Canal surfaces with generalized 1-type Gauss map

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 = fG+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.


Introduction
In the 1970's, Chen introduced the notion of finite-type submanifolds in Euclidean space or pseudo-Euclidean space, which was extended to differential maps, in particular to Gauss maps of submanifolds. The notions of finite-type immersion and finite-type Gauss map are very useful tools in investigating and characterizing many important submanifolds [3,1,2]. The simplest finite-type Gauss map is of 1-type, i.e., the Gauss map G of a submanifold M in Euclidean space or pseudo-Euclidean space satisfies ∆G = λ(G + C) for some constant λ ∈ R (λ = 0) and a constant vector C, where ∆ denotes the Laplace operator defined on M, given by where (x 1 , . . . , x n ) is a local coordinate system of M,g ij the components of the inverse matrix of the first fundamental form of M, and G the determinant of the first fundamental form of M. Planes, circular cylinders, and spheres in Euclidean 3-space are typical examples of surfaces with 1-type Gauss maps [12]. As a generalization of 1-type Gauss map, the definition of submanifold with pointwise 1-type Gauss map is proposed in [9], which takes the form ∆G = f (G+C) 200 JINHUA QIAN, MENGFEI SU, AND YOUNG HO KIM for a non-zero smooth function f and a constant vector C. For example, a helicoid, a catenoid, and a right cone in Euclidean 3-space are typical examples of surfaces with pointwise 1-type Gauss maps. Some related works have been done; for example, surfaces of revolution with pointwise 1-type Gauss map were studied in [5], and ruled submanifolds and hypersurfaces of Euclidean space with pointwise 1-type Gauss map were discussed in [6,7,9].
Very recently, the authors of [12] considered a cylindrical surface parameterized by  Indeed, the notion of generalized 1-type Gauss map can be regarded as a generalization of both 1-type Gauss map and pointwise 1-type Gauss map, since if both f and g are non-zero constants, then (1.1) can be written as ∆G = λ(G + C), (0 = λ ∈ R). In this case, the Gauss map is just of 1-type in the usual sense. If the function f is equal to g, (1.1) can be expressed as ∆G = f (G + C), which is called the Gauss map of pointwise 1-type. More precisely, the pointwise 1-type Gauss map is said to be of the first kind when C = 0, and of the second kind otherwise. If f and g vanish identically, then G is said to be harmonic.
Based on the definition of generalized 1-type Gauss map, the authors of [12] completely classified developable surfaces in Euclidean 3-space with a generalized 1-type Gauss map, i.e., cylindrical surfaces, conical surfaces, and tangent developable surfaces. Naturally, this idea of generalized 1-type Gauss map of submanifolds can be extended into many other submanifolds both in Euclidean space and in pseudo-Euclidean space.
The class of surfaces formed by sweeping a sphere was first investigated by Monge in 1850, who named them canal surfaces. Canal surfaces may be generated either by sweeping a sphere along a path, or by sweeping a particular circular cross-section of the sphere along the same path [8]. Canal surfaces are useful for representing long thin objects, e.g., pipes, poles, ropes, 3D fonts or intestines; they are also frequently used in solid and surface modelling for CAD/CAM. Representative examples are natural quadrics, tori, tubular surfaces, and Dupin cyclides. In 2016, the authors of [8] investigated the geometric properties of canal surfaces in Euclidean 3-space, and in 2019 they classified canal surfaces with pointwise 1-type Gauss map in [11]. Based on the conclusions achieved in [11] and the definition of generalized 1-type Gauss map, we would like to discuss in this work the canal surfaces with generalized 1-type Gauss map. This paper is organized as follows. In Section 2, some fundamental facts of canal surfaces are briefly recalled. In Section 3, surfaces of revolution and canal surfaces with generalized 1-type Gauss map are surveyed. Finally, some examples with generalized 1-type Gauss map are shown in Section 4.
Throughout this paper, we assume that all objects are smooth and all surfaces are connected, unless otherwise stated.

