pullback of a $k$-form

If $X$ is a manifold, let ${\mathrm{\Omega }}^k(X)$ be the vector space of $k$-forms on $X$.

Definition Suppose $X$ and $Y$ are smooth manifolds, and suppose$f$ is a smooth mapping $f:X\to Y$. Then the pullbackinduced by $f$ is the mapping $f^{\ast }:{\mathrm{\Omega }}^k(Y)\to {\mathrm{\Omega }}^k(X)$ defined asfollows: If $\omega \in {\mathrm{\Omega }}^k(Y)$, then $f^{\ast }(\omega )$ is the $k$-form on $X$defined by the formula

$${(f^{*}\omega )}_x(X_1,\mathrm{\dots },X_k)={\omega }_{f(x)}({(Df)}_x(X_1),\mathrm{\dots },{(Df)}_x(X_k))$$

where $x\in X$, $X_1,\mathrm{\dots },X_k\in T_x(X)$, and $Df$ is thetangent map $Df:TX\to TY$.

0.0.1 Properties

Suppose $X$ and $Y$ are manifolds.

• If ${\text{id}}_X$ is the identity map on $X$, then ${({\text{id}}_X)}^{\ast }$is the identity map on ${\mathrm{\Omega }}^k(X)$.

• If $X,Y,Z$ are manifolds, and $f,g$ are mappings$f:X\to Y$ and $g:Y\to Z$, then

  $${(g\circ f)}^{\ast }=f^{\ast }\circ g^{\ast }.$$

• If $f$ is a diffeomorphism $f:X\to Y$, then $f^{\ast }$ is a diffeomorphismwith inverse

  $${(f^{-1})}^{\ast }={(f^{\ast })}^{\ast }.$$

• If $f$ is a mapping $f:X\to Y$, and $\omega \in {\mathrm{\Omega }}^k(Y)$, then

  $$d{f}^{\ast }\omega =f^{\ast }d\omega ,$$

  where $d$ is the exterior derivative.

• Suppose $f$ is a mapping $f:X\to Y$, $\omega \in {\mathrm{\Omega }}^k(Y)$, and $\eta \in {\mathrm{\Omega }}^l(Y)$. Then

  $$f^{\ast }(\omega \wedge \eta )=f^{\ast }(\omega )\wedge f^{\ast }(\eta ).$$

• If $g$ is a $0$-form on $Y$, that is, $g$ is a real valued function $g:Y\to ℝ$, and $f$ is a mapping $f:X\to Y$,then $f^{\ast }(g)=f\circ g$.

• Suppose $U$ is a submanifold (or an open set) in an manifold $X$, and$\iota :U\hookrightarrow X$ is the inclusion mapping. Then ${\iota }^{\ast }$restricts $k$-forms on $X$ to $k$-forms on $U$.