Multiple positive solutions for system of mixed Hadamard fractional boundary value problems with $(p_{1}, p_{2})$-Laplacian operator

1.   Introduction

Chen ^{[11,12,13,14]} originally proposed the notion of submanifolds of finite order immersed in $m$-space $\mathbb{E}^{m}$ or pseudo-Euclidean $m$ -space $\mathbb{E}_{\nu }^{m}$ employing a finite number of eigenfunctions of their Laplacian. This subject has subsequently undergone thorough investigation.

Takahashi established that a Euclidean submanifold is classified as $1$-type if and only if it is minimal or minimal within a hypersphere of $\mathbb{E} ^{m}$. The study of $2$-type submanifolds on closed spheres was conducted by ^[9,10,12]. Garay further ^[26] examined Takahashi's theorem in $\mathbb{E}^{m}$. Cheng and Yau ^[18] focused on hypersurfaces with constant curvature, while Chen and Piccinni ^[15] concentrated on submanifolds with a Gauss map of finite type in $\mathbb{E}^{m}$. Dursun ^[22] introduced hypersurfaces with a pointwise 1-type Gauss map in $\mathbb{E}^{n+1}$. Aminov ^[2] delved into the geometry of submanifolds. Within the domain of space forms, Chen et al. ^[16] dedicated four decades to the investigation of 1-type submanifolds and the 1-type Gauss map.

In ${\mathbb{E}}^{3}$, Takahashi ^[43] explored the concept of minimal surfaces, where spheres and minimal surfaces with $\Delta r = \lambda r,$ $\lambda \in \mathbb{R}$ are the exclusive types of surfaces. Ferrandez et al. ^[23] identified that surfaces $\Delta H = A_{3\times 3}H$ are either the minimal sections of a sphere or a right circular cylinder. Choi and Kim ^[19] examined the minimal helicoid with a pointwise 1-type Gauss map of the first kind. Garay ^[25] derived a category of finite type surfaces that are revolution-based. Dillen et al. ^[20] investigated the unique surfaces characterized by $\Delta r = A_{3\times 3}r+B_{3\times 1}$, which include minimal surfaces, spheres, and circular cylinders. Stamatakis and Zoubi ^[42] established the properties of surfaces of revolution defined by $\Delta ^{III}x = A_{3\times 3}x$. Kim et al. ^[36] focused on the Cheng-Yau operator and the Gauss map of surfaces of revolution.

In ${\mathbb{E}}^{4}$, Moore ^[40,41] conducted two studies on general rotational surfaces. Hasanis and Vlachos ^[35] examined hypersurfaces with a harmonic mean curvature vector field. Cheng and Wan ^[17] focused on complete hypersurfaces with constant mean curvature. Arslan et al. ^[3] explored the Vranceanu surface with a pointwise 1-type Gauss map. Arslan et al. ^[4] investigated generalized rotational surfaces and ^[5] introduced tensor product surfaces with a pointwise 1-type Gauss map. Yoon ^[44] established certain relations involving the Clifford torus. Güler et al. ^[30] delved into helicoidal hypersurfaces, while Güler et al. ^[29] studied the Gauss map and the third Laplace-Beltrami operator of rotational hypersurfaces. Güler ^[28] investigated rotational hypersurfaces characterized by $\Delta ^{I}R = A_{4\times 4}R$. Furthermore, Güler ^[27] obtained the fundamental form $IV$ and curvature formulas of the hypersphere.

In Minkowski 4-space $\mathbb{E}_{1}^{4}$, Ganchev and Milousheva ^[24] explored the analogous surfaces to those in ^[40,41]. Arvanitoyeorgos et al. ^[8] investigated the mean curvature vector field, where they established $\Delta H = \alpha H$ with a constant $\alpha$. Arslan and Milousheva ^[6] focused on meridian surfaces of elliptic or hyperbolic type with a pointwise 1-type Gauss map. Arslan et al. ^[7] examined rotational $\lambda$-hypersurfaces in Euclidean spaces. Güler et al. ^{[31,32,33,34]} worked the concept of bi-rotational hypersurfaces. Li and Güler studied a family of hypersurfaces of revolution distinguished by four parameters in the five-dimensional pseudo-Euclidean space $\mathbb{E}^5_2$ ^[39].

