classical geometry

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1、ClassicalgeometryDannyCalegari5/4/2005,Version0.14Contents1Acrashcourseingrouptheory31.1Groups..................................31.2Subgroups,quotients,homomorphisms.................61.3Semidirectproducts..........................81.4Groupsdefinedbygeneratorsandrelations...............82F

2、undamentalideasingeometry112.1KleinsErlangerProgramm......................112.2Categorytheory.............................112.3Topology................................122.4Metricgeometry.............................142.5Differentialgeometry..........................162.6Axiomaticdefinitio

3、nofageometry...................213Modelgeometries223.1Euclideangeometry...........................223.2Sphericalgeometry...........................273.3Hyperbolicgeometry..........................364Discretegroups564.1Acrashcourseinalgebraictopology..................564.2Tessellations...

4、............................705Riemanniangeometry865.1Liouvillestheorem...........................861ListofFigures1.1AnelementϕoftheKlein4groupinterchangestwopairsofoppositeedges,andflipstheorientationoftheothertwoedges..........41.2Thecompositionω−1rω=r,butrω−1ω=r,soDisnotcom-acaa3mutati

5、ve.................................5−12.1Thecompositionρ=φ2φ1ontheoverlapφ1(U1∩U2)ofchartsshouldbesmoothandlocallyinvertible.................173.1ThreelunescoverhalfofS2andoverlapexactlyonthetriangleT...294.1TwoknotdiagramsrepresentthesametopologicalknotifftheyarerelatedbyasequenceofReid

6、emeistermoves..............664.2ThelinkingnumberofK1andK2isthesumoverundercrossingsofK2underK1ofthesignofthecrossing,definedasinthisfigure....674.3Thissequenceofmovesdemonstratestheequivalenceofthetwoman-ifoldsobtainedby1and−1surgeryonthefigure8knotandrighthandedtrefoilrespectively.........

7、..............684.4TheresultofslidingK2withframingnoverK1withframingm,bothintegers.................................692Chapter1Acrashcourseingrouptheory1.1Groups1.1.1DefinitionofagroupMathematicalobjectsaredefinedintermsofthepropertiesthattheysatisfy.Astructureonanobjectisal