A Note on a Class of Finsler Metrics of Isotropic S-Curvature

An $(alpha,beta)$-metric is defined by a Riemannian metric and $1$-form. In this paper, we investigate the known characterization for $(alpha,beta)$-metrics of isotropic S-curvature. We show that such a characterization should hold in dimension $nge 3$, and for the 2-dimensional case, there is one more class of isotropic S-curvature than the higher dimensional ones. Further, we construct corresponding examples for every two-dimensional class, especially for the class that the norm of $beta$ with respect to $alpha$ is not a constant.