Abstract
In this paper, we study (α, β)-metrics of scalar flag curvature on a manifold M of dimension n (n ≥ 3). Suppose that an (α, β)-metric F is not a Finsler metric of Randers type, that is, F ≠ \( k_1 \sqrt {\alpha ^2 + k_2 \beta ^2 } \) + k 3 β, where k 1 > 0, k 2 and k 3 are scalar functions on M. We prove that F is of scalar flag curvature and of vanishing S-curvature if and only if the flag curvature K = 0 and F is a Berwald metric. In this case, F is a locally Minkowski metric.
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Supported by National Natural Science Foundation of China (Grant No. 10971239)
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Cheng, X.Y. On (α, β)-metrics of scalar flag curvature with constant S-curvature. Acta. Math. Sin.-English Ser. 26, 1701–1708 (2010). https://doi.org/10.1007/s10114-010-8472-1
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DOI: https://doi.org/10.1007/s10114-010-8472-1