# tangent map

###### Definition 1.

Suppose $X$ and $Y$ are smooth manifolds with tangent bundles $TX$ and $TY$, and suppose $f:X\to Y$ is a smooth mapping. Then the tangent map of $f$ 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 $v$ by some curve $c\colon I\to X$ with $c(0)=x$ and $I=(-1,1)$. Now $(Df)(v)$ is defined as the tangent vector in $T(Y)$ 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 $X$ and $Y$ are a smooth manifolds.

• If $\operatorname{id}_{X}$ is the identity mapping on $X$, then $D\mbox{id}_{X}$ is the identity mapping on $TX$.

• Suppose $X,Y,Z$ are smooth manifolds, and $f,g$ are mappings $f\colon X\to Y$, $g\colon Y\to Z$. Then

 $D(f\circ g)=(Df)\circ(Dg).$
• If $f\colon X\to Y$ is a diffeomorphism, then the inverse of $Df$ is a diffeomorphism, and

 $(Df)^{-1}=D(f^{-1}).$

## Notes

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

 $Df\colon TX\to TY,$

whereas the pullback (http://planetmath.org/PullbackOfAKForm) of $f$ is a mapping

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

For this reason, the tangent map is also sometimes called the pushforward map. That is, a pullback takes objects from $Y$ to $X$, and a pushforward takes objects from $X$ to $Y$.

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

Title tangent map TangentMap 2013-03-22 14:06:19 2013-03-22 14:06:19 matte (1858) matte (1858) 7 matte (1858) Definition msc 53-00 push forward map pushforward pushforward map PullbackOfAKForm FlowBoxTheorem