A circle can be drawn in the plane, but the plane is not part of what makes the circle a one-dimensional smooth object. Near each point of the circle, a small arc behaves like an interval, so calculus ought to make sense there. The problem is that no single interval coordinate describes the whole circle without either repeating a point or cutting the circle open.

Smooth manifolds solve this by replacing global coordinates with compatible local coordinates. The local pieces look like open subsets of Euclidean space, and the compatibility condition says that moving from one coordinate description to another is governed by ordinary smooth changes of variables. In this way, multivariable calculus becomes available on spaces that are globally curved, wrapped, or assembled from several coordinate patches.

[example: Why One Coordinate Does Not Cover $S^1$] Let $S^1=\{(x,y)\in\mathbb R^2:x^2+y^2=1\}$, and define $\gamma:(0,2\pi)\to S^1$ by $\gamma(t)=(\cos t,\sin t)$. Since $\cos$ and $\sin$ are smooth real functions, both coordinate functions of $\gamma$ are smooth. Also $\gamma(t)\in S^1$ because \begin{align*} \cos^2 t+\sin^2 t=1. \end{align*} The point $(1,0)$ is not in the image, since $\cos t=1$ and $\sin t=0$ force $t=0$ or $t=2\pi$ modulo $2\pi$, neither of which lies in $(0,2\pi)$. Conversely, every point of $S^1\setminus\{(1,0)\}$ has a unique polar angle $t\in(0,2\pi)$, so $\gamma$ parametrizes exactly the punctured circle. This parametrization cannot be made into one coordinate chart on all of $S^1$ by merely adding an endpoint. If $0$ and $2\pi$ were both allowed, then \begin{align*} \gamma(0)=(\cos 0,\sin 0)=(1,0) \end{align*} and \begin{align*} \gamma(2\pi)=(\cos 2\pi,\sin 2\pi)=(1,0), \end{align*} so the map would not be one-to-one. If one instead keeps an open interval as the coordinate domain, then one endpoint of the interval is missing, and that missing endpoint is precisely the point where the circle closes. The local-coordinate remedy is to use more than one chart. Let $N=(0,1)$ and $S=(0,-1)$, and define stereographic coordinates \begin{align*} \varphi_N(x,y)=\frac{x}{1-y} \end{align*} on $S^1\setminus\{N\}$ and \begin{align*} \varphi_S(x,y)=\frac{x}{1+y} \end{align*} on $S^1\setminus\{S\}$. For $t\in\mathbb R$, the inverse of $\varphi_N$ is \begin{align*} \varphi_N^{-1}(t)=\left(\frac{2t}{1+t^2},\frac{t^2-1}{1+t^2}\right), \end{align*} because \begin{align*} \left(\frac{2t}{1+t^2}\right)^2+\left(\frac{t^2-1}{1+t^2}\right)^2=\frac{4t^2+(t^2-1)^2}{(1+t^2)^2}=\frac{t^4+2t^2+1}{(1+t^2)^2}=1. \end{align*} On the overlap $S^1\setminus\{N,S\}$, the coordinate $t=\varphi_N(x,y)$ is nonzero, and the transition map is \begin{align*} (\varphi_S\circ\varphi_N^{-1})(t)=\frac{\frac{2t}{1+t^2}}{1+\frac{t^2-1}{1+t^2}}=\frac{\frac{2t}{1+t^2}}{\frac{2t^2}{1+t^2}}=\frac{1}{t}. \end{align*} The reverse transition is the same formula with the variables renamed, so both transition maps are smooth on $\mathbb R\setminus\{0\}$. Thus $S^1$ is not naturally described by one global interval coordinate; it is described by compatible local coordinates whose changes of coordinate are ordinary smooth functions. [/example]

This local viewpoint changes what differentiability means. A real-valued function on the circle is smooth when its expression in each allowed coordinate is smooth. A map between curved spaces is smooth when all its coordinate representatives are smooth Euclidean maps. The theory therefore needs a precise way to say which coordinate systems are allowed and how they agree.

