1. Introduction
Twisted hypersurfaces, including helical or helicoidal ones, and related objects such as rotational, ruled, and minimal hypersurfaces, are of interest to mathematicians and have been studied extensively for a long time.
Obata [
1] offered a relation for the manifold isometric to the sphere. Takahashi [
2] served a Euclidean sub-manifold as a one-type if it is minimal or minimal of a hypersphere in
. Chern et al. [
3] studied the minimal sub-manifolds of a sphere. Lawson [
4] researched the minimal sub-manifolds and the Laplace–Beltrami operator.
In space forms, Chen et al. [
5] served the 40 years of one-type sub-manifolds and a one-type Gauss map.
In
, Bour [
6] determined the deformation of helical rotational surfaces. Kenmotsu [
7] described the rotational surfaces having prescribed mean curvature. Do Carmo and Dajczer [
8] studied the helical surfaces. Ferrandez et al. [
9] considered the surfaces supplying
,
A denoting a matrix of order three. Baikoussis and T. Koufogiorgos [
10] focused the helical surfaces having prescribed mean or Gaussian curvature. Ikawa [
11] served the Bour’s theorem and the Gauss map. Choi and Kim [
12] researched the minimal helicoid. Garay [
13] investigeted the surfaces of revolution. Dillen et al. [
14] focused the surfaces supplying
where
A is
, and
B is a
matrix. Güler et al. [
15] worked Bour’s theorem on a Gauss map. Stamatakis and Zoubi [
16] described the surfaces of revolution supplying
. Kim et al. [
17] researched the Cheng–Yau operator of the surfaces of revolution.
In Minkowski 3-space
, Dillen and Kühnel [
18] worked the ruled Weingarten surfaces. Ikawa [
19] determined Bour’s theorem. Beneki et al. [
20] studied the helical surfaces. Güler and Turgut Vanlı [
21] served Bour’s theorem. Güler [
22] worked the helical surfaces with a light-like generating curve. Mira and Pastor [
23] presented the helical maximal surfaces. Kim and Yoon [
24,
25,
26] considered the ruled and rotation surfaces. The readers can see [
2,
27,
28] for details.
In
, Moore [
29,
30] introduced the rotational surfaces in a general form. Hasanis and Vlachos [
31] focused the hypersurfaces holding the mean curvature of the harmonic.
In Minkowski 4-space
, Ganchev and Milousheva [
32] determined the corresponding surfaces of Moore [
29,
30]. Arvanitoyeorgos et al. [
33] introduced
(
H denotes the mean curvature,
is a constant). Güler [
34] introduced the helical hypersurface determined by a space-like axis in
. Li and Güler [
35,
36] studied a hypersurfaces of revolution family in pseudo-Euclidean spaces
and
. Other related works can be found in [
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47].
The aim of this study is to investigate the properties of twisted (i.e., helical) hypersurfaces in five-dimensional Euclidean space with a -rotating axis. Specifically, we focus on determining the fundamental forms, the Gauss map, and the shape operator of these hypersurfaces, as well as describing their curvatures using the Cayley–Hamilton theorem. We also address the open problem of finding solutions to the differential equations governing the curvatures of these hypersurfaces. Furthermore, we examine the umbilicality and minimality conditions associated with the curvatures of the helical hypersurfaces. Finally, we aim to establish the Laplace–Beltrami operator relation of , providing further insights into the geometric properties of these intriguing hypersurfaces.
We focus the twisted hypersurfaces
constructed by the
rotating axis in Euclidean 5-space
. We offer some properties of
in
Section 2. We formulate the components of the fundamental forms, the Gauss map, the shape operator of any hypersurface of
. We describe the twisted hypersurfaces
of
in
Section 3.
By way of the theorem of Cayley–Hamilton, we obtain all the formulas of curvatures of any hypersurface, and also compute the curvatures of twisted hypersurfaces
. We also determine some relations for curvatures
of
. We present the umbilical relations to the hypersurfaces in
Section 4.
Moreover, in
Section 5, we obtain
, where
is the
matrix. We serve some examples to all findings. In the last section, we offer a conclusion.
2. Preliminaries
We assume to be a hypersurface in Euclidean space , denotes its shape operator, and x its position vector. We suppose to be a local orthonormal frame consisting the principal directions of corresponding with the principal curvature for .
We consider
, where
is the
jth elementary symmetric function defined by
The following notation works:
with
,
. Function
denotes the
kth mean curvature,
and
denote the mean and Gauss–Kronecker curvatures of
, respectively. If
on
,
is named
j-minimal. The readers can refer to Alias and Gürbüz [
41], Kühnel [
46] for details.
