대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제56권1호
  • /
  • Pages.229-243
  • /
  • 2019
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

SOME INTEGRATIONS ON NULL HYPERSURFACES IN LORENTZIAN MANIFOLDS

  • Massamba, Fortune (School of Mathematics Statistics and Computer Science University of KwaZulu-Natal) ;
  • Ssekajja, Samuel (School of Mathematics Statistics and Computer Science University of KwaZulu-Natal)
  • 투고 : 2018.03.03
  • 심사 : 2018.07.20
  • 발행 : 2019.01.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We use the so-called pseudoinversion of degenerate metrics technique on foliated compact null hypersurface, $M^{n+1}$, in Lorentzian manifold ${\overline{M}}^{n+2}$, to derive an integral formula involving the r-th order mean curvatures of its foliations, ${\mathcal{F}}^n$. We apply our formula to minimal foliations, showing that, under certain geometric conditions, they are isomorphic to n-dimensional spheres. We also use the formula to deduce expressions for total mean curvatures of such foliations.

키워드

과제정보

연구 과제 주관 기관 : National Research Foundation of South Africa

참고문헌

  1. K. Andrzejewski and P. G. Walczak, The Newton transformation and new integral formulae for foliated manifolds, Ann. Global Anal. Geom. 37 (2010), no. 2, 103-111. https://doi.org/10.1007/s10455-009-9175-7
  2. C. Atindogbe, Scalar curvature on lightlike hypersurfaces, Appl. Sci. 11 (2009), 9-18.
  3. C. Atindogbe, J.-P. Ezin, and J. Tossa, Pseudoinversion of degenerate metrics, Int. J. Math. Math. Sci. 2003 (2003), no. 55, 3479-3501. https://doi.org/10.1155/S0161171203301309
  4. C. Atindogbe and H. T. Fotsing, Newton transformations on null hypersurfaces, Commun. Math. 23 (2015), no. 1, 57-83.
  5. J. L. M. Barbosa, K. Kenmotsu, and G. Oshikiri, Foliations by hypersurfaces with constant mean curvature, Math. Z. 207 (1991), no. 1, 97-107. https://doi.org/10.1007/BF02571378
  6. I. Burdujan, On minimal lightlike Monge hypersurfaces of a Lorentz space, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 47 (2001), no. 1, 95-103 (2002).
  7. C. Calin, Contributions to geometry of CR-submanifold, Ph.D. Thesis, University of Iasi (Romania), 1998.
  8. J. Dong and X. Liu, Totally umbilical lightlike hypersurfaces in Robertson-Walker space-times, ISRN Geom. 2014 (2014), Art. ID 974695, 10 pp.
  9. K. L. Duggal and A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications, Mathematics and its Applications, 364, Kluwer Academic Publishers Group, Dordrecht, 1996.
  10. K. L. Duggal and B. Sahin, Differential Geometry of Lightlike Submanifolds, Frontiers in Mathematics, Birkhauser Verlag, Basel, 2010.
  11. D. H. Jin, Ascreen lightlike hypersurfaces of an indefinite Sasakian manifold, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 20 (2013), no. 1, 25-35.
  12. D. N. Kupeli, Singular Semi-Riemannian Geometry, Mathematics and its Applications, 366, Kluwer Academic Publishers Group, Dordrecht, 1996.
  13. Y. Li, The Gauss-Bonnet-Chern theorem on Riemannian manifolds, arXiv:1111.4972 [math.DG].
  14. F. Massamba and S. Ssekajja, Quasi generalized CR-lightlike submanifolds of indefinite nearly Sasakian manifolds, Arab. J. Math. (Springer) 5 (2016), no. 2, 87-101. https://doi.org/10.1007/s40065-016-0146-0
  15. B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, 103, Academic Press, Inc., New York, 1983.