manifold Manifold 2002-02-15 10:52:30 2007-10-14 10:45:59 Definition coordinate chart chart local coordinates atlas change of coordinates differential structure transition function smooth structure diffeomorphism diffeomorphic topological manifold real-analytic manifold \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newtheorem{proposition}{Proposition} \newcommand{\reals}{\mathbb{R}} \newcommand{\natnums}{\mathbb{N}} \newcommand{\cnums}{\mathbb{C}} \newcommand{\lp}{\left(} \newcommand{\rp}{\right)} \newcommand{\lb}{\left[} \newcommand{\rb}{\right]} \newcommand{\cA}{\mathcal{A}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cT}{\mathcal{T}} \paragraph{Summary.} A {\em manifold} is a space that is locally like $\reals^n$, however lacking a preferred system of coordinates. Furthermore, a manifold can have global topological properties, such as non-contractible \PMlinkname{loops}{Curve}, that distinguish it from the topologically trivial $\reals^n$. \paragraph{Standard Definition.} An $n$-dimensional topological manifold $M$ is a second countable, Hausdorff topological space\footnote{For connected manifolds, the assumption that $M$ is second-countable is logically equivalent to $M$ being paracompact, or equivalently to $M$ being metrizable. The topological hypotheses in the definition of a manifold are needed to exclude certain counter-intuitive pathologies. Standard illustrations of these pathologies are given by the {\em long line} (lack of paracompactness) and the {\em forked line} (points cannot be separated). These pathologies are fully described in Spivak. See \PMlinkname{this page}{BibliographyForDifferentialGeometry}.} that is locally homeomorphic to open subsets of $\reals^n$. A differential manifold is a topological manifold with some additional structure information. A \emph{chart}, also known as a \emph{system of local coordinates}, is a mapping $\alpha: U \to \reals^n$, such that the domain $U\subset M$ is an open set, and such that $U$ is homeomorphic to the image $\alpha(U)$. Let $\alpha: U_\alpha \rightarrow \reals^n,$ and $\beta:U_\beta\rightarrow\reals^n$ be two charts with overlapping \PMlinkname{domains}{Function}. The continuous injection $$\beta\circ\alpha^{-1}: \alpha(U_\alpha\cap U_\beta)\rightarrow\reals^n$$ is called a \emph{transition function}, and also called a \emph{a change of coordinates}. An {\em atlas} $\cA$ is a collection of charts $\alpha:U_\alpha\rightarrow\reals^n$ whose domains cover $M$, i.e. $$M = \bigcup_{\alpha} U_\alpha.$$ Note that each transition function is really just $n$ real-valued functions of $n$ real variables, and so we can ask whether these are continuously differentiable. The atlas $\cA$ defines a differential structure on $M$, if every transition function is continuously differentiable. More generally, for $k=1,2,\ldots,\infty,\omega$, the atlas $\cA$ is said to define a $\cC^k$ differential structure, and $M$ is said to be of class $\cC^k$, if all the transition functions are $k$-times continuously differentiable, or real analytic in the case of $\cC^\omega$. Two differential structures of class $\cC^k$ on $M$ are said to be isomorphic if the union of the corresponding atlases is also a $\cC^k$ atlas, i.e. if all the new transition functions arising from the merger of the two atlases remain of class $\cC^k$. More generally, two $\cC^k$ manifolds $M$ and $N$ are said to be diffeomorphic, i.e. have equivalent differential structure, if there exists a homeomorphism $\phi:M\to N$ such that the atlas of $M$ is equivalent to the atlas obtained as $\phi$-pullbacks of charts on $N$. The atlas allows us to define differentiable mappings to and from a manifold. Let $$f:U\rightarrow\reals,\quad U\subset M$$ be a continuous function. For each $\alpha\in \cA$ we define $$f_\alpha: V \rightarrow \reals,\quad V\subset\reals^n,$$ called the representation of $f$ relative to chart $\alpha$, as the suitably restricted composition $$f_\alpha = f\circ \alpha^{-1}.$$ We judge $f$ to be differentiable if all the representations $f_\alpha$ are differentiable. A path $$\gamma: I\rightarrow M,\quad I\subset\reals$$ is judged to be differentiable, if for all differentiable functions $f$, the suitably restricted composition $f\circ\gamma$ is a differentiable function from $\reals$ to $\reals$. Finally, given manifolds $M, N$, we judge a continuous mapping $\phi:M\rightarrow N$ between them to be differentiable if for all differentiable functions $f$ on $N$, the suitably restricted composition $f\circ\phi$ is a differentiable function on $M$.