In
, the characteristic polynomial Equation
of the shape operator
is determined by
Here, describes the identity matrix of order The curvature formulas are given by . Here, (by definition), . Also,
Next, we present some notions to Riemannian geometry. The readers can see Kühnel [
46] for details. A vector with its transpose is regarded identical in this paper. We let
be an immersion from
to
.
Definition 1. In , a Euclidean inner product of two vectors and of is described by Definition 2. A Euclidean vector product of of is defined by Here, denotes the generator elements of .
Definition 3. In , any hypersurface has the following matrices, respectively:where the components are indicated by
etc.; the Gauss map of is denoted by Definition 4. In , hypersurface supplies the following relations:where denotes the shape operator, and describe the fundamental forms of . Definition 5. In ,determines the characteristic polynomial of , and denotes the identity matrix. The curvature formulas are described by . Here, , , denote the mean, Gauss–Kronecker curvatures, respectively. Definition 6. When , on a hypersurface , is named j-minimal.
See [
45] for details of
, and dimension four.
Next, we reval the formulas of the curvatures of in .
Theorem 1. The formulas of the curvature of a hypersurface in a five-space are given, respectively, by (by definition),where Here,
Proof. By using Definition 3, Definition 4, and Definition 5, and by direct computations, the characteristic polynomial is obtained. Then, values are found. □
3. Twisted Hypersurfaces Family with the Rotating Axis in
We describe the twisted hypersurfaces family. The readers can refer to Do Carmo and Dajczer [
42] for some results about the rotational hypersurfaces of Riemannian spaces.
Definition 7. We let I be an open interval , be a curve in plane Π, and ℓ be a line in Π. A rotational hypersurface is determined by a generating curve γ rotating about line (named axis) ℓ. While the generating curve γ rotates about ℓ, it concurrently replaces parallel lines orthogonal to ℓ. The speed of rotating commensurates to the speed of replacement. The construcing hypersurface is named the twisted hypersurfaces family with axis ℓ and pitches .
Readers can see Kühnel [
46] for details. Next, we describe the twisted hypersurfaces of
.
The rotation matrix
obtained by rotating axis
in
is described by
Here,
,
,
, and
holds:
The generating curve is determined by
Here,
denote the differentiable functions of
. In
, the twisted hypersurface
determined by
is described by
where
,
The parametric representation of twisted hypersurfaces
is determined by
We note that we describe the following different hypersurfaces in lower dimensions.
If we have a twisted surface with a axis in .
When we obtain a rotational surface with a axis in .
When we obtain a twisted hypersurface with a axis in .
If we find a rotational hypersurface with a axis in .
Next, we describe the curvature formulas for hypersurface in .
Theorem 2. Hypersurface in Euclidean 5-space has the following curvatures:where describes the characteristic polynomial of the shape operator matrix , , , and , denote the fundamental form matrices described by Definition 3. Proof. determines
of
in
. We compute
of
. Hence, we obtain
jth curvatures
:
denotes the principal curvatures of
where
. □
See [
44,
45] for the cases of dimension four.
The curvatures of the twisted hypersurfaces with rotating axis are given by the following theorem.
Theorem 3. In , the curvatures of the twisted hypersurfaces described by Equation (3) are, respectively, given bywhere, , , , , , , etc., Proof. Regarding Definition 3, and by using the first derivatives w.r.t.
of the hypersurface determined by (3), we find the components of
:
Hence,
Regarding the Gauss map formula indicated by Definition 3, we find the Gauss map
of the twisted hypersurface
given by Equation (3).
Next, regarding the Gauss map of
given by Equation (7), taking the second derivatives of
depending on
, we find the components of
given by Definition 3:
gives the following:
We obtain the characteristic polynomial of (8) as follows:
where
Here,
Finally, we obtain the components of
:
From Definition 5, the curvatures of the twisted hypersurfaces with the rotating axis described by (3) in the five-dimensional Euclidean space are obtained. □
Next, we offer some corollaries for the curvatures of the twisted hypersurfaces defined by Equation (3) with the rotating axis.
Corollary 1. By taking we obtain the following curvatures of the twisted hypersurfaces determined by Equation (3) with the rotating axis: Corollary 2. By choosing we find the following curvatures of the twisted hypersurfaces given by Equation (3) with the rotating axis: Here, described by Theorem 3. Then, We present a condition to the curvatures determined by Theorem 2 with the fundamental forms decribed by Definition 4.
