# 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)

What anti-symmetry do you expect and why?

\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.