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 $n\ge 3$, 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.