tangent map

Definition 1.

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

Properties

Suppose $X$ and $Y$ are a smooth manifolds.

• If ${\mathrm{id}}_X$ is the identity mapping on $X$, then $D{\text{id}}_X$ is the identity mapping on $TX$.

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

  $$D(f\circ g)=(Df)\circ (Dg).$$

• If $f:X\to Y$ is a diffeomorphism, then the inverse of $Df$ is a diffeomorphism, and

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

Notes

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

$$Df:TX\to TY,$$

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

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

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 ${ℝ}^n$ and ${ℝ}^m$, then $Df$ is simply the Jacobian of $f$.

Title	tangent map
Canonical name	TangentMap
Date of creation	2013-03-22 14:06:19
Last modified on	2013-03-22 14:06:19
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	7
Author	matte (1858)
Entry type	Definition
Classification	msc 53-00
Synonym	push forward map
Synonym	pushforward
Synonym	pushforward map
Related topic	PullbackOfAKForm
Related topic	FlowBoxTheorem