Bulletin of the
Korean Mathematical Society BKMS

In this paper, we study Einstein-type manifolds generalizing static spaces and $V$-static spaces. We prove that if an Einstein-type manifold has non-positive complete divergence of its Weyl tensor and non-negative complete divergence of Bach tensor, then $M$ has harmonic Weyl curvature. Also similar results on an Einstein-type manifold with complete divergence of Riemann tensor are proved.

Keywords: Einstein-type manifold, complete divergence, harmonic Weyl curvature

MSC numbers: Primary 53C25; Secondary 53C20

Supported by: The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education(NRF-2018R1D1A1B05042186), and the second author by the Ministry of Education(NRF-2019R1A2C1004948).