Smooth Manifolds

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 11 February 2012

Smooth Manifolds

A manifold, or topological manifold, is a topological space $X$ which is locally homeomorphic to ${ℝ}^n$. Locally homeomorphic to ${ℝ}^n$ means that if $x\in X$ there exists an open neighbourhood $U$ of $x$, an open set $V$ in ${ℝ}^n$ 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 ${ℝ}^n$ 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 $f:U\to ℝ$ 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 $$f\circ {{\phi }_{\alpha }}^{-1}:V_{\alpha }\to ℝ$$ is $C^{\infty }$.

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

Notes and References

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).

References

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

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