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# geom_DmitryFuchs - Dmitry Fuchs DIFFERENTIAL GEOMETRY(240A...

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Dmitry Fuchs DIFFERENTIAL GEOMETRY (240A) 1. Manifolds. 1.1 Definition of a manifold. 1.1.1. Charts. Let M be a set. An n -dimensional chart on M is a pair ( U, ϕ ) where U is an open subset of R n and ϕ is a 1–1 map of U into M . Two n -dimensional charts on M, ( U, ϕ ) and ( V, ψ ), are called compatible , if (1) the set ϕ - 1 ( ψ ( V )) U is open (in U in R n ); (2) the set ψ - 1 ( ϕ ( U )) V is open (in V in R n ); (3) the map ϕ - 1 ( ψ ( V )) u 7→ ψ - 1 ϕ ( u ) ------→ ψ - 1 ( ϕ ( U )) is smooth (= C continuous); (4) the map ψ - 1 ( ϕ ( U )) v 7→ ϕ - 1 ψ ( v ) ------→ ϕ - 1 ( ψ ( V )) is smooth. In particular, ( U, ϕ ) and ( V, ψ ) are compatible if ϕ ( U ) ψ ( V ) = . 1.1.2. Atlases. A set { ( U α , ϕ α ) | α A } of n -dimensional charts on M is called an ( n -dimensional) atlas if (1) [ α A ϕ α ( U α ) = M ; (2) For any α, β A the charts ( U α , ϕ α ) , ( U β , ϕ β ) are compatible. Two n -dimensional atlases on M , A and B are called equivalent, if their union A ∪ B 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 n -dimensional atlas A . A subset B of M is called open (with respect to A ), if for any chart ( U, ϕ ) ∈ A the set ϕ - 1 ( B ) is open (in U in R n ). (In particular, the sets ϕ ( 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,ψ ) B ( B ψ ( V )) = [ ( V,ψ ) B ψ ( ψ - 1 ( B )) . If B is open with respect to B , then all the sets ψ - 1 ( B ) are open. Let ( U, ϕ ) be a chart of A . Then ϕ - 1 ( B ) = ϕ - 1 [ ( V,ψ ) B ψ ( ψ - 1 ( B )) = [ ( V,ψ ) B ϕ - 1 ψ ( ψ - 1 ( B )) = [ ( V,ψ ) B ( ψ - 1 ϕ ) - 1 ( ψ - 1 ( B )) . 1

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Since ψ - 1 ϕ is continuous (see condition (3) in 1.1.1), the last formula shows that ϕ - 1 ( B ) is a union of open set; hence it is also open. Since this is true for any chart ( U, ϕ ) ∈ 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 n -dimensional atlases on M is called an n -dimensional 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 . Charts of atlases from D are called simply charts of D . A set M with n -dimensional differential structure is called a (smooth) n -dimensional manifold .
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geom_DmitryFuchs - Dmitry Fuchs DIFFERENTIAL GEOMETRY(240A...

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