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1、ComparisonGeometryMSRIPublicationsVolume30,1996PrefaceOneofthemainactivitiesduringthe1993{94specialyearindi erentialge-ometryatMSRIwasfocusedonthesubjectbaptizedComparisonGeometry"duringtheplanningphaseoftheworkshops.Althoughanamehasbeenlackingforthisbeautifulandmostgeometricb
2、ranchofriemanniangeometry,itshistorycanbetracedbacktothenineteenthcentury.Itdidnottakeroot,however,untilthe1930's,throughtheworkofH.Hopf,Morse{Schoenberg,Myers,andSynge.Therealbreakthroughcameinthe1950'swiththepioneeringworkofRauchandthefoundationalworkofAlexandrov,Toponogovand
3、Bishop.Sincethen,thesimpleideaofcompar-ingthegeometryofanarbitraryriemannianmanifoldwiththegeometriesofconstantcurvaturespaceshasseenatremendousevolution: rstinconjunctionwithMorsetheoryandconvexity,thenwithcriticalpointtheoryfordistancefunctions,andmostrecentlywiththeGromov{Ha
4、usdor topologyonspacesofriemannianmanifolds,andthegeometryofsingularspaces.Asaresult,ourunderstandingofrelationsbetweenthegeometryandtopologyofriemannianmanifoldshasgainedtremendousbreadthandconsistsnolongerofjustashortstringofpearls.Attheoutsetitisworthmentioningthatthe?avoran
5、dcharacterofproblemsandtechniquesrelatedtoupperratherthanlowercurvatureboundstoalargeextentareremarkablydi erent.Thisvolumeisanup-to-datere?ectionoftheabovementioneddevelopmentregardingspaceswithlower,ortwo-sided,curva-turebounds.Thesubjectofmanifoldswithnegativeornonpositivecu
6、rvature,withitsrami cationstodynamicsandnumbertheory,isnotrepresentedhere.Thecontentofthevolumere?ectssomeofthemostexcitingactivitiesonComparisonGeometryduringtheyear,andespeciallyoftheworkshopdevotedtothesubject.Asaconsequence,thebookfeaturessurveyarticles(byAbreschandMeyer,An
7、derson,Colding,Greene,Otsu,Petersen,andZhu)andresearcharticles(byPerelmanandPetrunin).Eachofthesurveyarticlesstemsfromrecentinterestingdevelopmentsconcerningeitherclassicalormorerecentim-portantproblemsandtwoofthemarelecturenotesfromone-quartercoursesxixiiPREFACEtaughtbytheauth
8、ors.Completeproofsareoftenprovided,andinonecaseanewuni
