PlanetMath: pullback of a $k$-form

(more info)

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pullback of a $k$

-form

(Definition)

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

Definition Suppose and are smooth manifolds, and suppose is a smooth mapping $f:X\to Y$ . Then the pullback induced by 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 -form on defined by the formula

$\displaystyle (f^*\omega)_x(X_1,\ldots ,X_k)=\omega_{f(x)}\big((Df)_x(X_1),\ldots ,(Df)_x(X_k))$

where $x\in X$ , $X_1,\ldots, X_k\in T_x(X)$ , and

is the tangent map $Df:TX\to TY$ .

Properties

Suppose

and

are manifolds.

If id is the identity map on , then id $_X)^\ast$ is the identity map on $\Omega^k(X)$ .
If are manifolds, and are mappings $f:X\to Y$ and $g:Y\to Z$ , then
$\displaystyle (g\circ f)^\ast = f^\ast\circ g^\ast.$
If is a diffeomorphism $f:X\to Y$ , then $f^\ast$ is a diffeomorphism with inverse
$\displaystyle (f^{-1})^\ast = (f^\ast)^\ast.$
If is a mapping $f:X\to Y$ , and $\omega\in \Omega^k(Y)$ , then
$\displaystyle df^\ast \omega = f^\ast d\omega,$
where is the exterior derivative.
Suppose is a mapping $f:X\to Y$ , $\omega\in \Omega^k(Y)$ , and $\eta\in \Omega^l(Y)$ . Then
$\displaystyle f^\ast(\omega\wedge \eta)= f^\ast(\omega)\wedge f^\ast(\eta).$
If is a 0-form on , that is, is a real valued function $g:Y\to \mathbb{R}$ , and is a mapping $f:X\to Y$ , then $f^\ast(g) = f\circ g$ .
Suppose is a submanifold (or an open set) in an manifold , and $\iota:U\hookrightarrow X$ is the inclusion mapping. Then $\iota^\ast$ restricts -forms on to -forms on .