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1、ComparisonGeometryMSRIPublicationsVolume30,1996PrefaceOneofthemainactivitiesduringthe1993{94specialyearindi erentialge-ometryatMSRIwasfocusedonthesubjectbaptizedComparisonGeometry"duringtheplanningphaseoftheworkshops.Althoughanamehasbeenlackingforthisbeautifulandmostgeomet
2、ricbranchofriemanniangeometry,itshistorycanbetracedbacktothenineteenthcentury.Itdidnottakeroot,however,untilthe1930's,throughtheworkofH.Hopf,Morse{Schoenberg,Myers,andSynge.Therealbreakthroughcameinthe1950'swiththepioneeringworkofRauchandthefoundationalworkofAlexandrov,Topo
3、nogovandBishop.Sincethen,thesimpleideaofcompar-ingthegeometryofanarbitraryriemannianmanifoldwiththegeometriesofconstantcurvaturespaceshasseenatremendousevolution: rstinconjunctionwithMorsetheoryandconvexity,thenwithcriticalpointtheoryfordistancefunctions,andmostrecentlywith
4、theGromov{Hausdor topologyonspacesofriemannianmanifolds,andthegeometryofsingularspaces.Asaresult,ourunderstandingofrelationsbetweenthegeometryandtopologyofriemannianmanifoldshasgainedtremendousbreadthandconsistsnolongerofjustashortstringofpearls.Attheoutsetitisworthmentioni
5、ngthatthe?avorandcharacterofproblemsandtechniquesrelatedtoupperratherthanlowercurvatureboundstoalargeextentareremarkablydi erent.Thisvolumeisanup-to-datere?ectionoftheabovementioneddevelopmentregardingspaceswithlower,ortwo-sided,curva-turebounds.Thesubjectofmanifoldswithneg
6、ativeornonpositivecurvature,withitsrami cationstodynamicsandnumbertheory,isnotrepresentedhere.Thecontentofthevolumere?ectssomeofthemostexcitingactivitiesonComparisonGeometryduringtheyear,andespeciallyoftheworkshopdevotedtothesubject.Asaconsequence,thebookfeaturessurveyartic
7、les(byAbreschandMeyer,Anderson,Colding,Greene,Otsu,Petersen,andZhu)andresearcharticles(byPerelmanandPetrunin).Eachofthesurveyarticlesstemsfromrecentinterestingdevelopmentsconcerningeitherclassicalormorerecentim-portantproblemsandtwoofthemarelecturenotesfromone-quartercourse
8、sxixiiPREFACEtaughtbytheauthors.Completeproofsareoftenprovided,andinonecaseanewuni
