An extension of biconservative timelike hypersurfaces in Einstein space

• Pashaie, Firooz ^[1]
  1. [1] University of Maragheh

     University of Maragheh

     Irán

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 41, Nº. 1, 2022, págs. 335-351
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-5002
• Enlaces
  • Texto completo
• Resumen

  • A well-known conjecture of Bang-Yen Chen says that the only biharmonic Euclidean submanifolds are minimal ones, which affirmed by himself for surfaces in 3-dimensional Euclidean space, E³. We consider an extended version of Chen conjecture (namely, Lk-conjecture) on Lorentzian hypersurfaces of the pseudo-Euclidean space E⁴₁ (i.e. the Einstein space). The biconservative submanifolds in the Euclidean spaces are submanifolds with conservative stress-energy with respect to the bienergy functional. In this paper, we consider an extended condition (namely, Lk-biconservativity) on non-degenerate timelike hypersurfaces of the Einstein space E⁴₁ . A Lorentzian hypersurface x : M³₁ → E⁴₁ is called Lk-biconservative if the tangent part of L²k x vanishes identically. We show that Lk-biconservativity of a timelike hypersurface of E⁴₁  (with constant kth mean curvature and some additional conditions) implies that its (k + 1) th mean curvature is constant.

• Referencias bibliográficas
  • K. Akutagawa, and S. Maeta, “Biharmonic properly immersed submanifolds in Euclidean spaces”, Geometriae Dedicata, vol. 164, pp. 351-355, 2013.
  • L. J. Alias, and N Gürbüz, “An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures”, Geometriae...
  • A. Arvanitoyeorgos, F. Defever, G. Kaimakamis, and B. J. Papantoniou, “Biharmonic Lorentz hypersurfaces in E⁴₁”, Pacific Journal of Mathematics,...
  • A. Arvanitoyeorgos, F. Defever, and G. Kaimakamis, “Hypersurfaces in E⁴ s with proper mean curvature vector”, Journal of the Mathematical...
  • B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Singapure: World Scientific Publishing, 2014.
  • B. Y. Chen, “Some open problems and conjetures on submanifolds of finite type”, Soochow Journal of Mathematics, vol. 17, pp. 169-188, 1991.
  • F. Defever, “Hypersurfaces of E⁴ satisfying ∆H = λH”, Michigan Mathematical Journal, vol. 44, pp. 355-363, 1997.
  • F. Defever, “Hypersurfaces of E⁴ with harmonic mean curvature vector”, Mathematische Nachrichten, vol. 196, pp. 61-69, 1998.
  • I. Dimitrić, “Submanifolds of En with harmonic mean curvature vector”, Bulletin of Institute of Mathematics, Academia sinica. New series,...
  • T. Hasanis, and T. Vlachos, “Hypersurfaces in E⁴ with harmonic mean curvature vector field”, Mathematische Nachrichten, vol. 172, pp. 145-169,...
  • S. M. B. Kashani,, “On some L1-finite type (hyper)surfaces in Rn+1”, Bulletin of the Korean Mathematical Society, vol. 46, no. 1, pp....
  • P. Lucas, and H. F. Ramirez-Ospina, “Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ = Aψ + b”, Geometriae Dedicata, vol....
  • M. A. Magid, “Lorentzian isoparametric hypersurfaces”, Pacific Journal of Mathematics, vol. 118, no. 1, pp. 165-197, 1985.
  • F. Pashaie, and A. Mohammadpouri, “Lk-biharmonic spacelike hypersurfaces in Minkowski 4-space E⁴₁", Sahand Communications in Mathematical...
  • B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity. New York: Academic Press, 1983.
  • F. Pashaie, and S. M. B. Kashani, “Spacelike hypersurfaces in Riemannian or Lorentzian space forms satisfying Lkx = Ax + b”, Bulletin...
  • F. Pashaie, and S. M. B. Kashani, “Timelike hypersurfaces in the Lorentzian standard space forms satisfying Lkx = Ax+b”, Mediterranean...
  • A. Z. Petrov, Einstein Spaces. New York: Pergamon Press, 1969.
  • R. C. Reilly, “Variational properties of functions of the mean curvatures for hypersurfaces in space forms”, Journal of Differential Geometry,...