On Ruled Surfaces of Coordinate Finite Type

HASSAN AL-ZOUBI, HAMZA ALZAAREER, AMJED ZRAIQAT, TAREQ HAMADNEH,

WASEEM AL-MASHALEH

Department of Mathematics,

Al-Zaytoonah University of Jordan,

P.O. Box 130, Amman 11733,

JORDAN

Abstract: - This article. in the introduction, gives a brief historic description on surfaces of finite Chen-type and

of coordinate finite Chen-type according to the first, second and third fundamental form of a surface in the

Euclidean space E3. Then, an important class of surfaces was introduced, namely, the ruled surfaces were

classified according to its coordinate finite Chen type with respect to the second fundamental form.

Key-Words: - Ruled surfaces, Surfaces in the Euclidean 3-space, Surfaces of coordinate finite Chen-type,

Laplace-Beltrami operator.

Received: October 14, 2021. Revised: September 11, 2022. Accepted: October 14, 2022. Published: November 4, 2022.

1 Introduction

Euclidean immersions of finite type were defined by

B.-Y. Chen about forty years ago and since then

research concerning this topic has become active by

many differential geometers. Many results on this

subject have been collected in the book [7]. Let Mn

be an n-dimensional submanifold of an arbitrary

dimensional Euclidean space Em. Denote by



I the

Beltrami- Laplace operator on Mn with respect to the

first fundamental form I of Mn. A submanifold Mn is

said to be of finite type with respect to the first

fundamental form I, if the vector field x of Mn can

be written as a finite sum of nonconstant ei-

genvectors of the Laplacian ΔI, that is,

x = c+





, (1)

where ΔIxi = λixi, i = 1,, k, c is a constant vector

and λ1, λ2, …, λk are eigenvalues of ΔI. Moreover, if

there are exactly k nonconstant eigenvectors x1, …,

xk appearing in (1) which all belong to different

eigenvalues λ1, …, λk, then Mn is said to be of I-type

k. However, if λi = 0 for some i = 1, , k, then Mn is

said to be of null I-type k , otherwise Mn is said to be

of infinite type.

The class of finite type submanifolds in an arbitrary

dimensional Euclidean space is very large,

meanwhile results about surfaces of finite type in

the Euclidean 3-space with respect to the first

fundamental form is very little known. Actually, so

far, minimal surfaces, the circular cylinders, and the

spheres are the only known surfaces of finite type in

the Euclidean 3-space. So in [8] B.-Y. Chen

mentions the following problem

Problem1. Determine all surfaces of finite type in

E3.

Important families of surfaces were studied by

different authors by proving that finite type ruled

surfaces, [10], finite type quadrics, [9], finite type

tubes [6], finite type cyclides of Dupin, [11], [12],

finite type cones, [13], and finite type spiral surfaces

[5] are surfaces of the only known examples in E3.

However, for surfaces of revolution, translation

surfaces as well as helicoidal surfaces, the

classification of its finite type surfaces is not known

yet.

In this area, S. Stamatakis and H. Al-Zoubi studied

the notion of surfaces of finite type with respect to

the second or third fundamental forms. Based on

this view, we raise the following questions:

Problem 2. Classify all surfaces of finite II-type in

E3.

Problem 3. Classify all surfaces of finite III-type in

E3.

According to problem 2, ruled surfaces [1] and

tubes are the only families that were studied

according to their finite type classification.

However, for all other classical families of surfaces,

the classification of its finite type surfaces is not

known yet.

This type of study can be also extended to any

smooth map, not necessary for the position vector of

the surface, for example, the Gauss map of a

surface. Here again, we give the following other two

problems

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.87

Hassan Al-Zoubi, Hamza Alzaareer,

Amjed Zraiqat, Tareq Hamadneh,

Waseem Al-Mashaleh

E-ISSN: 2224-2880

765

Volume 21, 2022

Problem 4. Classify all surfaces of finite II-type

Gauss map in E3.

Problem 5. Classify all surfaces of finite III-type

Gauss map in E3.

On one hand, an interesting theme within this

context is to study surfaces in E3 for which the

position vector x satisfies the condition ΔJx = Ax, J

= I, II, and A is a square matrix of order 3. Surfaces

satisfying this condition are said of coordinate finite

type. So we are led to the following problems

Problem 6. Classify all surfaces of coordinate finite

II-type in E3.

Problem 7. Classify all surfaces of coordinate finite

III-type in E3.

