Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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SOME METRIC ON EINSTEIN LORENTZIAN WARPED PRODUCT MANIFOLDS

  • Received : 2018.11.16
  • Accepted : 2019.12.24
  • Published : 2019.12.30
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Abstract

In this paper, let M = B×f2 F be an Einstein Lorentzian warped product manifold with 2-dimensional base. We study the geodesic completeness of some metric with constant curvature. First of all, we discuss the existence of nonconstant warping functions on M. As the results, we have some metric g admits nonconstant warping functions f. Finally, we consider the geodesic completeness on M.

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References

  1. A.L. Besse, Einstein manifold, 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.K. Beem, P.E. Ehrlich, and Th.G. Powell, Warped product manifolds in relativity, Selected Studies (Th.M.Rassias, eds.), North-Holland, 1982, 41-56.
  5. R.L. Bishop and O'Neill, Manifolds if negative curvature, Trans., Am. Math. Soc. 145 (1969), 1-49. https://doi.org/10.1090/S0002-9947-1969-0251664-4
  6. J. Case, Y.J. Shu, and G. Wei, Rigidity of quasi-Einstein metricc, Diff. Geo. and its applications 29 (2011), 93-100. https://doi.org/10.1016/j.difgeo.2010.11.003
  7. E.H. Choi, Y.H. Yang, and S.Y. Lee, The nonexistence of warping functuins on Riemannian warped product manifolds, J. Chungcheong Math. Soc. 24 (2) (2011),171-185.
  8. P.E. Ehrlich, Y.T. Jung, and S.B. Kim, Constant scalar curvatures on warped product manifolds, Tsukuba J. Math. 20 (1) (1996), 239-256.
  9. C. He, P.Petersen, and W. Wylie, On the classification of warped product Einstein metrics, math.DG. 24, Jan.(2011).
  10. C. He, P.Petersen, and W. Wylie, Uniqueness of warped product Einstein metrics and applications, math.DG.4, Feb.(2013).
  11. Y.T. Jung, Partial differential equations on semi-Riemmannian manifolds, J. Math. Anal. Appl. 241 (2000), 238-253. https://doi.org/10.1006/jmaa.1999.6640
  12. Y.T. Jung, S.H. Chae, and S.Y. Lee, The existence of some metrics on Riemannian warped product manifolds with fiber manifold of class (B), Korean J. Math, 23 (4) (2015), 733-740. https://doi.org/10.11568/kjm.2015.23.4.733
  13. Y.T. Jung, E.H. Choi, and S.Y. Lee, Nonconstant warping functions on Einstein Lorentzian warping product manifold, Honam Math. J. 39 (3) (2018), 447-456.
  14. Y.T. Jung, A.R. Kim, and S.Y. Lee, The existence and the completeness of some metrics on Lorentzian warped product manifolds with ber manifold of class (B), Honam Math. J. 39 (2) (2017), 187-197. https://doi.org/10.5831/HMJ.2017.39.2.187
  15. Y.T. Jung, Y.J. Kim, S.Y. Lee, and C.G. Shin, Scalar curvature on a warped product manifold, Korean Annals of Math. 15 (1998), 167-176.
  16. Y.T. Jung, Y.J. Kim, S.Y. Lee, and C.G. Shin, Partial differential equations and scalar curvature on semi-Riemannian manifolds(I), J. Korea Soc. Math. Educ. Ser. B: Pre Appl. Math. 5 (2) (1998), 115-122.
  17. Y.T. Jung, Y.J. Kim, S.Y. Lee, and C.G. Shin, Partial differential equations and scalar curvature on semi-Riemannian manifold(II), J. Korea Soc. Math. Educ. Ser. B: Pre Appl. Math. 6 (2) (1999), 95-101.
  18. Y.T. Jung, G.Y. Lee, A.R. Kim, and S.Y. Lee, The existence of warping functions on Riemannian warped product manifolds with fiber manifold of class (B), Honam Mathematical J. 36 (3) (2014), 597-603. https://doi.org/10.5831/HMJ.2014.36.3.597
  19. Y.T. Jung, J.M. Lee, and G.Y. Lee, The completeness of some metrics on Lorentzian warped product manifolds with fiber manifold of class (B), Honam Math. J. 37 (1) (2015), 127-134. https://doi.org/10.5831/HMJ.2015.37.1.127
  20. D.S. Kim, Einstein warped product spaces, Honam Mathematical J. 22 (1) (2000), 107-111.
  21. D.S. Kim, Compact Einstein warped product spaces, Trends in Mathematics, Information center for Mathematical Sciences, 5 (2) December (2002), 1-5.
  22. D.S. Kim and Y.H. Kim, Compact Einstein warped product spaces with nonpositive scalar curvature, proceeding of A.M.S. 131 (8) (2003), 2573-2576.
  23. W.Kuhnel and H.B. Rademacher, Conformally Einstein product spaces, math.DG. 12, Jul.(2016).
  24. S.Y. Lee, Nonconstant warping functions on Einstein warped product manifolds with 2-dimensional base, Korean J. Math. 26 (1) (2018), 75-85. https://doi.org/10.11568/KJM.2018.26.1.75
  25. B. O'Neill, Semi-Riemannian Geometry, Academic, New York, 1983.
  26. T.G. Powell, Lorentzian manifolds with non-smooth metrics and warped products, ph. D. thesis, Univ. of Missouuri-Columbia (1982).
  27. B. Unal, Multiply warped products, J. Geom. Phys. 34 (2000), 287-301. https://doi.org/10.1016/S0393-0440(99)00072-8