ON CONFORMALLY FLAT POLYNOMIAL (α, β)-METRICS WITH WEAKLY ISOTROPIC SCALAR CURVATURE

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 $m{\geq}2$, 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 $a_{m-1}a_m{\neq}0$, then such metric is either locally Minkowskian or Riemannian.