PlanetMath: tangent map

(more info)

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tangent map

(Definition)

Definition 1 Suppose and are smooth manifolds with tangent bundles and , and suppose $f:X\to Y$ is a smooth mapping. Then the tangent map of is the map $Df\colon TX\to TY$ defined as follows: If $v\in T_x(X)$ for some $x\in X$ , then we can represent by some curve $c\colon I \to X$ with and . Now is defined as the tangent vector in represented by the curve $f\circ c\colon I \to Y$ . Thus, since $(f\circ c)(0)=f(x)$ , it follows that $(Df)(v)\in T_{f(x)}(Y)$ .

Properties

Suppose

and

are a smooth manifolds.

If $\operatorname{id}_X$ is the identity mapping on , then id is the identity mapping on .
Suppose are smooth manifolds, and are mappings $f\colon X\to Y$ , $g\colon Y\to Z$ . Then
$\displaystyle D(f\circ g) = (Df)\circ (Dg).$
If $f\colon X\to Y$ is a diffeomorphism, then the inverse of is a diffeomorphism, and
$\displaystyle (Df)^{-1}=D(f^{-1}).$

Notes

Note that if $f\colon X\to Y$ is a mapping as in the definition, then the tangent map is a mapping

$\displaystyle Df\colon TX\to TY,$

whereas the pullback of

is a mapping

$\displaystyle f^\ast\colon \Omega^k(Y)\to \Omega^k(X).$

For this reason, the tangent map is also sometimes called the pushforward map. That is, a pullback takes objects from

, and a pushforward takes objects from

Sometimes, the tangent map of is also denoted by $f_\ast$ . However, the motivation for denoting the tangent map by is that if and are open subsets in $\mathbb{R}^n$ and $\mathbb{R}^m$ , then is simply the Jacobian of .