On timelike hypersurfaces of the Minkowski 4-space with 1-proper second mean curvature vector

Document Type : Research Paper

Authors

Department of Mathematics, University of Maragheh, P.O.Box 55181-83111, Maragheh, Iran

Abstract

The mean curvature vector field of a submanifold in the Euclidean n-space is said to be proper if it is an eigenvector of the Laplace operator Δ. It is proven that every hypersurface with proper mean curvature vector field in the Euclidean 4-space E4 has constant mean curvature. In this paper, we study an extended version of the mentioned subject on timelike (i.e., Lorentz) hypersurfaces of Minkowski 4-space E14. Let x:M13→E14 be the isometric immersion of a timelike hypersurface M13 in E14. The second mean curvature vector field H2 of M13 is called {\it 1-proper} if it is an eigenvector of the Cheng-Yau operator C (which is the natural extension of Δ). We show that each M13 with 1-proper H2 has constant scalar curvature. By a classification theorem, we show that such a hypersurface is C-biharmonic, C-1-type or  null-C-2-type. Since the shape operator of M13 has four possible matrix forms, the results will be considered in four different cases.

Keywords

References

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History

  • Receive Date: 18 March 2022
  • Revise Date: 17 August 2022
  • Accept Date: 27 September 2022

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