On the other hand, the last two problems mentioned

above can be applied for the Gauss map of a

surface, that is

Problem 8. Classify all surfaces of coordinate finite

II-type Gauss map in E3.

Problem 9. Classify all surfaces of coordinate finite

III-type Gauss map in E3.

Here also some results concerning the last two

problems can be found in [2] and [3].

In [4] the authors classified surfaces of revolution in

the Lorentz-Minkowski space, while in [15]

translation surfaces in Sol3 were studied.

2 Fundamentals

Let x = x(u1, u2) be a parametric representation of a

surface S in the Euclidean space E3 with non

vanishing Gauss curvature. Let I = gijduiduj, II =

bijduiduj and III = eijduiduj be the thee well-known

fundamental forms of S. For sufficient differentiable

functions f(u1, u2) and g(u1, u2) on S, the first

differential parameter of Beltrami with respect to

the fundamental form J = I, II, III is defined by

J(f,g): = aij f/i g/j, (2)

where f/i: =



and (aij) denotes the inverse tensor

of (gij), (bij) and (eij) for J = I, II and III respectively.

The second differential parameter of Beltrami with

respect to the fundamental form J = I, II, III of M is

defined by

ΔJf: = –aij

J

f/j, (3)

where f is a sufficiently differentiable function,

J

is the covariant derivative in the ui direction with

respect to the fundamental form J and (αij) stands, as

in definition (2), for the inverse tensor of (gij), (bij)

and (eij) for J = I, II and III respectively. Applying

(3) for the position vector x of S we have

IIx = –

gradIIIK – 2n. (4)

From (4) we obtain the following result:

Theorem 1 A surface S in E3 is of II-type 1 if and

only if S is part of a sphere.

Interesting research also one can follow the idea in

[14] by defining the first and second Laplace

operator using the definition of the fractional vector

operators.

Up to now, the only known surfaces of finite II-type

in E3 are parts of spheres. In this paper we will pay

attention to surfaces of finite II-type. Firstly, we will

establish a formula for IIx. Further, we continue

our study by proving coordinate finite type ruled

surfaces in the Euclidean 3-space, that is, their

position vector x satisfies:

ΔIIx = x (5)

3 Main Result

In the three-dimensional Euclidean space E3 let S be

a ruled Cr-surface, r ≥ 3, of nonvanishing Gaussian

curvature defined by an injective Cr-immersion x =

x(s, t) on a region U: = I  R (I  R open interval)

of R2. The surface S can be expressed in terms of a

directrix curve



: α = α (s) and a unit vector field β

(s) pointing along the rulings as follows

S: x(s,t) = α(s) + tβ(s), sJ, –  t   (6)

Moreover, we can take the parameter s to be the arc

length along the spherical curve β(s). Thus for the

curves α, β we have

 α', β  = 0,  β, β  = 1,  β', β'  = 1,

where the differentiation with respect to s is denoted

by a prime and  ,  denotes the standard scalar

product in E3. It can be easily verified that the first

and the second fundamental forms of S are given by

Ι = qds2 + dt2,

II =

ds2 +

dtds,

where

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.87

Hassan Al-Zoubi, Hamza Alzaareer,

Amjed Zraiqat, Tareq Hamadneh,

Waseem Al-Mashaleh

E-ISSN: 2224-2880

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Volume 21, 2022

q =  α', α' + 2 α', β' t + t2,

p = (α', β, α") + [(α', β, β") + (β', β, α")]t

+ (β', β, β")t2,

Α = (α', β, β').

If, for simplicity, we put

κ: =  α', α', λ: =  α', β',

μ: = (β', β, β"), ν: = (α', β, β") + (β', β, α"),

ρ: = (α', β, α"),

then we obtain the following relations

q = t2 + 2λt + κ, p = μt2 + νt + ρ.

Furthermore, the Gaussian curvature K of S is given

K =



Since S does not contain parabolic points, therefore

Α  0, sJ.

The Beltrami operator with respect to the second

fundamental form can be expressed as follows

ΔII =

























tsA

, (7)

where pt: =



Applying (7) for the position vector x we find













ββ

II =x

(8)

Let (x1, x2, x3) be the component functions of x.

