Pushforward (for tangent vectors) Let $v \in T_p M$ be a tangent vector at point $p \in M$. Then:
$$ d \varphi_p: T_p M \rightarrow T_{\varphi(p)} N $$
is the pushforward map (also called the differential of $\varphi$ ).
$$ \left(d \varphi_p v\right)(f)=v(f \circ \varphi) $$
for all $f \in C^{\infty}(N)$.
Pullback (for cotangent vectors, i.e. 1-forms) Let $\omega \in T_{\varphi(p)}^* N$ be a covector (e.g. $d f$ ). Then:
$$ \varphi^* \omega \in T_p^* M $$
is the pullback of $\omega$.
$$ \left(\varphi^* \omega\right)(v):=\omega\left(d \varphi_p v\right) $$
first push forward the vector $v$, then apply $\omega$ to it. pullback reverses the direction of maps:
- $\varphi: M \rightarrow N$
- $\varphi^*: \Omega^k(N) \rightarrow \Omega^k(M)$