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$x\mathrm{\in }X$, thenwe 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 inthe definition, then the tangent map isa 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 $X$, anda 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 simplythe Jacobian of $f$.