Geometric characterizations of canal surfaces with Frenet center curves

: In this work, we study the canal surfaces foliated by pseudo hyperbolic spheres H 20 along a Frenet curve in terms of their Gauss maps in Minkowski 3-space. Such kind of surfaces with pointwise 1-type Gauss maps are classiﬁed completely. For example, an oriented canal surface that has proper pointwise 1-type Gauss map of the ﬁrst kind satisﬁes ∆ G = − 2 K G , where K and G is the Gaussian curvature and the Gauss map of the canal surface, respectively.


Introduction
The idea of finite type immersion of Riemannian manifolds into Euclidean space (resp. pseudo Euclidean space) was introduced by B.Y. Chen in the late 1970's, which was extended to the differential maps on the submanifolds such as the Gauss maps. A submanifold M in Euclidean space (resp. pseudo Euclidean space) whose Gauss map G satisfies ∆G = f (G+C) is said to have a proper pointwise 1-type Gauss map for a non-zero smooth function f and a constant vector C, where ∆ is the Laplacian defined on M and in local coordinates given by where g i j are the components of the inverse matrix of the first fundamental form of M. Specially, it is said to be of the first kind or the second kind when the vector C is zero or non-zero, respectively. Furthermore, G is said to be of proper pointwise 1-type if the function f is not constant, otherwise a non-proper pointwise 1-type Gauss map is just of ordinary 1-type. When the smooth function f vanishes, G is said to be harmonic [1,2,5].
In the theory of surfaces, a canal surface is formed by moving a family of spheres whose centers lie on a space curve in Euclidean 3-space. The geometric characteristics of such surfaces have been studied by many experts and geometers [4,8,13]. For example, the authors of [4] investigated the geometric properties of such surfaces, including the Gaussian curvature, the mean curvature and their relationships. In recent years, the construction idea of canal surfaces in Euclidean 3-space is extended to Lorentz-Minkowski space. In Minkowski 3-space, a canal surface can be formed as the envelope of a family of pseudo-Riemannian space forms, i.e., pseudo spheres S 2 1 , pseudo hyperbolic spheres H 2 0 and lightlike cones Q 2 [3,9,12]. Let p be a fixed point, r > 0 be a constant in E 3 1 . The pseudo-Riemannian space forms, i.e., the de-Sitter space S 2 1 (p, r), the hyperbolic space H 2 0 (p, r) and the lightlike cone Q 2 1 (p) are defined by When r = 1 and the center p is the origin, we write them by S 2 1 , H 2 0 and Q 2 , simply. According to the classification of curves in Minkowski space, there are nine types of canal surfaces in Minkowski 3space whose fundamental geometric properties have been achieved by discussing the linear Weingarten canal surfaces in [3,9].
Based on the conclusions obtained in [4], a canal surface with pointwise 1-type Gauss map is discussed in [8]. In order to do further geometric investigation for canal surfaces in Minkowski 3space, in this work we study surfaces foliated by pseudo hyperbolic spheres H 2 0 along Frenet curves. In section 2, the Frenet formulas of Frenet curves, the parameterized equations and the relationships between the Gaussian curvatures and the mean curvatures of three types of canal surfaces are recalled. In section 3, three types of canal surfaces with pointwise 1-type Gauss maps are classified completely.
The surfaces which are discussed here are smooth, regular and topologically connected unless otherwise stated.

Preliminaries
Let E 3 1 be a Minkowski 3-space with natural Lorentzian metric in terms of the natural coordinate system (x 1 , x 2 , x 3 ). It is well known that a vector υ∈E 3 1 is called to be spacelike if υ, υ > 0 or υ = 0; timelike if υ, υ < 0; null (lightlike) if υ, υ = 0, respectively. The norm of a vector υ is given by υ = √ | υ, υ |. The timelike or lightlike vector is said to be causal [6]. Due to the causal character of the tangent vectors, the curves are classified into spacelike curves, timelike curves or lightlike (null) curves. What's more, the spacelike curves are classified into the first and the second kind of spacelike curves or the null type spacelike curves (pseudo null curves) according to their normal vectors are spacelike, timelike or lightlike, respectively.
where T is the tangent vector, N and B is the normal vector and the binormal vector of c(s), respectively. When c(s) is a timelike curve, 1 = 2 = −1; when c(s) is a spacelike curve of the first kind, 1 is called a canal surface which is formed as the envelope of a family of pseudo hyperbolic spheres H 2 0 (resp. pseudo spheres S 2 1 or lightlike cones Q 2 ) whose centers lie on a space curve c(s) framed by {T, N, B}. Then M can be parameterized by where λ, µ and ω are differential functions of s and θ, x(s, θ) − c(s) 2 = r 2 (s), ( = ±1 or 0). The curve c(s) is called the center curve and r(s) is called the radial function of M.
Precisely, if M is foliated by pseudo hyperbolic spheres H 2 0 (resp. pseudo spheres S 2 1 or lightlike cones Q 2 ), then = −1(resp. 1 or 0) and M is said to be of type M − (resp. M + or M 0 ). As well, the canal surfaces of type M − can be classified into M 1 − (resp. M 2 − or M 3 − ) when c(s) is spacelike (resp. timelike or null). Moreover, when c(s) is the first kind spacelike curve, the second kind spacelike curve and the pseudo null curve, M Remark 2.5. By Proposition 2.4, the principal curvatures κ 1 , κ 2 of the canal surface M 11 − (resp.
From now on, we concern on the classifications of three kinds of canal surfaces in terms of their Gauss maps. We only prove the results for M 11 − and omit the proofs for M 12 − and M 2 − since they can be similarly done to those of M 11 − .

