It seems I should have specified the frame in the connection form as well
as in the display (object? - I have no idea about OOP having been a
functional programmer for the last 30 years).
In [13]:
omega(0,0,e).display(e), omega(0,1,e).display(e)
Out [13]:
(nabla_g connection 1-form (0,0) = 0,
nabla_g connection 1-form (0,1) = y e^0 - x e^1)
In [14]:
omega(1,0,e).display(e), omega(1,1,e).display(e)
Out [14]:
(nabla_g connection 1-form (1,0) = -y e^0 + x e^1,
nabla_g connection 1-form (1,1) = 0)
On Tuesday 2 January 2024 at 17:28:43 UTC Dominic Steinitz wrote:
> 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)
>>>
>>>
--
You received this message because you are subscribed to the Google Groups
"sage-support" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to sage-support+unsubscr...@googlegroups.com.
To view this discussion on the web visit
https://groups.google.com/d/msgid/sage-support/790bc2d1-221c-4413-80c6-1e5fda28d63dn%40googlegroups.com.
