Dual Associate Null Scrolls with Generalized 1-Type Gauss Maps

Abstract

In this work, a pair of dual associate null scrolls are defined from the Cartan Frenet frame of a null curve in Minkowski 3-space. The fundamental geometric properties of the dual associate null scrolls are investigated and they are related in terms of their Gauss maps, especially the generalized 1-type Gauss maps. At the same time, some representative examples are given and their graphs are plotted by the aid of a software programme.

1. Introduction

In differential geometry, the associate curves and associate surfaces such as the Bertrand curve, the Mannheim curve, evolute-involute pair, the parallel surfaces and the focal surfaces etc. compose a large class of fascinating subjects in the curve and surface theory not only in Euclidean space but also in pseudo-Euclidean space, such as Minkowski space [1,2,3,4]. However, due to the difference of metric between Euclidean space and Minkowski space, there are cases where there are differences. For example, the Bertrand curve could be involved by the directional associated curve of a space curve due to the causal character of vectors in Minkowski space [5].

As the simplest foliated submanifolds, ruled surfaces are divided into five types according to the causal character of the base curve and the ruling flow [6]. Among of them, for the ruled surfaces with lightlike rulings, the base curves can be null curves or non-null curves obviously. It is worth noting that the ruled surfaces with non-null base curves are equivalent to the ones with null base curves via the appropriate transformation as stated in [7]. Without loss of generality, we always can choose a null curve as the base curve of a ruled surface with lightlike ruling and the normalization condition is satisfied at the same time, which is said to be a null scroll [7,8,9,10]. Furthermore, the null scroll under Cartan Frenet frame is called a B-scroll [11].

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. The Gauss mapGof a submanifoldMis of generalized 1-type if the Gauss mapGofMsatisfies

ΔG=fG+gC

for some non-zero functions(f,g)onMand a constant vector C, whereΔdenotes the Laplacian defined onM, which is given by

Δ=−1|G|∑i,j∂∂^xi(|G|^gij∂∂^xj),

where(_x1,…,_xn)is a local coordinate system ofM,^gijthe components of the inverse matrix of the first fundamental form ofMandGthe determinant of the first fundamental form ofM [17]. Especially, if both f and g are non-zero constants, (1) can be written byΔG=μ(G+C),(μ∈R−{0}) . In this case, the Gauss map is just of 1-type in the usual sense. If the function f is equal to g, (1) can be expressed byΔG=f(G+C). The Gauss map is said to be of pointwise 1-type. More precisely, the pointwise 1-type Gauss map is said to be of the first kind whenC=0, or else the second kind. If f and g vanish identically, thenGis said to be harmonic.

The paper is organized as follows. In Section 2, some basic facts including the Frenet formula and the structure function of null curves are reviewed, then a pair of associate curves on lightlike cone and a dual associate null scrolls are defined. In Section 3, the geometric properties such as the Gaussian curvatures, mean curvatures and the Laplacians of the Gauss maps are shown and the generalized 1-type Gauss maps are discussed, respectively. The relationships between the dual associate null scrolls are explored and summarized.

Throughout this paper, all the geometric objects under consideration are smooth and all surfaces are connected unless otherwise stated. 2. Preliminaries

Let_E13be the Minkowski 3-space with natural Lorentzian metric

〈·,·〉=d_x12+d_x22−d_x32

in terms of the natural coordinate system(_x1,_x2,_x3). A vector v in_E13is said to be spacelike, timelike and lightlike (null) if〈v,v〉>0orv=0,〈v,v〉<0and〈v,v〉=0(v≠0), respectively. The norm of a vector v is defined by∥v∥=|〈v,v〉|. An arbitrary curve r is spacelike, timelike or lightlike if its tangent vector^r′is spacelike, timelike or lightlike, correspondingly. At the same time, a surface is said to be timelike, spacelike or lightlike if its normal vector is spacelike, timelike or lightlike, respectively.

