# Connection Forms not Anti-Symmetric

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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1

What anti-symmetry do you expect and why?

( 2024-01-01 09:27:16 +0200 )edit
1

\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.

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

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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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1

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).

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