两类Cartan张量有界的(α,β)-度量（英文）
|
摘要
本文研究了两类重要的(α,β)-度量-(广义)Kropina度量和Matsumoto度量的Cartan张量.证明了这两类(α,β)-度量的Cartan张量是有界的,并由此给出了这两类度量的一些刚性结果及其应用.
In this paper,we study the Cartan torsion of two classes of important(α,β)-metrics including(generalized) Kropina metric and Matsumoto metric.We find that they have bounded Cartan torsion.Moreover,we give some applications and rigid results on these two classes of metrics.
In this paper,we study the Cartan torsion of two classes of important(α,β)-metrics including(generalized) Kropina metric and Matsumoto metric.We find that they have bounded Cartan torsion.Moreover,we give some applications and rigid results on these two classes of metrics.
引文
[1]Antonelli,P.L.,Ingarden,R.S.and Matsumoto,M.,The theory of Sprays and Finsler Spaces With Applications in Physics and Biology,FTPH 58,Dordrecht:Kluwer Academic Publishers,1993.
[2]Bao,D.,Chern,S.S.and Shen,Z.,An Introduction to Riemannian-Finsler Geometry,New York:SpringerVerlag,2000.
[3]Burago,D.and Ivanov,S.,Isometric embedding of Finsler manifolds,Algebra i Analiz,1993,5(1):179-192.
[4]Cheng,X.,Wang,H.and Wang,M.,(α,β)-metrics with relatively isotropic mean Landsberg curvature,Publ.Math.Debrecen,2008,72(3/4):475-485.
[5]Chern,S.S.and Shen,Z.,Riemann-Finsler Geometry,Singapore:World Scientific,2005.
[6]Deicke,A.,Uber die Finsler-Raume mit A_i=0,Arch.Math.,1953,4:45-51(in German).
[7]Finsler,P.,Ueber Kurven und Flachen in allgeneinen Raumen,Basel:Birkhauser-Verlag,1951(in German).
[8]Gu,C.H.,imbedding of Finsler manifold in Minkowski space,Acta.Math.Sin.,Chin.Ser.,1957,7:215-232.
[9]Gu,C.H.,imbedding of Finsler manifold in Minkowski space,Acta.Math.Sin.,Chin.Ser.,1958,8:282-285.
[10]Ingarden,R.S.,Uber die Einbetting eines Finslerschen Rammes in einan Minkowskiischen Raum,Bull.Acad.Polon.Sci.,1954,2:305-308(in German).
[11]Kropina,V.K,On projective two-dimensional Finsler spaces with special metric,Tridy Sem.Vector.Tensor.Anal.,1961,11:277.
[12]Matsumoto,M.,On C-reducible Finsler spaces,Tensor(N.S.),1972,24:29-37.
[13]Matsumoto,M.,Theory of Finsler space with(α,β)-metric,Reports on Mathematical Physics,1992,31(1):43-83.
[14]Matsumoto,M.,A slope of a moutain is a Finsler surface with respect to a time measure,J.Math.Kyoto Univ.,1989,29(1):17-25.
[15]Matsumoto,M.,Projectively flat Finsler spaces with(α,β)metric,Rep.Math.Phys.,1991,30(1):15-20.
[16]Mo,X.H.and Zhou,L.F.,A class of Finsler metrics with bounded Cartan torsion,Canad.Math.Bull.,2010,53(1):122-132.
[17]Nash,J.,The immedding problem for Riemannian manifolds,Ann.Math.,1956,63(1):20-63.
[18]Narasimhamurthy,S.K.,Latha Kumari,G.N.and Bagewadi,C.S.,Projectively flat Kropina metric with K=0,Appl.Math.Sci.,2011,5(12):585-593.
[19]Shen,Y.B.and Zhang,X.L.,On Einstein Kropina metrics,Diff.Geom.Appl.,2013,31(1):80-92.
[20]Shen,Z.M.,On Finsler geometry of submanifolds,Math.Ann.,1998,311(3):549-576.
[21]Shen,Z.M.,On R-quadratic Finsler spaces,Publ.Math.Debrecen,2001,58(1/2):263-274.
[22]Shen,Z.M.,Landsberg curvature,S-curvature and Riemann curvature,In:A Sampler of Finsler Geometry,Math.Sci.Res.Inst.Publ.,Vol.50,Cambridge:Cambridge Univ.Press,2004.