Proposition 1

([5]). Letr(s)be a null curve parameterized by the null arc length s (i.e.,∥^r″(s)∥=1)in_E13. Then there exists a unique Cartan frame{^r′(s)=T(s),N(s),B(s)}such that

^T′(s)=N(s),^N′(s)=κ(s)T(s)−B(s),^B′(s)=−κ(s)N(s),

where〈T(s),T(s)〉=〈B(s),B(s)〉=〈T(s),N(s)〉=〈B(s),N(s)〉=0,〈T(s),B(s)〉=〈N(s),N(s)〉=1andT(s)=N(s)×T(s),N(s)=T(s)×B(s),−B(s)=N(s)×B(s).

In the sequence,T(s),N(s)andB(s)is called the tangent, principal normal and binormal vector field ofr(s) , respectively. From (3), it is easy to know thatκ(s)=−12〈^r‴(s),^r‴(s)〉. The functionκ(s)is called the null curvature ofr(s) , which is an invariant under Lorentzian transformations [18].

Considering the relationship between the tangent vector fieldT(s)and the binormal vector fieldB(s)of a null curver(s), i.e.,〈T(s),B(s)〉=1, we could define a pair of associate curves on lightlike cone as follows:

Definition 1.

Letr(s)be a null curve framed by{T(s),N(s),B(s)}in_E13. Then_b1(s)=λ(s)T(s),_b2(s)=1λ(s)B(s)is called a generalized T-associate curve and a generalized B-associate curve ofr(s)for some non-zero smooth functionλ(s), respectively._b1(s)and_b2(s)are called dual associate curves ofr(s)on lightlike cone.

In [5], the authors introduced the structure function and the representation formula of a null curve. Namely,

Proposition 2

([5]). Letr(s):I→_E13be a null curve. Thenr(s)can be written as

r(s)=∫12^f′(^f2−1,2f,^f2+1)ds,(^f′=dfds),

wheref=f(s)is called the structure function ofr(s). And the null curvatureκ(s)ofr(s)can be expressed by

κ(s)=12^{[(logf′)′]2}−^(logf′)″.

Definition 2

([6]). Leta(s):_I1→_E13be a null curve parameterized by null arc length andb(s)a transversal null vector field alonga(s). Then the immersion

X(s,t)=a(s)+tb(s),(t∈_I2⊂R)

is called a null scroll which satisfies〈^a′(s),^a′(s)〉=0,〈b(s),b(s)〉=0and the normalization condition〈^a′(s),b(s)〉=1.

According to the definitions of generalized T-associate curve, generalized B-associate curve of a null curve and the definition of null scrolls, we want to construct a pair of null scrolls which satisfy the same normalization condition. This idea motivate the following definition.

Definition 3.

Letr(s)be a null curve framed by{T(s),N(s),B(s)}in_E13,_b1(s)and_b2(s)dual associate curves ofr(s). Then

_X1(s,_t1)=∫_b2(s)ds+_{t1 b1}(s)

is called a null scroll with generalized T-lightlike ruling;

_X2(s,_t2)=∫_b1(s)ds+_{t2 b2}(s)

is called a null scroll with generalized B-lightlike ruling. The null scrolls_X1(s,_t1)and_X2(s,_t2)are called dual associate null scrolls.

As the straightforward conclusion of Proposition 2, Definitions 1 and 3, we have

Proposition 3.

Letr(s)be a null curve with structure functionf=f(s)in_E13. Then

• the dual associate curves_bi(s)(i=1,2)ofr(s)can be expressed by

  _b1(s)=λ2^f′(^f2−1,2f,^f2+1),_b2(s)=1λ(−^f″24^f′3(^f2−1,2f,^f2+1)+^{f″ f′}(f,1,f)−^f′(1,0,1));

• the dual associate null scrolls_Xi(s)(i=1,2)can be expressed by

  _X1(s,_t1)=∫1λ(−^f″24^f′3(^f2−1,2f,^f2+1)+^{f″ f′}(f,1,f)−^f′(1,0,1))ds+_t1λ2^f′(^f2−1,2f,^f2+1),_X2(s,_t2)=∫λ2^f′(^f2−1,2f,^f2+1)ds+_t21λ(−^f″24^f′3(^f2−1,2f,^f2+1)+^{f″ f′}(f,1,f)−^f′(1,0,1)).

