**Helicoidal Hypersurfaces of Dini-Type with**

**Spacelike Axis in Minkowski 4-Space**

**Erhan Güler1*, Ömer Ki¸si2**

1 _{Bartın}_{University,}_{Faculty}_{of}_{Sciences,}_{Department}_{of}_{Mathematics,}_{74100}_{Bartın;}_{eguler@bartin.edu.tr}

2 _{Bartın}_{University,}_{Faculty}_{of}_{Sciences,}_{Department}_{of}_{Mathematics,}_{74100}_{Bartın;}_{okisi@@bartin.edu.tr}

***** Correspondence:eguler@bartin.edu.tr;Tel.:+90-378-5011000-1553

**Abstract: **Inthispaper,wedefineUlisseDini-typehelicoidalhypersurfacewithspacelikeaxisin

1

Minkowski 4-spaceE41. We compute the Gaussian and the mean curvature of the hypersurface. 2

Moreover, we obtain some special symmetry to the curvatures when they are flat and maximal.

3

**Keywords:**Minkowski space; Dini-type helicoidal hypersurface; Gauss map, spacelike axis

4

**1. Introduction**

5

The notion of finite type immersion of submanifolds of a Euclidean space has been used in

6

classifying and characterizing well known Riemannian submanifolds [4]. Chen posed the problem of

7

classifying the finite type surfaces in the 3-dimensional Euclidean spaceE3. An Euclidean submanifold

8

is said to be of Chen finite type if its coordinate functions are a finite sum of eigenfunctions of its

9

Laplacian∆[4]. Further, the notion of finite type can be extended to any smooth functions on a

10

submanifold of a Euclidean space or a pseudo-Euclidean space. Then the theory of submanifolds of

11

finite type has been studied by many geometers.

12

Takahashi [28] states that minimal surfaces and spheres are the only surfaces inE3satisfying the

13

condition∆r=*λ*r, whereris the position vector,*λ*∈R. Ferrandez, Garay and Lucas [11] prove that

14

the surfaces ofE3satisfying∆H=AH, whereHis the mean curvature andA∈ Mat(3, 3), are either

15

minimal, or an open piece of sphere or of a right circular cylinder. Choi and Kim [7] characterize the

16

minimal helicoid in terms of pointwise 1-type Gauss map of the first kind.

17

Dillen, Pas and Verstraelen [8] prove that the only surfaces inE3satisfying∆r= Ar+B,A ∈

18

Mat(3, 3),B∈Mat(3, 1)are the minimal surfaces, the spheres and the circular cylinders. Senoussi and

19

Bekkar [27] study helicoidal surfacesM2inE3which are of finite type in the sense of Chen with respect

20

to the fundamental formsI,I IandI I I, i.e., their position vector fieldr(u,v)satisfies the condition

21

∆J_{r}_{=}_{Ar}_{,}_{J}_{=} _{I}_{,}_{I I}_{,}_{I I I}_{, where}_{A}_{= (a}

ij)is a constant 3×3 matrix and∆Jdenotes the Laplace operator 22

with respect to the fundamental formsI,I IandI I I.

23

In classical surface geometry in Euclidean space, it is well known that the right helicoid (resp.

24

catenoid) is the only ruled (resp. rotational surface) which is minimal. If we focus on the ruled

25

(helicoid) and rotational characters, we have Bour’s theorem in [3].

26

About helicoidal surfaces in Euclidean 3-space, do Carmo and Dajczer [9] proved that, by using

27

a result of Bour [3], there exists a two-parameter family of helicoidal surfaces isometric to a given

28

helicoidal surface.

29

Lawson [20] give the general definition of the Laplace-Beltrami operator in his lecture notes.

30

Magid, Scharlach and Vrancken [22] introduce to the affine umbilical surfaces in 4-space. Vlachos

31

[30] consider hypersurfaces inE4with harmonic mean curvature vector field. Scharlach [26] study

32

on affine geometry of surfaces and hypersurfaces in 4-space. Cheng and Wan [5] consider complete

33

hypersurfaces of 4-space with constant mean curvature. Arslan, Deszcz and Yaprak [1] study on Weyl

34

pseudosymmetric hypersurfaces.

