Honam Mathematical Journal (호남수학학술지)

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THE EXISTENCE AND THE COMPLETENESS OF SOME METRICS ON LORENTZIAN WARPED PRODUCT MANIFOLDS WITH FIBER MANIFOLD OF CLASS (B)

  • Jung, Yoon-Tae (Department of Mathematics, Chosun University) ;
  • Kim, A-Ryong (Department of Mathematics, Graduate School of Education, Chosun University) ;
  • Lee, Soo-Young (Department of Mathematics, Chosun University)
  • Received : 2016.11.23
  • Accepted : 2017.05.16
  • Published : 2017.06.25
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Abstract

In this paper, we prove the existence of warping functions on Lorentzian warped product manifolds and the completeness of the resulting metrics with some prescribed scalar curvatures.

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References

  1. T. Aubin, Nonlinear analysis on manifolds, Monge-Ampere equations, Springer-Verlag, New York, Heidelberg, Berlin (1982).
  2. J.K. Beem and P.E. Ehrlich, Global Lorentzian geometry, Pure and Applied Mathematics, 67(1981), Dekker, NewYork.
  3. 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.
  4. F. Dobarro and E. Lami Dozo, Positive scalar curvature and the Dirac operater on complete Riemannian manifolds, Publ. Math.I.H.E.S. 58(1983), 295-408.
  5. 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. https://doi.org/10.21099/tkbjm/1496162996
  6. M. Gromov and H.B. Lawson, Positive scalar curvature and the Dirac operater on complete Riemannian manifolds, Math. I.H.E.S. 58(1983), 295-408.
  7. 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
  8. 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.
  9. 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.
  10. 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.
  11. 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
  12. 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
  13. J.L. Kazdan and F.W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Diff. Geo 10(1975), 113-134. https://doi.org/10.4310/jdg/1214432678
  14. J.L. Kazdan and F.W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature, Ann. of Math. 101(1975), 317-331. https://doi.org/10.2307/1970993
  15. J.L. Kazdan and F.W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. 99(1974), 14-74. https://doi.org/10.2307/1971012
  16. M.C. Leung, Conformal scalar curvature equations on complete manifolds, Commum. Partial Diff. Equation 20(1995), 367-417. https://doi.org/10.1080/03605309508821100
  17. M.C. Leung, Conformal deformation of warped products and scalar curvature functions on open manifolds, Bulletin des Science Mathematiques. 122(1998), 369-398. https://doi.org/10.1016/S0007-4497(98)80342-4
  18. T.G. Powell, Lorentzian manifolds with non-smooth metrics and warped products, Ph. D. thesis, Univ. of Missouuri-Columbia (1982).