Preliminaries
Let M be a hypersurface in the Euclidean (n + 1)-space E n+1 . We denote the Levi-Civita connections of E n+1 and M by∇ and ∇, respectively. Let X, Y be vector fields tangent to M and let ξ be a unit normal vector field of M. Then the Gauss and Weingarten formulas are given respectively bỹ Here, h is the second fundamental form and A ξ is the shape operator (or the For a hypersurface in the Euclidean space E n+1 , the following lemma plays an important role.
and ∇H is the gradient of the mean curvature H.
Note that, for a surface in E 3 , the gradient ∇H can be obtained by the following lemma.
where {s, t} is a local coordinate system of M such that ∂s, ∂s = g 11 , ∂s, ∂t = g 12 , and ∂t, ∂t = g 22 , and f s , f t are the partial derivatives of f with respect to s and t, respectively.
In order to classify completely the canal surfaces with generalized 1-type Gauss map, we review some basic facts of canal surfaces in E 3 according to [8].
A canal surface M in E 3 is an immersed surface swept out by a sphere moving along an arbitrary curve c = c(s) or by a particular circular cross-section of the sphere along the same path. Due to the generating process, the parametrization of M can be given as Initially, from (2.2) and the Frenet formula of a regular space curve c in E 3 with curvature κ and torsion τ , the quantities of the first fundamental form are given by Then, we have Meanwhile, the Gauss map G of M is given by Based on (2.4) and (2.7), the shape operator of M is obtained as (2.9) Note that, from (2.5), P = 0 everywhere due to the regularity of M. From (2.8), we see that the Gaussian curvature K and the mean curvature H are written, respectively, as At the same time, we have the following result.

Main results
In this section, we will focus on the surfaces of revolution and canal surfaces which have generalized 1-type Gauss maps.
3.1. Surfaces of revolution with generalized 1-type Gauss map. Let M be a surface of revolution in E 3 parameterized by for some smooth functions ψ and φ. Without loss of generality, we assume that the profile curve is of unit speed, i.e., ψ 2 + φ 2 = 1. A direct computation shows that the Gauss map G of M is Then, the Laplacian ∆G of the Gauss map G can be written as Explicitly, it can be expressed as The second and third of these equations obviously imply that C 2 = C 3 = 0. And where C 1 is a non-zero constant. Conversely, if we use the above information with the given functions ψ and φ, a surface of revolution with unit speed profile curve satisfies ∆G = f G + gC for such non-zero functions (f, g) and a constant vector C. Thus, we have the following result. 3.2. Canal surfaces with generalized 1-type Gauss map. In [11], the authors obtained the Laplacian of the Gauss map G for a canal surface M. We will recall the relevant process as follows.

Theorem 3.2. 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:
(1) a surface of revolution such as
Proof. Suppose that an oriented canal surface M satisfies ∆G = f G+gC. Without loss of generality, we may decompose the constant vector C as where C 1 = C, T , C 2 = C, N , C 3 = C, B . Substituting (2.6), (3.9), and (3.10) into ∆G = f G + gC, we obtain (3.11) From the first equation of (3.11), we have the following two cases: Case 1: r = − cos ϕ = 0. In this case, we have Taking (3.12) into the second and third equations of (3.11), we have (3.13) According to (3.13), we have Rearranging (3.14) with the help of (2.4) and (3.8), we get Furthermore, by comparing the coefficients of the highest degree of the power of sin 3θ in (3.17), we obtain that C 1 = 0, then C = (0, 0, 0). In this situation, M has pointwise 1-type Gauss map of the first kind, i.e., ∆G = f G. In this case, O is part of a catenoid or a surface of revolution, i.e., κ = 0 (for the details, see [11]). Therefore, O is empty and κ ≡ 0, i.e., M is a surface of revolution. Simplifying Since P, D, U are all functions of s when κ = 0, (3.19) yields that the functions f, g depend on s only, i.e., f = f (s), g = g(s). Then, simplifying (3.19) with the help of (3.5), (3.6), and (3.8), we get (3.20) It is obvious that M is a surface of revolution with generalized 1-type Gauss map. Without loss of generality, we may assume the center curve c(s) = (s, 0, 0) and T = (1, 0, 0), N = (0, 1, 0), B = (0, 0, 1). Then, M can be represented by
First of all, suppose that C 1 = 0. Then, from the first equation of (3.11) we get (3.24) Taking (3.24) into the second and third equations of (3.11), we obtain According to this equation, we have Therefore, κ = c (0 = c ∈ R) and τ = 0; then the center curve c of M is a circle. This, together with the fact that r is constant, implies that M is a torus. Furthermore, from the second and third equations of (3.11), we have . (3.27) Since K, H are all functions of θ only when r = 0, (3.27) yields that the functions (f, g) depend only on θ. Therefore, the Gauss map G of M is of generalized 1-type for some non-zero smooth functions (f, g) given by (3.27) and a constant vector C = (0, C 2 , C 3 ), where C 2 2 + C 2 3 = 0. Conversely, suppose that M is a torus or a surface of revolution parameterized by x(s, θ) = (r(s) cos ϕ(s) + s, r(s) sin ϕ(s) cos θ, r(s) sin ϕ(s) sin θ) satisfying (3.22). One can easily check that ∆G = f G+gC is satisfied for some nonzero smooth functions (f, g) given by (3.23) and (3.27) with the constant vectors C = (0, C 2 , C 3 ) and C = (C 1 , 0, 0), respectively. This completes the proof.
As immediate consequences of the above theorem, we can easily get the following conclusions.