Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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NONCONSTANT WARPING FUNCTIONS ON EINSTEIN WARPED PRODUCT MANIFOLDS WITH 2-DIMENSIONAL BASE

  • Received : 2017.11.04
  • Accepted : 2018.03.20
  • Published : 2018.03.30
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Abstract

In this paper, we study nonconstant warping functions on an Einstein warped product manifold $M=B{\times}_{f^2}F$ with a warped product metric $g=g_B+f(t)^2g_F$. And we consider a 2-dimensional base manifold B with a metric $g_B=dt^2+(f^{\prime}(t))^2du^2$. As a result, we prove the following: if M is an Einstein warped product manifold with a 2-dimensional base, then there exist generally nonconstant warping functions f(t).

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References

  1. A.L. Besse, Einstein manifolds, Springer-Verlag, New York, 1987.
  2. J.K. Beem and P.E. Ehrlich, Global Lorentzian geometry, Pure and Applied Mathematics, Vol. 67, Dekker, New York, 1981.
  3. J.K. Beem, P.E. Ehrlich and K.L. Easley, Global Lorentzian Geometry (2nd ed.), Marcel Dekker, Inc., New York (1996).
  4. J. Case, Y.J. Shu, and G. Wei, Rigidity of quasi-Einstein metrics, Diff. Geo. and its applications 29 (2011), 93-100. https://doi.org/10.1016/j.difgeo.2010.11.003
  5. F.E.S. Feitosa, A.A. Freitas, and J.N.V. Gomes, On the construction of gradient Ricci soliton warped product, math.DG. 26, May, (2017).
  6. C. He, P.Petersen, and W. Wylie, On the classification of warped product Einstein metrics, math.DG. 24, Jan.(2011).
  7. C. He, P. Petersen, and W. Wylie, Uniqueness of warped product Einstein metrics and applications, math. DG. 4, Feb.(2013).
  8. Dong-Soo Kim, Einstein warped product spaces, Honam Mathematical J. 22 (1) (2000), 107-111.
  9. Dong-Soo Kim, Compact Einstein warped product spaces, Trends in Mathematics, Information center for Mathematical Sciences, 5 (2) (2002) (2002), 1-5.
  10. Dong-Soo Kim and Young-Ho Kim, Compact Einstein warped product spaces with nonpositive scalar curvature, Proc. Amer. Soc. 131 (8) (2003), 2573-2576. https://doi.org/10.1090/S0002-9939-03-06878-3
  11. B. O'Neill, Semi-Riemannian Geometry, Academic, New York, 1983.

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