Journal of the
Korean Mathematical Society JKMS

In this paper, we study conformally flat ($\alpha,\beta$)-metrics in the form $F=\alpha(1+\sum_{j=1}^m a_j(\frac{\beta}{\alpha})^j)$ with $m\geq2$, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form on a smooth manifold $M$. We prove that if such conformally flat ($\alpha,\beta$)-metric $F$ is of weakly isotropic scalar curvature, then it must has zero scalar curvature. Moreover, if $a_{m-1} a_m\neq0$, then such metric is either locally Minkowskian or Riemannian.

Keywords: ($\alpha,\beta$)-metric, conformally flat, weakly isotropic scalar curvature

MSC numbers: 53B40, 53C60

Supported by: The first author was supported by the National Natural Science Foundation of China (no. 11471246, 11101307)