## Definition

A smooth manifold is the object that keeps exactly the coordinate information needed for calculus and discards the accidental coordinates used to describe it. The definition packages two ingredients: a space that is locally Euclidean, and a maximal rule saying which local coordinates are smoothly compatible with one another.

[definition: Smooth Manifold] A smooth $n$-manifold is a pair $(M,\mathcal A)$ where $M$ is a topological $n$-manifold and $\mathcal A$ is a maximal smooth atlas on $M$. [/definition]

When the atlas is understood, one writes $M$ rather than $(M,\mathcal A)$. The boundaryless smooth manifold is the parent notion here; a smooth manifold with boundary changes the local Euclidean model and appears later as a related specialization. The rest of this section unpacks the supporting notions that make the compact definition usable.

## Atlases and Coordinate Compatibility

### Local Euclidean Spaces

Before smoothness can be defined, the underlying space must at least have Euclidean neighbourhoods. The Hausdorff condition prevents distinct points from being topologically inseparable, while second countability keeps the space within the usual scope of local-to-global arguments. These hypotheses isolate the spaces on which coordinate calculus has a reasonable chance of behaving like ordinary calculus.

[definition: Topological $n$-Manifold] A topological $n$-manifold is a second-countable Hausdorff [topological space](/page/Topological%20Space) $M$ such that for every point $p\in M$ there exist an open neighbourhood $U$ of $p$ in $M$ and a homeomorphism $\varphi:U\to\varphi(U)$ onto an open subset of $\mathbb R^n$. [/definition]

A topological manifold gives the stage, but calculus needs named local measuring devices. We must be able to refer to a chosen neighbourhood and its coordinate map as a single object, because all later compatibility and smoothness tests are phrased in terms of such local measurements.

[definition: Coordinate Chart] Let $M$ be a topological $n$-manifold. A coordinate chart on $M$ is a pair $(U,\varphi)$ where $U$ is open in $M$ and $\varphi:U\to\varphi(U)$ is a homeomorphism onto an open subset of $\mathbb R^n$. [/definition]

If $(U,\varphi)$ is a chart and $\varphi(p)=(x_1(p),\ldots,x_n(p))$, the functions $x_i:U\to\mathbb R$ are local coordinate functions. A chart by itself is too small to describe most manifolds, and many charts may overlap. The next definition answers the essential question: when do two coordinate systems describe the same differentiable structure?

[definition: Smoothly Compatible Charts] Let $M$ be a topological $n$-manifold. Two coordinate charts $(U,\varphi)$ and $(V,\psi)$ on $M$ are smoothly compatible if either $U\cap V=\varnothing$, or the transition maps $\psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V)$ and $\varphi\circ\psi^{-1}:\psi(U\cap V)\to\varphi(U\cap V)$ are smooth maps between open subsets of $\mathbb R^n$. [/definition]

Smooth compatibility is the mechanism that makes coordinate calculations independent of the chosen chart. Still, a manifold usually needs a cover by many charts rather than an isolated compatible pair. We therefore gather compatible charts into a single coordinate system covering the whole space.

[definition: Smooth Atlas] Let $M$ be a topological $n$-manifold. A smooth atlas on $M$ is a collection $\mathcal A=\{(U_i,\varphi_i):i\in I\}$ of coordinate charts such that the sets $U_i$ cover $M$ and every two charts in $\mathcal A$ are smoothly compatible. [/definition]

An atlas can often be described using only a few charts, but there are many additional charts compatible with those choices. To make the smooth structure independent of this presentation, we need a canonical enlargement that contains every chart allowed by the same compatibility rule.

[definition: Maximal Smooth Atlas] Let $M$ be a topological $n$-manifold. A maximal smooth atlas on $M$ is a smooth atlas $\mathcal A$ such that every coordinate chart on $M$ that is smoothly compatible with every chart in $\mathcal A$ already belongs to $\mathcal A$. [/definition]