In this work, we model two event horizons associated with a null Cartan curve as two lightlike hypersurfaces, respectively. We define a lightlike surface and a spacelike surface whose images coincide with the sets of critical values of two event horizons, and meanwhile we present two curves whose images coincide with the sets of critical values of these two surfaces, respectively. Using the singularity theory, we characterize the local topological structures of two event horizons, two surfaces and two curves at their singularities by means of two new invariants. Moreover, we also present a spacelike braneworld model along the particle as a spacelike surface in hyperbolic 3-space. An important fact shows that from the viewpoint of Legendrian dualities, this surface is ${\mathrm{\Delta }}_2$-dual to the tangent trajectory $L(t)$ of the null Cartan curve in Lorentz–Minkowski space-time. Meanwhile, we also consider a curve whose image is the set of critical values of this surface in hyperbolic 3-space. The third invariant of the null Cartan curve characterizes the singularities of the surface $\mathcal{ℒ}\mathcal{𝒮}$ and the curve $\mathcal{𝒫}$ in hyperbolic 3-space. A result indicates that surface $\mathcal{ℒ}\mathcal{𝒮}$ is locally diffeomorphic to the swallowtail $SW$ or cuspidal edge $CE$ and $\mathcal{𝒫}$ is locally diffeomorphic to the $(2,3,4)$-cusp at certain a singular point. It is also shown that there exist deep relationships between the singularities of the surface $\mathcal{ℒ}\mathcal{𝒮}$ and the curve $\mathcal{𝒫}$ and the order of contact between $L(t)$ and elliptic quadric ${ℰ}{𝒬}(v_0,-1)$ or the order of contact between $L(t)$ and spacelike hyperplane $\mathrm{\text{HP}}(v_0,-1)$. Finally, we present several examples to describe the main results.

在这项工作中，我们将与零嘉当曲线相关的两个事件视界分别建模为两个类光超曲面。我们定义了一个类光表面和一个类空间表面，它们的图像与两个事件视界的临界值集重合，同时我们给出了两条曲线，它们的图像分别与这两个表面的临界值集重合。利用奇点理论，我们通过两个新的不变量在奇点处刻画了两个事件视界、两个曲面和两条曲线的局部拓扑结构。此外，我们还提出了一个沿粒子的类空间膜世界模型，作为双曲 3 空间中的类空间表面。一个重要的事实表明，从 Legendrian 对偶的观点来看，这个表面是${\mathrm{\Delta }}_2$-对切线轨迹$\mathrm{大号}(吨)$Lorentz-Minkowski 时空中的零嘉当曲线。同时，我们还考虑了一条曲线，其图像是该曲面在双曲 3 空间中的一组临界值。零嘉当曲线的第三个不变量表征了曲面的奇异性$\mathcal{ℒ}\mathcal{𝒮}$和曲线$\mathcal{𝒫}$在双曲 3 空间中。结果表明，表面$\mathcal{ℒ}\mathcal{𝒮}$与燕尾局部微分同胚$\mathrm{小号}W$或尖瓣边缘$C乙$和$\mathcal{𝒫}$是局部微分同胚的$(2,3,4)$-尖点在某个奇异点。还表明，表面的奇点之间存在着深厚的关系$\mathcal{ℒ}\mathcal{𝒮}$和曲线$\mathcal{𝒫}$以及之间的联系顺序$\mathrm{大号}(吨)$和椭圆二次曲线${ℰ}{𝒬}(v_0,-1)$或之间的联系顺序$\mathrm{大号}(吨)$和类空间超平面$\mathrm{\text{生命值}}(v_0,-1)$. 最后，我们提出了几个例子来描述主要结果。