Obviously, the dual associate curves and the dual associate null scrolls are decided by the functionλ(s)and the structure functionf(s), completely. The following example can explain the construction of the defined dual associate null scrolls.

Example 1.

Consider a null curver(s)with null curvatureκ(s)=4^s−2. From Proposition 2, we have

12^{[(logf′)′]2}−^(logf′)″=4^s−2.

Solving the above differential equation, we havef(s)=^s3orf(s)=^s−3. From Proposition 3, we have

• whenλ(s)=13, the dual associate curves and the dual associate null scrolls can be written as (see Figure 1 and Figure 2)

  _b1(s)=s18(^s3−^s−3,2,^s3+^s−3),_b2(s)=(−4^s2+^s−4,4^s−1,−4^s2−^s−4)

  and

  _X1(s,_t1)=13(−4^s3−^s−3,12logs,−4^s3+^s−3)+_t1s18(^s3−^s−3,2,^s3+^s−3),_X2(s,_t2)=118(^s55+^s−1,^s2,^s55−^s−1)+_t2(−4^s2+^s−4,4^s−1,−4^s2−^s−4);

• whenλ(s)=s, the dual associate curves and the dual associate null scrolls can be written as (see Figure 3 and Figure 4)

  _b1(s)=^s26(^s3−^s−3,2,^s3+^s−3),_b2(s)=13s(−4^s2+^s−4,4^s−1,−4^s2−^s−4)

  and

  _X1(s,_t1)=13(−2^s2−14^s−4,−4^s−1,−2^s2+14^s−4)+_t1 ^s26(^s3−^s−3,2,^s3+^s−3),_X2(s,_t2)=136(^s6−6logs,4^s3,^s6+6logs)+_t23s(−4^s2+^s−4,4^s−1,−4^s2−^s−4).

3. Main Result We will discuss the geometric properties of the dual associate null scrolls and the Laplacians of their Gauss maps. 3.1. The Null Scroll with Generalized T-Lightlike Ruling

To meet the requirements of discussion, we prepare some basic elements of_X1(s,_t1) . Initially, from (6) and Proposition 1, we have

_X1s=_t1 ^λ′T+_t1λN+1λB,_X1t1=λT.

Based on above equations, the components of the first fundamental form_gij(i,j=1,2)are

_g11=〈_X1s,_X1s〉=_t12 ^λ2+2_t1 ^λ′λ,_g12=〈_X1s,_X1t1〉=1,_g22=〈_X1t1,_X1t1〉=0,

then, we have_{g11 g22}−_g122=−1.Meanwhile, the Gauss map_G1of_X1is given by

_G1=_X1s×_X1t1∥_X1s×_X1t1∥=_t1 ^λ2T−N

which satisfies〈_G1,_G1〉=1. Furthermore, by (9), we have

_G1s=(2_t1λ^λ′−κ)T+_t1 ^λ2N+B,_G1t1=^λ2T.

Then, the components of the second fundamental form_hij(i,j=1,2)are

_h11=−〈_X1s,_G1s〉=κλ−3_t1 ^λ′−_t12 ^λ3,_h12=−〈_X1t1,_G1s〉=−λ,_h22=−〈_X1t1,_G1t1〉=0.

By (8) and (11), the Gaussian curvature_K1and the mean curvature_H1of_X1are given by, respectively

_K1=_{h11 h22}−_{h122g11 g22}−_g122=^λ2,

_H1=_{g11 h22}−2_{g12 h12}+_{g22 h11}2(_{g11 g22}−_g122)=−λ.

Obviously, the Gaussian curvature_K1and the mean curvature_H1of_X1satisfy

_H12=_K1=^λ2.

From now on, we compute the Laplacian of Gauss map_G1and discuss the null scroll_X1with generalized 1-type Gauss map.

By (2), the Laplacian_Δ1of the null scroll_X1is obtained by

_Δ1=−2∂∂s∂∂_t1+2(^λ2 _t1+^λ′λ)∂∂_t1+(^λ2 _t12+2^λ′ _t1λ)^∂2∂_t12.