The aim of this paper is to present a family of hypersurfaces of revolution in the seven-dimensional pseudo-Euclidean space ${\mathbb{E}}_{3}^{7}$. This family, denoted as $\mathfrak{r}$, is characterized by six parameters. The paper focuses on computing various matrices associated with $\mathfrak{r}$, including the fundamental form, Gauss map, and shape operator. The Cayley-Hamilton theorem is employed to determine the curvatures of $\mathfrak{r}$. Furthermore, the paper establishes equations that describe the relationship between the mean curvature and Gauss-Kronecker curvature of $\mathfrak{r}$. Additionally, the paper explores the connection between the Laplace-Beltrami operator of $\mathfrak{r}$ and a $7\times 7$ matrix.

In Section 2, we provide an explanation of the fundamental concepts of seven-dimensional pseudo-Euclidean geometry.

Section 3 is dedicated to presenting the curvature formulas of a hypersurface in ${\mathbb{E}}_{3}^{7}$.

In Section 4, we offer a comprehensive definition of the hypersurfaces of revolution family, focusing on their properties and characteristics.

In Section 5, we discuss the Laplace-Beltrami operator of a smooth function in ${\mathbb{E}}_{3}^{7}$ and utilize the previously discussed family to compute it.

Finally, we serve a conclusion in the last section.

2.   Preliminaries

In this paper, we use the following notations, formulas, eqations, etc.

For clarity, $\mathbb{E}_{v}^{m}$ represents a pseudo-Euclidean $m$-space with coordinates denoted as $(x_{1}, x_{2}, \cdots, x_{m})$ with index $v$. The canonical pseudo-Euclidean metric tensor on $\mathbb{E}_{v}^{m}$ is represented by $\widetilde{g}$ and defined as $\widetilde{g} = \langle \, \ \rangle = -\sum\limits_{i = 1}^{v}dx_{i}^{2}+\sum\limits_{i = v+1}^{m}dx_{i}^{2}$. Let $\widetilde{M}$ be an $m$-dimensional semi-Riemannian submanifold, and is embedded in $\mathbb{E}_{v}^{m}$, and the Levi-Civita connections ^[38] associated with $M$ are denoted as $\widetilde{\nabla }, \nabla$, respectively. We utilize $X, Y, Z,$ and $W$ to denote vector fields tangent to $M$, and $\xi, \varsigma$ to represent vector fields normal to $M$.

The Gauss formula and the Weingarten formula is given by

where $h$ represents the second fundamental form of $M$, $A$ denotes the shape operator, and $D$ corresponds to the normal connection of $M$. The shape operator $A_{\xi }$ is a symmetric endomorphism of the tangent space $T_{p}M$ at each point $p\in M$ for each $\xi \in T_{p}^{\bot }M$. The shape operator and the second fundamental form are related by the equation.

The Gauss equation is determined by

where $R$ describes the curvature tensor associated with the Levi-Civita connection $\nabla$, and $h$ denotes the second fundamental form of $M$. The Codazzi equation is given by

where $\bar{\nabla}h$ denotes the covariant derivative of $h$ w.r.t. the Levi-Civita connection $\nabla$, and $X, Y, Z$ represent tangent vector fields on $M$. The curvature tensor $R^{D}$ associated with the normal connection $D$ is not explicitly mentioned in the given equations. The covariant derivative of $h$ is defined by

where $D$ represents the normal connection of $M$.

Let $M$ be an oriented hypersurface in $\mathbb{E}^{n+1}$ with its shape operator $\mathcal{S}$, and position vector $x$. Consider a local orthonormal frame field $\{e_{1}, e_{2}, \cdots, e_{n}\}$ consisting of principal directions of $M$ coinciding with the principal curvature $k_{i}$ for $i = 1, 2, \cdots, n$. Let the dual basis of this frame field be $\{f_{1}, f_{2}, \cdots, f_{n}\}$. Then, the first structural equation of Cartan is determined by

where $\omega _{ij}$ indicates the connection forms coinciding with the chosen frame field. By the Codazzi equation, we derive the equations.

for different $i, j, l = 1, 2, \cdots, n$.

