Pullback of differential forms

Let $F:M\to N$ be a smooth map between smooth manifolds.

A differential $k$-form on $N$ assigns to each point $q\in N$ an alternating $k$-linear map on $T_qN$. The differential $dF_p:T_pM\to T_{F(p)}N$ transports tangent vectors, and the pullback transports covariant objects in the opposite direction.

Definition (pullback of a $k$-form). If $\omega$ is a $k$-form on $N$, the pullback $F^*\omega$ is the $k$-form on $M$ defined by

for $p\in M$ and $v_1,\dots,v_k\in T_pM$.

For $k=1$ this agrees with the pullback of covectors.

Key properties.

• Functoriality: for smooth maps $M\xrightarrow{F}N\xrightarrow{G}P$ one has $(G\circ F)^* = F^*\circ G^*$.
• Compatibility with wedge: for forms $\alpha,\beta$ on $N$,
  $$F^*(\alpha\wedge \beta)=F^*\alpha\wedge F^*\beta,$$
  using the wedge product.
• Compatibility with exterior derivative: for any form $\omega$ on $N$,
  $$F^*(d\omega)=d(F^*\omega),$$
  where $d$ is the exterior derivative. In particular, pullback preserves being closed and being exact, and therefore induces maps on de Rham cohomology.

If $F$ is a diffeomorphism, then $F^*$ is an isomorphism on forms with inverse $(F^{-1})^*$. If $i:Z\hookrightarrow N$ is a smooth embedding, then $i^*\omega$ is the restriction of $\omega$ to tangent vectors along $Z$.

Examples

1. Polar coordinates and the area form. Let $\Phi:(0,\infty)\times (0,2\pi)\to \mathbb{R}^2\setminus\{0\}$ be $\Phi(r,\theta)=(r\cos\theta,r\sin\theta)$. For the standard $2$-form $dx\wedge dy$ on $\mathbb{R}^2$, one computes
   $$\Phi^*(dx\wedge dy)= r\,dr\wedge d\theta.$$
   This is the differential-form version of the Jacobian factor in change of variables.

1. Pullback along an inclusion can kill high-degree forms. Let $i:S^1\hookrightarrow\mathbb{R}^2$ be the inclusion. Since $S^1$ is $1$-dimensional, any $2$-form on $\mathbb{R}^2$ pulls back to $0$ on $S^1$. In particular,
   $$i^*(dx\wedge dy)=0.$$

1. Projection from a product. Let $\pi_1:M\times N\to M$ be the projection. For any $k$-form $\omega$ on $M$, the pullback $\pi_1^*\omega$ is the $k$-form on $M\times N$ that ignores tangent directions in the $N$-factor and evaluates $\omega$ on the $M$-components.