35

Arvanitoyeorgos, Kaimakamais and Magid [2] show that if the mean curvature vector field ofM_{1}3

36

satisfies the equation∆H=*α*H(αa constant), thenM3_{1}has constant mean curvature in Minkowski

37

4-spaceE41. This equation is a natural generalization of the biharmonic submanifold equation∆H=0. 38

General rotational surfaces as a source of examples of surfaces in the four dimensional Euclidean

39

space were introduced by Moore [23,24]. Ganchev and Milousheva [12] consider the analogue of these

40

surfaces in the Minkowski 4-space. They classify completely the minimal general rotational surfaces

41

and the general rotational surfaces consisting of parabolic points.

42

Verstraelen, Valrave and Yaprak [29] study on the minimal translation surfaces inEnfor arbitrary

43

dimensionn.

44

Kim and Turgay [19] study surfaces with L1-pointwise 1-type Gauss map inE4. Moruz and

45

Munteanu [25] consider hypersurfaces in the Euclidean spaceE4defined as the sum of a curve and a

46

surface whose mean curvature vanishes. They call them minimal translation hypersurfaces inE4and

47

give a classification of these hypersurfaces. Kim·et al [18] focus on Cheng-Yau operator and Gauss

48

map of surfaces of revolution.

49

Güler Magid and Yaylı [15] study Laplace Beltrami operator of a helicoidal hypersurface in

50

E4. Güler, Hacisalihoglu and Kim [13] work on the Gauss map and the third Laplace-Beltrami

51

operator of rotational hypersurface inE4. Güler, Kaimakamis and Magid [14] introduce the helicoidal

52

hypersurfaces in Minkowski 4-spaceE41. Güler and Turgay [16] study Cheng-Yau operator and Gauss 53

map of rotational hypersurfaces inE4. Moreover, Güler, Turgay and Kim [17] considerL2operator and 54

Gauss map of rotational hypersurfaces inE5.

55

In this paper, we study the Ulisse Dini-type helicoidal hypersurface with spacelike axis in

56

Minkowski 4-spaceE41. We give some basic notions of four dimensional Minkowskian geometry, 57

and define helicoidal hypersurface in section 2. Moreover, we obtain Ulisse Dini-type helicoidal

58

hypersurface, and calculate its curvatures in the last section.

59

**2. Preliminaries**

60

In this section we would like to describe the notaion that we will use in the paper after we give

61

some of basic facts and basic definitions.

62

LetEmdenote the Minkowskianm-space with the canonical Euclidean metric tensor given by

e

g=h, i=

m−1

### ∑

i=1

dx2_{i} −dx2m,

where(x1,x2, . . . ,xm)is a coordinate system inEm1. 63

Consider ann-dimensional Riemannian submanifold of the spaceEm. We denote Levi-Civita

64

connections ofE1mandMby∇e and∇, respectively. We shall use lettersX, Y, Z, W(resp.,*ξ,* *η) to*
65

denote vectors fields tangent (resp., normal) toM. The Gauss and Weingarten formulas are given,

66

respectively, by

67

e

∇XY = ∇XY+h(X,Y), (1)

e

∇X*ξ* = −A* _{ξ}*(X) +DX

*ξ,*(2)

whereh,DandAare the second fundamental form, the normal connection and the shape operator of

68

M, respectively.

69

For each*ξ* ∈ T⊥_{p}M, the shape operatorA* _{ξ}* is a symmetric endomorphism of the tangent space
TpMatp∈M. The shape operator and the second fundamental form are related by

The Gauss and Codazzi equations are given, respectively, by

70

hR(X,Y,)Z,Wi = hh(Y,Z),h(X,W)i − hh(X,Z),h(Y,W)i, (3)

(∇¯Xh)(Y,Z) = (∇¯Yh)(X,Z), (4)

whereR, RDare the curvature tensors associated with connections∇andD, respectively, and ¯∇his defined by

(∇¯Xh)(Y,Z) =DXh(Y,Z)−h(∇XY,Z)−h(Y,∇XZ).

2.1. Hypersurfaces of Euclidean space

71

Now, letMbe an oriented hypersurface in the Euclidean spaceEn+1,Sits shape operator andxits position vector. We consider a local orthonormal frame field{e1,e2, . . . ,en}of consisting of principal

direction ofMcorresponding from the principal curvaturekifori=1, 2, . . .n. Let the dual basis of

this frame field be{*θ*1,*θ*2, . . . ,*θ*}. Then the first structural equation of Cartan is

d*θ*i=
n

### ∑

i=1

*θ*j∧*ω*ij, i=1, 2, . . . ,n (5)

where*ω*ij denotes the connection forms corresponding to the chosen frame field. We denote the
72

Levi-Civita connection ofMandEn+1by∇and∇e, respectively. Then, from the Codazzi equation (3) 73

we have

74

ei(kj) = *ω*ij(ej)(ki−kj), (6)

*ω*ij(el)(ki−kj) = *ω*il(ej)(ki−kl) (7)

for distincti,j,l=1, 2, . . . ,n,.

