Astrohelicoidal Hypersurfaces in 4-space

Abstract

We consider an astrohelicoidal hypersurface which its profile curve has astroid curve in the four dimensional Euclidean space ${\mathbb{E}}^{4}$. We also calculate Gaussian curvature and the mean curvature, and Weingarten relation of the hypersurface. Moreover, projecting hypersurface into 3-spaces, we draw some figures.

Keywords

• astrohelicoidal hypersurface,Gauss map,Gaussian curvature,mean curvature

References

1. Arslan, K., K\i l\i \c{c} Bayram, B., Bulca, B., \"{O}zt\"{u}rk, G., \textit{Generalized rotation surfaces in $\mathbb{E}^{4}$}, Results Math., \textbf{61}(2012), 315--327.
2. Eisenhart, L.P., A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications, N.Y., 1909.
3. Forsyth, A.R., Lectures on the Differential Geometry of Curves and Surfaces, Cambridge Un. press, 2nd ed. 1920.
4. Ganchev, G., Milousheva, V., \textit{General rotational surfaces in the 4-dimensional Minkowski space}, Turk. J. Math., \textbf{38}(2014), 883--895.
5. Gray, A., Salamon, S., Abbena, E., Modern Differential Geometry of Curves and Surfaces with Mathematica, Third ed. Chapman \& Hall/CRC Press, Boca Raton, 2006.
6. G\"{u}ler, E., Hac\i saliho{\u{g}}lu, H.H., Kim, Y.H., \textit{The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-Space}, Symmetry, \textbf{10(9)}(2018), 1--11.
7. G\"{u}ler, E., Magid, M., Yayl\i, Y., \textit{Laplace Beltrami operator of a helicoidal hypersurface in four space}, J. Geom. Sym. Phys., \textbf{41}(2016), 77--95.
8. G\"{u}ler, E., Turgay, N.C., {\em Cheng-Yau operator and Gauss map of rotational hypersurfaces in 4-space}, Mediterr. J. Math., \textbf{16(3)}(2019), 1--16.

9. Hac\i saliho{\u{g}}lu, H.H., Diferensiyel Geometri I, Ankara {\"U}n., Ankara, 1982.
10. Nitsche, J.C.C., Lectures on Minimal Surfaces. Vol. 1, Introduction, Fundamentals, Geometry and Basic Boundary Value Problems, Cambridge Un. Press, Cambridge, 1989.