Substituting (9) into (15), we get

_{Δ1 G1}=(2^λ4 _t1−2λ^λ′)T−2^λ2N=(2_{K12 t1}−_K1′)T−2_K1N.

Suppose that_X1has generalized 1-type Gauss map, i.e.,_{Δ1 G1}=_{f1 G1}+_{g1 C1}. Without loss of generality, we may decompose the constant vector_C1via the Cartan frame{T,N,B}of the null curver(s)as

_C1=_C11T+_C12N+_C13B,

where_C11=〈_C1,B〉,_C12=〈_C1,N〉,_C13=〈_C1,T〉.

Substituting (9), (16) and (17) into_{Δ1 G1}=_{f1 G1}+_{g1 C1}, we obtain the following equation system

2^λ4 _t1−2λ^λ′=^λ2 _{t1 f1}+_{g1 C11},−2^λ2=_{g1 C12}−_f1,_{g1 C13}=0.

Since_g1is a non-zero smooth function, then_C13=0 from the last equation of (18). Furthermore, by the first two equations of (18), we have

_f1(s,_t1)=_C12(2^λ4 _t1−2λ^λ′)+2^λ2 _C11C12 ^λ2 _t1+_C11=_C12(2_{K12 t1}−_K1′)+2_{K1 C11C12 K1 t1}+_C11,_g1(s,_t1)=−2λ^λ′_C12 ^λ2 _t1+_C11=−_{K1′C12 K1 t1}+_C11,

where_C112+_C122≠0.Meanwhile,λis a non-constant function since_g1is a non-zero smooth function.

Conversely, if we use the above information with the given functions(_f1,_g1) in (19) and the constant vector_C1, then the null scroll_X1satisfies_{Δ1 G1}=_{f1 G1}+_{g1 C1}.

Theorem 1.

Let_X1be a null scroll with generalized T-lightlike ruling in_E13. Then_X1has generalized 1-type Gauss map if and only if the Gauss map_G1of_X1satisfies

_{Δ1 G1}=_{f1 G1}+_{g1 C1}

for some non-zero smooth functions(_f1,_g1)as

_f1=_f1(s,_t1)=_C12(2^λ4 _t1−2λ^λ′)+2^λ2 _C11C12 ^λ2 _t1+_C11=_C12(2_{K12 t1}−_K1′)+2_{K1 C11C12 K1 t1}+_C11,_g1=_g1(s,_t1)=−2λ^λ′_C12 ^λ2 _t1+_C11=−_{K1′C12 K1 t1}+_C11

and a constant vector_C1=(_C11,_C12,0). Where_C112+_C122≠0,λ=λ(s)is a non-constant smooth function.

Remark 1.

Particularly, when_C1=(_C11,0,0), (_C11≠0), the functions(_f1,_g1)only depend on s, i.e.,

_f1=_f1(s)=2^λ2=2_K1,_g1=_g1(s)=−2λ^λ′_C11=−_{K1′ C11}.

Corollary 1.

Let_X1be a null scroll with generalized T-lightlike ruling in_E13. Then_X1has pointwise 1-type Gauss map of the second kind if and only if the Gauss map_G1of_X1satisfies

_{Δ1 G1}=_f1(_G1+_C1)

for some non-zero smooth function_f1as

_f1=_f1(s)=2^λ2=_C02 ^e−2_C11s,(_C0∈R−{0})

and a constant vector_C1=(_C11,0,0), (_C11≠0).

Proof of Corollary 1.

Suppose that the null scroll_X1has pointwise 1-type Gauss map of the second kind. It means that_f1=_g1in Theorem 1. Thus, we have

^λ4 _{C12 t1}−λ^λ′ _C12+λ^λ′=−^λ2 _C11.

From (20), we can get the following equation system

^λ4 _{C12 t1}=0,−λ^λ′ _C12+λ^λ′=−^λ2 _C11.

Sinceλis a non-zero smooth function,_C12=0 is concluded from the first equation of (21). By the second equation of (21), we get differential equationλ^λ′=−^λ2 _C11.Solving this equation, we have

λ=_C02^e−_C11s(_C0∈R−{0}).