75

We putsj =*σ*j(k1,k2, . . . ,kn), where*σ*jis thej-th elementary symmetric function given by

*σ*j(a1,a2, . . . ,an) =

### ∑

1≤i1<i2<...,ij≤n

ai_{1}ai2. . .aij.

We also use the following notation

r_{i}j=*σ*j(k1,k2, . . . ,ki−1,ki+1,ki+2, . . . ,kn).

By the definition, we haver0_{i} =1 andsn+1=sn+2=· · ·=0.
76

On the other hand, we will call the functionskas thek-th mean curvature ofM. We would like to 77

note that functionsH= _{n}1s1andK=snare called the mean curvature and Gauss-Kronecker curvature
78

ofM, respectively. In particular,Mis said to bej-minimal ifsj≡0 onM. 79

2.2. Helicoidal hypersurfaces with spacelike axis in Minkowskian spaces

80

In this section, we will obtain the helicoidal hypersurfaces in Minkowski 4-space. Before we

81

proceed, we would like to note that the definition of rotational hypersurfaces in Riemannian space

82

forms were defined in [10]. A rotational hypersurfaceM⊂En1generated by a curveCaround an axis 83

**r**that does not meetCis obtained by taking the orbit ofCunder those orthogonal transformations of

84

En1that leaves**r**pointwise fixed (See [10, Remark 2.3]).
85

Suppose that when a curveCrotates around the axis**r**, it simultaneously displaces parallel lines

86

orthogonal to the axis**r**, so that the speed of displacement is proportional to the speed of rotation.

87

Then the resulting hypersurface is called thehelicoidal hypersurfacewith axis**r**and pitchesa,b∈_{R}\{0}.

Consider the particular casen=4 and letCbe the curve parametrized by

*γ*(u) = (*ϕ*(u),f(u), 0, 0,). (8)
If**r**is the spacelikex1-axis, then an orthogonal transformations ofEn1that leaves**r**pointwise fixed has
the formZ(v,w)as follows:

Z(v,w) =

1 0 0 0

0 coshw 0 sinhw 0 sinhvsinhw coshv coshwsinhv 0 coshvsinhw sinhv coshvcoshw

, (9)

wherev,w∈_{R}. Therefore, the parametrization of the rotational hypersurface generated by a curveC
around an axis**r**is

**H(u**,v,w) =Z(v,w)*γ*(u)t+ (av+bw) (1, 0, 0, 0)t, (10)
whereu∈ I,v,w∈[0, 2π],a,b∈_{R}\{0}.

89

Clearly, we write helicoidal hypersurface with spacelike axis as follows:

**H(u**,v,w) =

*ϕ*(u) +av+bw
usinhw
usinhvcoshw
ucoshvcoshw

. (11)

Whenw=0, we have helicoidal surface with spacelike axis inE41. 90

In the rest of this paper, we shall identify a vector (a,b,c,d) with its transpose (a,b,c,d)T. Now we

91

give some basic elements of the Minkowski 4-spaceE41. 92

Let**M**=**M**(u,v,w)be an isometric immersion of a hypersurfaceM3_{in}

E41. The inner product of

−→_{x} _{= (x}

1,x2,x3,x4),−→y = (y1,y2,y3,y4)is defined as follows:

−→

x · −→y = 3

### ∑

i=1

xiyi−x4y4,

and the vector product of−→x × −→y × −→z is defined as follows:

det

e1 e2 e3 −e4 x1 x2 x3 x4 y1 y2 y3 y4 z1 z2 z3 z4

.

