If $X$ is a manifold, let $\Omega^k(X)$ 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 pullback induced by $f$ is the mapping $f^\ast:\Omega^k(Y)\to\Omega^k(X)$ defined as follows: If $\omega\in \Omega^k(Y)$ , then $f^\ast(\omega)$ is the $k$ -form on $X$ defined by the formula $$(f^*\omega)_x(X_1,\ldots ,X_k)=\omega_{f(x)}\big((Df)_x(X_1),\ldots ,(Df)_x(X_k))$$ where $x\in X$ , $X_1,\ldots, X_k\in T_x(X)$ , and $Df$ is the tangent map $Df:TX\to TY$ .

Properties

• If $\mbox{id}_X$ is the identity map on $X$ , then $(\mbox{id}_X)^\ast$ is the identity map on $\Omega^k(X)$ .
• 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 diffeomorphism with inverse $$(f^{-1})^\ast = (f^\ast)^\ast.$$
• If $f$ is a mapping $f:X\to Y$ , and $\omega\in \Omega^k(Y)$ , then $$df^\ast \omega = f^\ast d\omega,$$ where $d$ is the exterior derivative.
• Suppose $f$ is a mapping $f:X\to Y$ , $\omega\in \Omega^k(Y)$ , and $\eta\in \Omega^l(Y)$ . 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 $g:Y\to \mathbb{R}$ , 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 $U$ .