If one uses the method stereographic_coordinates to initialize steographic coordinates (instead of as via the constructor argument), then there are some inconsistencies.

For example,

sage: M = manifolds.Sphere(2)
sage: XN = M.stereographic_coordinates(pole='north')
sage: g = M.metric()
sage: U = XN.domain(); U
Open subset S^2-{NP} of the 2-sphere S^2 of radius 1 smoothly embedded in the 
 Euclidean space E^3
sage: g.restrict(U).display()
g = (cos(theta)^2 - 2*cos(theta) + 1) dy1⊗dy1
  + (cos(theta)^2 - 2*cos(theta) + 1) dy2⊗dy2

One would certainly expect the components of g to be expressed in terms of (y1,y2), i.e. the chart XN, which is the only chart that covers entirely U = S^2-{NP}.

CC: @tscrim @nthiery @mjungmath @egourgoulhon

Component: manifolds

Keywords: sphere

Author: Eric Gourgoulhon

Branch/Commit: 20b9cb0

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/32953