Hypersurfaces with Generalized 1-Type Gauss Maps

Abstract

In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, E n, is said to be of generalized 1-type if, for the Laplace operator, D, on the submanifold, it satisfies D G = fG + gC, where C is a constant vector and f and g are some functions. The notion of a generalized 1-type Gauss map is a generalization of both a 1-type Gauss map and a pointwise 1-type Gauss map. With the new definition, first of all, we classify conical surfaces with a generalized 1-type Gauss map in E 3. Second, we show that the Gauss map of any cylindrical surface in E 3 is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in E 3, except planes. Finally, we show that cylindrical hypersurfaces in E n + 2 always have generalized 1-type Gauss maps.