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

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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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Authors and Affiliations

  1. Departamento de Matemática, PUC-Rio, Rio de Janeiro, RJ, Brasil

    Marcos Craizer

  2. Instituto de Matemática e Estatística- UFG, Goiânia, GO, Brasil

    Ronaldo Garcia

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  1. Marcos Craizer
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  2. Ronaldo Garcia
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Correspondence to Marcos Craizer.

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The authors want to thank CNPq, PRONEX/ CNPq/ FAPEG 2017 10 26 7000 508, and CAPES (Finance Code 001) for financial support during the preparation of this manuscript.

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Craizer, M., Garcia, R. Dual relations between line congruences in \(\mathbb {R}^3\) and surfaces in \(\mathbb {R}^4\). Res Math Sci 11, 30 (2024). https://doi.org/10.1007/s40687-024-00445-y

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