We let $s_{j} = \sigma _{j}(k_{1}, k_{2}, \cdots, k_{n})$, where $\sigma _{j}$ denotes the $j$-th elementary symmetric function defined by

We consider the notation

According to the given definition, we have $r_{i}^{0} = 1$ and $s_{n+1} = s_{n+2} = \cdots = 0$. The function $s_{k}$ is referred to as the $k$ -th mean curvature of the oriented hypersurface $M$. The mean curvature $H = \frac{1}{n}s_{1}$ is also defined, and the Gauss-Kronecker curvature of $M$ is $K = s_{n}$. If $s_{j}\equiv 0,$ the hypersurface $M$ is known as $j$ -minimal.

In Euclidean $(n+1)$-space$,$ getting the $i$-th curvature formulas $\mathcal{K}_{i}$ (see ^[1,37] for details)$,$ where $i = 0, \cdots, n,$ we have the following characteristic polynomial equation $P_{\mathcal{S }}(\lambda) = 0$ of $\mathcal{S}$:

Here $i = 0, \cdots, n,$ $\mathcal{I}_{n}$ indicates the identity matrix$.$ Hence, we reveal the curvature formulas as $\binom{n}{i}\mathcal{K} _{i} = s_{i}$.

Let $\mathfrak{r} = \mathfrak{r}(u, v, w, \alpha, \beta, \gamma)$ be an immersion from $M^{6}\subset \mathbb{E}^{6}$ to $\mathbb{E}_{3}^{7}$.

Definition 1. An inner product of ${{\bf{ \upsilon}} ^{1}} = (\upsilon _{1}^{1}, \upsilon _{2}^{1}, \cdots, \upsilon _{7}^{1}), \cdots,$ ${{\bf{ \upsilon}} ^{2}} = (\upsilon _{1}^{2}, \upsilon _{2}^{2}, \cdots, \upsilon _{7}^{2})$ of $\mathbb{E}_{3}^{7}$ is determined by

Definition 2. A sixtuple vector product of ${{\bf{ \upsilon}} ^{1}} = (\upsilon _{1}^{1}, \upsilon _{2}^{1}, \cdots, \upsilon _{7}^{1}),$ ${{\bf{ \upsilon}} ^{2}} = (\upsilon _{1}^{2}, \upsilon _{2}^{2}, \cdots, \upsilon _{7}^{2}), \cdots,$ ${{\bf{ \upsilon}} ^{6}} = (\upsilon _{1}^{6}, \upsilon _{2}^{6}, \cdots, \upsilon _{7}^{6})$ of $\mathbb{ E}_{3}^{7}$ is defined by

Definition 3. The product matrix $\left(\mathfrak{g}_{ij}\right) ^{-1}$· $\left(\mathfrak{h}_{ij}\right)$ describes the shape operator matrix $\mathcal{S}$ of hypersurface $\mathfrak{r}$ in pseudo-Euclidean 7-space $\mathbb{E}_{3}^{7}$, $\ $where, $\left(\mathfrak{g} _{ij}\right) _{6\times 6}$ and $\left(\mathfrak{h}_{ij}\right) _{6\times 6}$ describe the first and the second fundamental form matrices, respectively, and $\mathfrak{g}_{ij} = \langle \mathfrak{r}_{i}, \mathfrak{r} _{j}\rangle,$ $\mathfrak{h}_{ij} = \langle \mathfrak{r}_{ij}, { \mathcal{G}}\rangle,$ $i, j = 1, 2, \cdots, 6,$ $\mathfrak{r}_{u} = \frac{ \partial \mathfrak{r}}{\partial u}$ when $i = 1,$ $\mathfrak{r}_{uv} = \frac{ \partial ^{2}\mathfrak{r}}{\partial u\partial v}$ when $i = 1$ and $j = 2,$ etc., $e_{k}$ denotes the natural base elements of $\mathbb{E}^{7}.$ Here,

determines the Gauss map of the hypersurface $\mathfrak{r}$.

