Canal surfaces with generalized 1-type Gauss map

Abstract

This work considers a kind of classification of canal surfaces in terms of their Gauss map G in Euclidean 3-space. We introduce the notion of generalized 1-type Gauss map for a submanifold that satisfies ∆G = f G + gC, where ∆ is the Laplace operator, C is a constant vector, and (f, g) are non-zero smooth functions. First of all, we show that the Gauss map of any surface of revolution with unit speed profile curve in Euclidean 3-space is of generalized 1-type. At the same time, the canal surfaces with generalized 1-type Gauss map are discussed.

Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map

“…Theorem 16.32. [141] An oriented canal surface M in E 3 has generalized 1-type Gauss map if and only if it is one of the following surfaces:…”

Dual Associate Null Scrolls with Generalized 1-Type Gauss Maps

“…Considering the normalization condition of a null scroll and the Frenet frame of a null curve, in the present work, a pair of null scrolls satisfying the same normalization condition are constructed, i.e., the tangent vector field of the base curve of the first null scroll is set as the ruling flow of the second null scroll and the tangent vector field of the base curve of the second null scroll is set as the ruling flow of the first null scroll. Since the 1970's, many research works about the classification of submanifolds respect to the Laplacian of Gauss maps have been done in Euclidean space and Minkowski space, which are very useful tools in investigating and characterizing many important submanifolds [12][13][14][15]. Based on the fundamental geometric properties of the null scroll pair, we aim at the Laplacian of the Gauss maps of the dual associate null scrolls according to the current progress for the classifications of submanifolds respect to the Gauss maps proposed in [16], i.e., the generalized 1-type Gauss map, which can be regarded as a generalization of both 1-type Gauss map and pointwise 1-type Gauss map.…”