Q:

Open immersion pulls back symplectic form to symplectic form?

If $M$ is symplectic, and $f: W \to M$ is an open immersion, i.e. an immersion where $W$ and $M$ have the same dimension, does $f$ necessarily pull back a symplectic form on $M$ to a symplectic form on $W$?

A:

A symplectic form is a 2-form $\omega$ on a $2n$-manifold such that $\omega^n$ is pointwise nonzero and $d\omega = 0$. Clearly $df^*(\omega) = f^*(d\omega) = 0$ and $f^*(\omega^n) = (f^*\omega)^n$; then the only point to make is that $f^*(\omega^n)$ is never zero, but this follows because you assumed $f$ is an immersion.

