Journal of the Korean Mathematical Society (대한수학회지)

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

  • Chen, Bin (School of Mathematical Sciences Tongji University) ;
  • Xia, KaiWen (School of Mathematical Sciences Tongji University)
  • Received : 2018.03.15
  • Accepted : 2018.06.26
  • Published : 2019.03.01
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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 $m{\geq}2$, 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.

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Acknowledgement

Supported by : National Natural Science Foundation of China

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