Pullback

Let $f:M\to N$ be a smooth map. For each $p\in M$, the differential $\mathrm{d}f_p:T_pM\to T_{f(p)}N$ induces a dual linear map on cotangent spaces,

This is the pullback of covectors, and it assembles fiberwise into a bundle map between cotangent bundles .

More generally, for a differential $k$-form $\omega\in\Omega^k(N)$, the pullback form $f^*\omega\in\Omega^k(M)$ is defined by

Pullback is functorial and unital: $(g\circ f)^*=f^*\circ g^*$ and $\mathrm{id}^*=\mathrm{id}$. It also commutes with the exterior derivative : for all forms $\omega$ on $N$, one has $f^*(\mathrm{d}\omega)=\mathrm{d}(f^*\omega)$.

Examples

1. Pulling back a 1-form by a map. Let $f:\mathbb{R}^2\to\mathbb{R}^2$ be $f(u,v)=(u^2,v)$. Then $f^*(\mathrm{d}x)=2u\,\mathrm{d}u$ and $f^*(\mathrm{d}y)=\mathrm{d}v$.
2. Restriction to a submanifold. If $i:S^1\hookrightarrow\mathbb{R}^2$ is the inclusion and $\alpha=x\,\mathrm{d}y-y\,\mathrm{d}x$ on $\mathbb{R}^2$, then $i^*\alpha$ is the standard angular 1-form on $S^1$ (in angle coordinate $\theta$, it equals $\mathrm{d}\theta$).
3. Pullback along a projection. For $\pi:M\times N\to M$ and any form $\omega$ on $M$, the form $\pi^*\omega$ on $M\times N$ is the “constant along $N$” lift of $\omega$.