characteriztions of space forms by circles on their geod. sphers

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1、No.7]Proc.JapanAcad.,78,Ser.A(2002)143Characterizationsofspaceformsbycirclesontheirgeodesicspheres∗)∗∗)ByToshiakiAdachiandSadahiroMaeda(CommunicatedbyHeisukeHironaka,m.j.a.,Sept.12,2002)Abstract:Inthispaperwecharacterizespaceformsbyobservingtheextrinsicshapeofcirclesonthe

2、irgeodesicspheres.Keywords:Curvesoforder2;planecurves;geodesics;circles;geodesicspheres;spaceforms.1.Introduction.AsmoothcurveγonaumbilicbutnottotallygeodesichypersurfacewithcompleteRiemannianmanifoldMparametrizedbyparallelsecondfundamentalform.Thistellsusthatitsarclength

3、iscalledacurveoforder2ifitsatisfieseverycircleoneachgeodesicsphereisacircleinthethefollowingnonlineardifferentialequation:ambientmanifoldM(c).Motivatedbythisfact,wehereestablishcharacterizationsofspaceformsfrom(C)*+theviewpointoftheirgeodesicspheres.Inthepre-∇γ˙2∇∇γ˙+∇γ˙2γ˙

4、=∇γ,˙∇∇γ˙∇γ,˙γ˙γ˙γ˙γ˙γ˙γ˙γ˙γ˙cedingpaper[AM],wecharacterizespaceformsbyobservingtheextrinsicshapeofgeodesicsontheirwhere∇γ˙denotesthecovariantdifferentiationalonggeodesicspheres.OurresultsareextensionsofthisγwithrespecttotheRiemannianconnection∇onresult.M.Typicalexamples

5、ofcurvesoforder2arecir-clesandplanecurves.Wecallasmoothcurveγ2.Curvesoforder2.Wedevotethissec-parametrizedbyitsarclengthacircleifitsatisfiestiontostudysomefundamentalpropertiesofcurves∇∇γ˙=−k2γ˙withsomenonnegativeconstantk.oforder2.Asmoothcurveγ=γ(s)parametrizedγ˙γ˙Thiscon

6、ditionisequivalenttotheconditionthatbyitsarclengthsiscalledaFrenetcurveoforder2thereexistanonnegativeconstantkandafieldofinthewidersenseifthereexistasmoothunitvec-unitvectorsYalongthiscurvewhichsatisfy∇γ˙γ˙=torfieldYalongγwhichisorthogonalto˙γandakYand∇γ˙Y=−kγ˙.Theconstantk

7、iscalledthesmoothfunctionκsatisfyingcurvatureofγ.Asweseek=∇γ˙γ˙,wefindcir-(F)clesarecurvesoforder2.Alsoweseegeodesicsaretreatedascirclesofnullcurvature.Asmoothcurve∇γ˙γ˙(s)=κ(s)Y(s)and∇γ˙Y(s)=−κ(s)˙γ(s).issaidtobeaplanecurveifitislocallycontainedWeshallcallκthecurvaturefun

8、ction.Whenweonsomereal2-dimensionaltotallygeodesicsubman-cantakeκasapositivefunction,wecallγaFre