三维反德西特空间中伪球面标架浸入的焦集奇点
Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space
AI总结：
本文在三维反德西特时空中引入伪球面非零标架曲线，利用活动标架定义渐屈线和焦曲面，揭示渐屈线恰为焦曲面的奇点集，并通过勒让德奇点理论将其解释为波前。
摘要：
我们引入三维反德西特时空中的伪球面非零标架曲线，并建立这些曲线的存在性与唯一性。然后，我们给出沿伪球面标架曲线的活动标架，这些活动标架即使在曲线的奇点处也有良好定义。这些活动标架使我们能够定义伪球面标架浸入的渐屈线和焦曲面。我们研究这些渐屈线和焦曲面的奇点性质。随后我们揭示，伪球面标架浸入的渐屈线是其焦曲面的奇点集合。我们还将渐屈线和焦曲面解释为某些高度函数的判别式集和次级判别式集，这使我们能够从勒让德奇点理论的观点将渐屈线和焦曲面解释为波前。文中提供例子以充实我们的结果，并使用双曲Hopf映射来可视化这些例子。
英文摘要：
We introduce pseudo-spherical non-null framed curves in the three-dimensional anti-de Sitter spacetime and establish the existence and uniqueness of these curves. We then give moving frames along pseudo-spherical framed curves, which are well-defined even at singular points of the curve. These moving frames enable us to define evolutes and focal surfaces of pseudo-spherical framed immersions. We investigate the singularity properties of these evolutes and focal surfaces. We then reveal that the evolute of a pseudo-spherical framed immersion is the set of singular points of its focal surface. We also interpret evolutes and focal surfaces as the discriminant and the secondary discriminant sets of certain height functions, which allows us to explain evolutes and focal surfaces as wavefronts from the viewpoint of Legendrian singularity theory. Examples are provided to flesh out our results, and we use the hyperbolic Hopf map to visualize these examples.
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