余弦度量与近拟余弦度量

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1、AbstractFinslerGeometryincludingitsspecialcaseRiemannGeometryisanimpor-tantforwardsubjectinmodernmathematics.Roughlyspeaking,FinslerGeom-etryisakindofRiemannGeometrywhosemetrichasnolimitsofquadraticforms.IncludingRandermetricasitsspecialcase,(α,β)−metricisanimpor-tantkindofFinslermetric,a

2、ndthiskindofmetrichasbeenwidespreadappliedinphysicsandbiology.Inthisthesis,theauthorstudiesthegeometricalproper-βtiesoftwokindsofspecial(α,β)−metric,i.e,cosinemetricF=αcosandthenαPnβ)2i−2orderapproximatecosinemetricF=α(−1)i−1(α(n=2,3,······),wheren(2i−2)!pi=1α(x,y)=a(x)yiyjisaRiemannmetri

3、candβ(x,y)=b(x)yiisone-formonijiasmoothmanifoldM.Thesufficientandnecessaryconditionsforthemtobelo-callyprojectivelyflat,especiallyforcosinemetrictobeofisotropicS−curvaturehavebeendiscussedinthisthesis.Weconcludethatcosinemetricislocallypro-jectivelyflatifandonlyifαislocallyprojectivelyflatand

4、βisparallelwithrespecttoα,andthesameresultisobtainedfromthenorder(n≥3)approximatecosinemetric.Whilethesufficientandnecessaryconditionforthesecondorderapprox-β2imatecosinemetricF2=α(1−2α2)tobelocallyprojectivelyflatisweaker,weneedβisexact,r=τ[2(1−b2)α2+3β2]andGi=ηyi+τα2bi.Here,τ=τ(x)is00αasc

5、alarfunctionandη=η(x)yiisaone-formonthemanifoldM.Mainconclu-isionsareasfollows:βTheorem3.2OnsmoothmanifoldM,cosinemetricF=αcosislocallypro-αjectivelyflatifandonlyif(1)βisparallelwithrespecttoα,(2)αislocallyprojectivelyflat,i.e,αhasconstantsectionalcurvature.Theorem4.2OnsmoothmanifoldM,these

6、condorderapproximatecosinemet-β2ricF2=α(1−2α2)islocallyprojectivelyflatifandonlyifβisclosedandiv8¹v(1)r=τ[2(1−b2)α2+3β2],00(2)Gi=ηyi+τα2bi.αHere,τ=τ(x)isascalarfunctiononM,andη=η(x)yiisaone-formonM.iTheorem5.2OnsmoothmanifoldM,thenorderapproximatecosinemetricPnβ)2i−2F=α(−1)i−1(α(n=3,4,····

7、··)islocallyprojectivelyflatifandonlyifn(2i−2)!i=1(1)βisparallelwithrespecttoα,(2)αislocallyprojectivelyflat,i.e,αhasconstantsectionalcurvature.βTheorem6.1LetF=αcosbeacosinemetriconasmoothmanifoldMwithαdimensionnandn≥3.Thenthefollowingconditionsareequivalent:(a)Fisofi