Dual relations between line congruences in $$\mathbb {R}^3$$ and surfaces in $$\mathbb {R}^4$$

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-04-09 DOI:10.1007/s40687-024-00445-y

Marcos Craizer, Ronaldo Garcia

引用次数: 0

Abstract

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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$$\mathbb {R}^3$$ 中的线全等与 $$\mathbb {R}^4$$ 中的面之间的双重关系

在 $\mathbb {R}^3$ 中的线段全等和在$\mathbb {R}^4$ 中的曲面之间有一种自然的对偶性，它将主线送入渐近线。同样的对应关系会把线段全等的判别曲线转化为对偶曲面的抛物曲线。此外，它还将脊曲线带入平脊曲线，而将线段全等的副抛物曲线带入曲面上的某些曲线，我们称之为平副抛物曲线。在本文中，我们将讨论这些关系，并描述副抛物线在线段全等的判别曲线（或等价于对偶曲面的抛物线）上的一般行为。我们还讨论了二重性视角下的卢瓦纳猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.