Abstract
In the present work, the notion of generalized Cheng–Yau 1-type Gauss map is proposed, which is similar to the idea of generalized 1-type Gauss maps. Based on this concept, the surfaces of revolution and the canal surfaces in the Euclidean three-space are classified. First of all, we show that the Gauss map of any surfaces of revolution with a unit speed profile curve is of generalized Cheng–Yau 1-type. At the same time, an oriented canal surface has a generalized Cheng–Yau 1-type Gauss map if, and only if, it is an open part of a surface of revolution or a torus.
Highlights
The finite-type immersion and finite-type Gauss map proposed by B
An oriented submanifold M is of a generalized Cheng–Yau 1-type Gauss map in the Euclidean space Em if its Gauss map G satisfies
Based on previous works about such surfaces [10,12,13], we focus on the canal surfaces of generalized Cheng–Yau 1-type Gauss maps in this work
Summary
The finite-type immersion and finite-type Gauss map proposed by B. 1-type Gauss map fulfills ∆G = f (G + C ) for a constant vector C and a non-zero smooth function f. The authors of [5] completely classified the developable surfaces, in Euclidean three-space, of the generalized 1-type Gauss map. The concepts of finite-type and pointwise 1-type Gauss maps for the submanifolds in Euclidean space have been extended and have taken the place of the Laplace operator ∆ with the Cheng–Yau operator 2. A submanifold M is an L1 -pointwise 1-type Gauss map when its Gauss map can be expressed as 2G = f (G + C ) for a constant vector C and a non-zero smooth function f. The surfaces discussed here are regular, smooth and topologically connected
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