Gradient almost Ricci solitons with vanishing conditions on Weyl tensor and Bach tensor
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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.
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.
참고문헌 (19)
- R. Bach, "Zur Weylschen Relativitatstheorie und der Weylschen Erweiterung des Krummungstensorbegris", Math. Z, 9(1-2), pp.110~135 (1921)[DOI]
- A. Barros, "Some characterizations for compact almost Ricci solitons", Proc. Amer. Math. Soc, 140(3), pp.1033~1040 (2012)[DOI]
- A. L. Besse, "Einstein Manifolds" (1987), Springer-Verlag
- H. -D. Cao, "Bach-at gradient steady Ricci solitons", Calc. Var. Partial Dierential Equations, 49(1-2), pp.125~138 (2014)[DOI]
- H.-D. Cao, "Recent developments on the Hamilton’s Ricci Flow", Surveys in Differential Geometry, 12(1), pp.47~112 (2007), International Press of Boston[DOI]
- H. -D. Cao, "On locally conformally at gradient steady Ricci solitons", Trans. Amer. Math. Soc, 364(5), pp.2377~2391 (2012)[DOI]
- H. -D. Cao, "On Bach-at gradient shrinking Ricci solitons", Duke Math. J, 162(6), pp.1149~1169 (2013)[DOI]
- H. -D. Cao, "On complete gradient shrinking Ricci solitons", J. Dierential Geom, 85(2), pp.175~185 (2010)
- G. Catino, "Generalized quasi-Einstein manifolds with harmonic Weyl tensor", Math. Z, 271(3-4), pp.751~756 (2012)[DOI]
- G. Catino, "Gradient Ricci solitons with vanishing conditions on Weyl", J. Math. Pures Appl, 108(1), pp.1~13 (2017)[DOI]
- B.-L. Chen, "Strong uniqueness of the Ricciow", J. Dierential Geom, 82(2), pp.363~382 (2009)
- B. Chow, "The Ricci ow: techniques and applications. Part I" (2007), American Mathematical Society
- M. Fernandez-Lopez, "Rigidity of shrinking Ricci solitons", Math. Z, 269(1-2), pp.461~466 (2011)[DOI]
- S. Hwang, "Ridigity of Ricci solitons with weakly harmonic Weyl tensors", Math. Nachr, 291(5-6), pp.897~907 (2018)[DOI]
- M. Listing, "Conformally invariant Cotton and Bach tensor in N-dimensions", arX-ive:math, 2004
- O. Munteanu, "On gradient Ricci solitons", J. Geom. Anal, 23(2), pp.539~561 (2013)[DOI]
- S. Pigola, "Ricci almost solitons", Ann. Sc. Norm. Super. Pisa Cl. Sci., 10(4), pp.757~799 (2011)
- G. Yun, "Bach-at h-almost gradient Ricci solitons", Pacic J. Math, 288(2), pp.475~488 (2017)[DOI]
- Z. -H. Zhang, "Gradient shrinking solitons with vanishing Weyl tensor", Pacic J. Math, 242(1), pp.189~200 (2009)[DOI]