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The classification of static spaces and related problems

Volume 151 / 2018

Guangyue Huang, Fanqi Zeng Colloquium Mathematicum 151 (2018), 189-202 MSC: Primary 53C24; Secondary 53C25. DOI: 10.4064/cm7035-2-2017 Published online: 1 December 2017

Abstract

We treat the gradient Ricci almost soliton equation, the static equation and the critical point equation in the same way. In all three cases, on a given Riemannian manifold $(M^n,g)$, there exists a smooth solution $f$ satisfying the equation $$f_{,ij}=f^pR_{ij}+\lambda g_{ij},$$ where $p\geq 0$ is a nonnegative integer, $R_{ij}$ denotes the Ricci curvature and $\lambda $ is a $C^1$ function. Using some ideas of Cao and Chen (2012, 2013), we obtain some classifications under the assumption that the Bach tensor vanishes, which generalize some results of Qing and Yuan (2013) and Cao and Chen (2013). In particular, for $n=3$, we prove that $(M^3, g)$ is locally conformally flat for any $p$ if the Bach tensor is divergence-free.

Authors

  • Guangyue HuangDepartment of Mathematics
    Henan Normal University
    453007 Xinxiang, China
    and
    Henan Engineering Laboratory for
    Big Data Statistical
    Analysis and Optimal Control
    e-mail
  • Fanqi ZengDepartment of Mathematics
    Tongji University
    200092 Shanghai, China
    e-mail

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