On a class of almost regular Landsberg metrics
|
摘要
There is a long existing "unicorn" problem in Finsler geometry: whether or not any Landsberg metric is a Berwald metric? Some classes of metrics were studied in the past and no regular non-Berwaldian Landsberg metric was found. However, if the metric is almost regular(allowed to be singular in some directions),some non-Berwaldian Landsberg metrics were found in the past years. All of them are composed by Riemannian metrics and 1-forms. This motivates us to ?nd more almost regular non-Berwaldian Landsberg metrics in the class of general(α, β)-metrics. In this paper, we ?rst classify almost regular Landsberg general(α, β)-metrics into three cases and prove that those regular metrics must be Berwald metrics. By solving some nonlinear PDEs,some new almost regular Landsberg metrics are constructed which have not been described before.
There is a long existing "unicorn" problem in Finsler geometry: whether or not any Landsberg metric is a Berwald metric? Some classes of metrics were studied in the past and no regular non-Berwaldian Landsberg metric was found. However, if the metric is almost regular(allowed to be singular in some directions),some non-Berwaldian Landsberg metrics were found in the past years. All of them are composed by Riemannian metrics and 1-forms. This motivates us to ?nd more almost regular non-Berwaldian Landsberg metrics in the class of general(α, β)-metrics. In this paper, we ?rst classify almost regular Landsberg general(α, β)-metrics into three cases and prove that those regular metrics must be Berwald metrics. By solving some nonlinear PDEs,some new almost regular Landsberg metrics are constructed which have not been described before.
There is a long existing "unicorn" problem in Finsler geometry: whether or not any Landsberg metric is a Berwald metric? Some classes of metrics were studied in the past and no regular non-Berwaldian Landsberg metric was found. However, if the metric is almost regular(allowed to be singular in some directions),some non-Berwaldian Landsberg metrics were found in the past years. All of them are composed by Riemannian metrics and 1-forms. This motivates us to ?nd more almost regular non-Berwaldian Landsberg metrics in the class of general(α, β)-metrics. In this paper, we ?rst classify almost regular Landsberg general(α, β)-metrics into three cases and prove that those regular metrics must be Berwald metrics. By solving some nonlinear PDEs,some new almost regular Landsberg metrics are constructed which have not been described before.
引文
1 Asanov G S.Finsleroid-Finsler space with Berwald and Landsberg conditions.Rep Math Phys,2006,58:275-300
2 Asanov G S.Finsleroid-Finsler space and geodesic spray coefficients.Publ Math Debrecen,2007,71:397-412
3 B′acs′o S.Reduction theorems of certain Landsberg spaces to Berwald spaces.Publ Math Debrecen,1996,48:357-366
4 Bao D.On two curvature-driven problems in Finsler geometry.Adv Stud Pure Math,2007,48:19-71
5 Cheng X,Li H,Zou Y.On conformally flat(α,β)-metrics with relatively isotropic mean Landsberg curvature.Publ Math Debrecen,2014,85:131-144
6 Crampin M.On Landsberg spaces and the Landsberg-Berwald problem.Houston J Math,2011,37:1103-1124
7 He Y,Zhong C P.Strongly convex weakly complex Berwald metrics and real Landsberg metrics.Sci China Math,2018,61:535-544
8 Li B L,Shen Z M.On a class of weak Landsberg metrics.Sci China Ser A,2007,50:573-589
9 Li B L,Shen Z M.On a class of locally projectively flat Finsler metrics.Internat J Math,2016,27:1650052
10 Matveev V S.On“Regular Landsberg metrics are always Berwald”by Z.I.Szab′o.Balkan J Geom Appl,2009,14:50-52
11 Mo X,Zhou L.The curvatures of spherically symmetric Finsler metrics in Rn.ArXiv:1202.4543v4,2014
12 Najafi B,Saberali S.On a class of isotropic mean Landsberg metrics.Differ Geom Dyn Syst,2016,18:72-80
13 Randers G.On an asymmetric metric in the four-space of general relativity.Phys Rev,1941,59:195-199
14 Shen Z.On a class of Landsberg metrics in Finsler geometry.Canad J Math,2009,61:1357-1374
15 Shibata C,Shimada H,Azuma M,et al.On Finsler spaces with Randers’metric.Tensor(NS),1977,31:219-226
