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On timelike hypersurfaces of the Minkowski 4-space with 1-proper second mean curvature vector | ||
| Journal of Mahani Mathematical Research | ||
| دوره 12، شماره 2 - شماره پیاپی 25، مرداد 2023، صفحه 217-233 اصل مقاله (548.52 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22103/jmmr.2022.19202.1222 | ||
| نویسندگان | ||
| Firooz Pashaie* ؛ Naser Tanoomand Khooshmehr؛ Asghar Rahimi؛ Leila Shahbaz | ||
| Department of Mathematics, University of Maragheh, P.O.Box 55181-83111, Maragheh, Iran | ||
| چکیده | ||
| 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 $\Delta$. It is proven that every hypersurface with proper mean curvature vector field in the Euclidean 4-space ${\Bbb E}^4$ 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 ${\Bbb E}^4_1$. Let ${\textbf x}:M_1^3\rightarrow{\Bbb E}_1^4$ be the isometric immersion of a timelike hypersurface $M^3_1$ in ${\Bbb E}_1^4$. The second mean curvature vector field ${\textbf H}_2$ of $M_1^3$ is called {\it 1-proper} if it is an eigenvector of the Cheng-Yau operator $\mathcal{C}$ (which is the natural extension of $\Delta$). We show that each $M^3_1$ with 1-proper ${\textbf H}_2$ has constant scalar curvature. By a classification theorem, we show that such a hypersurface is $\mathcal{C}$-biharmonic, $\mathcal{C}$-1-type or null-$\mathcal{C}$-2-type. Since the shape operator of $M^3_1$ has four possible matrix forms, the results will be considered in four different cases. | ||
| کلیدواژه‌ها | ||
| Weak convex؛ Lorentz hypersurface؛ Biharmonic؛ $mathcal{C}$-harmonic | ||
| مراجع | ||
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