In this paper, the generalized helical hypersurfaces $\mathbf{x}=\mathbf{x}(u,v,w)$ with a time-like axis in Minkowski spacetime ${\mathbb{E}}_1^4$ are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of $\mathbf{x}$ are calculated. Moreover, the curvatures of the generalized helical hypersurface $\mathbf{x}$ are obtained by using the Cayley–Hamilton theorem. The umbilical conditions for the curvatures of $\mathbf{x}$ are given. Finally, the Laplace–Beltrami operator of the generalized helical hypersurface with a time-like axis is presented in ${\mathbb{E}}_1^4$.

中文翻译：

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闵可夫斯基时空中具有类时轴的广义螺旋超曲面

在本文中，广义螺旋超曲面$\mathbf{X}=\mathbf{X}(你,v,w)$在 Minkowski 时空中具有类时轴${\mathbb{乙}}_1^4$被考虑。第一和第二基本形式矩阵、高斯图和形状算子矩阵$\mathbf{X}$被计算。此外，广义螺旋超曲面的曲率$\mathbf{X}$通过使用 Cayley-Hamilton 定理获得。曲率的脐带条件$\mathbf{X}$给出。最后，具有类时轴的广义螺旋超曲面的拉普拉斯-贝尔特拉米算子在${\mathbb{乙}}_1^4$.