Abstract
In this paper, we study a special class of Finsler metrics, called (α, β)-metrics, which are defined by F = αϕ(β/α), where α is a Riemannian metric and β is a 1-form. We show that if ϕ = ϕ(s) is a polynomial in s, it is Einstein if and only if it is Ricci-flat. We also determine the Ricci-flat (α, β)-metrics which are not of the type F = (α + ɛβ)2/α.
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S. Bácsó, X. Cheng and Z. Shen, Curvature properties of (α, β)-metrics, Advanced Studies in Pure Mathematics, Mathematical Society of Japan 48 (2007), 73–110.
D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry, in A Sampler of Finsler Geometry, Mathematical Sciences Research Institute Publications 50, Cambridge University Press, 2004, pp. 197–259.
X. Cheng and Z. Shen, A class of Finsler metrics with isotropic S-curvature, Israel Journal of Mathematics, 169 (2009), 317–340.
S. S. Chern and Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, Vol. 6, World Scientific, Singapore, 2005.
Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht-Boston-London, 2001.
L. Zhou, A local classification of a class of (α, β)-metrics with constant flag curvature, Differential Geometry and its Applications 28 (2010), 170–193.
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Supported by NNSF of China (10971239).
Supported in part by a NSF grant (DMS-0810159).
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Cheng, X., Shen, Z. & Tian, Y. A class of Einstein (α, β)-metrics. Isr. J. Math. 192, 221–249 (2012). https://doi.org/10.1007/s11856-012-0036-x
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DOI: https://doi.org/10.1007/s11856-012-0036-x