Given a differential form
$$x\,dy\wedge dz-y\,dx\wedge dz+z\,dx\wedge dy$$
I am supposed to prove that the it's pullback by a linear map of determinant one leaves it invariant. For example, if $\phi$ is a linear map and $\omega$ is a differential form then $\phi^*\omega=\omega$. Also, I was wondering, can we also say that any differential 2-form that is invariant under pull back of $\phi$ is scalar multiple of $\omega$?
I know the definition that for a linear map, we have that $\phi^*\omega(V)=\omega(\phi(V))$
but I don't know how to proceed.
Best Answer
Here's a hint: For any vectors $\mathbf v,\mathbf w\in\Bbb R^3$, $$\omega(\mathbf x)(\mathbf v,\mathbf w) = \det(\mathbf x,\mathbf v,\mathbf w).$$ Here I'm writing $\mathbf x=(x,y,z)$.
ADDENDUM: Here's a suggestion to address your uniqueness question. Consider two types of matrices with determinant $1$: $$\begin{bmatrix} c^2 & 0 & 0 \\ 0 & 1/c & 0 \\ 0 & 0 & 1/c \end{bmatrix} \qquad\text{and}\qquad \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}$$ and start with a $2$-form like $f dx\wedge dy$.