Pullback of differential forms

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

• Functoriality: for smooth maps $M\xrightarrow{F}N\xrightarrow{G}P$, one has $(G\circ F)^*=F^*\circ G^*$.
• Compatibility with wedge products: for forms $\alpha,\beta$ on $N$,
  $$F^*(\alpha\wedge \beta)=F^*\alpha\wedge F^*\beta,$$
  using the wedge product.
• Compatibility with the 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 $Z$.

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 $dx\wedge dy$, 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, every $2$-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 $\pi_1^*\omega$ ignores tangent directions in the $N$-factor and evaluates $\omega$ on the $M$-components.