3.   Curvatures in $\mathbb{E}_{3}^{7}$

In this section, we reveal the curvature formulas of any hypersurface $\mathfrak{r} = \mathfrak{r}(u, v, w, \alpha, \beta, \gamma)$ in $\mathbb{E}_{3}^{7}.$

Theorem 1. A hypersurface $\mathfrak{r}$ in $\mathbb{E}_{3}^{7}$ has the following curvature formulas, $\mathcal{K}_{0} = 1$ by definition,

where $\mathfrak{a}_{6}\lambda ^{6}+\mathfrak{a}_{5}\lambda ^{5}+ \mathfrak{a}_{4}\lambda ^{4}+\mathfrak{a}_{3}\lambda ^{3}+\mathfrak{a} _{2}\lambda ^{2}+\mathfrak{a}_{1}\lambda +\mathfrak{a}_{0} = 0$ denotes the characteristic polynomial equation $P_{\mathcal{S}}(\lambda) = 0$ of the shape operator matrix $\mathcal{S}$, $\mathfrak{a} _{6} = \det \left(\mathfrak{g}_{ij}\right)$, $\mathfrak{a}_{0} = \det \left(\mathfrak{h}_{ij}\right)$, and $\left(\mathfrak{g} _{ij}\right)$, $\left(\mathfrak{h}_{ij}\right)$ are the first, and the second fundamental form matrices, respectively$.$

Proof. The solution matrix $\left(\mathfrak{g}_{ij}\right) ^{-1}$ · $\left(\mathfrak{h}_{ij}\right)$ supplies the shape operator matrix $\mathcal{S}$ of hypersurface $\mathfrak{r}$ in pseudo-Euclidean 7-space $\mathbb{E}_{3}^{7}$. In $\mathbb{E} _{3}^{7}$, computing the curvature formula $\mathcal{K}_{i}$, where $i = 0, 1, \cdots, 6,$ we reveal the characteristic polynomial equation $\det (\mathcal{S} -\lambda \mathcal{I}_{6}) = 0$ of $\mathcal{S}$. Then, we obtain

Definition 4. A space-like hypersurface $\mathfrak{r}$ is called $j$-maximal if $\mathcal{K}_{j} = 0$, where $j = 1, ..., 6.$

Theorem 2. A hypersurface $\mathfrak{r} = \mathfrak{r} (u, v, w, \alpha, \beta, \gamma)$ in $\mathbb{E}_{3}^{7}$ has the following relation

where $\mathbb{I}, \mathbb{II}, \cdots, \mathbb{VII}$ determines the fundamental form matrices, $\mathcal{O}_{6}$ represents the zero matrix having order $6\times 6$ of the hypersurface$.$

Proof. Regarding $n = 6$ in $\left(2.1\right)$, it works.

4.   Hypersurfaces of revolution family in $\mathbb{E}_{3}^{7}$

In this section, we define the hypersurfaces of revolution family (HRF), then find its differential geometric properties in pseudo-Euclidean 7-space $\mathbb{E}_{3}^{7}$. An HR in Riemannian space forms were given in ^[21].

An HRF $M$ of Euclidean $\left(n+1\right)$-space constructed by a hypersurface $\hbar$ around rotating axis $\mathcal{\ell }$ does not meet $\hbar$ is acquired by taking the orbit of $\mathcal{\ell }$ under the orthogonal transformations of $\left(n+1\right)$-space.

