Characterization of Tubular Surfaces in Terms of Finite III-type
HASSAN AL-ZOUBI1, HAMZA ALZAAREER1, MOHAMMAD AL-RAWAJBEH2,
MOHAMMAD AL-KAFAWEEN1
1Department of Mathematics,
Al-Zaytoonah University of Jordan,
P.O. Box 130, Amman 11733,
JORDAN
2Department of Computer Science,
Al-Zaytoonah University of Jordan,
P.O. Box 130, Amman 11733,
JORDAN
Abstract: - In this paper, we first define relations regarding the first and the second Laplace operators
corresponding to the third fundamental form III of a surface in the Euclidean space E3. Then, we will
characterize the tubular surfaces in terms of their coordinate finite type.
Key-Words: - Surfaces in the Euclidean 3-space, Surfaces of finite Chen-type, Laplace operator, Tubular
surfaces, third fundamental form, Anchor ring.
Received: February 25, 2023. Revised: December 15, 2023. Accepted: February 13, 2024. Published: April 5, 2024.
1 Introduction
Tubular surfaces are a fascinating class of
geometric objects that arise in differential
geometry. This kind of surface can be seen as the
result of sweeping a curve through the Euclidean
space, constructing a "tube" around the curve which
is considered as the direction of the tube. Studying
this class of surfaces, namely the tubular surfaces, is
fundamental in many branches of mathematics and
is applicable in physics, engineering, and computer
graphics.
Normal bundle is one of the essential concepts
associated with tubular surfaces. In the Euclidean 3-
space, the normal bundle of a curve consists of
vectors that are orthogonal to the curve at each
point. By extending these vectors, one can create a
tubular neighborhood along the curve. The radius of
this neighborhood, or the size of the tube, is a
crucial parameter that influences the geometry of
the tubular surface.
Immersions of finite Chen type, introduced by
B.-Y. 50 years ago, [1] and has become a significant
topic of active research in the field of differential
geometry. Surfaces of finite Chen type encompass
diverse surfaces that exhibit certain geometric
properties. Examples of surfaces that fall under this
category include immersions with vanishing Gauss
curvature, minimal surfaces, and various special
classes of surfaces, such as tubes [2], quadrics [3],
[4], [5], translation surfaces [6], [7], ruled surfaces
[8], [9], [10], surfaces of revolution [11], [12], [13],
[14], [15], spiral surfaces [16], cyclides of Dupin
[17], [18] and helicoidal surfaces [19], [20]. These
classes represent various special cases of surfaces
that fall under the umbrella of finite Chen type.
Each of these classes has its distinctive geometric
features.
For a connected surface Q in Euclidean 3-space
E3, described by coordinates v1, and v2, the 1st, 2nd,
and 3rd fundamental forms are represented by (gij),
(bij), and (eij) respectively.
The first fundamental form (gij) is associated
with the metric tensor of the surface, representing
lengths and angles on the surface. The second
fundamental form (bij) is related to the shape
operator and provides information about the
extrinsic curvature of the surface. The third
fundamental form (eij) is associated with the
derivatives of the unit normal vector to the surface.
The 1st differential parameter of Beltrami is a
mathematical quantity associated with surfaces in
differential geometry. Now, let's consider two
functions γ and δ defined on the surface Q. The 1st
differential parameter of Beltrami concerning the
fundamental form J = I, II, III between these two
functions is defined as:
J(
,
): = cij
/i
/j,
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where
/i: =
i
v
and (cij) represents the inverse
tensor of (gij), (bij), and (eij). The 2nd differential
parameter of Beltrami regarding the fundamental
form J of Q is defined by:
ΔJ
: = -
/j/i )(
1
ij
cc
c
, c = det(cij).
For the position vector z = z(v1,v2), of Q in E3 we
have the following relation:
IIIz = -ΙI(
K
H2
, z) -
K
H2
N,
where N is the unit normal vector field, K is the
Gauss curvature, and H is the mean curvature of Q.
It was subsequently demonstrated that a surface
meeting this criterion:
ΔΙIΙz = λz, λΙR,
i.e. the statement asserts that if Q: z = z(v1,v2)
satisfies this condition, where all coordinate
functions are eigenfunctions of ΔΙIΙ with eigenvalue
λ is the same, then Q is either a part of a sphere
(with λ = 2) or a minimals (with λ = 0). In other
words, this condition provides a geometric
characterization of surfaces based on the behavior of
their Laplace-Beltrami eigenfunctions. The
eigenvalue λ being equal to 0 suggests a minimal
surface, which is a surface with mean curvature
equal to zero, while λ being equal to 2 suggests a
spherical geometry.
2 Fundamentals
Consider the parametric representation
1 , 2 , 3 , ( )
Of a surface Q. Denote by:
rx =
, ry =
, rxx =
, …
For the metric I of Q, it’s known that:
I = Edx2 + 2Fdxdy + Gdy2.
