This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Dmitry Fuchs DIFFERENTIAL GEOMETRY (240A) 1. Manifolds. 1.1 Definition of a manifold. 1.1.1. Charts. Let M be a set. An ndimensional chart on M is a pair ( U, ) where U is an open subset of R n and is a 11 map of U into M . Two ndimensional charts on M, ( U, ) and ( V, ), are called compatible , if (1) the set  1 ( ( V )) U is open (in U in R n ); (2) the set  1 ( ( U )) V is open (in V in R n ); (3) the map  1 ( ( V )) u 7  1 ( u )  1 ( ( U )) is smooth (= C continuous); (4) the map  1 ( ( U )) v 7  1 ( v )  1 ( ( V )) is smooth. In particular, ( U, ) and ( V, ) are compatible if ( U ) ( V ) = . 1.1.2. Atlases. A set { ( U , )  A } of ndimensional charts on M is called an ( ndimensional) atlas if (1) [ A ( U ) = M ; (2) For any , A the charts ( U , ) , ( U , ) are compatible. Two ndimensional atlases on M , A and B are called equivalent, if their union AB is also an atlas (in other words, if any chart of A is compatible with any chart of B . 1.1.3. Topology. Let M be a set with an ndimensional atlas A . A subset B of M is called open (with respect to A ), if for any chart ( U, ) A the set  1 ( B ) is open (in U in R n ). (In particular, the sets ( U ) are open.) Proposition. If the atlases A and B are equivalent, then a set B M is open with respect to A if and only if it is open with respect to B . Proof . For any B M , B = [ ( V, ) B ( B ( V )) = [ ( V, ) B (  1 ( B )) . If B is open with respect to B , then all the sets  1 ( B ) are open. Let ( U, ) be a chart of A . Then  1 ( B ) =  1 [ ( V, ) B (  1 ( B )) = [ ( V, ) B  1 (  1 ( B )) = [ ( V, ) B (  1 ) 1 (  1 ( B )) . 1 Since  1 is continuous (see condition (3) in 1.1.1), the last formula shows that  1 ( B ) is a union of open set; hence it is also open. Since this is true for any chart ( U, ) A , the set B is open with respect to A . It is easy to check (left to the reader) that sets, open with respect to an atlas, form a topology (that is, and M are open, any union and any finite intersection of open sets is open). This proposition shows that an equivalence class of atlases on M makes M a topo logical space, and we can speak of its purely topological properties like compactness or connectedness. actually, the two axioms given below in 1.1.4 are of topological nature: they are called in topology Second Countability Axiom and Hausdorff axiom. 1.1.4. Manifolds. A class D of equivalent ndimensional atlases on M is called an ndimensional differ ential structure on M , if the following two additional conditions hold: (1) the class D contains an at most countable atlas; (2) for any different p,q M there exist open U,V M such that p U,p / V,q / U,q V ....
View
Full
Document
This document was uploaded on 02/26/2010.
 Spring '09
 Geometry

Click to edit the document details