Using differential forms---within SageManifolds

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Given an affine connection, there is not a unique connection form but a set of them specified by two indices and a vector frame. Hence the method connection_form requires the indices as argument and, as an option, the vector frame with respect to which the connection form is defined (by default, the manifold's default frame). Here is a full example, yielding to the connection 1-form Omega^0_1 with respect to the vector frame (d/dx, d/dy) in the hyperbolic plane (Poincaré half-plane model):

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart('x y:(0,+oo)')
sage: g = M.metric('g')
sage: g[0,0], g[1,1] = 1/y, 1/y
sage: g.display()
g = 1/y dx*dx + 1/y dy*dy
sage: nab = g.connection()
sage: omega = nab.connection_form(0,1)
sage: omega.display()
nabla_g connection 1-form (0,1) = -1/2/y dx

As you can see from the display, the object returned by nab.connection_form(0,1) is not some component, but a genuine 1-form, which is colinear to dx. You can check this further:

sage: print(omega)
1-form nabla_g connection 1-form (0,1) on the 2-dimensional differentiable manifold M
sage: omega.category()
Category of elements of Free module /\^1(M) of 1-forms on the 2-dimensional differentiable manifold M

Note that the result is stored internally in some data attribute of the connection nab. So you can access it again by nab.connection_form(0,1) without triggering any new computation:

sage: nab.connection_form(0,1) is omega
True

Similar considerations hold for the torsion and curvature 2-forms.