Geodesic
On a Class of Landsberg Metrics in Finsler Geometry
Canadian journal of mathematics, Tome 61 (2009) no. 6, pp. 1357-1374

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper, we study a long existing open problem on Landsberg metrics in Finsler geometry. We consider Finsler metrics defined by a Riemannian metric and a 1-form on a manifold. We show that a regular Finsler metric in this form is Landsbergian if and only if it is Berwaldian. We further show that there is a two-parameter family of functions, $\phi \,=\,\phi \left( s \right)$ , for which there are a Riemannian metric $\alpha $ and a 1-form $\beta $ on a manifold $M$ such that the scalar function $F\,=\,\alpha \phi \left( \beta /\alpha\right)$ on $TM$ is an almost regular Landsberg metric, but not a Berwald metric.
DOI : 10.4153/CJM-2009-064-9
Mots-clés : 53B40, 53C60 
Shen, Zhongmin. On a Class of Landsberg Metrics in Finsler Geometry. Canadian journal of mathematics, Tome 61 (2009) no. 6, pp. 1357-1374. doi: 10.4153/CJM-2009-064-9
@article{10_4153_CJM_2009_064_9,
     author = {Shen, Zhongmin},
     title = {On a {Class} of {Landsberg} {Metrics} in {Finsler} {Geometry}},
     journal = {Canadian journal of mathematics},
     pages = {1357--1374},
     year = {2009},
     volume = {61},
     number = {6},
     doi = {10.4153/CJM-2009-064-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2009-064-9/}
}
TY  - JOUR
AU  - Shen, Zhongmin
TI  - On a Class of Landsberg Metrics in Finsler Geometry
JO  - Canadian journal of mathematics
PY  - 2009
SP  - 1357
EP  - 1374
VL  - 61
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2009-064-9/
DO  - 10.4153/CJM-2009-064-9
ID  - 10_4153_CJM_2009_064_9
ER  - 
%0 Journal Article
%A Shen, Zhongmin
%T On a Class of Landsberg Metrics in Finsler Geometry
%J Canadian journal of mathematics
%D 2009
%P 1357-1374
%V 61
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2009-064-9/
%R 10.4153/CJM-2009-064-9
%F 10_4153_CJM_2009_064_9

[1] [1] Asanov, G. S., Finsleroid-Finsler space with Berwald and Landsberg conditions. arXiv:math0603472. Google Scholar

[2] [2] Asanov, G. S., Finsleroid-Finsler space and spray coefficients. arXiv:math0604526. Google Scholar

[3] [3] Kitayama, M., Azuma, M., and Matsumoto, M., On Finsler spaces with (, )-metric. Regularity, geodesics and main scalars. J. Hokkaido Univ. Ed. Sect. II A 46(1995), 1–10. Google Scholar

[4] [4] Kikuchi, S., On the condition that a space with (, )-metric be locally Minkowskian. Tensor 33(1979), no. 2, 242–246. Google Scholar

[5] [5] Matsumoto, M., On Finsler spaces with Randers metric and special forms of important tensors. J. Math. Kyoto Univ.14(1974), 477–498. Google Scholar

[6] [6] Matsumoto, M., Finsler spaces with (, )-metric of Douglas type. Tensor 60(1998), no. 2, 123–134. Google Scholar

[7] [7] Shibata, C., Shimada, H., Azuma, M., and Yosuda, H., On Finsler spaces with Randers’ metric. Tensor 31(1977), no. 2, 219–226. Google Scholar

[8] [8] Szabó, Z.I., Positive definite Berwald spaces. Structure theorems on Berwald spaces. Tensor 35(1981), no. 1, 25-39. Google Scholar

Cité par Sources :