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DOI QR Code

THE EXISTENCE OF WARPING FUNCTIONS ON RIEMANNIAN WARPED PRODUCT MANIFOLDS WITH FIBER MANIFOLDS OF CLASS (A)

  • Received : 2015.09.14
  • Accepted : 2015.11.05
  • Published : 2015.11.15

Abstract

In this paper, we prove the existence and the nonexistence of warping functions on Riemannian warped product manifolds with some prescribed scalar curvatures according to the fiber manifolds of class (A).

Keywords

Acknowledgement

Supported by : Chosun University

References

  1. P. Aviles and R. McOwen, Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds, DiR. Geom. 27 (1998), 225-239.
  2. J. Bland and M. Kalka, Negative scalar curvature metrics on non-compact manifolds, Trans. A.M.S 316 (1989), 433-446. https://doi.org/10.1090/S0002-9947-1989-0987159-2
  3. R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. A.M.S 145 (1969), 1-49. https://doi.org/10.1090/S0002-9947-1969-0251664-4
  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. 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.
  6. 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
  7. H. Kitabara, H. Kawakami, and J. S. Pak, On a construction of completely simply connected Riemmannian manifolds with negative curvature, Nagoya Math. J. 113 (1980), 7-13.
  8. 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
  9. 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
  10. 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
  11. M. C. Leung, Conformal scalar curvature equations on complete manifolds, Comm. in P.D.E. 20 (1995), 367-417. https://doi.org/10.1080/03605309508821100
  12. M. C. Leung, Conformal deformation of warped products and scalar curvature functions on open manifolds, Bulletin des Science Math-ematiques. 122 (1998), 369-398. https://doi.org/10.1016/S0007-4497(98)80342-4
  13. H. B. Lawson and M. Michelsohn, Spin geometry, Princeton University Press, Princeton, (1989).
  14. X. Ma and R. C. Mcown, The Laplacian on complete manifolds with warped cylindrical ends, Commum. Partial Diff. Equation 16(1991), 1583-1614. https://doi.org/10.1080/03605309108820814
  15. S. Zucker, $L_2$ cohomology of warped products and arithmetric groups, Invent. Math. 70 (1982), 169-218. https://doi.org/10.1007/BF01390727