By Remark 1,_f1=2^λ2=_C02 ^e−2_C11s, (_C11≠0). Conversely, if we use the above information with the given function_f1and constant vector_C1, the null scroll_X1satisfies_{Δ1 G1}=_f1(_G1+_C1).  □

Corollary 2.

There does not exist the null scroll with generalized T-lightlike ruling in_E13which has 1-type Gauss map of the second kind.

Proof of Corollary 2.

Suppose that the null scroll_X1has 1-type Gauss map of the second kind, i.e.,_{Δ1 G1}=μ(_G1+_C1),(μ∈R−{0}). It means that_f1=μ, then_C11=0which contradicts with_C11≠0in Corollary 1.  □

Corollary 3.

Let_X1be a null scroll with generalized T-lightlike ruling in_E13. Then_X1has pointwise 1-type Gauss map of the first kind if and only if one of the following statements holds:

• _X1has 1-type Gauss map of the first kind;

• _X1has non-zero constant Gaussian curvature or non-zero constant mean curvature.

Proof of Corollary 3

Suppose that the null scroll_X1has pointwise 1-type Gauss map of the first kind, i.e.,_{Δ1 G1}=_{f1 G1}. From Corollary 1, we can get_C11=0and_f1=2^λ2=_C02.Therefore,_X1has 1-type Gauss map of the first kind since_{Δ1 G1}=_{C02 G1},(_C0∈R−{0}) . By (12) and (13), the Gaussian curvature_K1and the mean curvature_H1are non-zero constant. Conversely, if one of the statements holds, it follows thatλis a non-zero constant. This completes the proof.  □

Corollary 4.

There does not exist the null scroll with generalized T-lightlike ruling in_E13which has harmonic Gauss map.

Proof of Corollary 4

Suppose that the null scroll_X1has harmonic Gauss map, i.e.,_{Δ1 G1}=0. Then, we haveλ=0and it is impossible.  □

3.2. The Null Scroll with Generalized B-Lightlike Ruling

To meet the requirements of discussion, we prepare some basic elements of_X2 . Initially, from (7) and Proposition 1, we have

_X2s=λT−_t2κλN−_t2 ^λ′λ2B,_X2t2=1λB.

Based on above equations, the components of the first fundamental form_gij(i,j=1,2)are

_g11=〈_X2s,_X2s〉=_t22 ^κ2λ2−2_t2 ^λ′λ,_g12=〈_X2s,_X2t2〉=1,_g22=〈_X2t2,_X2t2〉=0,

then, we have_{g11 g22}−_g122=−1.Meanwhile, the Gauss map_G2of_X2is given by

_G2=_X2s×_X2t2∥_X2s×_X2t2∥=N+_t2κ^λ2B

which satisfies〈_G2,_G2〉=1. Furthermore, by (23), we have

_G2s=κT−_t2 ^κ2λ2N+(_t2 ^(κλ2)′−1)B,_G2t2=κ^λ2B.

Then, the components of the second fundamental form_hij(i,j=1,2)are

_h11=−〈_X2s,_G2s〉=λ(1−_t2 ^(κλ2)′)−_t22 ^κ3λ3+_t2κ^λ′λ2,_h12=−〈_X2t2,_G2s〉=−κλ,_h22=−〈_X2t2,_G2t2〉=0.

By (22) and (25), the Gaussian curvature_K2and the mean curvature_H2of_X2are given by, respectively

_K2=_{h11 h22}−_{h122g11 g22}−_g122=^{κ2 λ2},

_H2=_{g11 h22}−2_{g12 h12}+_{g22 h11}2(_{g11 g22}−_g122)=−κλ.

It is evident that the Gaussian curvature_K2and the mean curvature_H2of_X2satisfy

_H22=_K2=^{κ2 λ2}.

From now on, we compute the Laplacian of Gauss map_G2and discuss the null scroll_X2with generalized 1-type Gauss map.

