variational problems in geometry, seiki nishikawa, ams, 2002

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1、TranslationsofMATHEMATICALMONOGRAPHSVolume205VariationalProblemsinGeometrySe''NishikawakAmericanMathematicalsocietyVariationalProblemsinGeometryTranslationsofMATHEMATICALMONOGRAPHSVolume205VariationalProblemsinGeometrySeikiNishikawaTranslatedbyKinetsuAbeAmerl=nM

2、athematical8ocistyProvidence.RhodeIslandEditorialBoardShoshichiKobayashi(Chair)MasamichiTakesaki6fol0:FL16M**93KIKAGAKUTEKIHENBUNMONDAIbySeikiNishikawaOriginallypublishedinJapanesebyIwanamiShoten,Publishers,Tokyo,1998TanslatedfromtheJapanesebyKinetsuAbe2000Mathe

3、maticsSubjectCiasaication.Primary53-01,53C21,53C43,58E20,58J25.LibraryofCongressCataloging-In-PublicationDataNishikawa,Seiki.[Kikigakutekihenbunmondai.English]Variationalproblemsingeometry/SeildNishikawa;translatedbyKinetsuAbep.cm.-(Translationsofmathematicalmon

4、ographs,ISSN0065-9282;v.205)(Iwanamiseriesinmodemmathematics)Includesbibliographicalreferencesandindex.ISBN0-8218-1356-0(acid-freepaper)1.Harmonicmaps.2.Variationalinequalities(Mathematics).3.Riemann-ianmanifolds.I.Title.It.Series.III.Series:Iwanamiseriesinmoder

5、nmath-ematicsQA614.73.N57132001514'.74-dc212001046350©2002bytheAmericanMathematicalSociety.Allrightsreserved.TheAmericanMathematicalSocietyretainsallrightsexceptthosegrantedtotheUnitedStatesGovernment.PrintedintheUnitedStatesofAmerica.®Thepaperusedinthisbookisac

6、id-freeandfallswithintheguidelinesestablishedtoensurepermanenceanddurability.Informationoncopyingandreprintingcanbefoundinthebackofthisvolume.VisittheAMShomepageatURL:http://vvv.ams.org/10987654321070605040302ContentsPrefacetotheEnglishEditionPrefaceixOutlinesan

7、dObjectivesoftheTheoryChapter1.AreLengthofCurvesandGeodesics11.1.Arelengthandenergyofcurves11.2.Euler'sequation91.3.Connectionsandcovariantdifferentiation161.4.Geodesics281.5.Minimallengthpropertyofgeodesics38Summary43Exercises44Chapter2.FirstandSecondVariationF

8、ormulas472.1.Thefirstvariationformula472.2.Curvaturetensor542.3.Thesecondvariationformula652.4.Existenceofminimalgeodesics692.5.ApplicationstoRiemanniangeometry77Summ