1. Introduction
The concept of finite-type immersion of submanifolds of a Euclidean space has been known in classifying and characterizing Riemannian submanifolds [
1]. Chen proposed the problem of classifying these kinds surfaces in the three-dimensional Euclidean space
. A Euclidean submanifold is called Chen finite-type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian
[
1]. Hence, the idea of finite-type can be enlarged to any smooth functions on a submanifold of Euclidean or pseudo-Euclidean spaces.
Takahashi [
2] obtained spheres and the minimal surfaces are the unique surfaces in
satisfying the condition
where
r is the position vector,
Ferrandez, Garay and Lucas [
3] showed the surfaces of
providing
. Here
H is the mean curvature and
are either of a right circular cylinder, or of an open piece of sphere, or minimal. Choi and Kim [
4] worked the minimal helicoid with pointwise 1-type Gauss map of the first type.
Dillen, Pas, and Verstraelen [
5] studied the unique surfaces in
providing
are the spheres, the circular cylinder, the minimal surfaces. Senoussi and Bekkar [
6] obtained helicoidal surfaces in
by using the fundamental forms
and
In classical surface geometry, it is well known a pair of the right helicoid and the catenoid is the unique ruled and rotational surface, which is minimal. When we look at ruled (i.e., helicoid) and rotational surfaces, we meet Bour’s theorem in [
7]. By using a result of Bour [
7], Do Carmo and Dajczer [
8] worked isometric helicoidal surfaces.
Lawson [
9] defined the generalized Laplace-Beltrami operator. Magid, Scharlach and Vrancken [
10] studied the affine umbilical surfaces in 4-space. Vlachos [
11] introduced hypersurfaces with harmonic mean curvature in
. Scharlach [
12] gave the affine geometry of surfaces and hypersurfaces in 4-space. Cheng and Wan [
13] studied complete hypersurfaces of 4-space with CMC. Arslan, Deszcz and Yaprak [
14] obtained Weyl pseudosymmetric hypersurfaces. Arvanitoyeorgos, Kaimakamais and Magid [
15] wrote that if the mean curvature vector field of
satisfies the equation
(
a constant), then
has constant mean curvature in Minkowski 4-space
. This equation is a natural generalization of the biharmonic submanifold equation
General rotational surfaces in the four-dimensional Euclidean space were originated by Moore [
16,
17]. Ganchev and Milousheva [
18] considered the analogue of these surfaces in
. Verstraelen, Valrave, and Yaprak [
19] studied the minimal translation surfaces in
for arbitrary dimension
Kim and Turgay [
20] studied surfaces with
-pointwise 1-type Gauss map in
. Moruz and Munteanu [
21] considered hypersurfaces defined as the sum of a curve and a surface whose mean curvature vanishes in
.
Yoon [
22] considered rotational surfaces which has a finite-type Gauss map in
Dursun [
23] introduced hypersurfaces of pointwise 1-type Gauss map in Minkowski space. Dursun and Turgay [
24] studied minimal, pseudo-umbilical rotational surfaces in
. Arslan, Bulca and Milousheva [
25] focused pointwise 1-type Gauss map of meridian surfaces in
. Aksoyak and Yaylı [
26] worked boost-invariant surfaces with pointwise 1-type Gauss map in
Also they [
27] considered generalized rotational surfaces of pointwise 1-type Gauss map in
Güler, Magid and Yaylı [
28] defined helicoidal hypersurface with the Laplace-Beltrami operator in
. Furthermore, Güler, Hacısalihoğlu and Kim [
29] worked rotational hypersurface with the III Laplace-Beltrami operator and the Gauss map in
.
There are few works in the literature about Italian Mathematician Ulisse Dini’s helicoidal surface [
30] in
. Moreover, Güler and Kişi [
31] introduced helicoidal hypersurfaces of Dini-type with spacelike axis in
In this paper, we study the Ulisse Dini-type helicoidal hypersurface with timelike axis in Minkowski 4-space
. We give some basic notions of Minkowskian geometry, and define helicoidal hypersurface in
Section 2. Moreover, we obtain the Dini-type helicoidal hypersurface timelike axis, and calculate its curvatures in the
Section 3. We obtain some special symmetries in the last section.
2. Preliminaries
In this section, we will describe the notation that will be used in the paper, after we give some basic facts and basic definitions.
Let
be the Minkowski
m-space with the Euclidean metric denoted by
where
is a coordinate system in
.
Consider an
n-dimensional Minkowskian submanifold of the space
. We denote Levi-Civita connections of
and
M by
and ∇, respectively. We will use letters
(resp.,
) to show vector fields tangent (resp., normal) to
M. The Gauss and the Weingarten formulas are defined by as follows:
where
h,
D, and
A are the second fundamental form, the normal connection and the shape operator of
M, respectively.
