DOI QR코드

DOI QR Code

GRADIENT ALMOST RICCI SOLITONS WITH VANISHING CONDITIONS ON WEYL TENSOR AND BACH TENSOR

  • Co, Jinseok (Department of Mathematics Chung-Ang University) ;
  • Hwang, Seungsu (Department of Mathematics Chung-Ang University)
  • Received : 2019.03.08
  • Accepted : 2019.04.24
  • Published : 2020.03.01

Abstract

In this paper we consider gradient almost Ricci solitons with weak conditions on Weyl and Bach tensors. We show that a gradient almost Ricci soliton has harmonic Weyl curvature if it has fourth order divergence-free Weyl tensor, or it has divergence-free Bach tensor. Furthermore, if its Weyl tensor is radially flat, we prove such a gradient almost Ricci soliton is locally a warped product with Einstein fibers. Finally, we prove a rigidity result on compact gradient almost Ricci solitons satisfying an integral condition.

Keywords

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. Barros and E. Ribeiro, Jr., Some characterizations for compact almost Ricci solitons, Proc. Amer. Math. Soc. 140 (2012), no. 3, 1033-1040. https://doi.org/10.1090/S0002-9939-2011-11029-3
  3. A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987. https://doi.org/10.1007/978-3-540-74311-8
  4. H.-D. Cao, G. Catino, Q. Chen, C. Mantegazza, and L. Mazzieri, Bach-at gradient steady Ricci solitons, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 125-138. https://doi.org/10.1007/s00526-012-0575-3
  5. H.-D. Cao, B.-L. Chen, and X.-P. Zhu, Recent developments on Hamilton's Ricci flow, in Surveys in differential geometry. Vol. XII. Geometric flows, 47-112, Surv. Differ. Geom., 12, Int. Press, Somerville, MA, 2008. https://doi.org/10.4310/SDG.2007.v12.n1.a3
  6. H.-D. Cao and Q. Chen, On locally conformally at gradient steady Ricci solitons, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2377-2391. https://doi.org/10.1090/S0002-9947-2011-05446-2
  7. H.-D. Cao and Q. Chen, On Bach-at gradient shrinking Ricci solitons, Duke Math. J. 162 (2013), no. 6, 1149-1169. https://doi.org/10.1215/00127094-2147649
  8. H.-D. Cao and D. Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175-185. http://projecteuclid.org/euclid.jdg/1287580963
  9. G. Catino, Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271 (2012), no. 3-4, 751-756. https://doi.org/10.1007/s00209-011-0888-5
  10. G. Catino, P. Mastrolia, and D. D. Monticelli, Gradient Ricci solitons with vanishing conditions on Weyl, J. Math. Pures Appl. (9) 108 (2017), no. 1, 1-13. https://doi.org/10.1016/j.matpur.2016.10.007
  11. B.-L. Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363-382. http://projecteuclid.org/euclid.jdg/1246888488
  12. B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knoph, P. Lu, F. Luo, and L. Ni, The Ricci flow: techniques and applications. Part I, Mathematical Surveys and Monographs, 135, American Mathematical Society, Providence, RI, 2007.
  13. M. Fernandez-Lopez and E. Garcia-Rio, Rigidity of shrinking Ricci solitons, Math. Z. 269 (2011), no. 1-2, 461-466. https://doi.org/10.1007/s00209-010-0745-y
  14. S. Hwang and G. Yun, Ridigity of Ricci solitons with weakly harmonic Weyl tensors, Math. Nachr. 291 (2018), no. 5-6, 897-907. https://doi.org/10.1002/mana.201600285
  15. M. Listing, Conformally invariant Cotton and Bach tensor in N-dimensions, arXive:math/0408224v1[math.DG] 17 Aug 2004.
  16. O. Munteanu and N. Sesum, On gradient Ricci solitons, J. Geom. Anal. 23 (2013), no. 2, 539-561. https://doi.org/10.1007/s12220-011-9252-6
  17. S. Pigola, M. Rigoli, M. Rimoldi, and A. G. Setti, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), no. 4, 757-799.
  18. G. Yun, J. Co, and S. Hwang, Bach-flat h-almost gradient Ricci solitons, Pacific J. Math. 288 (2017), no. 2, 475-488. https://doi.org/10.2140/pjm.2017.288.475
  19. Z.-H. Zhang, Gradient shrinking solitons with vanishing Weyl tensor, Pacific J. Math. 242 (2009), no. 1, 189-200. https://doi.org/10.2140/pjm.2009.242.189