[23]Shen,Z.M.,On a class of Landsberg metrics in Finsler Geometry,Canad.J.Math.,2009,61(6):1357-1374.
[24]Yoshikawa,R.and Okubo,K.,Kropina spaces of constant curvature,Tensor(N.S.),2007,68(2):190-203.
[25]Yoshikawa,R.and Okubo,K.,Constant curvature conditions for Kropina spaces,Balkan J.Geom.Appl.,2012,17(1):115-124.
[2]Bao,D.,Chern,S.S.and Shen,Z.,An Introduction to Riemannian-Finsler Geometry,New York:SpringerVerlag,2000.
[3]Burago,D.and Ivanov,S.,Isometric embedding of Finsler manifolds,Algebra i Analiz,1993,5(1):179-192.
[4]Cheng,X.,Wang,H.and Wang,M.,(α,β)-metrics with relatively isotropic mean Landsberg curvature,Publ.Math.Debrecen,2008,72(3/4):475-485.
[5]Chern,S.S.and Shen,Z.,Riemann-Finsler Geometry,Singapore:World Scientific,2005.
[6]Deicke,A.,Uber die Finsler-Raume mit A_i=0,Arch.Math.,1953,4:45-51(in German).
[7]Finsler,P.,Ueber Kurven und Flachen in allgeneinen Raumen,Basel:Birkhauser-Verlag,1951(in German).
[8]Gu,C.H.,imbedding of Finsler manifold in Minkowski space,Acta.Math.Sin.,Chin.Ser.,1957,7:215-232.
[9]Gu,C.H.,imbedding of Finsler manifold in Minkowski space,Acta.Math.Sin.,Chin.Ser.,1958,8:282-285.
[10]Ingarden,R.S.,Uber die Einbetting eines Finslerschen Rammes in einan Minkowskiischen Raum,Bull.Acad.Polon.Sci.,1954,2:305-308(in German).
[11]Kropina,V.K,On projective two-dimensional Finsler spaces with special metric,Tridy Sem.Vector.Tensor.Anal.,1961,11:277.
[12]Matsumoto,M.,On C-reducible Finsler spaces,Tensor(N.S.),1972,24:29-37.
[13]Matsumoto,M.,Theory of Finsler space with(α,β)-metric,Reports on Mathematical Physics,1992,31(1):43-83.
[14]Matsumoto,M.,A slope of a moutain is a Finsler surface with respect to a time measure,J.Math.Kyoto Univ.,1989,29(1):17-25.
[15]Matsumoto,M.,Projectively flat Finsler spaces with(α,β)metric,Rep.Math.Phys.,1991,30(1):15-20.
[16]Mo,X.H.and Zhou,L.F.,A class of Finsler metrics with bounded Cartan torsion,Canad.Math.Bull.,2010,53(1):122-132.
[17]Nash,J.,The immedding problem for Riemannian manifolds,Ann.Math.,1956,63(1):20-63.
[18]Narasimhamurthy,S.K.,Latha Kumari,G.N.and Bagewadi,C.S.,Projectively flat Kropina metric with K=0,Appl.Math.Sci.,2011,5(12):585-593.
[19]Shen,Y.B.and Zhang,X.L.,On Einstein Kropina metrics,Diff.Geom.Appl.,2013,31(1):80-92.
[20]Shen,Z.M.,On Finsler geometry of submanifolds,Math.Ann.,1998,311(3):549-576.
[21]Shen,Z.M.,On R-quadratic Finsler spaces,Publ.Math.Debrecen,2001,58(1/2):263-274.
[22]Shen,Z.M.,Landsberg curvature,S-curvature and Riemann curvature,In:A Sampler of Finsler Geometry,Math.Sci.Res.Inst.Publ.,Vol.50,Cambridge:Cambridge Univ.Press,2004.
[23]Shen,Z.M.,On a class of Landsberg metrics in Finsler Geometry,Canad.J.Math.,2009,61(6):1357-1374.
[24]Yoshikawa,R.and Okubo,K.,Kropina spaces of constant curvature,Tensor(N.S.),2007,68(2):190-203.
[25]Yoshikawa,R.and Okubo,K.,Constant curvature conditions for Kropina spaces,Balkan J.Geom.Appl.,2012,17(1):115-124.