Well apparently I should *not* be using `connection_form` but instead
`gam_e = nabla.coef(e)`
`gam_e[:]`
> [[[0, 0], [y, -x]], [[-y, x], [0, 0]]]
Which is what I would have expected. I'm still confused by what
`connection_form` does if it doesn't calculate the connection forms.
On Monday 1 January 2024 at 17:43:12 UTC Dominic Steinitz wrote:
> That should be
>
> `omega(1,1).display(e)`
>
> > nabla_g connection 1-form (1,1) = x e^0 + y e^1)
>
> On Sunday 31 December 2023 at 16:43:16 UTC Dominic Steinitz wrote:
>
>> I am calculating the connection forms for the case of a Poincaré Disk but
>> the diagonal elements appear to be non-zero
>>
>> `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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