tangent map TangentMap 2004-01-08 12:39:30 2004-09-13 14:28:42 Definition tangent space % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions % % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\sR}[0]{\mathbb{R}} \newcommand{\sC}[0]{\mathbb{C}} \newcommand{\sN}[0]{\mathbb{N}} \newcommand{\sZ}[0]{\mathbb{Z}} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand*{\norm}[1]{\lVert #1 \rVert} \newcommand*{\abs}[1]{| #1 |} \newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} \PMlinkescapeword{represent} \begin{defn} 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 {\bf 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 $(Df)(v)\in T_{f(x)}(Y)$. \end{defn} \subsubsection*{Properties} Suppose $X$ and $Y$ are a smooth manifolds. \begin{itemize} \item If $\operatorname{id}_X$ is the identity mapping on $X$, then $D\mbox{id}_X$ is the identity mapping on $TX$. \item Suppose $X,Y,Z$ are smooth manifolds, and $f,g$ are mappings $f\colon X\to Y$, $g\colon Y\to Z$. Then $$ D(f\circ g) = (Df)\circ (Dg). $$ \item If $f\colon X\to Y$ is a diffeomorphism, then the inverse of $Df$ is a diffeomorphism, and $$ (Df)^{-1}=D(f^{-1}). $$ \end{itemize} \subsubsection*{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 \PMlinkname{pullback}{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 $\sR^n$ and $\sR^m$, then $Df$ is simply the Jacobian of $f$.