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, En, is said to be of generalized 1-type if, for the Laplace operator, Δ, on the submanifold, it satisfies ΔG=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 E3. Second, we show that the Gauss map of any cylindrical surface in E3 is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in E3, except planes. Finally, we show that cylindrical hypersurfaces in En+2 always have generalized 1-type Gauss maps.
Highlights
The notion of finite type submanifolds in a Euclidean space or a pseudo-Euclidean space was introduced by Chen in the 1980s [1]
We study developable surfaces in E3 : cylindrical surfaces, conical surfaces and tangent developable surfaces
We prove that cylindrical hypersurfaces in En+2 always have generalized 1-type Gauss maps
Summary
The notion of finite type submanifolds in a Euclidean space or a pseudo-Euclidean space was introduced by Chen in the 1980s [1]. He extended this notion to a general differential map, namely, the Gauss map, on the submanifolds. As a generalization of a 1-type Gauss map, the first and third authors introduced the notion of a pointwise 1-type Gauss map of submanifolds in reference [15]. In [22], Dursun studied flat surfaces in E3 with a pointwise 1-type Gauss map and proved the following proposition. Throughout this paper, we assume that all objects are smooth and all surfaces are connected unless mentioned
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