Connection Forms not Anti-Symmetric

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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)