Let \(F\colon M\to N\) be smooth. Then the pullback \(F^*\) is defined by
\begin{equation*} (F^*\omega)_p(v)=\omega_{F(p)}(dF_pv). \end{equation*}
Remark
- If \(\omega\) is smooth so also \(F^*\omega\).
- Note, that compared to the pushforward no further assumptions on \(F\) is needed.
- indeed, the pullback is also well defined for non smooth covector fields