Abstract
We study a special class of Finsler metrics, namely, Matsumoto metrics \(F = \tfrac{{\alpha ^2 }} {{\alpha - \beta }}\), where α is a Riemannian metric and β is a 1-form on a manifold M. We prove that F is a (weak) Einstein metric if and only if α is Ricci flat and β is a parallel 1-form with respect to α. In this case, F is Ricci flat and Berwaldian. As an application, we determine the local structure and prove the 3-dimensional rigidity theorem for a (weak) Einstein Matsumoto metric.
Similar content being viewed by others
References
Antonelli P L, Ingarden R S, Matsumoto M. The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology. Boston: Kluwer Academic Publishers, 1993
Bao D, Robles C. Ricci and flag curvatures in Finsler geometry. In: A Sampler of Finsler Geometry. Cambridge: Cambridge University Press, 2004, 197–259
Cheng X, Shen Z, Tian Y. A class of Einstein (α, β)-metrics. Israel J Math, 2012, 192: 221–249
Chern S S, Shen Z. Riemannian-Finsler Geometry. Singapore: World Scientific, 2005
Rafie-Rad M, Rezaei B. On Einstein Matsumoto metrics. Nonlinear Anal, 2012, 13: 882–886
Rafie-Rad M, Rezaei B. Matsumoto metrics of constant flag curvature are trivial. Results Math, 2013, 63: 475–483
Sevim E, Shen Z, Zhao L. On a Class of Ricci-flat Douglas Metrics. Internat J Math, 2012, 23: 1250046
Xia Q. On Kropina metrics of scalar flag curvature. Diff Geom Appl, 2013, 31: 393–404
Zhang X, Shen Y. On Einstein Kropina metrics. Diff Geom Appl, 2013, 31: 80–92
Zhang X, Shen Y. On some Einstein Matsumoto metrics. ArXiv:1207.1937v1, 2012
Zhou L. A local classification of a class of (α, β)-metrics with constant flag curvature. Diff Geom Appl, 2010, 28: 170–193
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, X., Xia, Q. On Einstein Matsumoto metrics. Sci. China Math. 57, 1517–1524 (2014). https://doi.org/10.1007/s11425-014-4788-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-014-4788-0