Pseudo-Riemannian manifods

[egourgoulhon]

This ticket implements pseudo-Riemannian manifolds, i.e. real differentiable manifolds equipped with a metric tensor. Important subcases are of course Riemannian manifolds and Lorentzian manifolds. Taking into account that pseudo-Riemannian metric tensors are already implemented in Sage (see ​here), this ticket introduces

• the parent class PseudoRiemannianManifold, as a subclass of the existing class DifferentiableManifold, with the specific methods metric and volume_form
• new methods gradient, laplacian and dalembertian for scalar fields
• new methods divergence, laplacian and dalembertian for tensor fields
• new methods curl, dot, cross and norm for vector fields

For a greater generality, all these methods have an optional argument metric; if it is omitted, the metric of the underlying pseudo-Riemannian manifold is assumed. The follow-up ticket #24623 implements Euclidean spaces.

This work is part of the ​SageManifolds project, see #18528 for an overview.

[egourgoulhon]

This is work in progress...

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New commits:

​8e1148c	First version of pseudo-Riemannian manifold class
​8e56cf3	Pseudo-Riemannian manifolds constructed by the generic function Manifold
​ce3a617	Divergence of a tensor field + new section on pseudo-Riemannian manifolds in the reference manual
​06f8fd3	Add doctests for pseudo-Riemannian manifolds

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​55f356c

Improve documentation of pseudo-Riemannian manifolds

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​0fc661a

Remove method set_metric and improve doc of pseudo-Riemannian manifolds

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​b4dddf7

Add operators Laplacian, d'Alembertian and curl on pseudo-Riemannian manifolds

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​040b01c

Add dot product and cross product of vector fields

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​46b8f66

Add norm of vector fields.