Riemanniangeometryischaracterized,andresearchisorientedtowardsandshapedbyconcepts(geodesics,connections,curvature,...)andobjectives,inparticulartounderstandcertainclassesof(compact)Riemannianmanifoldsdefinedbycurvatureconditions(constantorpositiveornegativecurvature,...).Bywayofcontrast,geometricanalysisisaperhapssomewhatlesssystematiccollectionoftechniques,forsolvingextremalproblemsnaturallyarisingingeometryandforinvestigatingandcharacterizingtheirsolutions.Itturnsoutthatthetwofieldscomplementeachotherverywell;geometricanalysisofferstoolsforsolvingdifficultproblemsingeometry,andRiemanniangeometrystimulatesprogressingeometricanalysisbysettingambitiousgoals.
ItistheaimofthisbooktobeasystematicandcomprehensiveintroductiontoRiemanniangeometryandarepresentativeintroductiontothemethodsofgeometricanalysis.ItattemptsasynthesisofgeometricandanalyticmethodsinthestudyofRiemannianmanifolds.
ThepresentworkisthesixtheditionofmytextbookonRiemanniangeometryandgeometricanalysis.IthasdevelopedonthebasisofseveralgraduatecoursesItaughtattheRuhr~UniversityBochumandtheUniversityofLeipzig.ThemainnewfeatureofthepresenteditionisasystematicpresentationofthespectrumoftheLaplaceoperatoranditsrelationwiththegeometryoftheunderlyingRiemannianmarufold.Naturally,Ihavealsoincludedseveralsmalleradditionsandminorcorrections(forwhichIamgratefultoseveralreaders).Moreover,theorganizationofthechaptershasbeensystematicallyrearranged.