Let $M$ be a compact, connected, and orientable smooth manifold of dimension 6. Let $\alpha$ and $\beta$ be 2-forms on $M$. Show that there is a point of $M$ where $d\alpha \wedge d\beta = 0$. Hint: integrate
I am stuck on how to approach this function. It seems like Stokes' Theorem would be the best approach, but we know nothing about the boundary of the manifold. In addition, if $d\alpha \wedge d\beta$ turns out to be an orientation form, we would have that there is no such point in $M$, so how do we know that this doesn't happen? Any help is appreciated. Thanks!