Calculating the Pullback of a 1-Form on S1 by a Differentiable Map

[WWGD]

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.

 

[micromass]

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.

Shouldn't this be

$$f_*(\frac{\partial}{\partial x^i}) = \frac{\partial (\theta\circ f)}{\partial x^i} \frac{\partial}{\partial \theta}$$

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.