To consctruct an HRF, we start with the generating hypersurface given by $\hbar = \hbar \left(u, v, w\right) = \left(\eta, 0, \psi, 0, \phi, 0, \varphi \right),$ and apply the rotation matrix $\mathfrak{R} =$ diag$\left(\mathcal{R}_{\alpha }, \mathcal{R}_{\beta }, \mathcal{R}_{\gamma }, 1\right)$ with the elements given by $\mathcal{R}_{k} = \left(\begin{array}{cc} \cosh k & \sinh k \\ \sinh k & \cosh k \end{array} \right),$ $k = \alpha, \beta, \gamma,$ respectively, and $\mathfrak{R}{\text{ · }}\mathcal{\ell } = \mathcal{\ell }$, $\det \mathfrak{R} = 1.$ Therefore, we state the HRF given by $\mathfrak{r} = \mathfrak{R}$ · $\hbar ^{T}$ when $\hbar$ rotates about axis $\mathcal{ \ell } = \overrightarrow{e_{7}} = (0, 0, 0, 0, 0, 0, 1).$ We then present the following.

Definition 5. An HRF is an immersion $\mathfrak{r}$ : $M^{6}$ $\subset \mathbb{E}^{6}\longrightarrow$ $\mathbb{E}_{3}^{7}$ with rotating axis $\overrightarrow{e_{7}}$, defined by

where $\eta, \psi, \phi, \varphi$ denote the differentiable functions, depend on $u, v, w\in \mathbb{R}$, $0\leq \alpha, \beta, \gamma < 2\pi.$

Considering the first derivatives of HRF given by Eq $\left(4.1\right)$ w.r.t. $u, v, w, \alpha, \beta, \gamma,$ respectively, we find the symmetical first fundamental form matrix

where

and $\eta _{u} = \frac{\partial \eta }{\partial u},$ $\eta _{v} = \frac{\partial \eta }{\partial v},$ $\eta _{u}^{2} = \frac{\partial ^{2}\eta }{\partial u^{2}},$ etc. Hence, ${\bf{\hat{g}}} = \det \left(\mathfrak{g}_{ij}\right) = \eta ^{2}\psi ^{2}\phi ^{2}\mathcal{Q},$ where

and

Since ${\bf{\hat{g}}} > 0$, the HRF given by Eq $\left(4.1 \right)$ is a space-like hypersurface.

Using $\left(2.2\right),$ we obtain the following Gauss map of the HRF determined by Eq $\left(4.1\right)$:

With the help of the second derivatives w.r.t. $u, v, w, \alpha, \beta, \gamma,$ of HRF described by Eq $\left(4.1\right),$ and by using the Gauss map given by Eq $\left(4.3\right)$, we reveal the following symmetical second fundamental form matrix

where

and $\eta _{uu} = \frac{\partial ^{2}\eta }{\partial u^{2}},$ $\eta _{uv} = \frac{\partial ^{2}\eta }{\partial u\partial v},$ ect.$.$ By using $\left(4.2\right)$ and $\left(4.4\right)$, we compute the following shape operator matrix of $\left(4.1\right)$:

with the following components

Finally, using $\left(3.1\right)$, with $\left(4.2\right)$, $\left(4.4\right)$, respectively, we find the curvatures of the HRF defined by Eq $\left(4.1\right)$ as follows.

Theorem 3. Let $\mathfrak{r}$ be an HRF determined by Eq $\left(4.1\right)$ in $\mathbb{E}_{3}^{7}$. $\mathfrak{r}$ contains the following curvatures

Here, $\mathcal{K}_{1}$ represents the mean curvature, $\mathcal{K}_{6}$ denotes the Gauss-Kronecker curvature.

Proof. By using the Cayley-Hamilton theorem, we reveal the following characteristic polynomial equation $P_{\mathcal{S}}(\lambda) = 0$ of $\mathcal{S}$:

The curvatures $\mathcal{K}_{1}$ and $\mathcal{K}_{6}$ of $\mathfrak{r}$ are obtained by the above equation.

Corollary 1. Let $\mathfrak{r}$ be an HRF defined by Eq $\left(4.1\right)$ in $\mathbb{E}_{3}^{7}$. $\mathfrak{r}$ is a $1$-maximal (i.e., has zero mean curvature) iff the following partial differential equation appears

Corollary 2. Let $\mathfrak{r}$ be a HRF given by Eq $\left(4.1\right)$ in $\mathbb{E}_{3}^{7}$. $\mathfrak{r}$ is a $6$-maximal (i.e., has zero Gauss-Kronecker curvature) iff the following partial differential equation occurs

Corollary 3. Let $\mathfrak{r}$ be a HRF defined by Eq $\left(4.1\right)$ in $\mathbb{E}_{3}^{7}$. $\mathfrak{r}$ has umbilical point (i.e., $\left(\mathcal{K} _{1}\right) ^{6} = \mathcal{K}_{6}$) iff the following partial differential equation holds

Hence, we find the following.