The shape operator
is a symmetric endomorphism of the tangent space
at
for each
. The second fundamental form and the shape operator are connected by
The Gauss and Codazzi equations are denoted, respectively, as follows:
Here,
are the curvature tensors related with connections ∇ and
D, respectively, and
is defined by
2.1. Hypersurfaces of Minkowski Space
Assume that
M be an oriented hypersurface in Minkowski space
,
its shape operator and
x its position vector. We think about a local orthonormal frame field
occurring of the principal directions of
M matching to the principal curvatures
for
. Let
be dual basis of this frame field. Then the first structural equation of Cartan is
Here,
demonstrates the connection forms matching to the chosen frame field. We show the Levi-Civita connection of
M and
by ∇ and
respectively. Then, from the Codazzi Equation (
2) we have
for distinct
.
We take
where
is the
j-th elementary symmetric function given by
We also use the following notation
By definition, we have and .
On the other hand, we will call the function as the k-th mean curvature of M. We would like to note that functions and are called the mean and the Gauss-Kronecker curvatures of M, respectively. Particularly, M is called j-minimal if on M.
2.2. Helicoidal Hypersurfaces with Timelike Axis in Minkowskian Spaces
In this subsection, we will obtain the helicoidal hypersurfaces with timelike axis in Minkowski 4-space . In the rest of this paper, we will identify a vector (a,b,c,d) with its transpose.
Before we proceed, we would like to note that the definition of rotational hypersurfaces in Riemannian space forms were defined in [
32]. A rotational hypersurface
generated by a curve
C about an axis
does not meet
C is generated by using the orbit of
C under those orthogonal transformations of
which leave
pointwise fixed (See [
32] remark 2.3).
A curve C rotates about the axis , and at the same time replaces parallel lines orthogonal to the axis , so that the speed of replacement is proportional to the speed of rotation. Finally, the resulting hypersurface is called the helicoidal hypersurface with axis .
Consider the particular case
and let
C be the curve parametrized by
where
f and
are differentiable functions. If
is the timelike vector
, then an orthogonal transformation of
that leaves
pointwise fixed has the form
as follows:
and the following relations hold:
Therefore, the parametrization of the rotational hypersurface obtained by a curve
C around an axis
is
where
and pitches
.
Clearly, an helicoidal hypersurface with timelike axis written as
When we have an helicoidal surface with timelike axis in .
Now we give some basic elements of the Minkowski 4-space
Let
be an isometric immersion of a hypersurface
in
. Using vectors
,
and
, the Minkowskian inner product and vector product are defined by as follows, respectively,
For a hypersurface
in
, the first fundamental form matrix is
and
and also the second fundamental form matrix is
and
where
and
and some partial differentials we represent are
is the Gauss map.
gives the matrix of shape operator (i.e., Weingarten map)
Therefore, we get the Gaussian and the mean curvature formulas, respectively, as follows:
and
3. Dini-Type Helicoidal Hypersurface with a Timelike Axis
Taking
in (3), we define Dini-type helicoidal hypersurface with a timelike axis in
, as follows:
where
and
Computing the first differentials of (6) depend on
we obtain the first quantities as follows:
and have
where
By using the second differentials depend on
we have the second quantities as follows:
and we get
The Gauss map of a helicoidal hypersurface with a timelike axis is
Finally, we have the Gaussian curvature of a helicoidal hypersurface with a timelike axis as follows:
where
:
Then we calculate the mean curvature of a helicoidal hypersurface with a timelike axis as follows:
where
Therefore, we get the following theorems about flatness and minimality of the hypersurface.
Theorem 1. Let:
⟶
be an isometric immersion given by (6)
. Then is flat if and only if Theorem 2. Let:
⟶
be an isometric immersion given by (6)
. Then is minimal if and only if Solving these two equations is an attractive problem.
In the next two propositions, we will use the function
as in Dini helicoidal surface used by Ulisse Dini in Euclidean 3-space, and its following derivatives
and
Proposition 1. Letis Dini-type flat hypersurface with a timelike axis (i.e.) in Minkowski 4-space. Using the function (9)
and its derivatives (10), (11)
and substituting them into the (7)
in Theorem 1, we obtainwhere
Proposition 2. Letis Dini-type minimal helicoidal hypersurface with a timelike axisin Minkowski 4-space. Using the function (9)
and its derivatives (10)
, (11)
and substituting them into the (8)
in Theorem 2, we getwhere
Corollary 1. From the Proposition 1, and the Proposition 2, we obtain following special symmetries of, respectively,andwhere “∼” means theandterm coefficients which ignored signs, respectively, are equal.