Characterizations of the Bertrand offsets of ruled surfaces with the Laplace operator in Euclidean 3-space $ \mathbb{E}^3 $

Duygu Çağlar Çay; Nurten Gürses; Duygu Çağlar Çay; Nurten Gürses

doi:10.3934/math.2026959

AIMS Mathematics

2026, Volume 11, Issue 8: 23817-23846. doi: 10.3934/math.2026959

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Research article

Characterizations of the Bertrand offsets of ruled surfaces with the Laplace operator in Euclidean 3-space $ \mathbb{E}^3 $

Duygu Çağlar Çay ^{1
,
,},
Nurten Gürses ²

1.
Department of Software Engineering, Faculty of Engineering and Architecture, Istanbul Gelisim University, Istanbul 34310, Türkiye
2.
Department of Mathematics, Faculty of Arts and Sciences, Yildiz Technical University, Istanbul 34220, Türkiye

Received: 09 May 2026 Revised: 18 July 2026 Accepted: 28 July 2026 Published: 04 August 2026
MSC : 47A75, 53A04, 53A05, 53A55

In the present paper, the Laplace operators with respect to the first and second fundamental forms of the Bertrand offset of a non-developable ruled surface were investigated, taking into account the structure functions of the non-developable ruled surface. For this purpose, the Bertrand offset of a non-developable ruled surface was considered, and its fundamental forms, mean curvature, and Gaussian curvature were calculated. Thus, finite (or infinite) type characterizations of the surface were obtained, and the relationships between Bertrand ruled surface pairs were established through the resulting Laplace operators.
- non-developable ruled surface,
- Bertrand offset,
- fundamental forms,
- Laplace operator,
- finite type surface
Citation: Duygu Çağlar Çay, Nurten Gürses. Characterizations of the Bertrand offsets of ruled surfaces with the Laplace operator in Euclidean 3-space $ \mathbb{E}^3 $[J]. AIMS Mathematics, 2026, 11(8): 23817-23846. doi: 10.3934/math.2026959

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Abstract

In the present paper, the Laplace operators with respect to the first and second fundamental forms of the Bertrand offset of a non-developable ruled surface were investigated, taking into account the structure functions of the non-developable ruled surface. For this purpose, the Bertrand offset of a non-developable ruled surface was considered, and its fundamental forms, mean curvature, and Gaussian curvature were calculated. Thus, finite (or infinite) type characterizations of the surface were obtained, and the relationships between Bertrand ruled surface pairs were established through the resulting Laplace operators.

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