Generalized Null 2-Type Surfaces in Minkowski 3-Space

For the mean curvature vector field H and the Laplace operator ∆ of a submanifold in the Minkowski space, a submanifold satisfying the condition ∆H = f H + gC is known as a generalized null 2-type, where f and g are smooth functions, and C is a constant vector. The notion of generalized null 2-type submanifolds is a generalization of null 2-type submanifolds defined by B.-Y. Chen. In this paper, we study flat surfaces in the Minkowski 3-space L3 and classify generalized null 2-type flat surfaces. In addition, we show that the only generalized null 2-type null scroll in L3 is a B-scroll.


Introduction
Let x : M −→ E m be an isometric immersion of an n-dimensional connected submanifold M in an m-dimensional Euclidean space E m .Denote by H and ∆, respectively, the mean curvature vector field and the Laplacian operator with respect to the induced metric on M induced from that of E m .Then, it is well known as ∆x = −nH.
By using (1), Takahashi [1] proved that minimal submanifolds of a hypersphere of E m are constructed from eigenfunctions of ∆ with one eigenvalue λ ( =0).In [2,3], Chen initiated the study of submanifolds in E m that are constructed from harmonic functions and eigenfunctions of ∆ with a nonzero eigenvalue.The position vector x of such a submanifold admits the following simple spectral decomposition: x = x 0 + x q , ∆x 0 = 0, ∆x q = λx q (2) for some non-constant maps x 0 and x q , where λ is a nonzero constant.A submanifold satisfying ( 2) is said to be of null 2-type [3].From the definition of null 2-type submanifolds and (1), it follows that the mean curvature vector field H satisfies the following condition: A result from [4] states that a surface in the Euclidean space E 3 satisfying (3) is either a minimal surface or an open part of an ordinary sphere or a circular cylinder.Ferrández and Lucas [5] extended it to the Lorentzian case.They proved that the surface satisfying (3) is either a minimal surface or an open part of a Lorentz circular cylinder, a hyperbolic cylinder, a Lorentz hyperbolic cylinder, a hyperbolic space, a de Sitter space or a B-scroll.Afterwards, several authors studied null 2-type submanifolds in the (pseudo-)Euclidean space [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].Now, we will give a generalization of null 2-type submanifolds in the Minkowski space.It is well known that a Lorentz circular cylinder S 1 (r) × R 1  1 is a null 2-type surface in the Minkowski 3-space L 3 satisfying ∆H = 1 r 2 H, where S 1 (r) is a circle with radius r and R 1 1 is a Lorentz straight line.However, the following surface has another property as follows: a parametrization is a cylindrical surface in L 3 .On the other hand, the mean curvature vector field H of the surface is given by and the surface satisfies , 0) .
Next, we consider another surface with a parametrization The surface is a conical surface in L 3 , and it satisfies the following equation for the mean curvature vector . Thus, based on the above examples, we give the definition: Definition 1.A submanifold M of the Minkowski space is said to be of generalized null 2-type if it satisfies the condition for some smooth functions f , g and a constant vector C.In particular, if the functions f and g are equal to each other in (4), then the submanifold M is called of generalized null 2-type of the first kind and of the second kind otherwise.
In [22], the authors recently classified generalized null 2-type flat surfaces in the Euclidean 3-space.Conical surfaces, cylindrical surfaces or tangent developable surfaces are developable surfaces (or flat surfaces) as ruled surfaces in the Minkowski 3-space L 3 .In this paper, we study developable surfaces in L 3 and completely classify generalized null 2-type developable surfaces, and give some examples.In addition, we investigate null scrolls in the Minkwoski 3-space L 3 satisfying the condition (4).