Then it is well-known that

ΔIIx = (ΔIIx1, ΔIIx2, ΔIIx3). (9)

Let (α1, α2, α3) and (β1, β2, β3) be the coordinate

functions of the vectors α, β respectively. From (8)

we have

ΔIIxi =











i

iβ

, i = 1, 2, 3

Denote by λij the entries of the matrix , i, j = 1, 2,

3, where all entries are real numbers. By using (6),

and (8) condition (5) is found to be equivalent to the

following system

iβ

2

= (λi1α1 + λi2α2 + λi3α3)

+ (λi1β1 + λi2β2 + λi3β3)t, (10)

i = 1, 2, 3.

We put

Χi: = λi1α1 + λi2α2 + λi3α3, i = 1, 2, 3,

Yi: = λi1β1 + λi2β2 + λi3β3, i = 1, 2, 3.

Then equations (10) reduce to

iβ

2

= Χi + tYi

We raise the last ratio to the square and we get

q(4

ii i i

2 4 3

β' +β 4 β 'β

A A A



) = Χi2 + 2tΧiYi + t2Yi2,

i = 1, 2, 3.

or 4Α2(t2 + 2λt +κ)βi'2 + (t2 + 2λt +κ)(4μ2t2 +

2μνt +ν2)βi2 – 4Α(t2 + 2λt +κ)(2μt + ν)βiβi' =

= Α4(Χi2 + 2tΧiYi + t2Yi2),

i = 1, 2, 3,

which can be written analytically as

4μ2βi2t4 + (2μνβi2 + 8λμ2βi2 – 8Αμβiβi')t3

+ (4Α2βi'2 + ν2βi2 + 4κμ2βi2 + 4λμνβi2 – 4Ανβiβi'

– 16Αλμβiβi'– Α4Yi2)t2 +

(8Α2λβi'2 + 2λν2βi2 + 2κμνβi2 –

8Ακμβiβi'– 8λνAβiβi' – 2Α4ΧiYi )t

+ 4A2κβi'2 + κν2βi2 – 4Ακνβiβi' – Α4Χi2 = 0, (11)

i = 1, 2, 3.

It’s easily verified that (11) are polynomials in t

with functions in s as coefficients for i = 1, 2, 3.

This means that the coefficients of the powers of t in

(11) must be zeros, and so we have the following

equations

4μ2βi2 = 0, (12)

2μνβi2 + 8λμ2βi2 – 8Αμβiβi' = 0, (13)

4Α2βi'2 + ν2βi2 + 4κμ2βi2 + 4λμνβi2 – 4Ανβiβi'

– 16Αλμβiβi' – Α4Yi2= 0, (14)

8Α2λβi'2 + 2λν2βi2 + 2κμνβi2 – 8Ακμβiβi' – 8λνAβiβi'

– 2Α4ΧiYi = 0, (15)

4A2κβi'2 + κν2βi2 – 4Ακνβiβi' – Α4Χi2 = 0, (16)

i = 1, 2, 3.

Since (12) holds true for each i = 1, 2, 3 we

conclude

μ = (β', β, β") = 0.

This means that β', β, β"are linearly dependent

vectors, so there exist two functions

1 = 1(s) and 2 = 2(s) such that

β" = σ1β + σ2β'. (17)

Differentiating the relation  β', β' = 1, we get

 β', β" = 0. (18)

From (17) and (18) we obtain

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.87

Hassan Al-Zoubi, Hamza Alzaareer,

Amjed Zraiqat, Tareq Hamadneh,

Waseem Al-Mashaleh

E-ISSN: 2224-2880

767

Volume 21, 2022

β" = σ1β.

Hence ν = (β', β, α"). Relations (14), (15) and (16)

become

4Α2βi'2 + ν2βi2– 4Ανβiβi'– Α4Yi2= 0, (20)

8Α2λβi'2 + 2λν2βi2 – 8λνAβiβi' – 2Α4ΧiYi = 0, (21)

4A2κβi'2 + κν2βi2 – 4Ακνβiβi' – Α4Χi2 = 0,

i = 1, 2, 3.

Multiplying (20) by 2λ, and from (21) one finds

Xi = Yi, i = 1, 2, 3.

Or in vector notation X =



Which can be written α =



β. Now we have the

following two cases:

Case I.  is the zero matrix. Then from (8) and

taking into account pt = ν, it can be easily verified

that 2Aβ' - νβ = 0, which is a contradiction since it

yields that A = 0.

