Affine translation hypersurfaces in Euclidean and isotropic spaces

2016/11/17 by Aydin, Muhittin Evren
#53A05 #53A35 #53B25 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.05608

Abstract

In this paper, we extend the notion of affine translation surfaces introduced by Liu and Yu (Proc. Japan Acad. Ser. A Math. Sci. 89, 111--113, 2013) in a Euclidean space R³ to higher dimensional ambient spaces. We provide that an affine translation hypersurface of constant Gauss-Kronocker curvature K₀ in Rⁿ⁺¹ is a cylinder, i.e. K₀=0. As further applications we describe such hypersurfaces in the isotropic spaces satisfying certain conditions on the isotropic curvatures and the Laplacian.

Related

• Translation hypersurfaces with constant curvature in 4-dimensional isotropic space
• Translation hypersurfaces with constant Sr curvature in the Euclidean space
• Translation surfaces in Euclidean space with constant Gaussian curvature
• Classifications of translation surfaces in isotropic geometry with constant curvature
• Classification of translation surfaces in ℝ³ with constant sectional curvature