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ON WARPED PRODUCT SPACES WITH A CERTAIN RICCI CONDITION

  • Received : 2012.12.04
  • Published : 2013.09.30

Abstract

In this paper, we obtain the criteria that the Riemannian manifold B is Einstein or a gradient Ricci soliton from the information of the second derivative of $f$ in the warped product space $R{\times}_fB$ with gradient Ricci solitons. Moreover, we construct new examples of non-Einstein gradient Ricci soliton spaces with an Einstein or non-Einstein gradient Ricci soliton leaf using our main theorems. Finally we also get analogous criteria for the Lorentzian version.

Keywords

Ricci curvature;Einstein metric;warped product space

References

  1. A. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987.
  2. H. D. Cao, Geometry of Ricci solitons, Lecture note, Lehigh Univ., 2008.
  3. M. Eminenti, G. La Nave, and C. Mantegazza, Ricci solitons: the equation point of view, Manuscripta Math. 127 (2008), no. 3, 345-367. https://doi.org/10.1007/s00229-008-0210-y
  4. M. Fernandez-Lopez and E. Garcia-Rio, A remark on compact Ricci solitons, Math. Ann. 340 (2008), no. 4, 893-896. https://doi.org/10.1007/s00208-007-0173-4
  5. 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.
  6. T. Ivey, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1993), no. 4, 301-307. https://doi.org/10.1016/0926-2245(93)90008-O
  7. T. Ivey, New examples of complete Ricci solitons, Proc. Amer. Math. Soc. 122 (1994), no. 1, 241-245. https://doi.org/10.1090/S0002-9939-1994-1207538-5
  8. B. H. Kim, Warped products with critical Riemannian metric, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), no. 6, 117-118. https://doi.org/10.3792/pjaa.71.117
  9. G. Perelman, Ricci flow with surgery on three manifolds, arXiv:math. DG/0303109.
  10. P. Petersen and W.Wylie, On gradient Ricci solitons with symmetry, Proc. Amer. Math. Soc. 137 (2009), no. 6, 2085-2092. https://doi.org/10.1090/S0002-9939-09-09723-8
  11. R. Pina and K. Tenenblat, On solutions of the Ricci curvature equation and the Einstein equation, Israel J. Math. 171 (2009), 61-76. https://doi.org/10.1007/s11856-009-0040-y

Cited by

  1. Gradient Ricci Solitons with Structure of Warped Product vol.71, pp.3-4, 2017, https://doi.org/10.1007/s00025-016-0583-2

Acknowledgement

Supported by : Kyung Hee University