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Helicoidal Hypersurfaces of Dini-Type with

Spacelike Axis in Minkowski 4-Space

Erhan Güler1*, Ömer Ki¸si2

1 BartınUniversity,FacultyofSciences,DepartmentofMathematics,74100Bartın;eguler@bartin.edu.tr

2 BartınUniversity,FacultyofSciences,DepartmentofMathematics,74100Bartın;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=λr, 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

∆Jr=Ar,J= I,I I,I I I, whereA= (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

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Arvanitoyeorgos, Kaimakamais and Magid [2] show that if the mean curvature vector field ofM13

36

satisfies the equation∆H=αH(αa constant), thenM31has 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

dx2i −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

∇Xξ = −Aξ(X) +DXξ, (2)

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

68

M, respectively.

69

For eachξ ∈ T⊥pM, the shape operatorAξ is a symmetric endomorphism of the tangent space TpMatp∈M. The shape operator and the second fundamental form are related by

(3)

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{θ1,θ2, . . . ,θ}. Then the first structural equation of Cartan is

dθi= n

i=1

θj∧ωij, i=1, 2, . . . ,n (5)

whereωij 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) = ωij(ej)(ki−kj), (6)

ωij(el)(ki−kj) = ωil(ej)(ki−kl) (7)

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

75

We putsj =σj(k1,k2, . . . ,kn), whereσjis thej-th elementary symmetric function given by

σj(a1,a2, . . . ,an) =

1≤i1<i2<...,ij≤n

ai1ai2. . .aij.

We also use the following notation

rij=σj(k1,k2, . . . ,ki−1,ki+1,ki+2, . . . ,kn).

By the definition, we haver0i =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= n1s1andK=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

rthat does not meetCis obtained by taking the orbit ofCunder those orthogonal transformations of

84

En1that leavesrpointwise fixed (See [10, Remark 2.3]). 85

Suppose that when a curveCrotates around the axisr, it simultaneously displaces parallel lines

86

orthogonal to the axisr, so that the speed of displacement is proportional to the speed of rotation.

87

Then the resulting hypersurface is called thehelicoidal hypersurfacewith axisrand pitchesa,b∈R\{0}.

(4)

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

γ(u) = (ϕ(u),f(u), 0, 0,). (8) Ifris the spacelikex1-axis, then an orthogonal transformations ofEn1that leavesrpointwise 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 axisris

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

LetM=M(u,v,w)be an isometric immersion of a hypersurfaceM3in

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 hypersurfaceMinE41, the first fundamental form matrix is as follows:

I= gij

3×3,

and

detI=det gij

,

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

II= hij

3×3,

and

detII=det hij

(5)

where 1≤i,j≤5,

g11 =MMu, g12=MMv, ..., g55 =MMw,

and

h11 =Muu·G, h12=Muv·G, ..., h55=Mww·G,

”·” means dot product, and some partial differentials that we represent areMu= Mu,Muw= 2M

uw,

G= MMMw

kMMMwk

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)Sas follows:

S= 1 detI

sij

3×3, (12)

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

K=det(S) =detII

detI, (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=

 

ϕ02−cosh2u aϕ0 bϕ0 aϕ0 sinh2ucosh2w+a2 ab

bϕ0 ab sinh2u+b2

,

and have

detI=nϕ02sinh2ucosh2w−

h

a2+b2+sinh2ucosh2wicosh2uosinh2u,

whereϕ= ϕ(u),ϕ0= dduϕ.

(6)

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

96

L = sinh

2ucoshw(

ϕ00coshu+ϕ0sinhu)

p

kdetIk ,

M = asinhucosh

2ucoshw p

kdetIk ,

N = sinh

2ucosh2w(

ϕ0sinhucoshw−bcoshusinhw)

p

kdetIk ,

P = bsinhucosh

2ucoshw p

kdetIk ,

T = asinh

2ucoshusinhw p

kdetIk ,

V = ϕ

0sinh3ucoshw p

kdetIk ,

and we get

detI I=

       

ϕ02ϕ00sinh8ucoshucosh5w+bϕ0ϕ00sinh7ucosh2usinhwcosh4w

+a2ϕ00sinh6ucosh3usinh2wcoshw+ϕ03cosh5wsinh9u

−bϕ02sinh8ucoshucosh4wsinhw

−ha2sinh2usinh2w+a2+b2cosh2wcosh2ucosh2wiϕ0sinh5ucosh2ucoshw

+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= (−ϕ0sinhusinhw−bcoshucoshw)sinhucoshw,

99

e3= (−ϕ0sinhusinhvcosh2w+acoshucoshv+bcoshusinhvsinhwcoshw)sinhu,

100

e4= (−ϕ0sinhucoshvcosh2w+acoshusinhv+bcoshucoshvsinhwcoshw)sinhu.

