On conformally flat polynomial $(\alpha,\beta)$-metrics with weakly isotropic scalar curvature

J. Korean Math. Soc. 2019 Vol. 56, No. 2, 329-352 Published online 2019 Mar 01

Bin Chen, KaiWen Xia Tongji University; Tongji University

Abstract : 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.