Example 1. Let $\mathfrak{r}$ be an HRF determined by Eq $\left(4.1\right)$ in $\mathbb{E}_{3}^{7}$. When the profile hypersurface $\gamma$ of $\mathfrak{r}$ is parametrized by the unit hypersphere: $\eta = \cos u\cos v\cos w$, $\psi = \sin u\cos v\cos w$, $\phi = \sin v\cos w$, $\varphi = \sin w$, then $\mathcal{S} = \mathcal{I}_{6}$ and the HRF has the following curvatures $\mathcal{K}_{i} = 1$, where $i = 0, 1, ..., 6.$

Example 2. Assume $\mathfrak{r}$ be an HRF denoted by Eq $\left(4.1\right)$ in $\mathbb{E}_{3}^{7}$. While the profile hypersurface $\gamma$ of $\mathfrak{r}$ is parametrized by the rational unit hypersphere: $\eta = \frac{ 1-u^{2}}{1+u^{2}}\frac{1-v^{2}}{1+v^{2}}\frac{1-w^{2}}{1+w^{2}}$, $\psi = \frac{2u}{1+u^{2}}\frac{1-v^{2}}{1+v^{2}}\frac{1-w^{2}}{1+w^{2}}$ , $\phi = \frac{2v}{1+v^{2}}\frac{1-w^{2}}{1+w^{2}}$, $\varphi = \frac{2w}{1+w^{2}}$, the HRF has the same results determined by Example 1.

Example 3. Let $\mathfrak{r}$ be an HRF defined by Eq $\left(4.1\right)$ in $\mathbb{E}_{3}^{7}$. When the generating hypersurface $\gamma$ of $\mathfrak{r}$ is parametrized by the Riemann hypersphere: $\eta = \frac{2u}{ u^{2}+v^{2}+w^{2}+1}$, $\psi = \frac{2v}{u^{2}+v^{2}+w^{2}+1}$ , $\phi = \frac{2w}{u^{2}+v^{2}+w^{2}+1}$, $\varphi = \frac{ u^{2}+v^{2}+w^{2}-1}{u^{2}+v^{2}+w^{2}+1}$, the HRF has $\mathcal{S} = -\mathcal{I}_{6},$ and has the following curvatures $\mathcal{K} _{i} = \left(-1\right) ^{i}$, where $i = 0, 1, ..., 6.$

Example 4.Considering the pseudo-hypersphere $\mathbb{S} _{3}^{6}(\rho): = \left\{ {\bf{p}}\in \mathbb{E}_{3}^{7}\mid \langle {\bf{p}}, {\bf{p}}\rangle = \rho ^{2}\right\},$ radius $\rho > 0$, parametrized by

we compute $\mathcal{S} = \frac{1}{\rho }\mathcal{I}_{6}.$ Hence, we find the following curvatures $\mathcal{K}_{i} = \frac{1}{\rho ^{i}}$, where $i = 0, 1, ..., 6.$ Then, the hypersurface ${\bf{p}}$ described by Eq $\left(4.5\right)$ is an umbilical hypersphere (i.e., it supplies $\left(\mathcal{K}_{1}\right) ^{6} = \mathcal{K}_{6}$) of $\mathbb{E}_{3}^{7}$.

5.   Hypersurfaces of revolution family satisfying $\Delta \mathfrak{r}= \mathcal{A}\mathfrak{r}$ in $\mathbb{E}_{3}^{7}$

In this section, our focus is on the Laplace-Beltrami operator of a smooth function in $\mathbb{E}_{3}^{7}$. We will proceed to compute it utilizing the HRF, which is defined by Eq $\left(4.1\right)$.

