This ticket implements pseudo-riemannian metrics on infinitely differentiable manifolds (i.e. smooth manifolds) over R. This is a follow-up of #19147 within the SageManifolds project (see the metaticket #18528 for an overview).

This ticket implements the following Python classes:

• PseudoRiemannianMetric for pseudo-Riemannian metrics on a real smooth manifold
  • PseudoRiemannianMetricParal for pseudo-Riemannian metrics on a real smooth parallelizable manifold
• LeviCivitaConnection for the Levi-Civita connection associated with a pseudo-Riemannian metric.

Various methods of the above classes allow for the computation of

• the connection coefficients and Christoffel symbols of the Levi-Civita connection associated with a
  given metric
• the Riemann and Ricci tensor of a given metric
• the Ricci scalar of a given metric
• the Weyl tensor of a given metric
• the volume n-form associated with a given metric on a n-dimensional manifold
• the metric duals of tensor fields (musical isomorphisms)

The user interface is via the method DifferentiableManifold.metric(). At the user choice, CPU-demanding computations (like that of the Riemann tensor) can be parallelized, thanks to #18100.

Documentation:
The reference manual is produced by
sage -docbuild reference/manifolds html
It can also be accessed online at http://sagemanifolds.obspm.fr/doc/19209/reference/manifolds/
More documentation (e.g. example worksheets) can be found here.

Depends on #18100
Depends on #19147

CC: @sagetrac-mbejger @man74cio

Component: geometry

Keywords: differentiable manifold, pseudo-Riemannian metric, Riemannian metric, Lorentzian metric, Levi-Civita connection

Author: Eric Gourgoulhon, Michal Bejger, Marco Mancini

Branch: c622eb9

Reviewer: Volker Braun

Issue created by migration from https://trac.sagemath.org/ticket/19209