<p>In this paper, we extend Kummer’s theory for line congruences to the case <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lbrace x, \xi \rbrace $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$x: U \rightarrow \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is a smooth map and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\xi : U \rightarrow \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is a proper frontal. We show that if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lbrace x, \xi \rbrace $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is a normal congruence, the equation of the principal surfaces is a multiple of the equation of the developable surfaces, furthermore, the zero set of this multiplicative factor is the singular set of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation>.</p>

Line Congruences on Singular Surfaces

期刊论文
发表日期：

2025年11月19日

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Débora Lopes da Silva,
Tito Alexandro Medina Tejeda,
Maria Aparecida Soares Ruas,
Igor Chagas Santos

摘要

In this paper, we extend Kummer’s theory for line congruences to the case $\lbrace x, \xi \rbrace $ ${x, ξ}$ , where $x: U \rightarrow \mathbb {R}^3$ $x : U \to R^{3}$ is a smooth map and $\xi : U \rightarrow \mathbb {R}^3$ $ξ : U \to R^{3}$ is a proper frontal. We show that if $\lbrace x, \xi \rbrace $ ${x, ξ}$ is a normal congruence, the equation of the principal surfaces is a multiple of the equation of the developable surfaces, furthermore, the zero set of this multiplicative factor is the singular set of $\xi $ $ξ$ .