On the translation hypersurfaces with Gauss map G satisfying ΔG=AG

Abstract

It is a known fact that a translation hypersurface is obtained by combination of any three curves in the 4-dimensional Euclidean space. We examine a special situation where the Gauss map of a translation hypersurface satisfies the condition G D AG where represents the Laplace operator and A is a 4 4-real matrix. Our result is that such a translation hypersurface is one of the following three hypersurfaces: the hypersurface of translation surface and a constant vector along this surface, the hyperplane, the hypersurface˙ R where˙is a translation surface.

Translation Hypersurfaces in Euclidean 4-Spaces

“…The idea of investigating the translation surfaces by considering them from various perspectives in different dimensional spaces is a remarkable area for the geometers, [21,7]. First of all, the translation surfaces in 3-dimensional Euclidean space were discussed [10,5,1], then these studies were generalized and carried up to 4-dimensional and n-dimensional [7,19,20]. In addition, there are many studies in non-Euclidean Geometry (especially Lorentz-Minkowski and Galilean spaces) [25,24,3,2,4].…”