A new characterization of Aminov surface with regards to its Gauss map in E^4

Year 2023, , 25 - 32, 22.03.2023

Sezgin Büyükkütük , Günay Öztürk

https://doi.org/10.17798/bitlisfen.1170647

Abstract

In this work, we focus on Aminov surface with regard to its Gauss map in E^4. Firstly, we write the covariant derivatives according to linear combinations of orthonormal vectors and separate the equalities using Gauss and Weingarten formulas. Then, we get the laplace of the Gauss map. After giving some conditions, we yield as main results: Aminov surfaces can not have harmonic Gauss map and can not have pointwise one-type Gauss map of I. kind in E^4. Further, we give an example of helical cylinder which is also congruent to an Aminov surface. Lastly, we obtain the conditions of having pointwise one-type Gauss map of II. kind.

Keywords

Aminov surface, Gauss map, Monge patch

References

• [1] B. Bulca, K. Arslan, " Surfaces Given with the Monge Patch in E^4 ," Journal of Mathematical Physics, Analysis, Geometry, vol. 9, pp. 435-447, 2013.
• [2] S. Büyükkütük, G. Öztürk, "A new type timelike surface given with Monge patch in E^4 , " TWMS J. App. and Eng. Math., vol. 11, 176–183, 2021.
• [3] B.Y. Chen, Geometry of Submanifolds. New York: Dekker, 1973.
• [4] B.Y. Chen, M. Choi, Y.H. Kim, "Surfaces of revolution with pointwise 1-type Gauss map.," J. Korean Math. Soc. vol . 42, 447–455, 2005.
• [5] G. Gray, "A Monge Patch.," Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton. FL: CRC Press. pp. 398–401, 1997.
• [6] J. Glymph, D. Schelden, C. Ceccato, J.Mussel, H. Schober, "A parametric strategy for free-form glass structures using quadrilateral planar facets," Automation in Construction, vol. 13, 187–202, 2004.
• [7] E.A. Pamuk, B. Bulca, "Translation-Factorable Surfaces in 4-dimensional Euclidean Space," Konuralp Journal of Mathematics. Vol. 9, 193–200, 2021.
• [8] L. Verstraelen, J. Walrave, S. Yaprak, "The Minimal Translation Surface in Euclidean Space," Schoow J. Math., vol. 20, 77–82, 1994.