Last updates: 11 February 2012

A **manifold**, or **topological manifold**,
is a topological space $X$ which is
locally homeomorphic to ${\mathbb{R}}^{n}$.
Locally homeomorphic to ${\mathbb{R}}^{n}$
means that if $x\in X$ there exists
an open neighbourhood $U$ of $x$,
an open set $V$ in
${\mathbb{R}}^{n}$
and a homeomorphism
$\phi :U\to V$.
The map
$\phi :U\to V$
is a **chart**. An **atlas** is an open covering
$\left({U}_{\alpha}\right)$ of
$X$, a set of open sets
$\left({V}_{\alpha}\right)$
of ${\mathbb{R}}^{n}$
and a collection of charts
${\phi}_{\alpha}:{U}_{\alpha}\to {V}_{\alpha}$.
Examples of manifolds are

the sphere

and the torus.

A **smooth manifold** is a manifold with an atlas
$\left({\phi}_{\alpha}\right)$
such that for each pair of charts
${\phi}_{\alpha},{\phi}_{\beta}$ the maps
$${\phi}_{\beta}\circ {{\phi}_{\alpha}}^{-1}:{\phi}_{\alpha}({U}_{\alpha}\cap {U}_{\beta})\to {\phi}_{\beta}({U}_{\alpha}\cap {U}_{\beta})$$
are smooth (i.e. ${C}^{\infty}$).
Let $M$ be a smooth manifold and let $U$ be an open
subset of $M$. The ring of smooth functions on
$U$ is the set of functions
$f:U\to \mathbb{R}$ that are
smooth at every point of $U$,
i.e. if $x\in U$ then there is a chart
${\phi}_{\alpha}:{U}_{\alpha}\to {V}_{\alpha}$ such that $x\in {U}_{\alpha}$ and $$f\circ {{\phi}_{\alpha}}^{-1}:{V}_{\alpha}\to \mathbb{R}$$ is ${C}^{\infty}$.

Let ${V}_{\alpha}$ be an open set of ${\mathbb{R}}^{n}$. For each open set $V$ of ${V}_{\alpha}$ let ${C}_{\alpha}^{\infty}\left(V\right)$ be the set of functions $f:V\to \mathbb{R}$ that are ${C}^{\infty}$ at every point of $V$. If $V\hookrightarrow {V}^{\prime}$ then we have a map $$\begin{array}{ccc}{C}_{\alpha}^{\infty}\left(V\prime \right)& \u27f6& {C}_{\alpha}^{\infty}\left(V\right)\\ f& \u27fc& f{|}_{V}.\end{array}$$ Thus $$\begin{array}{cccc}{C}_{\alpha}^{\infty}:& \left\{\text{open sets of}{V}_{\alpha}\right\}& \u27f6& \left\{\text{rings}\right\}\\ & V& \u27fc& {C}_{\alpha}^{\infty}\left(V\right)\end{array}$$ is a sheaf on ${V}_{\alpha}$ and $({V}_{\alpha},{C}_{\alpha}^{\infty})$ is a ringed space.

A **smooth manifold** is a Hausdorff topological space $M$
which is locally isomorphic to ${\mathbb{R}}^{n}$,
i.e. Hausdorff ringed space $(M,C)$
with an open cover $\left({U}_{\alpha}\right)$ such that each $({U}_{\alpha},{C}_{\alpha})$
is isomorphic (as a ringed space) to an open set $({V}_{\alpha},{C}_{\alpha}^{\infty})$ of ${\mathbb{R}}^{n}$.

These notes are a retyped version of page 2 of Chapter 4 of Representation Theory: Lecture Notes of Arun Ram from 17 January 2003 (Book2003/chap41.17.03.tex).

[Bou]
N. Bourbaki,
*Algèbre, Chapitre ?: ???????????*
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