Preliminaries
The Minkowski 3-space L 3 is a real space R 3 with the standard flat metric given by where (x 1 , x 2 , x 3 ) is a rectangular coordinate system of R 3 .An arbitrary vector x of L 3 is said to be space-like if x, x > 0 or x = 0, time-like if x, x < 0 and null if x, x = 0 and x = 0.A time-like or null vector in L 3 is said to be causal.Similarly, an arbitrary curve γ = γ(s) is space-like, time-like or null if all of its tangent vectors γ (s) are space-like, time-like or null, respectively.From now on, the "prime" means the partial derivative with respect to the parameter s unless mentioned otherwise.
We now put a 2-dimensional space form in L 3 as follows: We call S 2 1 and H 2 the de-Sitter space and the hyperbolic space, respectively.Let γ : I −→ L 3 be a space-like or time-like curve in the Minkowski 3-space L 3 parameterized by its arc-length s.Denote by {t, n, b} the Frenet frame field along γ(s).
If γ(s) is a space-like curve in L 3 , the Frenet formulae of γ(s) are given by [23]: where Here, the functions κ(s) and τ(s) are the curvature function and the torsion function of a space-like curve γ(s), respectively.If γ(s) is a time-like curve in L 3 , the Frenet formulae of γ(s) are given by [23]: where t, t = −1, n, n = b, b = 1.Here κ(s) and τ(s) are the curvature function and the torsion function of a time-like curve γ(s), respectively.If γ(s) is a space-like or time-like pseudo-spherical curve parametrized by arc-length s in Q 2 (ε), let t(s) = γ (s) and g(s) = γ(s) × γ (s).Then, we have a pseudo-orthonormal frame {γ(s), t(s), g(s)} along γ(s).It is called the pseudo-spherical Frenet frame of the pseudo-spherical curve γ(s).If γ is a space-like curve, then the vector g is time-like when γ is on S 2 1 , and the vector g is space-like when γ is on H 2 .Similarly, if the curve γ is time-like, then the vector g is space-like.The following theorem can be easily obtained.Theorem 1. ( [24,25]) Under the above notations, we have the following pseudo-spherical Frenet formulae of γ: (1) If γ is a pseudo-spherical space-like curve, (2) If γ is a pseudo-spherical time-like curve, The function κ g (s) is called the geodesic curvature of the pseudo-spherical curve γ.Now, we define a ruled surface M in L 3 .Let I and J be open intervals in the real line R. Let α = α(s) be a curve in L 3 and β = β(s) a vector field along α with α (s) × β(s) = 0 for every s ∈ J.Then, a ruled surface M is defined by the parametrization given as follows: x = x(s, t) = α(s) + tβ(s), s ∈ J, t ∈ I.
For such a ruled surface, α and β are called the base curve and the director curve respectively.In particular, if β is constant, the ruled surface is said to be cylindrical, and if it is not so, it is called non-cylindrical.Furthermore, we have five different ruled surfaces according to the characters of the base curve α and the director curve β as follows: if the base curve α is space-like or time-like, then the ruled surface M is said to be of type M + or type M − , respectively.In addition, the ruled surface of type M + can be divided into three types.In the case that β is space-like, it is said to be of type + if β is non-null or null, respectively.When β is time-like, β is space-like because of the causal character.In this case, M is said to be of type M 3 + .On the other hand, for the ruled surface of type M − , it is also said to be of type [26].However, if the base curve α is a light-like curve and the vector field β along α is a light-like vector field, then the ruled surface M is called a null scroll.In particular, a null scroll with Cartan frame is said to be a B-scroll [27].It is also a time-like surface.
A non-degenerate surface in L 3 with zero Gaussian curvature is called a developable surface.The developable surfaces in L 3 are the same as in the Euclidean space, and they are planes, conical surfaces, cylindrical surfaces and tangent developable surfaces [13].

Generalized Null 2-Type Cylindrical Surfaces
For a surface in the Minkowski 3-space L 3 , the next lemma is well known and useful.

Lemma 1. ([16]
) Let M be an oriented surface of L 3 .Then, the Laplacian of the mean curvature vector field H of M is given by where ε is the sign of the unit normal vector N of the surface M and ∇H, A are the gradient of the mean curvature H and the shape operator of M, respectively.
Theorem 2. All cylindrical surfaces in L 3 are of generalized null 2-type.
Proof.Let M be a cylindrical ruled surface in the Minkowski 3-space L 3 of type where the base curve α(s), which is a space-like or time-like curve with the arc-length parameter s, lies in a plane with a space-like or time-like unit normal vector β that is the director of M, that is, where κ(s) is the curvature function of α(s) and ε 3 (= ±1) is the sign of e 3 .From this, the mean curvature vector field H of M is given by and the Laplacian ∆H of H is expressed as Suppose that M is of generalized null 2-type.With the help of ( 4) and ( 12), we obtain the following equations: where In this case, C 1 is a constant, and C 2 , C 3 are functions of the variable s.
If g is identically zero, then, from ( 14), the curvature κ(s) is constant, and from (15), the function f is constant, say λ.Thus, M satisfies ∆H = λH, that is, it is either a Euclidean plane, a Minkowski plane, a Lorentz circular cylinder S 2 × R 1  1 , a hyperbolic cylinder H 1 × R or a Lorentz hyperbolic cylinder S 1  1 × R according to [16].
We now assume that g = 0.It follows from ( 13) that C 1 = 0.By using (10), we can show that the component functions of C satisfy the following equations: We may put from ( 16) where θ(s) = κ 0 + κ(s)ds for some constant κ 0 .Therefore, the constant vector C becomes Combining ( 14), ( 15) and (17), one also gets Thus, the mean curvature vector field H of the cylindrical surface M + 3 satisfies where f , g and C are given in ( 18) and ( 19), respectively.Case 2: Let M be of type M 1 + .In this case, ε 1 = 1, ε 2 = 1 and the constant vector C is space-like, time-like or null.
First of all, we consider the constant vector C is non-null.Then, from ( 16), we may put where θ(s) = − κ(s)ds + κ 0 with a constant κ 0 .
By using ( 14), ( 15) and ( 17), the functions f (s) and g(s) are determined by Thus, for the non-null constant vector C, the cylindrical surface where f , g and C are given by ( 20) and ( 21), respectively.Next, let the constant vector C be null, that is, η = 0.Then, we get We will consider the case s) , where θ(s) = − κ(s)ds + κ 0 for some constant κ 0 .In this case, we have and, for the null constant vector C, the surface satisfies the condition In this case, the constant vector C is space-like, time-like or null.Applying the same method as in Case 2, the functions f (s) and g(s) are determined by and the component functions of C are given by where θ(s) = κ(s)ds + κ 0 for some constant κ 0 .Thus, from Cases 1, 2 and 3, Theorem 2 is proved.
Example 1.We consider a surface defined by This parametrization is a cylindrical ruled surface of type M 1 + .In this case, the mean curvature vector field H of the surface is given by By a direct computation, the Laplacian ∆H of the mean curvature vector field H becomes and it can be rewritten in terms of the mean curvature vector field H and a constant vector C as follows: where C = ( 1 4 , 1 4 , 0) is a null vector.Thus, the cylindrical ruled surface defined by ( 25) is a generalized null 2-type surface of the first kind.Remark 1.A cylindrical surface in L 3 generated by the base curve α(s) with the curvature κ(s) = 1 s and a constant director β is a generalized null 2-type surface of the first kind if the constant vector C is null.

