$\mathbb{R}^3$ 中线汇与 $\mathbb{R}^4$ 中曲面之间的对偶关系
Dual relations between line congruences in $\mathbb{R}^3$ and surfaces in $\mathbb{R}^4$
AI总结：
本文研究 $\mathbb{R}^3$ 中线汇与 $\mathbb{R}^4$ 中曲面间的对偶关系，该对偶将主切线、判别曲线、脊线及次抛物曲线分别映射为渐近线、抛物曲线、平坦脊线及平坦次抛物曲线，并探讨次抛物曲线在判别曲线处的典型行为及 Loewner 猜想。
摘要：
$\mathbb{R}^3$ 中的线汇与 $\mathbb{R}^4$ 中的曲面之间存在一种自然的对偶关系，该关系将主切线映射为渐近线。同一对应关系将线汇的判别曲线映射为对偶曲面的抛物曲线。此外，它将脊线映射为平坦脊线，而线汇的次抛物曲线则被映射到曲面上的某些曲线，我们称之为平坦次抛物曲线。在本文中，我们讨论这些关系，并描述线汇的判别曲线（等价地，对偶曲面的抛物曲线）上次抛物曲线的典型行为。我们还从对偶的视角讨论 Loewner 猜想。
英文摘要：
There is a natural duality between line congruences in $\mathbb{R}^3$ and surfaces in $\mathbb{R}^4$ that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat ridge curves, while the subparabolic curves of a line congruence are taken to certain curves on the surface that we call flat subparabolic curves. In this paper, we discuss these relations and describe the generic behavior of the subparabolic curves at the discriminant curve of the line congruence, or equivalently, the parabolic curve of the dual surface. We also discuss Loewner's conjectures under the duality viewpoint.
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