Case II. α =



β. Differentiating this equation with

respect to s we find

α' =



'β +



β'

Taking the inner product of both sides of the above

equation with respect to β we find that



' = 0, that

is,



is constant. Hence we will get α' =



β' and this

leads us to that A = 0 a case which has been

excluded. Thus we have proved the following

Theorem 2. There are no ruled surfaces in the

Euclidean 3-space that satisfy the relation (5).

4 Conclusion

Firstly, we introduce the class of ruled surfaces

in the Euclidean 3-space. Then, we define a

formula for the Laplace operator regarding the

second fundamental form II. Finally, we

classify the ruled surfaces satisfying the relation

ΔIIx = x, for a real square matrix  of order 3.

We proved that there are no ruled surfaces in the

Euclidean 3-space that satisfy the relation ΔIIx = x.

An interesting research one can follow, if this

type of study can be applied to other families of

surfaces that have not been investigated yet

such as quadric surfaces, tubular surfaces, or

spiral surfaces.

References:

[1] H. Al-Zoubi, A. Dababneh, M. Al-Sabbagh,

Ruled surfaces of finite II-type, WSEAS

Trans. Math. 18 (2019), pp. 1-5.

[2] H. Al-Zoubi, H. Alzaareer, T. Hamadneh, M.

Al Rawajbeh, Tubes of coordinate finite type

Gauss map in the Euclidean 3-space, Indian J.

Math. 62 (2020), 171-182.

[3] H. Al-Zoubi, M. Al-Sabbagh, Anchor rings of

finite type Gauss map in the Euclidean 3-

space, Int. J. Math. Comput. Methods 5

(2020), 9-13.

[4] H. Al-Zoubi, A. K. Akbay, T. Hamadneh, and

M. Al-Sabbagh, Classification of surfaces of

coordinate finite type in the Lorentz-

Minkowski 3-space. Axioms 11 (2022).

[5] Ch. Baikoussis, L. Verstraelen, The Chen-

type of the spiral surfaces, Results. Math. 28,

214-223 (1995).

[6] B.-Y. Chen, Surfaces of finite type in

Euclidean 3-space, Bull. Soc. Math. Belg. Ser.

B 39 243-254 (1987).

[7] B.-Y. Chen, Total mean curvature and

submanifolds of finite type, Second edition,

World Scientific Publisher, (2014).

[8] B.-Y. Chen, Some open problems and

conjectures on submanifolds of finite type,

Soochow J. Math. 17 (1991), pp. 169-188.

[9] B.-Y. Chen, F. Dillen, Quadrics, of finite type,

J. Geom. 38, 16-22 (1990).

[10] B.-Y. Chen, F. Dillen, L. Verstraelen, L.

Vrancken, Ruled surfaces of finite type, Bull.

Austral. Math. Soc. 42, 447-553 (1990).

[11] F. Denever, R. Deszcz L. Verstraelen, The

compact cyclides of Dupin and a conjecture

by B.-Y Chen, J. Geom. 46, 33-38 (1993).

[12] F. Denever, R. Deszcz L. Verstraelen, The

Chen type of the noncompact cyclides of

Dupin, Glasg. Math. J. 36, 71-75 (1994).

[13] O. Garay, Finite type cones shaped on

spherical submanifolds, Proc. Amer. Math.

Soc. 104, 868-870 (1988).

[14] M. Mhailan, M. Abu Hammad, M. Al Horani,

R. Khalil, On fractional vector analysis, J.

Math. Comput. Sci. 10 (2020), 2320-2326.

[15] B. Senoussi, H. Al-Zoubi, Translation

surfaces of finite type in Sol3, Comment.

Math. Univ. Carolin. 61 (2020), pp. 237-256.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.87

Hassan Al-Zoubi, Hamza Alzaareer,

Amjed Zraiqat, Tareq Hamadneh,

Waseem Al-Mashaleh

E-ISSN: 2224-2880

768

Volume 21, 2022

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Hassan Al-Zoubi carried out the introduction and

the main result of the article.

Amjed Zraiqat, Hamza Alzaareer and Tareq

Hamadneh have improved section 2.

Waseem Al-Mashaleh has reviewed and checked

the calculations of this paper.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

This study did not receive any funding in any form.

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(Attribution 4.0 International, CC BY 4.0)

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Creative Commons Attribution License 4.0

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2022.21.87

Hassan Al-Zoubi, Hamza Alzaareer,

Amjed Zraiqat, Tareq Hamadneh,

Waseem Al-Mashaleh

E-ISSN: 2224-2880

769

Volume 21, 2022