Changes between Initial Version and Version 10 of Ticket #32953


Ignore:
Timestamp:
Dec 6, 2021, 5:35:44 PM (14 months ago)
Author:
gh-tobiasdiez
Comment:

Oh, then I misunderstood your comment above.

To circumvent it, you should use M = manifolds.Sphere(2, coordinates='stereographic')

This works perfectly, thanks!

I've updated the ticket description accordingly.

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  • Ticket #32953 – Description

    initial v10  
    1 Calculating the volume form on the sphere uses a 'strange' vector frame on the lower hemisphere.
     1If one uses the method `stereographic_coordinates` to initialize steographic coordinates (instead of as via the constructor argument), then there are some inconsistencies.
    22
    3 In more detail, the following test script
     3For example,
    44{{{
    5 import sage.all
    6 from sage.manifolds.differentiable.examples.sphere import Sphere
    7 
    8 M = Sphere(2)
    9 M.stereographic_coordinates()
    10 
    11 print("Metric")
    12 for restr in M.metric()._restrictions.values():
    13     print(restr)
    14     for comp in restr._components:
    15         print(comp)
    16 
    17 print("Volume form")
    18 for restr in M.metric().volume_form()._restrictions.values():
    19     print(restr)
    20     for comp in restr._components:
    21         print(comp)
    22 
    23 print("Metric")
    24 for restr in M.metric()._restrictions.values():
    25     print(restr)
    26     for comp in restr._components:
    27         print(comp)
     5sage: M = manifolds.Sphere(2)
     6sage: XN = M.stereographic_coordinates(pole='north')
     7sage: g = M.metric()
     8sage: U = XN.domain(); U
     9Open subset S^2-{NP} of the 2-sphere S^2 of radius 1 smoothly embedded in the
     10 Euclidean space E^3
     11sage: g.restrict(U).display()
     12g = (cos(theta)^2 - 2*cos(theta) + 1) dy1⊗dy1
     13  + (cos(theta)^2 - 2*cos(theta) + 1) dy2⊗dy2
    2814}}}
    29 produces the output
    30 {{{
    31 Metric
    32 Riemannian metric g on the Open subset A of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
    33 Coordinate frame (A, (∂/∂theta,∂/∂phi))
    34 Riemannian metric g on the Open subset S^2-{NP} of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
    35 Coordinate frame (S^2-{NP}, (∂/∂y1,∂/∂y2))
    36 Riemannian metric g on the Open subset S^2-{SP} of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
    37 Coordinate frame (S^2-{SP}, (∂/∂yp1,∂/∂yp2))
    38 
    39 Volume form
    40 2-form eps_g on the Open subset S^2-{SP} of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
    41 Coordinate frame (S^2-{SP}, (∂/∂yp1,∂/∂yp2))
    42 2-form eps_g on the Open subset S^2-{NP} of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
    43 Vector frame (S^2-{NP}, (f_1,f_2))
    44 
    45 Metric
    46 Riemannian metric g on the Open subset A of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
    47 Coordinate frame (A, (∂/∂theta,∂/∂phi))
    48 Riemannian metric g on the Open subset S^2-{NP} of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
    49 Coordinate frame (S^2-{NP}, (∂/∂y1,∂/∂y2))
    50 Vector frame (S^2-{NP}, (f_1,f_2))
    51 Riemannian metric g on the Open subset S^2-{SP} of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
    52 Coordinate frame (S^2-{SP}, (∂/∂yp1,∂/∂yp2))
    53 }}}
    54 As one can see, the volume form has components with respect to a coordinate frame on S2-{SP} but with respect to a vector frame on S2-{NP}.
    55 Also the metric itself is updated by the calculation of the volume form to have components with respect to both both frames.
    56 
    57 I'm not sure if that's a bug indeed or the expected behavior but I found it a bit strange. In the context of #30362 it leads to issues as I want to calculate contractions of certain forms, some of them only have components wrt to the vector frame (as they are derived from the volume form) while others have components only wrt to the coordinate frame.
     15One 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}.