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1、AnnalsofMathematicsMinimalImmersionsofSpheresintoSpheresAuthor(s):ManfredoP.DoCarmoandNolanR.WallachReviewedwork(s):Source:TheAnnalsofMathematics,SecondSeries,Vol.93,No.1(Jan.,1971),pp.43-62Publishedby:AnnalsofMathematicsStableURL:http://www.jstor.org/stable/1970752.Accessed:30/12/201103:55Yo
2、uruseoftheJSTORarchiveindicatesyouracceptanceoftheTerms&ConditionsofUse,availableat.http://www.jstor.org/page/info/about/policies/terms.jspJSTORisanot-for-profitservicethathelpsscholars,researchers,andstudentsdiscover,use,andbuilduponawiderangeofcontentinatrusteddigitalarchive.Weuseinformatio
3、ntechnologyandtoolstoincreaseproductivityandfacilitatenewformsofscholarship.FormoreinformationaboutJSTOR,pleasecontactsupport@jstor.org.AnnalsofMathematicsiscollaboratingwithJSTORtodigitize,preserveandextendaccesstoTheAnnalsofMathematics.http://www.jstor.orgMinimalimmersionsofspheresintospher
4、esByMANFREDOP.DOCARMO*andNOLANR.WALLACH1.Introduction1.1.InthispaperwewillconsiderisometricminimalimmersionsA:SkSfcR'+'ofthen-sphereSkwithconstantsectionalcurvaturekintotheunitsphereSIoftheeuclideanspaceR'+'.Forn=2thefollowingresult[2],[5];(cf.also?4)givesanicedescriptionofsuchimmersions.1.2.
5、THEOREM.Foreachpositiveintegersthereexistsanisometricmin-imalimmersionk2,8Sk(8)S2s,wherek(s)=21(s(s+1)).LetR:S2-SIbeoenarbitraryisometricminimalimmersionandassumethat9(S2)isnotcon-tainedinahyperplaneofR'+'.Then,mtoduloarigidnotion,9=*2,8sforsomes;inparticulark=k(s)and1=2s.Theminimalimmersions
6、*2,sinthestatementofTheorem1.2caneasilybegeneralizedtothen-dimensionalcase.In?2wewillshowhowtoassociatetoeachpositiveintegersanisometricminimalimmersion*k,,:Sk(s)S-+withk(s)=ns(s+n-1)=(2s-(s+n-2)!m(s)+n1)s!(n-1)!-theimmersions*k,swillbecalledthestandard(minimal)immersions.Itisthenanaturalques
7、tiontoaskiftheuniquenessofTheorem1.2canbeex-tendedtohigherdimensions.ItwillbeconvenienttosaythataminimalimmersionA:SScR1~'kisfullif(9(S1)isnotcontainedinahyperplaneofR1+'andthattwosuchim-mersions9192areequivalentifthereexistsanisometrypofSIsu
