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- Title
ON TIMELIKE HYPERSURFACES OF THE MINKOWSKI 4-SPACE WITH 1-PROPER SECOND MEAN CURVATURE VECTOR.
- Authors
PASHAIE, FIROOZ; TANOOMAND-KHOOSHMEHR, N.; RAHIMI, A.; SHAHBAZ, L.
- 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 E41. Let x: M³1 → E41 be the isometric immersion of a timelike hypersurface M³1 in E41. The second mean curvature vector field H2 of M³1 is called 1-proper if it is an eigenvector of the Cheng-Yau operator C (which is the natural extension of Δ). We show that each M³1 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 M³1 has four possible matrix forms, the results will be considered in four different cases.
- Subjects
HYPERSURFACES; MINKOWSKI space; VECTOR fields; EUCLIDEAN geometry; EIGENVECTORS
- Publication
Journal of Mahani Mathematical Research Center, 2023, Vol 12, Issue 2, p217
- ISSN
2251-7952
- Publication type
Article
- DOI
10.22103/jmmr.2022.19202.1222