pullback of a $k$-form

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 $Df:TX\to TY$.

0.0.1 Properties

Suppose $X$ and $Y$ are manifolds.

• 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 $U$.

Title pullback of a $k$-form PullbackOfAKform 2013-03-22 14:00:34 2013-03-22 14:00:34 bwebste (988) bwebste (988) 7 bwebste (988) Definition msc 53-00 Pullback2 TangentMap