Differential forms, projections I have a problem with one exercise from differential geometry. I don't even know how to start. Anyone could help with this problem?
Let $M$, $N$ be manifolds, $M$ connected. Let $\pi:M\times N \to N $ be a projection of second coordinate. Prove that $k-$form $\omega$ on $M\times N$ is of form ${\pi}^{*}\eta$ for some $k-$form $\eta$ on $N$ if and only if ${i}_{X}\omega={L}_{X}\omega=0$ for every field $X$ on $M\times N$ satisfying condition $d\pi \circ X=0$.

Note that this is a local problem, and choose some coordinate $x_1,...x_n$ on $M$, $y_1,....,y_m$ on $N$. A vector field satisfy your condition iff it is on the form $\sum X_i {\partial \over \partial x_i}$. So $i_X\omega =0$ for every such vector field implies that for all $i$, $i({\partial \over \partial x_i}\omega)=0$, i.e. $\omega$ contains no term of the form $ a_*d x_{i_1}..d x_{i_l} d y_{j_1}..d y_{j_n}$ and that $\omega = \sum a_{j_1....j_k} d y_{j_1}...d y_{j_k}$. Then $L_X \omega =0$ means, for $X={\partial \over \partial x_i}$ that the coefficients $a_{j_1...j_k}$ do not depends on $x_i$. 
Summing up $\omega$ satisfies the hypothesis iff in this coordinate system it is of the form  $\omega = \sum a_{j_1....j_k} (y) d y_{j_1}...d y_{j_k}$.
