Bertrand Curves in $n$-Dimensional Riemann-Otsuki Space

Year 2025, Issue: 50, 38 - 47, 28.03.2025

Jeta Alo , Leyla Zeren Akgün

https://doi.org/10.53570/jnt.1630419

Abstract

In this paper, we extend the classic properties of Bertrand curves in Euclidean 3-space to an $n$-dimensional Riemann-Otsuki space. We introduce the concept of infinitesimal deformations of curves within this space, and by applying the Frenet formulas concerning the contravariant component of the covariant derivative, we derive conditions under which a given deformation of a curve corresponds to a Bertrand curve in this $n$-dimensional space.

Keywords

General connections, Riemann-Otsuki space, Frenet frame, Bertrand curves

References

• T. Otsuki, Tangent Bundles of order 2 and general connections, Mathematical Journal of Okayama University 8 (2) (1958) 143--179.
• T. Otsuki, On tangent bundles of order 2 and affine connections, Proceedings of the Japan Academy 34 (6) (1958) 325--330.
• T. Otsuki, On general connections I, Mathematical Journal of Okayama University 9 (1960) 99--164.
• T. Otsuki, On general connections II, Mathematical Journal of Okayama University 10 (1961) 113--124.
• T. Otsuki, On metric general connections, Proceedings of the Japan Academy 37 (1961) 183--188.
• Dj. F. Nadj, The Frenet formula of Riemann-Otsuki space, Review of Research Faculty of Science University of Novi Sad Mathematics Series 16 (1) (1986) 95--106.
• Dj. F. Nadj, The Gauss, Codazzi and Kühne equations of Riemann-Otsuki spaces, Acta Mathematica Hungarica 44 (3-4) (1984) 255--260.
• A. Moor, Otsukische übertragung mit rekurrentem masstensor, Acta Scientiarum Mathematicarum 40 (1-2) (1978) 129--142.
• A. Moor, Über die veränderung der länge der vektoren in Weyl-Otsukischen räumen, Acta Scientiarum Mathematicarum 41 (1-2) (1979) 173--185.
• B. Pirinççi, Congruence of curves in Weyl-Otsuki spaces, Adıyaman University Journal of Science 14 (2) (2024) 123--139.
• H. A. Hayden, On a generalized helix in a Riemannian $n$-space , Proceedings of the London Mathematical Society s2-32 (1) (1931) 337--345.
• J. A. Schouten, E. R. van Kampen, Beitrage zur theorie der deformation, Prace Matematyczno-Fizyczne 41 (1) (1934) 1--19.
• K. Yano, K. Takano, Y. Tomonaga, On infinitesimal deformations of curves in spaces with linear connection, Proceedings of the Japan Academy 22 (10) (1946) 294--309.
• L. R. Pears, Bertrand curves in Riemannian space, Journal of the London Mathematical Society s1-10 (3) (1935) 180--183.
• J. Alo, Generalized helices on n-dimensional Riemann-Otsuki space, Beykent University Journal of Science and Engineering, 12 (1) (2019) 6--11.
• M. Y. Yilmaz, M. Bektaş, General properties of Bertrand curves in Riemann-Otsuki space, Nonlinear Analysis: Theory, Methods and Applications 69 (10) (2008) 3225--3231.
• Y. Li, A. Uçum, K. İlarslan, Ç. Camcı, A new class of Bertrand curves in Euclidean 4-space, Symmetry 14 (6) (2022) 1191.