On a classification of warped product spaces with gradient Ricci solitons

Sang Deok Lee, Byung Hak Kim, Jin Hyuk Choi


In this paper, we study Ricci solitons, gradient Ricci solitons in the warped product spaces and gradient Yamabe solitons in the Riemannian product spaces. We obtain the necessary and sufficient conditions for the Riemannian product spaces to be Ricci solitons. Moreover we classify the warped product space which admit gradient Ricci solitons under some conditions of the potential function.


Ricci curvature; warped product space; Ricci soliton

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A.Besse, Einstein manifolds, Springer-Verlag, Berlin, 1987.

H. D. Cao, Geometry of Ricci solitons, Lecture note, Lehigh Univ., 2008.

H. D. Cao, Recent progress on Ricci solitons, arXiv:0908.2006v1, 2008.

T. Chave, G. Valent, On a class of compact and non-compact quasi-Einstein metrics and their renoirmalizability properties, Nuclear Phys. B478 (1996), 758– 778.

M. Eminenti, G. La Nave and C. Mantegazza, Ricci solitons : The equation point of view, Menuscripta Math. 127 (2008) 345–367.

M. Fern ́andez-L ́opez and E. Garc ́ıa-R ́ıo, A remark on compact Ricci solitons, Math. Ann. 340 (2008) 893–896.

R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz CA, 1986), 237-262, Contemp. Math. 71, American Math. Soc., 1988. [8] T. Ivey, Ricci solitons on compact 3-manifolds, Differential Geom. Appl. 3 (1993) 301-307.

T. Ivey, New examples of complete Ricci solitons, Proc. AMS 122 (1994) 241–245. [10] B. H. Kim, Warped products with critical Riemannian metric , Proc. Japan Academy 71 (1995) 117–118.

B. H. Kim, S.D.Lee, J.H.Choi and Y.O.Lee, On warped product spaces with a certain Ricci condition , B. Korean Math. Soc. 50 (2013) (5), 1693–1710.

L. Ma and V. Miquel, Remarks on scalar curvature of Yamabe solitons, Ann. Glob. Ana., Geom. 42 (2012), 195–205.

G. Perelman, Ricci flow with surgery on three manifolds, arXiv:math. DG/0303109.

DOI: http://dx.doi.org/10.11568/kjm.2016.24.4.627


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