For a hypersurface**M**inE41, the first fundamental form matrix is as follows:

**I**= gij

3×3,

and

det**I**=det gij

,

and then, the second fundamental form matrix is as follows:

**II**= hij

3×3,

and

det**II**=det hij

where 1≤i,j≤5,

g11 =**M**u·**M**u, g12=**M**u·**M**v, ..., g55 =**M**w·**M**w,

and

h11 =**M**uu·**G**, h12=**M**uv·**G**, ..., h55=**M**ww·**G**,

”·” means dot product, and some partial differentials that we represent are**M**u= *∂ _{∂}*

**M**

_{u},

**M**uw=

*∂*2

_{M}*∂*u*∂*w,

**G**= **M**u×**M**v×**M**w

k**M**u×**M**v×**M**wk

is the Gauss map (i.e. the unit normal vector). The product matrices

gij

−1

. hij

,

gives the matrix of the shape operator (i.e. Weingarten map)**S**as follows:

**S**= 1
det**I**

sij

3×3, (12)

So, we get the formulas of the Gaussian curvature and the mean curvature, respectively, as follow:

K=det(**S**) =det**II**

det**I**, (13)

and

H= 1

5tr(S). (14)

**3. Dini-Type Helicoidal Hypersurface with Spacelike Axis**

93

Now, we consider Dini-type helicoidal hypersurface with spacelike axis inE41, as follows:

D(u,v,w) =

*ϕ*(u) +av+bw
sinhusinhw
sinhusinhvcoshw
sinhucoshvcoshw

. (15)

whereu∈R\{0}and 0≤v,w≤2π.

94

Using the first differentials of(15)with respect tou,v,w, we get the first quantities as follow:

I=

*ϕ*02−cosh2u a*ϕ*0 b*ϕ*0
a*ϕ*0 sinh2ucosh2w+a2 ab

b*ϕ*0 ab sinh2u+b2

,

and have

detI=n*ϕ*02sinh2ucosh2w−

h

a2+b2+sinh2ucosh2wicosh2uosinh2u,

where*ϕ*= *ϕ*(u),*ϕ*0= d_{du}*ϕ*.

Using the second differentials with respect tou,v,w, we have the second quantities as follow:

96

L = sinh

2_{u}_{cosh}_{w}_{(}_{−}

*ϕ*00coshu+*ϕ*0sinhu)

p

kdetIk ,

M = asinhucosh

2_{u}_{cosh}_{w}
p

kdetIk ,

N = sinh

2_{u}_{cosh}2_{w}_{(}

*ϕ*0sinhucoshw−bcoshusinhw)

p

kdetIk ,

P = bsinhucosh

2_{u}_{cosh}_{w}
p

kdetIk ,

T = asinh

2_{u}_{cosh}_{u}_{sinh}_{w}
p

kdetIk ,

V = *ϕ*

0_{sinh}3_{u}_{cosh}_{w}
p

kdetIk ,

and we get

detI I=

−*ϕ*02*ϕ*00sinh8ucoshucosh5w+b*ϕ*0*ϕ*00sinh7ucosh2usinhwcosh4w

+a2*ϕ*00sinh6ucosh3usinh2wcoshw+*ϕ*03cosh5wsinh9u

−b*ϕ*02sinh8ucoshucosh4wsinhw

−ha2sinh2usinh2w+a2+b2cosh2wcosh2ucosh2wi*ϕ*0sinh5ucosh2ucoshw

+b2a2+b2cosh2wsinh4ucosh5usinhwcosh2w

(detI)3/2 .

The Gauss map of the helicoidal hypersurface with spacelike axis is

eD= √1 detI

e1 e2 e3 e4

, (16)

where

97

e1=−sinh2ucoshucoshw, 98

e2= (−*ϕ*0sinhusinhw−bcoshucoshw)sinhucoshw,

99

e3= (−*ϕ*0sinhusinhvcosh2w+acoshucoshv+bcoshusinhvsinhwcoshw)sinhu,

100

e4= (−*ϕ*0sinhucoshvcosh2w+acoshusinhv+bcoshucoshvsinhwcoshw)sinhu.

101

Finally, we calculate the Gaussian curvature of the helicoidal hypersurface with spacelike axis as follows:

K= *α*1*ϕ*
02

*ϕ*00+*α*2*ϕ*0*ϕ*00+*α*3*ϕ*00+*α*4*ϕ*03+*α*5*ϕ*02+*α*6*ϕ*0+*α*7

(detI)5/2 ,

where

102

*α*1=−sinh8ucoshucosh5w,
103

*α*2=bsinh7ucosh2usinhwcosh4w,
104

*α*3=a2sinh6ucosh3usinh2wcoshw,
105

*α*4=sinh9ucosh5w,
106

*α*5=−bsinh8ucoshucosh4wsinhw,
107

*α*6=−
h

a2_{sinh}7_{u}_{sinh}2_{w}_{+}_{a}2_{+}_{b}2_{cosh}2_{w}_{sinh}5_{u}_{cosh}2_{u}_{cosh}2_{w}i_{cosh}2_{u}_{cosh}_{w}_{,}
108

*α*7=b

2a2+b2cosh2wsinh4ucosh5usinhwcosh2w.

