Abstract
In this paper, we study generalized Douglas–Weyl (α, β)-metrics. Suppose that a regular (α, β)-metric F is not of Randers type. We prove that F is a generalized Douglas–Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover, by ignoring the regularity, if F is not a Berwald metric, then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas–Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.
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Bácsó, S., Cheng, X., Shen, Z.: Curvature properties of (α, β)-metrics. Adv. Stud. Pure Math., 48, 73–110 (2007)
Bácsó, S., Matsumoto, M.: On Finsler spaces of Douglas type, a generalization of notion of Berwald space. Publ. Math. Debrecen, 51, 385–406 (1997)
Bácsó, S., Papp, I.: A note on a generalized Douglas space. Periodica. Math. Hungar., 48, 181–184 (2004)
Cheng, X.: On (α, β)-metrics of scalar flag curvature with constant S-curvature. Acta Math. Sin., Engl. Series, 26(9), 1701–1708 (2010)
Cheng, X., Lu, C.: Two kinds of weak Berwald metrics of scalar flag curvature. J. Math. Res. Exposition, 29(4), 607–614 (2009)
Cheng, X., Shen, Z.: A class of Finsler metrics with isotropic S-curvature. Israel J. Math., 169, 317–340 (2009)
Cui, N.: On the S-curvature of some (α, β)-metrics. Acta Math. Sci., Ser. A, 26(7), 1047–1056 (2006)
Matsumoto, M.: Theory of Finsler spaces with (α, β)-metric. Rep. Math. Phys., 31, 43–84 (1992)
Mo, X., Yu, C.: On the Ricci curvature of a Randers metrics of isotropic S-curvature. Acta Math. Sin., Engl. Series, 24, 991–996 (2008)
Najafi, B., Shen, Z., Tayebi, A.: On a projective class of Finsler metrics. Publ. Math. Debrecen, 70, 211–219 (2007)
Najafi, B., Shen, Z., Tayebi, A.: Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties. Geometriae Dedicata, 131, 87–97 (2008)
Piscoran, L-I.: From Finsler geometry to noncommutative geometry. Gen. Math., 12(4), 29–38 (2004)
Sakaguchi, T.: On Finsler spaces of scalar curvature. Tensor N. S., 38, 211–219 (1982)
Shen, Z.: Volume comparison and its applications in Riemann–Finsler geometry. Adv. Math., 128, 306–328 (1997)
Shen, Z.: Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001
Shen, Z.: Finsler manifolds with nonpositive flag curvature and constant S-curvature. Math. Z., 249(3), 625–639 (2005)
Shen, Z.: On a class of Landsberg metrics in Finsler geometry. Canad. J. Math., 61(6), 1357–1374 (2009)
Shen, Z., Xing, H.: On Randers metrics of isotropic S-curvature. Acta Math. Sin., Engl. Series, 24, 789–796 (2008)
Tayebi, A., Rafie-Rad, M.: S-curvature of isotropic Berwald metrics. Sci. China, Ser. A, 51, 2198–2204 (2008)
Tayebi, A., Sadeghi, H., Peyghan, E.: On generalized Douglas–Weyl spaces. Bull. Malays. Math. Sci. Soc., 36(3), 587–594 (2013)
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Tayebi, A., Sadeghi, H. On generalized Douglas–Weyl (α, β)-metrics. Acta. Math. Sin.-English Ser. 31, 1611–1620 (2015). https://doi.org/10.1007/s10114-015-3418-2
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DOI: https://doi.org/10.1007/s10114-015-3418-2