diff-manifolds2 - A very brief introduction to...

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Unformatted text preview: A very brief introduction to differentiable manifolds Tom Carter http://cogs.csustan.edu/˜ tom/diff-manifolds Santa Fe Institute Complex Systems Summer School June, 2001 1 Our general topics: Why differentiable manifolds 3 Topological spaces 4 Examples of topological spaces 11 Coordinate systems and manifolds 14 Manifolds 19 References 21 2 Why differentiable manifolds ← • Differentiable manifolds can generally be thought of as a generalization of R n . They are mathematical objects equipped with smooth (local) coordinate systems. Much of physics can be thought of as having a natural home in differentiable manifolds. A particularly valuable aspect of differentiable manifolds is that unlike traditional flat (Euclidean) R n , they can have (intrinsic) curvature. 3 Topological spaces ← • We need a way to talk about “nearness” of points in a space, and continuity of functions. We can’t (yet) talk about the “distance” between pairs of points or limits of sequences – we will use a more abstract approach. We start with: Def.: A topological space ( X , T ) is a set X together with a topology T on X . A topology on a set X is a collection of subsets of X (that is, T ⊂ P ( X )) satisfying: 1. If G 1 ,G 2 ∈ T , then G 1 ∩ G 2 ∈ T . 2. If { G α | α ∈ J } is any collection of sets in T , then [ α ∈ J G α ∈ T . 3. ∅ ∈ T , and X ∈ T . 4 • The sets G ∈ T are called open sets in X . A subset F ⊂ X whose complement is open is called a closed set in X . • If A is any subset of a topological space X , then the interior of A , denoted by A ◦ , is the union of all open sets contained in A . The closure of A , denoted by A , is the intersection of all closed sets containing A . • If x ∈ X , then a neighborhood of x is any subset A ⊂ X with x ∈ A ◦ ....
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This note was uploaded on 02/10/2012 for the course MATH 2112 taught by Professor Carter during the Fall '09 term at University of Central Florida.

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diff-manifolds2 - A very brief introduction to...

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