Charts

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

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

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

Charting a circle  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 A  1-sphere: S 1 real line: E 1

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

Circle Manifold  Manifold S 1  Two charts  : (  /2,  )  ’: (  /2, 2  )  Transition functions (  /2,  ) f:  ’= f(  ) =  (-  /2, 0) f:  ’= f(  ) =   1-sphere: S 1   

Sphere or Torus  Sphere S 2 2-dimensional space2-dimensional space Loops can shrink to a pointLoops can shrink to a point Simply-connectedSimply-connected  Torus S 1  S 1 2-dimensional space Some loops don’t reduce Multiply-connected

Torus Manifold  Manifold S 1  S 1  Four charts Treat as two circles { ,  }, { ,  ’}, {  ’,  }, {  ’,  ’}  : (  /2,  ),  : (  /2,  )  ’: (  /2, 2  ),  ’: (  /2, 2  )  Transition functions are similar to the circle manifold.  Torus: S 1  S 1  (0,0) chart 1 (- ,-  /8) chart 1 (15 ,15  /8) chart 4

Sphere Manifold  Manifold S 2  Chart 1 described in spherical coordinates:  : (  /4,  )  : (  /4, 7  )  Chart 2  ’,  ’ use same type of range as chart 1 Exclude band on chart 1 equator from  = [-  /4,  ] and  = [3  /4, 5  ] 2-sphere: S 2 next Chart 1 Chart 2