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Charts

Charts. A chart is a neighborhood W and function f . W  X f : W  E d f is a diffeomorphism The dimension of the chart is the dimension d of E d . Coordinate Chart. 2-sphere: S 2. x. X. W. f. real plane: E 2.

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Charts

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  1. Charts

  2. A chart is a neighborhood W and function f. WX f: WEd fis a diffeomorphism The dimension of the chart is the dimension d of Ed. Coordinate Chart 2-sphere: S2 x X W f real plane: E2

  3. A space is first countable if there is countable neighborhood basis. Includes all metric spaces A space is second countable if the whole topology has a countable basis. Euclidean space is second countable. Points with rational coordinates Open balls with rational radii Countability N x X

  4. A differentiable structure F of class Cn satisfies certain topological properties. Union of charts U Diffeomorphisms f Locally Euclidean space M A differentiable manifold requires that the space be second countable with a differentiable structure. Differentiable Manifold

  5. Open set around A looks like a line segment. Two overlapping segments Each maps to the real line Overlap regions may give different values Transformation converts coordinates in one chart to another Charting a circle 1-sphere: S1 A f real line: E1

  6. A manifold M of dimension d Closed MEn pM, WM. W is a neighborhood of p W is diffeomorphic to an open subset of Ed. The pairs (W, f) are charts. The atlas of charts describes the manifold. Charts to Manifolds Hausdorff requirement: two distinct points must have two distinct neighborhoods.

  7. Circle Manifold • Manifold S1 • Two charts • q: (-p/2, p) • q ’: (p/2, 2p) • Transition functions • (p/2, p) f: q ’= f(q) = q • (-p/2, 0) f: q ’= f(q) = q + 2p 1-sphere: S1 p/2 p 0 3p/2

  8. Sphere S2 2-dimensional space Loops can shrink to a point Simply-connected Torus S1S1 2-dimensional space Some loops don’t reduce Multiply-connected Sphere or Torus

  9. Torus Manifold • Manifold S1 S1 • Four charts • Treat as two circles • {q,f}, {q,f’}, {q ’,f}, {q ’, f’} • q: (-p/2, p), f: (-p/2, p) • q ’: (p/2, 2p), f’: (p/2, 2p) • Transition functions are similar to the circle manifold. Torus: S1 S1 q (0,0) chart 1 f (-p/8,-p/8) chart 1 (15p/8,15p/8) chart 4

  10. Sphere Manifold • Manifold S2 • Chart 1 described in spherical coordinates: • q: (-3p/4, 3p/4) • f: (p/4, 7p/4) • Chart 2 • q’, f’ use same type of range as chart 1 • Exclude band on chart 1 equator from q = [-p/4, p/4] and f = [3p/4, 5p/4] 2-sphere: S2 Chart 2 Chart 1 next

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