By (2), the Laplacian_Δ2of the null scroll_X2is obtained by

_Δ2=−2∂∂s∂∂_t2+2(^κ2 _t2^λ2−^λ′λ)∂∂_t2+(^κ2 _t22^λ2−2^λ′ _t2λ)^∂2∂_t22.

Substituting (23) into (29), we get

_{Δ2 G2}=2^κ2λ2N+(2κ^λ′λ3+2^κ3 _t2^λ4−2^κ′λ2)B=2_K2N+2_{K22 t2}−_K2′κB.

Suppose that_X2has generalized 1-type Gauss map, i.e.,_{Δ2 G2}=_{f2 G2}+_{g2 C2}. Without loss of generality, we may decompose the constant vector_C2via the Cartan frame{T,N,B}ofr(s)as

_C2=_C21T+_C22N+_C23B,

where_C21=〈_C2,B〉,_C22=〈_C2,N〉,_C23=〈_C2,T〉.

Substituting (23), (30) and (31) into_{Δ2 G2}=_{f2 G2}+_{g2 C2}, we obtain the following equation system

_{g2 C21}=0,_f2+_{g2 C22}=2^κ2λ2,_f2κ_t2^λ2+_{g2 C23}=2κ^λ′λ3+2^κ3 _t2^λ4−2^κ′λ2.

Since_g2 is a non-zero smooth function, from the first equation of (32), we conclude_C21=0 . By the last two equations of (32), we have

_f2(s,_t2)=_C22(2κλ^λ′+2^κ3 _t2−2^{κ′ λ2})−2_C23 ^{κ2 λ2λ2}(_C22κ_t2−_C23 ^λ2)=_C22(2_{K22 t2}−_K2′)−2_C23κ_{K2C22 K2 t2}−_C23κ,_g2(s,_t2)=2λ^(κλ)′_C22κ_t2−_C23 ^λ2=_{K2′C22 K2 t2}−_C23κ,

where_C222+_C232≠0. Meanwhile,κλis a non-constant function since_g2is a non-zero smooth function.

Conversely, if we use the above information with the given functions(_f2,_g2) in (33) and the constant vector_C2, the null scroll_X2satisfies_{Δ2 G2}=_{f2 G2}+_{g2 C2}.

Theorem 2.

Let_X2be a null scroll with generalized B-lightlike ruling in_E13. Then_X2has generalized 1-type Gauss map if and only if the Gauss map_G2of_X2satisfies

_{Δ2 G2}=_{f2 G2}+_{g2 C2}

for some non-zero smooth functions(_f2,_g2)as

_f2=_f2(s,_t2)=_C22(2κλ^λ′+2^κ3 _t2−2^{κ′ λ2})−2_C23 ^{κ2 λ2λ2}(_C22κ_t2−_C23 ^λ2)=_C22(2_{K22 t2}−_K2′)−2_C23κ_{K2C22 K2 t2}−_C23κ,_g2=_g2(s,_t2)=2λ^(κλ)′_C22κ_t2−_C23 ^λ2=_{K2′C22 K2 t2}−_C23κ

and a constant vector_C2=(0,_C22,_C23). Where_C222+_C232≠0andκλis a non-constant smooth function.

Remark 2.

In particular, when_C2=(0,0,_C23), (_C23≠0), the functions(_f2,_g2)only depend on s, i.e.,

_f2=_f2(s)=2^κ2λ2=2_K2,_g2=_g2(s)=2(κ^λ′−^κ′λ)_C23 ^λ3=−_K2′C23κ.

Corollary 5.

Let_X2be a null scroll with generalized B-lightlike ruling in_E13. Then_X2has pointwise 1-type Gauss map of the second kind if and only if the Gauss map_G2of_X2satisfies

_{Δ2 G2}=_f2(_G2+_C2)

for some non-zero smooth functions_f2as

_f2=2^κ2λ2=2_K2,

and a constant vector_C2=(0,0,_C23), (_C23≠0).

