Bulletin of the
Korean Mathematical Society BKMS

In this paper, we study Finsler metrics of constant $S$-curva\-ture. First we produce infinitely many Randers metrics with non-zero (constant) $S$-curvature which have vanishing $H$-curvature. They are counterexamples to Theorem 1.2 in [20]. Then we show that the existence of $(\alpha,\,\beta)$-metrics with arbitrary constant $S$-curvature in {\em each} dimension which is not Randers type by extending Li-Shen' construction.

Keywords: Finsler metric, $S$-curvature, $(\alpha,\,\beta)$-metric, existence, $H$-curva\-ture

MSC numbers: 58E20