16 Szab′o Z I.Positive definite Berwald spaces(structure theorem on Berwald spaces).Tensor(NS),1981,35:25-39
17 Szab′o Z I.All regular Landsberg metrics are Berwald.Ann Global Anal Geom,2008,34:381-386
18 Szab′o Z I.Correction to“All regular Landsberg metrics are Berwald”.Ann Global Anal Geom,2009,35:227-230
19 Tayebi A.On the class of generalized Landsberg manifolds.Period Math Hungar,2016,72:29-36
20 Tayebi A,Tabatabaeifar T.Unicorn metrics with almost vanishing H-andΞ-curvatures.Turkish J Math,2017,41:998-1008
21 Vincze C.An observation on Asanov’s unicorn metrics.Publ Math Debrecen,2017,90:251-268
22 Wang X,Li B.On Douglas general(α,β)-metrics.Acta Math Sin Engl Ser,2017,33:951-968
23 Yu C,Zhu H.On a new class of Finsler metrics.Differential Geom Appl,2011,29:244-254
24 Zhou L.Projective spherically symmetric Finsler metrics with constant flag curvature in Rn.Geom Dedicata,2012,158:353-364
25 Zhou S,Li B.On Landsberg general(α,β)-metrics with a conformal 1-form.ArXiv:1706.00533,2017
26 Zhu H.On a class of Finsler metrics with relatively isotropic mean Landsberg curvature.Publ Math Debrecen,2016,89:483-498
27 Zhu H.On general(α,β)-metrics with isotropic Berwald curvature.Bull Korean Math Soc,2017,54:399-416
28 Zohrehvand M,Maleki H.On general(α,β)-metrics of Landsberg type.Int J Geom Methods Mod Phys,2016,13:1650085
2 Asanov G S.Finsleroid-Finsler space and geodesic spray coefficients.Publ Math Debrecen,2007,71:397-412
3 B′acs′o S.Reduction theorems of certain Landsberg spaces to Berwald spaces.Publ Math Debrecen,1996,48:357-366
4 Bao D.On two curvature-driven problems in Finsler geometry.Adv Stud Pure Math,2007,48:19-71
5 Cheng X,Li H,Zou Y.On conformally flat(α,β)-metrics with relatively isotropic mean Landsberg curvature.Publ Math Debrecen,2014,85:131-144
6 Crampin M.On Landsberg spaces and the Landsberg-Berwald problem.Houston J Math,2011,37:1103-1124
7 He Y,Zhong C P.Strongly convex weakly complex Berwald metrics and real Landsberg metrics.Sci China Math,2018,61:535-544
8 Li B L,Shen Z M.On a class of weak Landsberg metrics.Sci China Ser A,2007,50:573-589
9 Li B L,Shen Z M.On a class of locally projectively flat Finsler metrics.Internat J Math,2016,27:1650052
10 Matveev V S.On“Regular Landsberg metrics are always Berwald”by Z.I.Szab′o.Balkan J Geom Appl,2009,14:50-52
11 Mo X,Zhou L.The curvatures of spherically symmetric Finsler metrics in Rn.ArXiv:1202.4543v4,2014
12 Najafi B,Saberali S.On a class of isotropic mean Landsberg metrics.Differ Geom Dyn Syst,2016,18:72-80
13 Randers G.On an asymmetric metric in the four-space of general relativity.Phys Rev,1941,59:195-199
14 Shen Z.On a class of Landsberg metrics in Finsler geometry.Canad J Math,2009,61:1357-1374
15 Shibata C,Shimada H,Azuma M,et al.On Finsler spaces with Randers’metric.Tensor(NS),1977,31:219-226
16 Szab′o Z I.Positive definite Berwald spaces(structure theorem on Berwald spaces).Tensor(NS),1981,35:25-39
17 Szab′o Z I.All regular Landsberg metrics are Berwald.Ann Global Anal Geom,2008,34:381-386
18 Szab′o Z I.Correction to“All regular Landsberg metrics are Berwald”.Ann Global Anal Geom,2009,35:227-230
19 Tayebi A.On the class of generalized Landsberg manifolds.Period Math Hungar,2016,72:29-36
20 Tayebi A,Tabatabaeifar T.Unicorn metrics with almost vanishing H-andΞ-curvatures.Turkish J Math,2017,41:998-1008
21 Vincze C.An observation on Asanov’s unicorn metrics.Publ Math Debrecen,2017,90:251-268
22 Wang X,Li B.On Douglas general(α,β)-metrics.Acta Math Sin Engl Ser,2017,33:951-968
23 Yu C,Zhu H.On a new class of Finsler metrics.Differential Geom Appl,2011,29:244-254
24 Zhou L.Projective spherically symmetric Finsler metrics with constant flag curvature in Rn.Geom Dedicata,2012,158:353-364
25 Zhou S,Li B.On Landsberg general(α,β)-metrics with a conformal 1-form.ArXiv:1706.00533,2017
26 Zhu H.On a class of Finsler metrics with relatively isotropic mean Landsberg curvature.Publ Math Debrecen,2016,89:483-498
27 Zhu H.On general(α,β)-metrics with isotropic Berwald curvature.Bull Korean Math Soc,2017,54:399-416
28 Zohrehvand M,Maleki H.On general(α,β)-metrics of Landsberg type.Int J Geom Methods Mod Phys,2016,13:1650085