# Computing Pullback of 1-Form

1. Jul 23, 2013

### 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?

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