DOI QR코드

DOI QR Code

RIGIDITY CHARACTERIZATIONS OF COMPLETE RIEMANNIAN MANIFOLDS WITH α-BACH-FLAT

  • Huang, Guangyue (College of Mathematics and Information Science Henan Normal University) ;
  • Zeng, Qianyu (College of Mathematics and Information Science Henan Normal University)
  • Received : 2020.02.17
  • Accepted : 2020.06.05
  • Published : 2021.03.01

Abstract

For complete manifolds with α-Bach tensor (which is defined by (1.2)) flat, we provide some rigidity results characterized by some point-wise inequalities involving the Weyl curvature and the traceless Ricci curvature. Moveover, some Einstein metrics have also been characterized by some $L^{\frac{n}{2}}$-integral inequalities. Furthermore, we also give some rigidity characterizations for constant sectional curvature.

Keywords

Acknowledgement

The research of author is supported by NSFC (Nos. 11971153, 11671121).

References

  1. R. Bach, Zur Weylschen Relativitatstheorie und der Weylschen Erweiterung des Krummungstensorbegriffs, Math. Z. 9 (1921), no. 1-2, 110-135. https://doi.org/10.1007/BF01378338
  2. A. L. Besse, Einstein Manifolds, reprint of the 1987 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2008.
  3. H.-D. Cao and Q. Chen, On Bach-flat gradient shrinking Ricci solitons, Duke Math. J. 162 (2013), no. 6, 1149-1169. https://doi.org/10.1215/00127094-2147649
  4. Y. Chu, A rigidity theorem for complete noncompact Bach-flat manifolds, J. Geom. Phys. 61 (2011), no. 2, 516-521. https://doi.org/10.1016/j.geomphys.2010.11.003
  5. Y. Chu and S. Fang, Rigidity of complete manifolds with parallel Cotton tensor, Arch. Math. (Basel) 109 (2017), no. 2, 179-189. https://doi.org/10.1007/s00013-017-1047-y
  6. Y. Chu and P. Feng, Rigidity of complete noncompact Bach-flat n-manifolds, J. Geom. Phys. 62 (2012), no. 11, 2227-2233. https://doi.org/10.1016/j.geomphys.2012.06.011
  7. H.-P. Fu and J. Peng, Rigidity theorems for compact Bach-flat manifolds with positive constant scalar curvature, Hokkaido Math. J. 47 (2018), no. 3, 581-605. https://doi. org/10.14492/hokmj/1537948832
  8. H.-P. Fu and L.-Q. Xiao, Einstein manifolds with finite Lp-norm of the Weyl curvature, Differential Geom. Appl. 53 (2017), 293-305. https://doi.org/10.1016/j. difgeo.2017.07.003
  9. H.-P. Fu, G.-B. Xu, and Y.-Q. Tao, Some remarks on Riemannian manifolds with parallel Cotton tensor, Kodai Math. J. 42 (2019), no. 1, 64-74. https://doi.org/10. 2996/kmj/1552982506 https://doi.org/10.2996/kmj/1552982506
  10. H.-P. Fu, G.-B. Xu, and Y.-Q. Tao, Some remarks on Bach-flat manifolds with positive constant scalar curvature, Colloq. Math. 155 (2019), no. 2, 187-196. https://doi.org/10.4064/cm7358-2-2018
  11. H. Y. He and H.-P. Fu, Rigidity theorem for compact Bach-flat manifolds with positive constant σ2, arXiv:1810.06302
  12. G. Huang, Rigidity of Riemannian manifolds with positive scalar curvature, Ann. Global Anal. Geom. 54 (2018), no. 2, 257-272. https://doi.org/10.1007/s10455-018-9600-x
  13. G. Huang and B. Ma, Rigidity of complete Riemannian manifolds with vanishing Bach tensor, Bull. Korean Math. Soc. 56 (2019), no. 5, 1341-1353. https://doi.org/10.4134/BKMS.b181193
  14. G. Huang and Q. Y. Zeng, Rigidity characterizations of Riemannian manifolds with generic linear combination of divergences of the Weyl tensor, to appear
  15. G. Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985), no. 1, 47-62. http://projecteuclid.org/euclid.jdg/1214439463 https://doi.org/10.4310/jdg/1214439463
  16. S. Kim, Rigidity of noncompact complete Bach-flat manifolds, J. Geom. Phys. 60 (2010), no. 4, 637-642. https://doi.org/10.1016/j.geomphys.2009.12.014
  17. A. M. Li and G. S. Zhao, Isolation phenomena of Riemannian manifolds with parallel Ricci curvature, Acta Math. Sinica 37 (1994), no. 1, 19-24.
  18. B. Ma, G. Huang, X. Li, and Y. Chen, Rigidity of Einstein metrics as critical points of quadratic curvature functionals on closed manifolds, Nonlinear Anal. 175 (2018), 237-248. https://doi.org/10.1016/j.na.2018.05.017
  19. R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47-71. https://doi.org/10.1007/BF01393992