Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON H2-PROPER TIMELIKE HYPERSURFACES IN LORENTZ 4-SPACE FORMS

  • Firooz Pashaie (Department of Mathematics Faculty of Basic Sciences University of Maragheh)
  • Received : 2023.03.26
  • Accepted : 2024.02.29
  • Published : 2024.07.31
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Abstract

The ordinary mean curvature vector field 𝗛 on a submanifold M of a space form is said to be proper if it satisfies equality Δ𝗛 = a𝗛 for a constant real number a. It is proven that every hypersurface of an Riemannian space form with proper mean curvature vector field has constant mean curvature. In this manuscript, we study the Lorentzian hypersurfaces with proper second mean curvature vector field of four dimensional Lorentzian space forms. We show that the scalar curvature of such a hypersurface has to be constant. In addition, as a classification result, we show that each Lorentzian hypersurface of a Lorentzian 4-space form with proper second mean curvature vector field is C-biharmonic, C-1-type or C-null-2-type. Also, we prove that every 𝗛2-proper Lorentzian hypersurface with constant ordinary mean curvature in a Lorentz 4-space form is 1-minimal.

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References

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