Smooth Manifolds

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 11 February 2012

Smooth Manifolds

A manifold, or topological manifold, is a topological space X which is locally homeomorphic to n. Locally homeomorphic to n means that if xX there exists an open neighbourhood U of x, an open set V in n and a homeomorphism φ:UV. The map φ:UV is a chart. An atlas is an open covering (Uα) of X, a set of open sets (Vα) of n and a collection of charts φα: Uα Vα . Examples of manifolds are

the sphere

and the torus.

A smooth manifold is a manifold with an atlas (φα) such that for each pair of charts φα, φβ the maps φβ φα-1 : φα( UαUβ) φβ( UαUβ) are smooth (i.e. C). 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 that are smooth at every point of U, i.e. if xU then there is a chart φα: UαVα such that xUα and f φα-1 :Vα is C.

Let Vα be an open set of n. For each open set V of Vα let Cα (V) be the set of functions f:V that are C at every point of V. If VV 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 Vα and (Vα, Cα ) is a ringed space.

A smooth manifold is a Hausdorff topological space M which is locally isomorphic to n, i.e. Hausdorff ringed space (M,C) with an open cover (Uα) such that each (Uα, Cα) is isomorphic (as a ringed space) to an open set ( Vα, Cα) of 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).


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

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