Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CONFORMAL VECTOR FIELDS AND TOTALLY UMBILIC HYPERSURFACES

  • Kim, Dong-Soo (Department of Mathematics, College of Natural Sciences, Chonnam National University) ;
  • Kim, Seon-Bu (Department of Mathematics, College of Natural Sciences, Chonnam National University) ;
  • Kim, Young-Ho (Department of Mathematics, College of Natural Sciences, Kyungpook National University) ;
  • Park, Seong-Hee (Department of Mathematics, College of Natural Sciences, Chonnam National University)
  • Published : 2002.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this article, we show that if a semi-Riemannian space form carries a conformal vector field V of which the tangential part $V^T$ on a connected hypersurface $M^N$ ecomes a conformal vector field and the normal part $V^N on $M^N$ does not vanish identically, then $M^N$ is totally umbilic. Furthermore, we give a complete description of conformal vector fields on semi-Riemannian space forms.

Keywords

References

  1. J. Korean Math. Soc. v.33 Totally umbilic Lorentzian submanifolds S. -S. Ahn;D. -S. Kim;Y. H. Kim
  2. Nagoya Math. J. v.55 On the zeros of a conformal vector field D. E. Blair
  3. Quart. J. Math. Oxford(2) v.36 On Riemannian manifolds which admit a parallel family of totally umbilical gypersurfaces J. Bolton https://doi.org/10.1093/qmath/36.1.1
  4. Geometry of submanifolds and its applications B. -Y. Chen
  5. Total mean curvature and submanifolds of finite type
  6. Symmetries of Spacetime and Riemannian Manifolds K. L. Duggal;R. Sharma
  7. Diff. Geom. and Appl. v.1 Symmetries and geometry in general relativity G. S. Hall https://doi.org/10.1016/0926-2245(91)90020-A
  8. J. Diff. Geom. v.11 Transformations conformes des varietes pseudo-riemanniannes Y. Kerbrat
  9. Thesis Conformal transformations fo pseudo-Riemannian Einstein manifolds M. G. Kerckhove
  10. Class. Quantum Grav. v.8 The structure of Einstein spaces admitting conformal motions https://doi.org/10.1088/0264-9381/8/5/007
  11. Transformation groups in differential geometry S. Kobayashi
  12. J. Math. Pures et Appl. v.74 Essential conformal fields in pseudo-Riema-nnian geometry W. Kuhnel;H. B. Rademacher
  13. Diff. Geom. and Appl. v.7 Conformal vector fields on pseudo-Riema-nnian spaces https://doi.org/10.1016/S0926-2245(96)00052-6
  14. Semi-Riemannian geometry with application to relativity B. O'Neill
  15. Cambridge monographs on Math. Physics v.1;2 Spinors and space time R. Penrose;W. Rindler
  16. Differential geometric structures W. A. Poor
  17. C. R. Math. Rep. Acad. Sci. Canada v.7 A characterization of an affine conformal vector field R. Sharma;K. L. Duggal
  18. J. Math. Soc. Japan v.19 Conformal transformationsin complete product Riemannian manifolds Y. Tashiro;K. Miyashita https://doi.org/10.2969/jmsj/01930328
  19. The theory of Lie derivatives and its applications The theory of Lie derivatives and its applications K. Yano

Cited by

  1. Infinitesimal rigidity of hyperquadrics in semi-Euclidean space vol.13, pp.02, 2016, https://doi.org/10.1142/S0219887816300014
  2. On the umbilicity of complete constant mean curvature spacelike hypersurfaces vol.360, pp.3-4, 2014, https://doi.org/10.1007/s00208-014-1049-z
  3. Characterizing spheres by an immersion in Euclidean spaces vol.23, pp.1, 2017, https://doi.org/10.1016/j.ajmsc.2016.09.002
  4. On the Gauss mapping of hypersurfaces with constant scalar curvature in ℍ n+1 vol.45, pp.1, 2014, https://doi.org/10.1007/s00574-014-0043-0
  5. On the Gauss map of Weingarten hypersurfaces in hyperbolic spaces vol.47, pp.4, 2016, https://doi.org/10.1007/s00574-016-0203-5
  6. Characterizations of immersed gradient almost Ricci solitons vol.288, pp.2, 2017, https://doi.org/10.2140/pjm.2017.288.289