ChartsCoordinate ChartCountabilityDifferentiable ManifoldCharting a circleCharts to ManifoldsCircle ManifoldSphere or TorusTorus ManifoldSphere ManifoldChartsChartsCoordinate ChartCoordinate ChartA chart is a neighborhood W A chart is a neighborhood W and function and function ..•WW  XX• : : WW  EEdd•  is a diffeomorphismis a diffeomorphismThe dimension of the chart is The dimension of the chart is the dimension the dimension dd of of EEdd..xXW2-sphere: S2real plane: E2CountabilityCountabilityA space is first countable if A space is first countable if there is countable there is countable neighborhood basis.neighborhood basis.•Includes all metric spacesIncludes all metric spacesA space is second countable A space is second countable if the whole topology has a if the whole topology has a countable basis.countable basis.XN xEuclidean space is second Euclidean space is second countable.countable.•Points with rational Points with rational coordinatescoordinates•Open balls with rational radiiOpen balls with rational radiiDifferentiable ManifoldDifferentiable ManifoldA differentiable structure A differentiable structure FF of of class class CCnn satisfies certain satisfies certain topological properties.topological properties.•Union of charts Union of charts UU•Diffeomorphisms Diffeomorphisms •Locally Euclidean space Locally Euclidean space MMA differentiable manifold A differentiable manifold requires that the space be requires that the space be second countable with a second countable with a differentiable structure.differentiable structure.  AUF :,MUAACn,, is 1FUACn),(Then , are andIf11Charting a circleCharting a circleOpen set around Open set around AA looks like looks like a line segment.a line segment.Two overlapping segmentsTwo overlapping segments•Each maps to the real lineEach maps to the real line•Overlap regions may give Overlap regions may give different valuesdifferent values•Transformation converts Transformation converts coordinates in one chart to coordinates in one chart to anotheranotherA1-sphere: S1real line: E1Charts to ManifoldsCharts to ManifoldsA manifold A manifold MM of dimension of dimension dd•Closed Closed MM  EEnn pp  MM, , WW  MM..•WW is a neighborhood of is a neighborhood of pp•WW is diffeomorphic to an is diffeomorphic to an open subset of open subset of EEdd..The pairs (The pairs (WW, , ) are charts.) are charts.•The atlas of charts The atlas of charts describes the manifold.describes the manifold.Hausdorff requirement: two distinct points must have two distinct neighborhoods.Circle ManifoldCircle ManifoldManifold Manifold SS11Two chartsTwo charts• : (: (/2, /2, ))• ’’: (: (/2, 2/2, 2))Transition functionsTransition functions•((/2, /2, ) ) ff: : ’= ’= ff(() = ) = •(-(-/2, 0) /2, 0) ff: : ’= ’= ff(() = ) =   1-sphere: S1Sphere or TorusSphere or TorusSphere Sphere SS22•2-dimensional space2-dimensional space•Loops can shrink to a pointLoops can shrink to a point•Simply-connectedSimply-connectedTorus Torus SS11  SS11•2-dimensional space2-dimensional space•Some loops don’t reduceSome loops don’t reduce•Multiply-connectedMultiply-connectedTorus ManifoldTorus ManifoldManifold Manifold SS11  SS11Four chartsFour charts•Treat as two circlesTreat as two circles•{{,,}, {}, {,,’}, {’}, {’,’,}, {}, {’, ’, ’}’}• : (: (/2, /2, ), ), : (: (/2, /2, ))• ’’: (: (/2, 2/2, 2), ), ’: (’: (/2, 2/2, 2))Transition functions are Transition functions are similar to the circle manifold.similar to the circle manifold.Torus: S1  S1(0,0) chart 1(-,-/8) chart 1(15,15/8) chart 4Sphere ManifoldSphere ManifoldManifold Manifold SS22Chart 1 described in Chart 1 described in spherical coordinates: spherical coordinates: • : (: (  /4, /4,   ))• : (: (/4, 7/4, 7))Chart 2Chart 2• ’’, , ’’ use same type of use same type of range as chart 1range as chart 1•Exclude band on chart 1 Exclude band on chart 1 equator from equator from = = [-[-/4, /4, ] ] and and = = [3[3/4, 5/4, 5] ] 2-sphere: S2nextChart 1Chart