Follow-up: provide 3-sphere with additional features

[mjungmath]

This is a follow-up of #30804. This ticket provides:

• a Lie group structure of S^3 (?)
• a global frame of S^3
• Hopf coordinates

CC: @egourgoulhon @tscrim @tobiasdiez

Component: manifolds

Issue created by migration from https://trac.sagemath.org/ticket/31617

[mjungmath]

Description changed:

--- 
+++ 
@@ -1,5 +1,5 @@
 This is a follow-up of #30804. This ticket provides:
 
+- a Lie group structure of S^3 (?)
 - a global frame of S^3
 - Hopf coordinates
-- Lie group structure (?)

[mkoeppe]

comment:2

Sage development has entered the release candidate phase for 9.3. Setting a new milestone for this ticket based on a cursory review.

[egourgoulhon]

comment:3

Just a reminder of some notebooks which might serve as starting points for this project:

• Hopf coordinates: https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Notebooks/SM_sphere_S3_Hopf.ipynb
• Lie group structure: https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Notebooks/SM_sphere_S3_vectors.ipynb
• Riemannian structure: https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Notebooks/SM_sphere_S3_metric.ipynb

[egourgoulhon]

comment:4

Also, regarding Lie groups in Sage, at the moment, we have only nilpotent Lie groups (cf. https://doc.sagemath.org/html/en/reference/groups/sage/groups/lie_gps/nilpotent_lie_group.html), which have been implemented through #26344.