Hypersurfaces with L_{r}-Pointwise 1-Type Gauss Map

Akram Mohammadpouri

University of Tabriz, Department of Pure Mathematics, Faculty of Mathematical Sci-
ences, Tabriz, Iran
E-mail: pouri@tabrizu.ac.ir

Received March 9, 2016, revised December 15, 2016.

Abstract

In this paper, we study hypersurfaces in 𝔼^{n+1} whose Gauss map G satisfies
the equation L_{r}G = f(G + C) for a smooth function f and a constant
vector C, where L_{r} is the linearized operator of the (r+1)-st mean curvature of the hypersurface, i.e., L_{r}(f) = Tr(P_{r} ○∇^{2}f) for f ∈ 𝐶^{∞}(M), where Pr is the r-th Newton transformation, ∇^{2}f is the Hessian of f, L_{r}G =
(L_{r}G_{1}, . . . ,L_{r}G_{n+1}) and G = (G_{1}, . . . ,G_{n+1}). We focus on hypersurfaces with constant (r + 1)-st mean curvature and constant mean curvature. We obtain some classification and characterization theorems for these classes of hypersurfaces.