Then we calculate the mean curvature of the helicoidal hypersurface with spacelike axis as follows:

H= *β*1*ϕ*
00_{+}

*β*2*ϕ*03+*β*3*ϕ*02+*β*4*ϕ*0+*β*5
3(detI)3/2 ,
where

110

*β*1=−
h

sinh6ucosh3w+a2+b2cosh2wsinh4ucoshwi*ϕ*00coshu

111

*β*2= +2ϕ03sinh5ucosh3w,
112

*β*3=−b*ϕ*02sinh4ucoshusinhwcosh2w,

113

*β*4=sinh7ucosh3w+

a2+b2cosh2wsinh5ucoshw−2 sinh5ucosh2ucosh3w

114

−3a2+b2cosh2wsinh3ucosh2ucoshw,

115

*β*5= +bsinh4ucosh3ucosh2wsinhw+b

2a2_{+}_{b}2_{cosh}2_{w}_{sinh}2_{u}_{cosh}3_{u}_{sinh}_{w}_{.}
116

**Theorem 1.**LetD: M3−→_{E}4_{be an isometric immersion given by}_{(}_{15}_{). Then M}3_{is flat if and only if}
*α*1*ϕ*02*ϕ*00+*α*2*ϕ*0*ϕ*00+*α*3*ϕ*00+*α*4*ϕ*03+*α*5*ϕ*02+*α*6*ϕ*0+*α*7=0.

117

**Theorem 2.**LetD: M3_{−→}

E4be an isometric immersion given by(15). Then M3is maximal if and only if

*β*1*ϕ*00+*β*2*ϕ*03+*β*3*ϕ*02+*β*4*ϕ*0+*β*5=0.

Solutions of these two eqs. are attracted problem for us.

118

**Proposition 1.** If Dis Dini-type maximal helicoidal hypersurface with spacelike axis (i.e. H =0)in
Minkowski 4-space,taking(as in Dini helicoidal surface in Euclidean 3-space)

*ϕ*(u) =coshu+log

tanhu 2

,

then we get

6

### ∑

i=0

Aitanhi

u

2

=0,

119

where

A6=−*β*2,

A5=2β1+6β2sinhu+2β3, A4=

3−12 sinh2u*β*2−8β3sinhu−4β4,
A3=8β1coshu+

8 sinh3u−12 sinhu*β*2
+8 sinh2u−4*β*3+8β4sinhu+8β5,
A2=

−3+12 sinh2u*β*2+8β3sinhu+4β4,
A1=−2β1+6β2sinhu+2β3,

A0=*β*2.

(17)

120

**Proposition 2.**If Dis Dini-type flat hypersurface with spacelike axis (i.e. K=0)in Minkowski 4-space,
taking (as in Dini helicoidal surface in Euclidean 3-space)

*ϕ*(u) =coshu+log

tanhu 2

then we get

8

### ∑

i=0

Bitanhi

u

2

=0,

where

B8=*α*1,

B7=−4α1sinhu−2α2−2α4, B6=

−2+4 sinh2u+4 coshu*α*1+4α2sinhu+4α3+12α4sinhu+4α5,
B5= (4 sinhu−16 coshusinhu)*α*1+ (2−8 coshu)*α*2+

6−24 sinh2u*α*4

−16α5sinhu−8α6,

B4=

−8 coshu+16 coshusinh2u*α*1+16α2coshusinhu+16α3coshu
+16 sinh3u−24 sinhu*α*4+

16 sinh2u−8*α*5+16α6sinhu+16α7,
B3= (4 sinhu+16 coshusinhu)*α*1+2α2+8α2coshu+

−6+24 sinh2u*α*4
+16α5sinhu+8α6,

B2=

2−4 sinh2u+4 coshu*α*1−4α2sinhu−4α3+12α4sinhu+4α5,
B1=−4α1sinhu−2α2+2α4,

B0=−*α*1.
121

**Corollary 1.**In Proposition 1, and Proposition 2, we see following special symmetries, respectively:
A6=−A0, A5∼A1, A4=−A2,

and

B8=−B0, B7∼B1, B6∼B2, B5∼B3,

where ”∼” means similar ignored sign (or equal without sign of coefficient).

122

**Competing Interests**

123

The authors declare that they have no competing interests.

124

**Authors’ Contributions**

125

All authors completed the writing of this paper and read and approved the final manuscript.

126

127

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