Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3

• Autorzy: Singh Kuljeet , Sharma Sandeep , Lone Mohamd Saleem

Identyfikatory

DOI

10.1515/dema-2023-0145

• Języki publikacji: EN

Abstrakty

EN

An osculating curve is a type of curve in space that holds significance in the study of differential geometry. In this article, we investigate certain geometric invariants of osculating curves on smooth and regularly immersed surfaces under conformal transformations in Euclidean space ℝ3. The primary objective of this article is to explore conditions sufficient for the conformal invariance of the osculating curve under both conformal transformations and isometries. We also compute the tangential and normal components of the osculating curves, demonstrating that they remain invariant under the isometry of the surfaces in ℝ3.

Słowa kluczowe

EN

osculating curve; conformal transformation; homothetic transformation; isometry of surfaces,; normal and tangential components

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2024
• Tom: Vol. 57, nr 1
• Strony: art. no. 20230145
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Singh Kuljeet

• School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, Jammu and Kashmir, India

autor

Sharma Sandeep

• School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, Jammu and Kashmir, India

autor

Lone Mohamd Saleem

• Department of Mathematics, School of Physical and Mathematical Sciences, University of Kashmir, Srinagar- 190006, Jammu and Kashmir, India

Bibliografia

• [1] M. P. do Carmo, Differential Geometry of Curves and Surfaces, Dover Publications, Mineola, New York, 2016.
• [2] B. Y. Chen, What does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly 110 (2003), no. 2, 147–152.
• [3] S. Deshmukh, B. Y. Chen, and S. H. Alshammari, On rectifying curves in Euclidean 3-space, Turk. J. Math. 42 (2018), no. 2, 609–620.
• [4] M. He, D. B. Goldgof, and C. Kambhamettu, Variation of Gaussian curvature under conformal mapping and its application, Comput. Math. Appl. 26 (1993), no. 1, 63–74.
• [5] B. Y. Chen and F. Dillen, Rectifying curve as centrode and extremal curve, Bull. Inst. Math. Acad. Sinica 33 (2005), no. 2, 77–90.
• [6] A. A. Shaikh and P. R. Ghosh, Rectifying and osculating curves on a smooth surface, Indian J. Pure Appl. Math. 51 (2020), no. 1, 67–75.
• [7] M. S. Lone, Geometric invariants of normal curves under conformal transformation in E3, Tamkang J. Math. 53 (2022), no. 1, 75–87.
• [8] A. I. Bobenko and C. Gunn, DVD-Video PAL, Springer VideoMATH, Heidelberg, Berlin, Germany, 2018, https://www.springer.com/us/book/9783319734736.
• [9] A. A. Shaikh and P. R. Ghosh, Rectifying curves on a smooth surface immersed in the Euclidean space, Indian J. Pure Appl. Math. 50 (2019), no. 4, 883–890.
• [10] A. A. Shaikh, M. S. Lone, and P. R. Ghosh, Rectifying curves under conformal transformation, J. Geom. Phys. 163 (2021), Article no. 104117.
• [11] A. A. Shaikh, M. S. Lone, and P. R. Ghosh, Normal curves on a smooth immersed surface, Indian J. Pure Appl. Math. 51 (2020), 1343–1355.
• [12] A. A. Shaikh, M. S. Lone, and P. R. Ghosh, Conformal image of an osculating curve on a smooth immersed surface, J. Geom. Phys. 151 (2020), Article no. 103625.
• [13] M. Obata, Conformal transformations of Riemannian manifolds, J. Differ. Geom. 4 (1970), no. 3, 311–333.
• [14] K. Ilarslan and E. Nešović, Timelike and null normal curves in Minkowski space E13, Indian J. Pure Appl. Math. 35 (2004), 881–888.
• [15] S. Sharma and K. Singh, Some aspects of rectifying curves on regular surfaces under different transformations, Int. J. Appl. 21 (2023), 78.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).