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Introduction to Smooth Manifolds.pdf

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Introduction to Smooth Manifolds.pdf

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Introduction to Smooth Manifolds.pdf

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文档介绍:,,WA98195-******@/˜-leveltextbookonthetheoryofsmoothmanifolds,forstudentswhoalreadyhaveasolidacquaintancewithgeneraltopology,thefundamentalgroup,andcoveringspaces,[Lee00].Thissubjectisoftencalled“differentialgeometry.”Ihavemostlyavoidedthisterm,however,becauseitappliesmoreproperlytothestudyofsmoothmanifoldsendowedwithsomeextrastructure,suchasaRiemannianmet-ric,asymplecticstructure,aLiegroupstructure,orafoliation,-thoughIdotreatallofthesesubjectsinthisbook,-terestingcanbesaid,,heless,,,5through7,presentthefirstoffourmajorfoundationaltheoremsonwhichallofsmoothmanifoldstheoryrests—theinversefunctiontheorem—andsomeapplicationsofit:tosubmanifoldthe-viPrefaceory,embeddingsofsmoothmanifoldsintoEuclideanspaces,approximationofcontinuousmapsbysmoothones,,8through11,focusontensorsandtensorfieldsonmanifolds,andprogressfromRiemannianmetricsthroughdifferentialforms,integration,andStokes’stheorem(thesecon