Properties of Berwald scalar curvature

doi:10.1007/s11464-020-0872-7

Front. Math. China

2020, Vol. 15

Issue (6) : 1143-1153 https://doi.org/10.1007/s11464-020-0872-7

RESEARCH ARTICLE

Properties of Berwald scalar curvature

Ming LI¹(

), Lihong ZHANG²

¹. Mathematical Science Research Center, Chongqing University of Technology, Chongqing 400054, China
². School of Sciences, Chongqing University of Technology, Chongqing 400054, China

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Abstract

We prove that a Finsler manifold with vanishing Berwald scalar curvature has zero E-curvature. As a consequence, Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds. For (α,β)-metrics on manifold of dimension greater than 2, if the mean Landsberg curvature and the Berwald scalar curvature both vanish, then the Berwald curvature also vanishes.

Keywords Landsberg curvature Berwald curvature E-curvature S-curvature Berwald scalar curvature

Corresponding Author(s): Ming LI

Issue Date: 05 February 2021

Cite this article:

Ming LI,Lihong ZHANG. Properties of Berwald scalar curvature[J]. Front. Math. China, 2020, 15(6): 1143-1153.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-020-0872-7
https://academic.hep.com.cn/fmc/EN/Y2020/V15/I6/1143

1	D Bao, S-S Chern, Z Shen. An Introduction to Riemann-Finsler Geometry. Berlin: Springer-Verlag, 2000 https://doi.org/10.1007/978-1-4612-1268-3
2	G Chen, Q He, S Pan. On weak Berwald (α, β)-metrics of scalar flag curvature. J Geom Phys, 2014, 86: 112–121 https://doi.org/10.1016/j.geomphys.2014.07.031
3	S-S Chern, Z Shen. Riemann-Finsler Geometry. Singapore: World Scientific, 2005 https://doi.org/10.1142/5263
4	M Crampin. A condition for a Landsberg space to be Berwaldian. Publ Math Debrecen, 2018, 93(1-2): 143–155 https://doi.org/10.5486/PMD.2018.8111
5	H Feng, M Li. Adiabatic limit and connections in Finsler geometry. Comm Anal Geom, 2013, 21(3): 607–624 https://doi.org/10.4310/CAG.2013.v21.n3.a6
6	M Li. Equivalence theorems of Minkowski spaces and applications in Finsler geometry. Acta Math Sinica (Chin Ser), 2019, 62(2): 177–190 (in Chinese)
7	X Mo. An Introduction to Finsler Geometry. Singapore: World Scientific, 2006 https://doi.org/10.1142/6095
8	R Schneider. Zur affnen differential geometrie im Groen. I. Math Z, 1967, 101: 375–406 https://doi.org/10.1007/BF01109803
9	Y Shen, Z Shen. Introduction to Modern Finsler Geometry. Singapore: World Scientific, 2016 https://doi.org/10.1142/9726
10	Z Shen. Volume comparison and its applications in Riemann-Finsler geometry. Adv Math, 1997, 128(2): 306–328 https://doi.org/10.1006/aima.1997.1630
11	Z Shen. Differential Geometry of Spray and Finsler Spaces. Dordrecht: Springer, 2001 https://doi.org/10.1007/978-94-015-9727-2
12	Z Shen. Lectures on Finsler Geometry. Singapore: World Scientific, 2001 https://doi.org/10.1142/4619
13	Z Shen. Landsberg curvature, S-curvature and Riemann curvature. In: Bao D, Bryant R L, Chern S-S, Shen Z M, eds. A Sampler of Riemann-Finsler Geometry. Math Sci Res Inst Publ, Vol 50. Cambridge: Cambridge Univ Press, 2004, 303–355
14	V Simon, S Schellschmidt, H Viesel. Introduction to the Affine Differential Geometry of Hypersurfaces. Tokyo: Science Univ Tokyo, 1991
15	W Zhang. Lectures on Chern-Weil Theory andWitten Deformations. Singapore: World Scientific,2001 https://doi.org/10.1142/4756
16	Y Zou, X Cheng. The generalized unicorn problem on (α, β)-metrics. J Math Anal Appl, 2014, 414(2): 574–589 https://doi.org/10.1016/j.jmaa.2014.01.011

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