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differential form pullback

asked 2018-01-15 11:22:09 -0600

danielvolinski gravatar image

Hi All,

I have a differential form defined in 3D Cartesian Coordinates

$\mathit{{\sigma}}=\frac{x\mathit{dy}\mathit{dz}}{a}-\frac{y\mathit{dx}\mathit{dz}}{a}$

and the following map

$g:[{a}{\ast}\cos(\theta), {a}{\ast}\sin(\theta), z]$

How do I calculate the pullback $g^{\ast}(\theta,z)$ of $\sigma$ in SageMath

I've seen all the documentation of pullback in "manifold.pdf" but I am unable to apply those examples to my case.

Thanks.

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answered 2018-01-15 14:19:08 -0600

eric_g gravatar image

Does the following correspond to what you want?

First we define the 2-form $\sigma$ on the 3-manifold $M$ covered by coordinates $(x,y,z)$:

sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: a = var('a', domain='real')
sage: sigma = M.diff_form(2)
sage: sigma[1,2] = x/a
sage: sigma[0,2] = -y/a
sage: sigma.display()
-y/a dx/\dz + x/a dy/\dz

Then we define the 2-manifold $N$ covered by coordinates $(\theta,z)$ and the map $g: N\to M$:

sage: N = Manifold(2, 'N')
sage: XN.<th,z> = N.chart(r'th:\theta:(0,2*pi) z')
sage: g = N.diff_map(M, [a*cos(th), a*sin(th), z])
sage: g.display()
N --> M
   (th, z) |--> (x, y, z) = (a*cos(th), a*sin(th), z)

The pullback of $\sigma$ by $g$ is then

sage: s = g.pullback(sigma)
sage: s
2-form on the 2-dimensional differentiable manifold N
sage: s.display()
a dth/\dz
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Hi Eric, Yes, this is exactly what I want. Thanks.

danielvolinski gravatar imagedanielvolinski ( 2018-01-15 18:05:41 -0600 )edit

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Asked: 2018-01-15 11:22:09 -0600

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Last updated: Jan 15