Applying the Laplacian operator ΔΙ, to a sufficiently
differentiable function φ(x, y) defined on the same
region D gives, [21]:
ΔΙφ = -
y
yx
x
yx
FEG
EF
FEG
FG
FEG 222
1
.
The metric II of Q is:
II = Ldx2 + 2Mdxdy + Ndy2.
The Laplacian ΔΙΙ is given by [21]:
ΔΙΙφ =
y
yx
x
yx
MLN
LM
MLN
MN
MLN 222
1
.
The metric III of Q is:
III = e11 dx2 + 2e12 dxdy + e22 dy2.
The Laplacian ΔΙIΙ is given by:
ΔΙΙΙφ: = -eik
kΙΙΙφ/i.
For any vector-valued function r = {r1, r2, r3},
defined on B , we have:
ΔJr = {ΔJr1, ΔJr2, ΔJr3}, J = I, II, III.
Certainly, let's elaborate on the definition of
immersions of coordinate finite type, Subsequently,
we can extend this investigation to a significant
category of surfaces known as tubular surfaces.
Definition 1. A surface Q is termed to be of
coordinate finite type concerning the metric III if the
position vector r of Q adheres to a specific relation
of the form
IIIr = Ar, (1)
where A is a square matrix of order 3.
3 Tubular Surfaces
A tubular surface is a surface that is formed by
sweeping a regular unit speed curve C: c = c(v),
v(a, b) of finite length in space along a given
direction. It can be thought of as a surface "wrapped
around" a curve. Let T, N, B be the Frenet frame of
the curve C and let
Κ
> 0 be its curvature. Then a
regular parametric representation of a tubular
surface Ѣ of radius s satisfies 0 < s < min
Κ
1
is
given by [7]:
Ѣ : r(v, ψ) = c + s cosψ N + s sinψ B. (2)
For the components gij of the first fundamental
form I = gijdvidψ j we have:
gij =
22
2222
ss
ss
,
while the components bij of the second fundamental
form are given by:
bij =
ss
ss
cos
2
Κ
,
where τ is the torsion of the curve c, and δ:= (1 - s
Κ
cosψ). For the Gauss curvature of Ѣ, we have:
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KG = -
s
cos
Κ
(3)
As we note before the Gauss curvature never
vanishes, so we must have
Κ
≠ 0. The Beltrami
operator corresponding to the metric III of Ѣ can be
found as follows:
Δ=
vcos
β
v)cos 2
2
2
Κ
Κ
(
1
2
2
222
2)cos(τ
v
2τ
Κ
)sincos'(
Κcos
Κ2
. (4)
where β: =
Κ
'cosψ +
Κ
τsinψ, and ': =
dv
d
.
Inserting the position vector of (2) in relation (4) we
get:
Δr =
3
)cos(
Κ
T + (2s cosψ -
2
cos
1
Κ
)N + 2ssinψ B.
(5)
Let r1, r2, r3 the component functions of the
parametric representation (5). We will examine
when will the surface Ѣ satisfies the relation (1).
Analytically, we have:
3
2
1
r
r
r
=
333231
232221
131211
3
2
1
r
r
r
. (6).
Let ci, Ti, Ni, Bi, i = 1, 2, 3, the component functions
of the vectors c, T, N, and B respectively. From (2)
and system (6) we have:
3
)cos(
Κ
Ti + (2s cosψ -
2
cos
1
Κ
)Ni + 2ssinψ Bi =
= λi1(c1 + s cosψ N1+ s sinψ B1) +
λi2(c2 + s cosψ N2+ s sinψ B2) +
+ λi3(c3 + s cosψ N3+ s sinψ B3), (7)
i = 1, 2, 3.
We have the following two cases:
Case I. β = 0. Then
Κ
' = 0 and
Κ
τ = 0. Thus τ = 0
and
Κ
= const. ≠ 0, therefore the curve c is a plane
circle and so, Ѣ is an anchor ring. In this case, a
regular parametric representation of an anchor ring
is:
Ѣ: r(u,v) ={(a + scosu)cosv , (a+scosu)sinv ,s sinu},
(8)
a s, a,sIR, 0 u 2π, 0 v 2π.
The first fundamental form becomes:
Ι: = s2du2 + (a +scosu)2dv2,
while the second is:
II: = sdu2 + (a + scosu)cosudv2.
The Laplacian corresponding to the metric III of Ѣ
can be found as follows:
Δ = -
2
2
22
2
vucos
1
ucosu
sinu
u
. (9)
Let r1, r2, r3 the component functions of the
parametric representation (2). Applying relation (9)
for the functions r1, r2, and r3 we get:
Δr1 = Δ[(a + scosu)cosv] =
ucos
acosv
2
+ 2scosucosv,
Δr2 = Δ[(a + rcosu)sinv] =
ucos
asinv
2
+ 2scosusinv,
Δr3 = Δ(ssinu) = 2ssinu.
