Abstract
Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general (α, β)-metrics, which are defined by a Riemannian metric \(\alpha = \sqrt {{a_{ij}}\left( x \right){y^i}{y^j}} \) and a 1-form β = b i (x)y i. We obtain the differential equations that characterizes these metrics with vanishing Douglas curvature. By solving the equivalent PDEs, the metrics in this class are totally determined. Then many new Douglas metrics are constructed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bácsó, S., Matsumoto, M.: On Finsler spaces of Douglas type. A generalization of the notion of Berwald space. Publ. Math. Debrecen, 51, 385–406 (1997)
Chen, B., Shen, Z., Zhao, L.: On a class of Ricci-flat Finsler metrics in Finsler geometry. J. Geom. Phys, 70, 30–38 (2013)
Chen, M.: On a class of Douglas Finsler metrics, master’s degree thesis in China, Zhejiang University, 2015
Douglas, J.: The general geometry of paths. Ann. of Math., 29, 143–168 (1927–1928)
Li, B., Shen, Y., Shen, Z.: On a class of Douglas metrics. Studia Sci. Math. Hungar., 46(3), 355–365 (2009)
Li, B., Shen, Z.: On a class of projectively flat Finsler metrics. Int. J. Math., 27(4), 1650052 (2016)
Mo, X., Solórzano, N. M., Tenenblat, K.: On spherically symmetric Finsler metrics with vanishing Douglas curvature. Differential Geom. Appl., 31, 746–758 (2013)
Randers, G.: On an asymmetric metric in the four-space of general relativity. Phys. Rev., 59, 195–199 (1941)
Sevim, E., Shen, Z., Zhao, L.: On a class of Ricci-flat Douglas metrics. Int. J. Math., 23(6), 1250046 (2012)
Shen, Z.: Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001
Shen, Z.: On Projectively flat (α, β)-metrics. Canad. Math. Bull., 52(1), 132–144 (2009)
Shen, Z., Xing, H.: On Randers Metrics with Isotropic S-Curvature. Acta Math. Sin., Engl. Ser., 24(5), 789–796 (2008)
Xia, Q.: On a class of projectively flat Finsler metrics. Differential Geom. Appl., 44, 1–16 (2016)
Yu, C., Zhu, H.: On a new class of Finsler metrics. Differential Geom. Appl., 29, 244–554 (2011)
Zhu, H.: On general (α, β)-metrics with vanishing Douglas curvature. Int. J. Math., 26(9), 1550076 (2015)
Acknowledgements
After the paper was submitted, the referee informed us that M. Chen studied Douglas general (α, β)-metrics and obtained some special Douglas metrics under the condition that β is closed in [3]. Here we prove β must be closed in Theorem 1.1 and obtain all the Douglas metrics in this class. The authors would like to thank the referees for their valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research is supported by NSFC (Grant No. 11371209), ZPNSFC (Grant No. LY13A010013) and K. C. Wong Magna Fund in Ningbo University
Rights and permissions
About this article
Cite this article
Wang, X.M., Li, B.L. On Douglas general (α, β)-metrics. Acta. Math. Sin.-English Ser. 33, 951–968 (2017). https://doi.org/10.1007/s10114-017-6333-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-017-6333-x