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Connection Forms not Anti-Symmetric

asked 2023-12-31 17:46:37 +0200

M = Manifold(2, 'M', r'\mathcal{M}')

c_xy.<x,y> = M.chart('x:(-1,1) y:(-1,1)', coord_restrictions=lambda x,y: x^2+y^2<1)

g = M.riemannian_metric('g')

g[0,0], g[1,1] = 4/(1 - x^2 - y^2)^2, 4/(1 - x^2 - y^2)^2

e1 = M.vector_field((1 - x^2 - y^2) / 2, 0)

e2 = M.vector_field(0, (1 - x^2 - y^2) / 2)

e = M.vector_frame('e', (e1, e2), non_coordinate_basis=True)

nabla = g.connection()

omega = nabla.connection_form

omega(0,0).display(e), omega(0,1).display(e)

(nabla_g connection 1-form (0,0) = x e^0 + y e^1, nabla_g connection 1-form (0,1) = y e^0 - x e^1)

omega(1,0).display(e), omega(0,1).display(e)

(nabla_g connection 1-form (1,0) = -y e^0 + x e^1, nabla_g connection 1-form (0,1) = y e^0 - x e^1)

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What anti-symmetry do you expect and why?

rburing gravatar imagerburing ( 2024-01-01 09:27:16 +0200 )edit
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\omega_{ij} = -\omega{ji} - the connection forms for the Levi-Civita connection are well known to be anti-symmetric so the diagonal elements should be 0. Anyhow see my answer.

idontgetoutmuch gravatar imageidontgetoutmuch ( 2024-01-03 10:30:47 +0200 )edit

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answered 2024-01-03 10:28:39 +0200

It seems one should add the frame as an argument to the connection form as well as to the request for display.

omega(0,0,e).display(e), omega(0,1,e).display(e)

(nabla_g connection 1-form (0,0) = 0, nabla_g connection 1-form (0,1) = ye^0 - x e^1)

omega(1,0,e).display(e), omega(1,1,e).display(e)

(nabla_g connection 1-form (1,0) = -ye^0 + x e^1, nabla_g connection 1-form (1,1) = 0)

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Indeed, if the frame argument is not provided, the connection 1-forms are computed for the manifold's default frame, see the documentation: https://doc.sagemath.org/html/en/refe...

The antisymmetry you referring to holds only for an orthonormal frame, which is the case of e, but not of the manifold's default frame, the latter being the coordinate frame (d/dx, d/dy).

eric_g gravatar imageeric_g ( 2024-01-03 15:15:39 +0200 )edit

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Asked: 2023-12-31 17:46:37 +0200

Seen: 313 times

Last updated: Jan 03