Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

EINSTEIN-TYPE MANIFOLDS WITH COMPLETE DIVERGENCE OF WEYL AND RIEMANN TENSOR

  • Hwang, Seungsu (Department of Mathematics Chung-Ang University) ;
  • Yun, Gabjin (Department of Mathematics Myong Ji University)
  • Received : 2021.09.08
  • Accepted : 2021.11.10
  • Published : 2022.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

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

Acknowledgement

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).

References

  1. A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, Berlin, 1987. https://doi.org/10.1007/978-3-540-74311-8
  2. G. Catino, P. Mastrolia, D. D. Monticelli, and M. Rigoli, On the geometry of gradient Einstein-type manifolds, Pacific J. Math. 286 (2017), no. 1, 39-67. https://doi.org/10.2140/pjm.2017.286.39
  3. F. Coutinho, R. Diogenes, B. Leandro, and J. Ribeiro, Static perfect fluid space-time on compact manifolds, Classical Quantum Gravity 37 (2020), no. 1, 015003, 23 pp. https://doi.org/10.1088/1361-6382/ab5402
  4. A. E. Fischer and J. E. Marsden, Manifolds of Riemannian metrics with prescribed scalar curvature, Bull. Amer. Math. Soc. 80 (1974), 479-484. https://doi.org/10.1090/S0002-9904-1974-13457-9
  5. S. Hwang and G. Yun, Vacuum static spaces with vanishing of complete divergence of Weyl tensor, J. Geom. Anal. 31 (2021), no. 3, 3060-3084. https://doi.org/10.1007/s12220-020-00384-4
  6. B. Leandro, Vanishing conditions on Weyl tensor for Einstein-type manifolds, Pacific J. Math. 314 (2021), no. 1, 99-113. https://doi.org/10.2140/pjm.2021.314.99
  7. J. Qing and W. Yuan, A note on static spaces and related problems, J. Geom. Phys. 74 (2013), 18-27. https://doi.org/10.1016/j.geomphys.2013.07.003
  8. F. Yang and L. Zhang, Rigidity of gradient shrinking Ricci solitons, Asian J. Math. 24 (2020), no. 4, 533-547. https://doi.org/10.4310/AJM.2020.v24.n4.a1