DOI QR코드

DOI QR Code

ON A CLASS OF FINSLER METRICS WITH ISOTROPIC BERWALD CURVATURE

  • Zhu, Hongmei (College of Mathematics and Information Science Henan Normal University)
  • Received : 2015.09.23
  • Published : 2017.03.31

Abstract

In this paper, we study a class of Finsler metrics called general (${\alpha},{\beta}$)-metrics, which are defined by a Riemannian metric ${\alpha}$ and a 1-form ${\beta}$. We show that every general (${\alpha},{\beta}$)-metric with isotropic Berwald curvature is either a Berwald metric or a Randers metric. Moreover, a lot of new isotropic Berwald general (${\alpha},{\beta}$)-metrics are constructed explicitly.

Acknowledgement

Supported by : National Natural Science Foundation of China, Youth Science Fund of Henan Normal University, doctoral scientific research foundation of Henan Normal University

References

  1. S. Bacso and M. Matsumoto, On Finsler spaces of Douglas type-a generalization of the notion of Berwald space, Publ. Math. Debrecen 51 (1997), no. 3-4, 385-406.
  2. D. Bao, C. Robles, and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Differential Geom. 66 (2004), no. 3, 377-435. https://doi.org/10.4310/jdg/1098137838
  3. R. Bryant, Some remarks on Finsler manifolds with constant flag curvature, Houston J. Math. 28 (2002), no. 2, 221-262.
  4. X. Chen, X. Mo, and Z. Shen, On the flag curvature of Finsler metrics of scalar curvature, J. London Math. Soc. 68 (2003), no. 3, 762-780. https://doi.org/10.1112/S0024610703004599
  5. X. Chen and Z. Shen, Randers metrics with special curvature properties, Osaka J. Math. 40 (2003), no. 1, 87-101.
  6. X. Chen and Z. Shen, On Douglas metrics, Publ. Math. Debreen 66 (2007), 503-512.
  7. X. Cheng and Z. Shen, A class of Finsler metrics with isotropic S-curvature, Israel J. Math. 169 (2009), 317-340. https://doi.org/10.1007/s11856-009-0013-1
  8. X. Cheng, Z. Shen, and Y. Tian, Einstein (${\alpha},\;{\beta}$)-metrics, Israel J. Math. 192 (2012), no. 1, 221-249. https://doi.org/10.1007/s11856-012-0036-x
  9. S. S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, 25 Vol. 6 (World Scientific Publishing, Hackensack, NJ, 2005), x+192 pp.
  10. E. Guo, H. Liu, and X. Mo, On spherically symmetric Finsler metrics with isotropic Berwald curvature, Int. J. Geom. Methods Mod. Phys. 10 (2013), no. 10, 1350054, 13 pp. https://doi.org/10.1142/S0219887813500540
  11. B. Li and Z. Shen, Projectively flat fourth root Finsler metrics, Can. Math. Bull. 55 (2012), no. 1, 138-145. https://doi.org/10.4153/CMB-2011-056-5
  12. X. Mo and H. Zhu, On a class of projectively flat Finsler metrics of negative constant flag curvature, Internat. J. Math. 23 (2012), no. 8, 1250084, 14 pp. https://doi.org/10.1142/S0129167X1250084X
  13. Z. Shen, Diffierential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, 2001.
  14. Z. Shen and C. Yu, On a class of Einstein Finsler metrics, Internat. J. Math. 25 (2014), no. 4, 1450030, 18 pp. https://doi.org/10.1142/S0129167X1450030X
  15. A. Tayebi and B. Najafi, On isotropic Berwald metric, Ann. Polon. Math. 103 (2012), no. 2, 109-121. https://doi.org/10.4064/ap103-2-1
  16. A. Tayebi and M. Rafie-Rad, S-curvature of isotropic Berwald metrics, Sci. China Ser. A 51 (2008), no. 12, 2198-2204. https://doi.org/10.1007/s11425-008-0095-y
  17. C. Yu and H. Zhu, On a new class of Finsler metrics, Differential Geom. Appl. 29 (2011), no. 2, 244-254. https://doi.org/10.1016/j.difgeo.2010.12.009
  18. C. Yu and H. Zhu, Projectively flat general (${\alpha},\;{\beta}$)-metrics with constant flag curvature, J. Math. Anal. Appl. 429 (2015), no. 2, 1222-1239. https://doi.org/10.1016/j.jmaa.2015.04.072
  19. H. Zhu, A class of Finsler metrics of scalar flag curvature, Differntial Geom. Appl. 40 (2015), 321-331. https://doi.org/10.1016/j.difgeo.2015.02.011
  20. H. Zhu, On general (${\alpha},\;{\beta}$)-metrics with vanishing Douglas curvature, Internat. J. Math. 26 (2015), no. 9, 1550076, 16 pp. https://doi.org/10.1142/S0129167X15500767

Cited by

  1. On a class of almost regular Landsberg metrics pp.1869-1862, 2019, https://doi.org/10.1007/s11425-017-9290-6