Properties of Berwald scalar curvature

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

References

[1] Bao D, Chern S-S, Shen Z. An Introduction to Riemann-Finsler Geometry. Berlin: Springer-Verlag, 2000

[2] Chen G, He Q, Pan S. On weak Berwald (α β)-metrics of scalar flag curvature. J Geom Phys, 2014, 86: 112–121

[3] Chern S-S, Shen Z. Riemann-Finsler Geometry. Singapore: World Scientific, 2005

[4] Crampin M. A condition for a Landsberg space to be Berwaldian. Publ Math Debrecen, 2018, 93(1-2): 143–155

[5] Feng H, Li M. Adiabatic limit and connections in Finsler geometry. Comm Anal Geom, 2013, 21(3): 607–624

[6] Li M. Equivalence theorems of Minkowski spaces and applications in Finsler geometry. Acta Math Sinica (Chin Ser), 2019, 62(2): 177–190 (in Chinese)

[7] Mo X. An Introduction to Finsler Geometry. Singapore: World Scientific, 2006

[8] Schneider R. Zur affnen differential geometrie im Groen. I. Math Z, 1967, 101: 375–406

[9] Shen Y, Shen Z. Introduction to Modern Finsler Geometry. Singapore: World Scientific, 2016

[10] Shen Z. Volume comparison and its applications in Riemann-Finsler geometry. Adv Math, 1997, 128(2): 306–328

[11] Shen Z. Differential Geometry of Spray and Finsler Spaces. Dordrecht: Springer, 2001

[12] Shen Z. Lectures on Finsler Geometry. Singapore: World Scientific, 2001

[13] Shen Z. 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] Simon V, Schellschmidt S, Viesel H. Introduction to the Affine Differential Geometry of Hypersurfaces. Tokyo: Science Univ Tokyo, 1991

[15] Zhang W. Lectures on Chern-Weil Theory andWitten Deformations. Singapore: World Scientific,2001

[16] Zou Y, Cheng X. The generalized unicorn problem on (α β)-metrics. J Math Anal Appl, 2014, 414(2): 574–589

RIGHTS & PERMISSIONS

Higher Education Press