对偶平坦Kropina度量（英文）
|
摘要
本文研究了一类具有奇性的芬斯勒度量——广义Kropina度量.文中给出了刻画广义Kropina度量的等价方程.进一步的研究工作表明,由共形1-形式β构造的Kropina度量是对偶平坦的,当且仅当其中的黎曼度量α是欧氏的,且该1-形式β是常向量场.还给出了一类很有意思的非平凡局部对偶平坦Kropina度量的例子.
In this paper,a class of Finsler metrics with singularity,called general Kropina metrics,are studied.Equivalent equations are given to characterize locally dually flat general Kropina metrics.Further research shows that a Kropina metric is dually flat with the conformal1-form β if and only if it is induced from a Euclidean metric α and a constant vector field β.A class of interesting non-trivial locally dually flat Kropina metrics is also given as an example.
In this paper,a class of Finsler metrics with singularity,called general Kropina metrics,are studied.Equivalent equations are given to characterize locally dually flat general Kropina metrics.Further research shows that a Kropina metric is dually flat with the conformal1-form β if and only if it is induced from a Euclidean metric α and a constant vector field β.A class of interesting non-trivial locally dually flat Kropina metrics is also given as an example.
引文
[1]Asanov,G.,Finsler Geometry,Relativity and Gauge Theories,Dordrecht:D.Reidel Publishing Company,1985.
[2]Cheng,X.Y.and Shen,Z.M.,Finsler Geometry—An Approach via Randers Spaces,Beijing:Science Press,2012.
[3]Cheng,X.Y.,Shen,Z.M.and Zhou,Y.S.,On locally dually flat Randers metrics,Intern.Math.J.,2010,21(11):1531-1543.
[4]Kropina,V.,Projective two-dimensional Finsler spaces with a special metric,Trudy Sem.Vektor.Tenzor.Anal.,1961,11:277-292(in Russian).
[5]Shen,Z.M.,Riemannian-Finsler geometry with applications to information geometry,Chinese Ann.Math.Ser.B,2006,27(1):73-94.
[6]Xia,Q.L.,On locally dually flat(α,β)-metrics,Differential Geom.Appl,2011,29(2):233-243.
[7]Yajima,T.and Nagahama,H.,Kawaguchi space,Zermelo's condition and seismic ray path,Nonlinear Anal.Real World Appl.,2007,8(1):130-135.
[8]Yu,C.T.,On dually flat Randers metrics,Nonlinear Anal,2014,95:146-155.
[9]Zhang,X.L.,On projectively flat Kropina metrics,2013,arXiv:1304.1612.
[2]Cheng,X.Y.and Shen,Z.M.,Finsler Geometry—An Approach via Randers Spaces,Beijing:Science Press,2012.
[3]Cheng,X.Y.,Shen,Z.M.and Zhou,Y.S.,On locally dually flat Randers metrics,Intern.Math.J.,2010,21(11):1531-1543.
[4]Kropina,V.,Projective two-dimensional Finsler spaces with a special metric,Trudy Sem.Vektor.Tenzor.Anal.,1961,11:277-292(in Russian).
[5]Shen,Z.M.,Riemannian-Finsler geometry with applications to information geometry,Chinese Ann.Math.Ser.B,2006,27(1):73-94.
[6]Xia,Q.L.,On locally dually flat(α,β)-metrics,Differential Geom.Appl,2011,29(2):233-243.
[7]Yajima,T.and Nagahama,H.,Kawaguchi space,Zermelo's condition and seismic ray path,Nonlinear Anal.Real World Appl.,2007,8(1):130-135.
[8]Yu,C.T.,On dually flat Randers metrics,Nonlinear Anal,2014,95:146-155.
[9]Zhang,X.L.,On projectively flat Kropina metrics,2013,arXiv:1304.1612.