Proof Corollary 5

Suppose that the null scroll_X2has pointwise 1-type Gauss map of the second kind, i.e.,_{Δ2 G2}=_f2(_G2+_C2). It means that_f2=_g2in Theorem 2. Thus, we get2^κ3 _C22=0. Since_f2is a non-zero smooth function, thenκ≠0and_C22=0.Therefore,

_f2=2^κ2λ2=2_K2.

Conversely, if we use the above information with the given function_f2and constant vector_C2, the null scroll_X2satisfies_{Δ2 G2}=_f2(_G2+_C2).  □

Corollary 6.

There does not exist the null scroll with generalized B-lightlike ruling in_E13which has 1-type Gauss map of the second kind.

Proof Corollary 6

Suppose that the null scroll_X2has 1-type Gauss map of the second kind, i.e.,_{Δ2 G2}=μ(_G2+_C2),(μ∈R−{0}). It means that_f2=2^κ2λ2=μ.Obviously,_H2 is a non-zero constant. By (32), we get_C22=0and_C23=−_H2′κ_H2=0which contradicts with_C23≠0in Corollary 5.  □

Corollary 7.

Let_X2be a null scroll with generalized B-lightlike ruling in_E13. The_X2has pointwise 1-type Gauss map of the first kind if and only if one of the following statements holds:

• _X2has 1-type Gauss map of the first kind;

• _X2has non-zero constant Gaussian curvature or non-zero constant mean curvature.

Proof Corollary 7

Suppose that the null scroll_X2has pointwise 1-type Gauss map of the first kind, i.e.,_{Δ2 G2}=_{f2 G2} . From the last two equations of (32), we have_f2=2^κ2λ2and^(κλ)′=0. Then the function_f2is a non-zero constant function, i.e.,_X2 has 1-type Gauss map of the first kind. By (26) and (27), the Gaussian curvature_K2and the mean curvature_H2are non-zero constant. Conversely, if one of the statements holds, thenκλis a non-zero constant. This completes the proof.  □

Corollary 8.

Let_X2be a null scroll with generalized B-lightlike ruling in_E13. The_X2has harmonic Gauss map if and only if the Gaussian curvature or the mean curvature of_X2vanishes.

Proof Corollary 8

Suppose that the null scroll_X2has harmonic Gauss map, i.e.,_{Δ2 G2}=0.Then we haveκ=0and the Gaussian curvature_K2and the mean curvature_H2 are equal to zero by (26) and (27). The converse is obvious.  □

3.3. The Relationship between the Dual Associate Null Scrolls In this section, we summarize and investigate the relations between the dual associate null scrolls. Meanwhile, the representations of some dual associate null scrolls are obtained according to their Gauss maps.

By (14) and (28), we have the following conclusion readily.

Theorem 3.

The Gaussian curvatures_Ki(i=1,2)and the mean curvatures_Hi(i=1,2)of the dual associate null scrolls_Xi(i=1,2)in_E13are related by

_{H12 H22}=_{K1 K2}=^κ2.

From (16) and (30), we get

Theorem 4.

The Laplacians of Gauss maps_{Δi Gi}(i=1,2)of the dual associate null scrolls_Xi(i=1,2)in_E13are related by

〈_{Δ1 G1},_{Δ1 G1}〉〈_{Δ2 G2},_{Δ2 G2}〉=16_{K12 K22}=16^κ4.

From Proposition 3 and Corollary 1, we can get the following result.

Corollary 9.

Let_X1be a null scroll with generalized T-lightlike ruling which has pointwise 1-type Gauss map of the second kind in_E13. Then the dual associate null scrolls can be expressed as

_X1=∫2^e_C11s_C0(−^f″24^f′3(^f2−1,2f,^f2+1)+^{f″ f′}(f,1,f)−^f′(1,0,1))ds+_t12_C04^{f′ e_C11s}(^f2−1,2f,^f2+1),_X2=∫2_C04^{f′ e_C11s}(^f2−1,2f,^f2+1)ds+_t22^e_C11s_C0(−^f″24^f′3(^f2−1,2f,^f2+1)+^{f″ f′}(f,1,f)−^f′(1,0,1)),

where_C0∈R−{0}and_C11≠0.

By the proof of Corollary 3, the following conclusion is obtained easily.

