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 $φβ∘ φα-1 : φα( Uα∩Uβ) →φβ( Uα∩Uβ)$ 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 $f∘ φα-1 :Vα→ℝ$ 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 $V↪{V}^{\prime }$ then we have a map $Cα∞ (V′) ⟶ Cα∞ (V) f ⟼ f|V.$ Thus $Cα∞: {open sets of Vα} ⟶ {rings} V ⟼ Cα∞ (V)$ is a sheaf on ${V}_{\alpha }$ and $\left({V}_{\alpha },{C}_{\alpha }^{\infty }\right)$ 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 $\left(M,C\right)$ with an open cover $\left({U}_{\alpha }\right)$ such that each $\left({U}_{\alpha },{C}_{\alpha }\right)$ is isomorphic (as a ringed space) to an open set $\left({V}_{\alpha },{C}_{\alpha }^{\infty }\right)$ 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?????.