Last updates: 11 February 2012
A manifold, or topological manifold, is a topological space which is locally homeomorphic to . Locally homeomorphic to means that if there exists an open neighbourhood of , an open set in and a homeomorphism . The map is a chart. An atlas is an open covering of , a set of open sets of and a collection of charts . Examples of manifolds are
and the torus.
A smooth manifold is a manifold with an atlas such that for each pair of charts the maps are smooth (i.e. ). Let be a smooth manifold and let be an open subset of . The ring of smooth functions on is the set of functions that are smooth at every point of , i.e. if then there is a chart such that and is .
Let be an open set of . For each open set of let be the set of functions that are at every point of . If then we have a map Thus is a sheaf on and is a ringed space.
A smooth manifold is a Hausdorff topological space which is locally isomorphic to , i.e. Hausdorff ringed space with an open cover such that each is isomorphic (as a ringed space) to an open set of .
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).