Generalized Null 2-Type Non-Cylindrical Flat Surfaces
In this section, we classify non-cylindrical flat surfaces satisfying It is well-known that a non-cylindrical flat surface in the Minkowski 3-space L 3 is an open part of a conical surface or a tangent developable surface.
First of all, we consider a conical surface M in L 3 .Then, we may give the parametrization of M by x(s, t) = α 0 + tβ(s), s ∈ I, t > 0, such that β (s), β (s) = ε 1 and β(s), β(s) = ε 2 , where α 0 is a constant vector.We take the orthonormal tangent frame {e 1 , e 2 } on M such that e 1 = 1 t ∂ ∂s and e 2 = ∂ ∂t .The unit normal vector of M is given by e 3 = e 1 × e 2 .By the Gauss and Weingarten formulas, we have where κ g (s) = β(s), β (s) × β (s) , which is the geodesic curvature of the pseudo-spherical curve β(s) in Q 2 (ε).From ( 27), the mean curvature vector field H of M is given by and the Laplacian ∆H of the mean curvature vector field H is expressed as Suppose that κ g is constant.If κ g = 0, by a rigid motion, the pseudo-spherical curve β(s) in Q 2 (ε) lies on yz-plane or xz-plane.Thus M is an open part of a Euclidean plane or a Minkowski plane.If κ g is a non-zero constant, from (27), we can obtain by a straightforward computation Case 1: ε 2 (κ 2 g (s) − ε 1 ) = k 2 for some real number k.Let ε 1 = 1.Without loss of generality, we may assume β (0) = (0, 1, 0).Thus, β (s) = k 2 β (s) implies β (s) = (B 1 sinh ks, cosh ks + B 2 sinh ks, B 3 sinh ks) for some constants B 1 , B 2 and B 3 .Since ε 1 = 1, we have B 2 1 − B 2 3 = 1 and B 2 = 0. From this, we can obtain We now change the coordinates by x, ȳ, z such that With respect to the coordinates ( x, ȳ, z), β(s) turns into for a constant Thus, up to a rigid motion M has the parametrization of the form We call such a surface a hyperbolic conical surface of the first kind, and it satisfies Next, let (ε 1 , ε 2 ) = (−1, 1).We now consider a initial condition β (0) = (1, 0, 0) of the ordinary differential equation (ODE) (30).Quite similarly as we did, we obtain With respect to the new coordinates ( x, ȳ, z), the vector β(s) becomes where We call such a surface generated by (33) a hyperbolic conical surface of the second kind and it satisfies Case 2: ε 2 (κ 2 g (s) − ε 1 ) = −k 2 for some real number k.
Let ε 1 = 1.We may give the initial condition by β (0) = (0, 1, 0) for the differential equation β (s) + k 2 β (s) = 0.Under such an initial condition, a vector field β(s) is given by where B 1 , B 3 , D 1 and D 3 are some constants satisfying If we take another coordinate system ( x, ȳ, z) such that then a vector β(s) takes the form where We call such a surface generated by (34) an elliptic conical surface and it satisfies for some constants k 1 , k 2 and k 3 .Since β (s), β (s) = ε 1 = 1, we may set (k 1 , k 2 , k 3 ) = (0, 1, 0) up to an isometry and hence for some constants c 1 , c 2 and c 3 .However, We call such a surface generated by (35) a quadric conical surface.
As shown in the Introduction, a quadric conical surface is of generalized null 2-type of the first kind.Let us suppose that κ g is a non-constant, i.e., κ g = 0 on an open interval.Suppose that M is of generalized null 2-type, that is, M satisfies the condition (4).Then, we have the following equations: where Combining ( 36) and (37), and using (40), we have where c is a constant of integration.Together with (37) and (42), we can find Substituting ( 42) into (39), we get Then, (38) and (44) lead to Furthermore, it follows from (41) and (42) that and its solution is given by for some constant a 1 .Combining (44) and (46), the geodesic curvature κ g satisfies the following equation: To solve the ODE, we put p = κ g .Then, (47) can be written of the form and it is a Bernoulli differential equation.Thus, the solution