differential geometry w notes from teacher_Part_5

differential geometry w notes from teacher_Part_5 - 9 1.2....

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1.2. MANIFOLDS 9 5. Thus F ( M ) is compact. ± Theorem 1.2.3 A continuous real-valued function f : M R on a compact topological space M is bounded. Proof : 1. f ( M ) is compact in R n . 2. Thus f ( M ) is closed and bounded. ± 1.2.2 Idea of a Manifold A manifold M of dimension n is a topological space that is locally homeo- morphic to R n . A manifold M is covered by a family of local coordinate systems { U α ; x 1 α , . . . , x n α } α A , called an atlas , consisting of open sets, called patches (or charts ), U α , and coordinates x α . A point p U α U β that lies in two coordinate patches has two sets of coordinates x α and x β related by smooth functions x i α = f i αβ ( x 1 β , . . . , x n β ) , i = 1 , . . . , n . The coordinates x α and x β are said to be compatible . If all the functions f αβ are smooth, then the manifold M is said to be smooth . If these functions are analytic, then the manifold is said to be
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differential geometry w notes from teacher_Part_5 - 9 1.2....

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