reviewer: Volker Braun

Issue created by migration from https://trac.sagemath.org/ticket/19147