From the last three equations and system (6) we
have:
ucos
acosv
2
+ 2scosucosv =
λ11(a + scosu)cosv + λ12(a + scosu)sinv +λ13ssinu,
(10)
ucos
asinv
2
+ 2scosusinv =
λ21(a + scosu)cosv + λ22(a + scosu)sinv +λ23ssinu,
(11)
2ssinu = λ31(a + scosu)cosv +
λ32(a + scosu)sinv + λ33ssinu. (12)
From (12) it can be easily seen that:
λ31 = λ32 = 0, λ33 = 2.
Deriving relation (10) twice with respect to the
parameter v we get:
ucos
acosv
2
+ 2scosucosv =
λ11(a + scosu)cosv + λ12(a + scosu)sinv. (13)
From (10) and (13) we find that λ13 = 0.
Similarly, we will get λ23 = 0, and relations (10) and
(11) finally become
acosv = λ11acos2ucosv + λ12acos2usinv +
(λ11 - 2)scos3ucosv + λ12scos3usinv, (14)
asinv = λ21acos2ucosv + λ22acos2usinv +
+ (λ22 - 2)scos3usinv + λ21scos3ucosv. (15)
Deriving (14) and (15) with respect to the parameter
v we have:
2λ11acosv + 2λ12asinv +
3(λ11 - 2)scosucosv + 3λ12scosusinv = 0, (16)
2λ21acosv + 2λ22asinv +
3(λ22 - 2)scosusinv + 3λ21scosucosv = 0.
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Deriving (16) with respect to the parameter v we
have:
-2λ11asinv + 2λ12acosv - 3(λ11 - 2)scosusinv
+ 3λ12scosucosv = 0. (17)
Multiplying (16) by sinv and (17) by cosv and
adding the resulting equations, we obtain:
λ12(2a + 3rcosu) = 0 λ12 = 0.
Following the same procedure, we also get λ21 = 0.
Thus relations (14) and (15) become:
a =λ11acos2u + (λ11 - 2)scos3u,
a =λ22acos2u + (λ22 - 2)scos3u.
From the last two equations, we conclude that
λ11, and λ22 depend on the parameter u and are not
constants, and hence relation (1) cannot be satisfied
so we proved
Proposition 1. The position vector of a parametric
representation of an anchor ring (8) does not satisfy
the relation ΔIIIr = Ar.
Case II. β ≠ 0.
Recalling equations (7), then we can write these
equations as follows:
βTi + 2s
Κ
3cos4ψNi -
Κ
2cosψNi + 2s
Κ
3sinψcos3ψBi -
-λi1
Κ
3(c1cos3ψ + scos4ψN1 + scos3ψsinψB1) -
-λi2
Κ
3(c2cos3ψ + scos4ψN2 + scos3ψsinψB2) -
-λi3
Κ
3(c3cos3ψ + scos4ψN3 + scos3ψsinψB3) = 0,
i =1, 2, 3,
We also rewrite it in terms of cosψ as follows:
βTi -
Κ
2cosψNi + s
Κ
3(2Ni - λi1N1 - λi2N2 -
λi3N3)cos4ψ +
+ s
Κ
3(2Bi - λi1B1 - λi2B2 - λi3B3)cos3ψsinψ –
Κ
3(λi1c1 + λi2c2 + λi3c3)cos3ψ = 0,
i = 1, 2, 3.
The above equations for i =1, 2, 3, are
polynomials of the variables cosψ, sinψ with
coefficients functions of the variable v. To be the
last equations satisfied for all i = 1, 2, 3, then the
coefficients functions of these polynomials must
equal zeros. So we must have:
λi1c1 + λi2c2 + λi3c3 = 0,
2Ni - λi1n1 - λi2N2 - λi3N3 = 0,
2Bi - λi1B1 - λi2B2 - λi3B3 = 0,
βTi -
Κ
2cosψ Ni = 0, (18)
i = 1, 2, 3.
Since relation (18) holds for all i = 1, 2, 3, then
we write (18) in vector notation as follows:
βT +
Κ
2cosψ N = 0,
from which we obtain that β = 0 and
Κ
= 0. Hence
Ѣ is an anchor ring, a case that has been investigated
previously. So we proved:
Theorem 1. There are no tubular surfaces in the
three-dimensional Euclidean space whose position
vector satisfies the relation
IIIr = Ar.
4 Conclusion
This research article was divided into three sections,
where after the introduction, the needed definitions
and relations regarding this interesting field of study
were given. Then a formula for the Laplace operator
corresponding to the first, second, and third
fundamental forms of a surface Q were defined.
Finally, we classified the tubular surfaces satisfying
the relation Δr = Ar, for a real square matrix A of
order 3. It is also interesting if this type of research
can be applied to other families of surfaces that have
not been studied yet such as spiral surfaces, or
cyclides of Dupin.
Acknowledgement:
We thank the anonymous referees for their
insightful comments which led to substantial paper
improvements.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
Authors acknowledge the financial support from
ZUJ under the grant number, 08/07/2022-2023.
Conflict of Interest
The author has no conflicts of interest to declare.
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