ON H2-PROPER TIMELIKE HYPERSURFACES IN LORENTZ 4-SPACE FORMS
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초록
The ordinary mean curvature vector field $\bH$ on a submanifold $M$ of a space form is said to be {\it proper} if it satisfies equality $\Delta\bH=a\bH$ 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 $\bH_2$-proper Lorentzian hypersurface with constant ordinary mean curvature in a Lorentz 4-space form is 1-minimal.
Abstract
The ordinary mean curvature vector field $\bH$ on a submanifold $M$ of a space form is said to be {\it proper} if it satisfies equality $\Delta\bH=a\bH$ 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 $\bH_2$-proper Lorentzian hypersurface with constant ordinary mean curvature in a Lorentz 4-space form is 1-minimal.
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