is given by , which is equivalent to for some constant a 2 .If we put where , and then we have for some constant a 3 .Thus, the geodesic curvature κ g is given by Furthermore, the constant vector C can be expressed as Conversely, for some constants a 1 , a 2 and c such that the function is well-defined on an open interval J ⊂ (0, ∞), we take an indefinite integral F(v) of the function ψ(v).Let I be the image of the function F. We can take an open subinterval J 1 ⊂ J such that F : J 1 → I is a strictly increasing function with F (v) = ψ(v).Let us consider the function ϕ defined by ϕ(s) = F −1 (±s + a 3 ) for some constant a 3 .Then, the function ϕ satisfies F(ϕ) = ±s + a 3 .
For any unit speed pseudo-spherical curve β(s) in Q 2 (ε) with geodesic curvature κ g (s) = ϕ(s), we consider the conical surface M in L 3 parametrized by where α 0 is a constant vector.Given any nonzero constant c, we put f and g the functions, respectively, given by For a nonzero constant c and the pseudo-orthonormal frame {e 1 , e 2 , e 3 } on L 3 such that e 1 = 1 t ∂ ∂s and e 2 = ∂ ∂t are tangent to M and e 3 normal to M, we put Note that it follows from the definition of ϕ that the function ϕ satisfies (47).Hence, using (27), it is straightforward to show that ∇ e 1 C = ∇ e 2 C = 0, which implies that C is a constant vector.Furthermore, the same argument as in the first part of this subsection yields the mean curvature vector field H of the conical surface M satisfies where f , g and C are given in (54) and (55), respectively.This shows that the conical surface is of generalized null 2-type.Thus, we have the following: Theorem 3. Let M be a conical surface in the Minkowski 3-space L 3 .Then, M is of generalized null 2-type if and only if it is an open part of one of the following surfaces: (1) a Euclidean plane; (2) a Minkowski plane; (3) a hyperbolic conical surface of the first kind; (4) a hyperbolic conical surface of the second kind; (5) an elliptic conical surface; (6) a quadric conical surface; (7) a conical surface parameterized by where α 0 is a constant vector and β(s) is a unit speed pseudo-spherical curve in Q 2 (ε) with the non-constant geodesic curvature κ g which is, for some indefinite integral F(v) of the function where a 3 is constant.
Next, we study tangent developable surfaces in the Minkowski 3-space L 3 .Proof.Let α(s) be a curve parameterized by arc-length s in L 3 with non-zero curvature κ(s).Then, a non-degenerate tangent developable surface M in L 3 is defined by x(s, t) = α(s) + tα (s), t = 0.
In the case, we can take the pseudo-orthonormal frame {e (58) Suppose that M is of generalized null 2-type, that is, M satisfies ∆H = f H + gC for some smooth functions f , g and a constant vector C.With the help of (57) and ( 58), ( 4) can be written in the form By combining the first and second equations of (59), we get This shows that we obtain 3ε 2 κτ 2 C 1 = 0, Consider the open set O = {p ∈ M|τ(p) = 0}.Suppose that O is a non-empty set.(63) shows that C 1 = 0 and C 2 = 0, and it follows from (61) that C 3 = 0.That is, C = 0 on O.In addition, (59) gives τ = 0, and it is a contradiction.Thus, the open set O is empty and τ is identically zero.Therefore, α(s) is a plane curve, and the surface M is an open part of a Euclidean plane or a Minkowski plane.
The converse of Theorem 4 follows a straightforward calculation.
we take a local pseudo-orthonormal frame {e 1 , e 2 , e 3 } on L 3 such that e 1 = ∂ ∂t and e 2 = ∂ ∂s are tangent to M, and e 3 normal to M. It follows that the Levi-Civita connection ∇ of L 3 is expressed as

Theorem 4 .
Let M be a tangent developable surface in the Minkowski 3-space L 3 .Then, M is of generalized null 2-type if and only if M is an open part of a Euclidean plane or a Minkowski plane.