The canal surfaces of type
where sinh ϕ(s) = r (s).
Due to f 0, the mean curvature cannot be constant. With the help of (3.6), Equation (3.19) can be rewritten as (3.23) Simplifying Eq (3.23) with the help of (3.22), the radial function r(s) satisfies where κ 1 is stated as (3.21). Conversely, if M 11 − is a surface of revolution satisfying (3.24), ∆G = f (G + C) can be satisfied for a non-zero function f as stated by (3.20) and a constant vector C = (C 1 , 0, 0) in which C 1 is a non-zero constant.
where H and K are given by (3.22), C 1 is a non-zero constant. − is a surface of revolution satisfying (3.24). By Corollary 3.3, we get From (3.24) and (3.25), we have where λ and C 1 are non-zero constants, κ 1 is stated as (3.21). The converse is straightforward.  With the help of Eq (3.3), we obtain P 2 1 H θ = 0. Therefore, H θ = 0 due to P 1 0. Furthermore, from the first two equations of (3.27), we get 2r 2 P 1 H s = 0. It is obvious that H s = 0. Then, the mean curvature of M 11 − is constant. By the Corollary 2 of [9], i.e., the canal surface M 11 − with non-zero constant mean curvature does not exist, thus the canal surface M 11 − is minimal. From the Theorem 4 of [9], it is a part of a surface of revolution with the following form where r(s) satisfies Looking back the Eq (3.27) with the conclusions obtained above, we have Conversely, suppose that M 11 − is a surface of revolution satisfying (3.29), M 11 − is minimal from the Theorem 4 of [9] and ∆G = f G is satisfied for a non-zero function f given by (3.30). − with proper pointwise 1-type Gauss map of the first kind satisfies Assume that a canal surface M 11 − satisfies ∆G = λG, (λ ∈ R − {0}). By Corollary 3.6, we have λ = − 2 r 2 is a constant, i.e., r is a constant. Thus, we have the following result.  where sinh ϕ(s) = r (s). Through direct calculations, we have where x 1 s = rr + cosh 2 ϕ + rκ cosh ϕ cosh θ; x 2 s = r cosh ϕ cosh θ + rr κ + rr ϕ cosh θ + rτ cosh ϕ sinh θ; x 3 s = r cosh ϕ sinh θ + rτ cosh ϕ cosh θ + rr ϕ sinh θ; where P 2 = rr + cosh 2 ϕ + rκ cosh ϕ cosh θ = rQ 2 + cosh 2 ϕ, From Eqs (4.2) and (4.3), the Gaussian curvature K and the mean curvature H of M 12 − are Remark 4.1. From g 11 g 22 − g 2 12 = r 2 P 2 2 , due to regularity, we see that P 2 0 everywhere. Next, we compute the Laplacian of the Gauss map G of M 12 − . First, from the first fundamental form of M 12 − , we have Substituting (4.1), (4.2) and (4.4) into (1.1), and by putting where H s = 2rr r cosh 2 ϕ + r 2 r cosh 2 ϕ − 4r 2 r r 2 − 2r 2 r κ 2 cosh 2 ϕ cosh 2 θ − 5r 2 r r κ cosh ϕ cosh θ after tedious tidying up, we get Do discussions similar to those of M 11 − , we have the following conclusions directly.
Corollary 5.8. The canal surface M 2 − with harmonic Gauss map does not exist.

Conclusions
Until now, the canal surfaces M 11 − , M 12 − and M 2 − foliated by pseudo hyperbolic spheres H 2 0 along the first kind spacelike curve, the second kind spacelike curve and a timelike curve, respectively have been classified in terms of their Gauss maps. The similar works for the canal surfaces M 11 + , M 12 + and M 2 + have been done in another recent work. The canal surfaces M 13 − , M 3 − (M 13 + , M 3 + ) foliated by pseudo hyperbolic spheres H 2 0 (resp. pseudo spheres S 2 1 ) along a null type spacelike curve or a null curve are to be investigated in the continued works.