Inspired by the optical phenomenon of conical refraction, discovered by Hamilton in 1832, we study the existence of singular optical phenomena associated with linear differential operators acting on vector fields on a surface. We do this by studying the singularities of the Fresnel hyper-surface associated with the differential operator and show that the existence of these singularities can be accounted for from purely topological considerations. We associate a topological number (an integer) to the space of singularities of the Fresnel hyper-surface, which can be used to “count” the number points at which singular optical phenomena occur.

中文翻译：

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用于表面微分算子的奇异几何光学

受汉密尔顿1832年发现的锥形折射光学现象的启发，我们研究了与作用于表面矢量场上的线性微分算子相关的奇异光学现象的存在。我们通过研究与微分算子相关的菲涅耳超曲面的奇异性来做到这一点，并表明这些奇异性的存在可以从纯粹的拓扑考虑中解决。我们将拓扑数（整数）与菲涅耳超曲面的奇异性空间相关联，该空间可用于“计数”发生奇异光学现象的数量点。