Theorem 4. Hypersurface in Euclidean space haswhere describe the fundamental form matrices, and determines the zero matrix with order four of . Proof. With the help of the theorem of Cayley–Hamilton, we obtain
Hence, we have
□
We note that the three dimension effects of Theorem 4 are determined by
and
Here,
describes the zero matrix of order two,
denotes the mean curvature,
determines the Gaussian curvature of a surface of dimension three. Also, the acts of dimension four of Theorem 4 are described as follows:
and
Here, denotes the zero matrix of order three.
4. The Umbilical Hypersurfaces in
Next, we present the umbilical acts of the hypersurfaces of .
From Theorem 1, the following occurs:
Then, we obtain the the following.
Corollary 3. In , the following holds: See [
34,
46] for details of umbilical facts.
Theorem 5. The twisted hypersurfaces given by Equation (3) have a umbilical point if the following comes out Proof. Twisted hypersurfaces constructed by the -rotation axis cover the umbilical point of , i.e., . □
Problem 1. Find the φ solutions of Equation determined by Theorem 5.
Now, we serve the minimality acts determined by Definition 6 of the twisted hypersurfaces defined by Equation (3).
Corollary 4. The twisted hypersurfaces defined by Equation (3) have zero mean curvature, i.e., one-minimal if the following occurs: Problem 2. Find the φ solutions of Equation given by Corollary 4.
Corollary 5. The twisted hypersurfaces determined by Equation (3) are two-minimal if the following holds: Problem 3. Find the φ solutions of Equation given by Corollary 5.
Corollary 6. The twisted hypersurfaces decribed by Equation (3) are three-minimal if the following Equation becomes Problem 4. Find the φ solutions of the Equation determined by Corollary 6.
Corollary 7. The twisted hypersurfaces defined by Equation (3) have a zero Gauss–Kronecker curvature, i.e., four-minimal if the following Equation comes out: Problem 5. Find the φ solutions of Equation described by Corollary 7.
5. Twisted Hypersurfaces with the Rotating Axis
Supplying in
We determine that the Laplace–Beltrami operator depends on of a smooth function in , and we find the Laplace–Beltrami operator of the twisted hypersurfaces given by (3).
Definition 8. In , the Laplace–Beltrami operator of a smooth function of class is described bywhere and We regard the inverse matrix of
determined by (3). Hence, the coefficients of
are denoted by
where
We compensate
with
and substitute it into (10). Therefore, by using the inverse matrix of (5), we have the following:
We obtain the information below.
Theorem 6. The Laplace–Beltrami operator of the twisted hypersurfaces determined by (3) has . Here, denotes the mean curvature determined by Theorem 3, and describes the Gauss map determined by (7) of the family.
Proof. With straight calculations of (3) on (10), we have . □
Next, we offer the following about and of the family determined by (3).
Theorem 7. We let : ⟶ be an immersion described by (3). , where is a square matrix of order five if , i.e., twisted hypersurfaces are one-minimal.
Proof. We use
then obtain the following Equations:
where
denotes a
matrix,
Derivativing the above ODEs twice w.r.t.
we obtain
Therefore, the following relations occur:
where
Regarding the fact that sin and cos are linear independent on
each of the coefficients of matrix
are 0.
then
. This means, from Definition 6, that hypersurface
determined by (3) is a one-minimal twisted hypersurface with a
rotating axis. □
Hence, we offer the following examples.
Example 1. In , by using to γ determined by (2), we construct the rotational hypersurfacewhere Then, we have Here, describes the identity matrix, describes the diagonal side of the matrix, .
We also apply the rational rotational hypersurface with the rotating axis to the following.
Example 2. We substitute rational functions , , into γ described by (2). We obtain the following rational rotational hypersurface:where in . Then, we obtain Here, denotes the identity matrix, denotes the diagonal side of the matrix, . The rational hypersphere with the rotating axis holds Equation determined by (9).
6. Conclusions
This research introduced twisted hypersurfaces in a five-dimensional Euclidean space with a rotating axis along . The fundamental forms, the Gauss map, and the shape operator of were computed, providing a comprehensive understanding of its geometric properties. By employing the Cayley–Hamilton theorem, the curvatures of were determined, highlighting their relationship with the curvatures of hypersurfaces in .
However, the solutions to the differential equations governing the curvatures of these hypersurfaces remain open problems, offering avenues for future research. The study also presented the umbilicality and minimality conditions for the curvatures of , contributing to the characterization of their geometric behavior. Furthermore, a significant result was obtained, establishing the Laplace–Beltrami operator relation , where is a square matrix of order five, further deepening the understanding of the geometric properties of .
Overall, these findings shed light on the intricate nature of twisted hypersurfaces in a five-dimensional space and provided a foundation for further investigations in this field.