Let $E(x, y, z) = (x, y, z)$ be a vector field on $\mathbb{R}^3$.
$α(x, y, z) = x dy ∧ dz − y dx ∧ dz + z dx ∧ dy$ is a 2-form.
Find $\phi^{t*}_E\alpha$.
The flow of X is $\phi^t = (x_0e^t,y_0e^t,z_0e^t)$.
I need to compute $d\phi_t^{*}$ as $\phi_t^{*}\alpha = \alpha_{\phi_t}(d \phi_{t})$.
1) Is the pullbak of a k-form always a k-form?
2) If we consider $\phi_{t}$ as a 1-form, $d\phi_{t}$ is a 2-form, how to compute it?
Thank you for your help!
I will partially answer my question.
For $2)$ :
Let $v,w$ be two vectors of $\mathbb{R}^3$. Let u be a point in $\mathbb{R}^3$.
The usual expression of the pullback gives:
$\phi_t^* \alpha(u)(v,w) = \alpha(\phi_t(u))(d \phi_t(u).v, d \phi_t(u).w)$
Which gives:
$\phi_t^* \alpha(u)(v,w) = \alpha(e^tu)(ve^t, we^t) = e^{3t}\alpha(u)(v,w)$.