ThesubjectmatterofthisbookhaditsgenesisinRiemanns1854"habil-itation"address："UberdieHypothesen，welchederGeometriezuGrundeliegen"（OntheHypotheses，whichlieattheFoundationsofGeometry）.VolumeIIofSpivaksDifferentialGeometrycontainsanEnglishtranslationofthisinfluentiallecture，withacommentarybySpivakhimself.Riemann，undoubtedlythegreatestmathematicianofthe19thcentury，aimedatintroducingthenotionofamanifoldanditsstructures.Theprob-leminvolvedgreatdifficulties.But，withhypothesesonthesmoothnessofthefunctionsinquestion，theissuescanbesettledsatisfactorilyandthereisnowacompletetreatment.Traditionally，thestructurebeingfocusedonistheRiemannianmetric，whichisaquadraticdifferentialform.Putanotherway，itisasmoothlyvaryingfamilyofinnerproducts，oneoneachtangentspace.Theresultinggeometry——Riemanniangeometry——hasundergonetremendousdevelop-mentinthiscentury.AreasinwhichithashadsignificantimpactincludeEinsteinstheoryofgeneralrelativity，andglobaldifferentialgeometry.InthecontextofRiemannslecture，thisrestrictiontoaquadraticdif-ferentialform