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tangent map
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(Definition)
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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 $(Df)(v)\in T_{f(x)}(Y)$ .
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 $$ (Df)^{-1}=D(f^{-1}). $$
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 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 $\sR^n$ and $\sR^m$ , then $Df$ is simply the Jacobian of $f$ .
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Cross-references: Jacobian, open subsets, objects, pullback, inverse, diffeomorphism, identity mapping, tangent vector, curve, map, smooth mapping, tangent bundles, smooth manifolds
There are 11 references to this entry.
This is version 4 of tangent map, born on 2004-01-08, modified 2004-09-13.
Object id is 5506, canonical name is TangentMap.
Accessed 7268 times total.
Classification:
| AMS MSC: | 53-00 (Differential geometry :: General reference works ) |
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Pending Errata and Addenda
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