101

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

K= α1ϕ 02

ϕ00+α2ϕ0ϕ00+α3ϕ00+α4ϕ03+α5ϕ02+α6ϕ0+α7

(detI)5/2 ,

where

102

α1=−sinh8ucoshucosh5w, 103

α2=bsinh7ucosh2usinhwcosh4w, 104

α3=a2sinh6ucosh3usinh2wcoshw, 105

α4=sinh9ucosh5w, 106

α5=−bsinh8ucoshucosh4wsinhw, 107

α6=− h

a2sinh7usinh2w+a2+b2cosh2wsinh5ucosh2ucosh2wicosh2ucoshw, 108

α7=b

2a2+b2cosh2wsinh4ucosh5usinhwcosh2w.

(7)

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

H= β1ϕ 00+

β2ϕ03+β3ϕ02+β4ϕ0+β5 3(detI)3/2 , where

110

β1=− h

sinh6ucosh3w+a2+b2cosh2wsinh4ucoshwiϕ00coshu

111

β2= +2ϕ03sinh5ucosh3w, 112

β3=−bϕ02sinh4ucoshusinhwcosh2w,

113

β4=sinh7ucosh3w+

a2+b2cosh2wsinh5ucoshw−2 sinh5ucosh2ucosh3w

114

−3a2+b2cosh2wsinh3ucosh2ucoshw,

115

β5= +bsinh4ucosh3ucosh2wsinhw+b

2a2+b2cosh2wsinh2ucosh3usinhw. 116

Theorem 1.LetD: M3−→E4be an isometric immersion given by(15). Then M3is flat if and only if α1ϕ02ϕ00+α2ϕ0ϕ00+α3ϕ00+α4ϕ03+α5ϕ02+α6ϕ0+α7=0.

117

Theorem 2.LetD: M3−→

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

β1ϕ00+β2ϕ03+β3ϕ02+β4ϕ0+β5=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=−β2,

A5=2β1+6β2sinhu+2β3, A4=

3−12 sinh2uβ2−8β3sinhu−4β4, A3=8β1coshu+

8 sinh3u−12 sinhuβ2 +8 sinh2u−4β3+8β4sinhu+8β5, A2=

−3+12 sinh2uβ2+8β3sinhu+4β4, A1=−2β1+6β2sinhu+2β3,

A0=β2.

(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

(8)

then we get

8

i=0

Bitanhi

u

2

=0,

where

B8=α1,

B7=−4α1sinhu−2α2−2α4, B6=

−2+4 sinh2u+4 coshuα1+4α2sinhu+4α3+12α4sinhu+4α5, B5= (4 sinhu−16 coshusinhu)α1+ (2−8 coshu)α2+

6−24 sinh2uα4

−16α5sinhu−8α6,

B4=

−8 coshu+16 coshusinh2uα1+16α2coshusinhu+16α3coshu +16 sinh3u−24 sinhuα4+

16 sinh2u−8α5+16α6sinhu+16α7, B3= (4 sinhu+16 coshusinhu)α1+2α2+8α2coshu+

−6+24 sinh2uα4 +16α5sinhu+8α6,

B2=

2−4 sinh2u+4 coshuα1−4α2sinhu−4α3+12α4sinhu+4α5, B1=−4α1sinhu−2α2+2α4,

B0=−α1. 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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128

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131

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(1994) 3-4, 171-204.

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135

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137

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138

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147

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148

14. Güler E., Kaimakamis G., Magid M. Helicoidal hypersurfaces in Minkowski 4-spaceE41(submitted). 149

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of Submanifolds, France (2007) 251-256.

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Babe¸s-Bolyai Math. 60-3 (2015) 437-448.

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References

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