Corollary 10.

Let_X1be a null scroll with generalized T-lightlike ruling which has pointwise 1-type Gauss map of the first kind in_E13. Then the dual associate null scrolls can be expressed by

_X1=∫1c(−^f″24^f′3(^f2−1,2f,^f2+1)+^{f″ f′}(f,1,f)−^f′(1,0,1))ds+_t1c2^f′(^f2−1,2f,^f2+1),_X2=∫c2^f′(^f2−1,2f,^f2+1)ds+_t21c(−^f″24^f′3(^f2−1,2f,^f2+1)+^{f″ f′}(f,1,f)−^f′(1,0,1)),

wherec∈R−{0}.

Theorem 5.

Let_X2be a null scroll with generalized B-lightlike ruling which has harmonic Gauss map in_E13. Then the dual associate null scrolls can be expressed by

_X1(s,_t1)=∫1λ(1a,0,−1a)ds+_t1λ(a2−^s22a,−s,a2+^s22a),_X2(s,_t2)=∫λ(a2−^s22a,−s,a2+^s22a)ds+_t21λ(1a,0,−1a).

wherea∈R−{0}.

Proof of Theorem 5

By the proof of Corollary 8, we haveκ=0 . From (5), we have

12^{[(logf′)′]2}−^(logf′)″=0.

Solving the above equation, the structure functionf(s)ofr(s)is given by

f(s)=−as,(a∈R−{0}).

By Proposition 3, the dual associate null scrolls can be expressed by

_X1(s,_t1)=∫1λ(1a,0,−1a)ds+_t1λ(a2−^s22a,−s,a2+^s22a),_X2(s,_t2)=∫λ(a2−^s22a,−s,a2+^s22a)ds+_t21λ(1a,0,−1a).

□

Example 2.

Consider a null curver(s)withκ=−12. From Proposition 2, we have

12^{[(logf′)′]2}−^(logf′)″=−12.

Solving the above differential equation, we havef(s)=2tans2. From Proposition 3, whenλ(s)=12, the dual associate curves and the dual associate null scrolls can be written as (see Figure 5 and Figure 6)

_b1(s)=(^sin2s2−14^cos2s2,12sins,^sin2s2+14^cos2s2),_b2(s)=(12^sin2s2−2^cos2s2,sins,−12^sin2s2−2^cos2s2)

and

_X1(s,_t1)=(−34s−54sins,−coss,−54s−34sins)+_t1(^sin2s2−14^cos2s2,12sins,^sin2s2+14^cos2s2),_X2(s,_t2)=(38s−58sins,−12coss,58s−38sins)+_t2(12^sin2s2−2^cos2s2,sins,−12^sin2s2−2^cos2s2).

Example 3.

In Theorem 5, supposinga=1and the functionλ(s)=^es , then the dual associate curves and the dual associate null scrolls can be written as (see Figure 7 and Figure 8).

_b1(s)=^es(12−^s22,−s,12+^s22),_b2(s)=^e−s(1,0,−1)

and

_X1(s,_t1)=(−^e−s,0,^e−s)+_t1 ^es(12−^s22,−s,12+^s22),_X2(s,_t2)=(−12^{es (s−1)2},^es(1−s),12^es(^(s−1)2+2))+_t2 ^e−s(1,0,−1).

Remark 3.

In Example 2, the dual associate null scrolls have pointwise 1-type Gauss map of the first kind; in Example 3, the null scroll_X1has pointwise 1-type Gauss map of the second kind and the null scroll_X2has harmonic Gauss map, respectively.

Author Contributions

J.Q. and X.F. set up the problem and computed the details. S.D.J. checked and polished the draft. All authors have read and agreed to the published version of the manuscript.

Funding

The first author was supported by NSFC (11801065) and the Fundamental Research Funds for the Central Universities (N2005012). The third author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2018R1A2B2002046) and the 2020 scientific promotion program funded by Jeju National University.

Acknowledgments

We thank H. Liu of Northeastern University and the referee for the valuable comments to improve this paper.

Conflicts of Interest

The authors declare no conflict of interest.