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Computing Pullback of 1-Form

  1. Jul 23, 2013 #1

    WWGD

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    Hi, All:
    I'm kind of rusty in my computations. I'm trying to compute the pullback of the form dθ on S1 by a differentiable map f: M→S1, where f is differentiable and M is a 3-manifold; please tell me if this is correct:

    0)Say we use the basis vectors {∂/∂x1,∂/∂x2, ∂/∂x3, }

    for TxM ;

    1)We compute the pushforwards of the three basis vectors, and get:

    f*(∂/∂xi)=∂f/∂xi∂/∂θ , for i=1,2,3.


    2)We evaluate dθ at each of the pushforwards of the basis vectors, to get:

    dθ (∂f/∂xi∂/∂θ)= (∂f/∂xi); i=1,2,3.


    3)We conclude :

    f*dθ = ∂f/∂x1dx+ ∂f/∂x2dy+ ∂f/∂x3dz

    Is this correct?

    Thanks for your comments.
     
  2. jcsd
  3. Jul 24, 2013 #2

    micromass

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    Shouldn't this be

    [tex]f_*(\frac{\partial}{\partial x^i}) = \frac{\partial (\theta\circ f)}{\partial x^i} \frac{\partial}{\partial \theta}[/tex]

    The rest look right. But there is a general result. That says that if ##G:M\rightarrow N## is smooth and if ##f:N\rightarrow \mathbb{R}## is smooth, then ##G^*(df) = d(f\circ G)##. This could make your calculations easier.
     
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