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 pullback induced by $f$ is the mapping ${f}^{\ast}:{\mathrm{\Omega}}^{k}(Y)\to {\mathrm{\Omega}}^{k}(X)$ defined as follows: 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 the tangent map $Df:TX\to TY$.
0.0.1 Properties
Suppose $X$ and $Y$ are manifolds.

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

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If $X,Y,Z$ are manifolds, and $f,g$ are mappings $f:X\to Y$ and $g:Y\to Z$, then
$${(g\circ f)}^{\ast}={f}^{\ast}\circ {g}^{\ast}.$$ 
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If $f$ is a diffeomorphism $f:X\to Y$, then ${f}^{\ast}$ is a diffeomorphism with inverse
$${({f}^{1})}^{\ast}={({f}^{\ast})}^{\ast}.$$ 
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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^{}.

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

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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 

Canonical name  PullbackOfAKform 
Date of creation  20130322 14:00:34 
Last modified on  20130322 14:00:34 
Owner  bwebste (988) 
Last modified by  bwebste (988) 
Numerical id  7 
Author  bwebste (988) 
Entry type  Definition 
Classification  msc 5300 
Related topic  Pullback2 
Related topic  TangentMap 