Definition 6. The Laplace-Beltrami operator of a smooth function $f = f(x^{1}, x^{2}, ..., x^{6})\mid _{\mathcal{D}}$ $(\mathcal{D} \subset {\mathbb{R}}^{6})$ of class $C^{6}$ depends on the first fundamental form $\left(\mathfrak{g}_{ij}\right)$ of a hypersurface $\mathfrak{r}$, and is the operator defined by

where $\left(\mathfrak{g}^{ij}\right) = \left(\mathfrak{g} _{kl}\right) ^{-1}$ and ${\bf{\hat{g}}} = \det \left(\mathfrak{g} _{ij}\right).$

By using the inverse matrix of the first fundamental form matrix $\left(\mathfrak{g}_{ij}\right) _{6\times 6},$ we have the following.

For an HRF given by Eq $\left(4.1\right),$ $\mathfrak{g}_{ij} = 0$ when $i\neq j$ except for $i, j < 4.$ Therefore, the Laplace-Beltrami operator of the HRF $\mathfrak{r} = \mathfrak{r}(u, v, w, \alpha, \beta, \gamma)$ is given by

By using the derivatives of the functions in $\left(5.2\right)$, w.r.t. $u, v, w, \alpha, \beta, \gamma,$ resp., we obtain the following.

Theorem 4. The Laplace-Beltrami operator of the HRF $\mathfrak{r}$ denoted by Eq $\left(4.1\right)$ is given by $\Delta \mathfrak{r} = 6\mathcal{K}_{1}\mathcal{G}$, where $\mathcal{K}_{1}$ denotes the mean curvature, $\mathcal{G}$ represents the Gauss map of $\mathfrak{r}$.

Proof. By directly computing $\left(5.2\right)$, we obtain $\Delta \mathfrak{r}$.

Theorem 5.Let $\mathfrak{r}$ be an HRF defined by Eq $\left(4.1\right)$. $\Delta \mathfrak{r} = \mathcal{A}\mathfrak{r }$, where $\mathcal{A}$ denotes the square matrix of order $7$ iff $\mathfrak{r}$ has $\mathcal{K}_{1} = 0$, i.e., it is a $1$-maximal hypersurface.

Proof. We found $6\mathcal{K}_{1}\mathcal{G} = \mathcal{A}\mathfrak{r}$, and then we have

where $\mathcal{A} = \left(a_{ij}\right)$ is the $7\times 7$ matrix, $\Upsilon = 6\mathcal{K}_{1}{\bf{\hat{g}}}^{-1/2},$ where ${\bf{\hat{g}}} = \eta ^{2}\psi ^{2}\phi ^{2}\mathcal{Q}.$ Derivating above ODEs twice w.r.t. $\alpha$, we obtain the following $a_{i7} = 0,$ $\Upsilon = 0,$ where $i = 1, 2, ..., 7.$ Then, we get $\left(a_{i1}\cosh \alpha +a_{i2}\sinh \alpha \right) \eta = 0,$ where $i = 1, 2, ..., 7.$ The functions $\cosh$ and $\sinh$ are linear independent on $\alpha$, then all the components of the matrix $\mathcal{A}$ are $0$. Since $\Upsilon = 6\mathcal{K}_{1}{\bf{\hat{g}}} ^{-1/2},$ then $\mathcal{K}_{1} = 0$. This means, $\mathfrak{r}$ is a $1$ -maximal HRF.

Therefore, we give the following.

Example 5. Let $\mathfrak{r}$ be an HRF given by Eq $\left(4.1\right)$, and let the generating hypersurface $\gamma$ of $\mathfrak{r}$ be parametrized by the unit hypersphere determined by Example 1. Then, an HRF $\mathfrak{r}$ supplies $\Delta \mathfrak{r} = \mathcal{A}\mathfrak{r}$, where $\mathcal{A} = -6\mathcal{I}_{7},$ $\mathcal{I}_{7}$denotes identity matrix$.$

Example 6. Let $\mathfrak{r}$ be an HRF denoted by Eq $\left(4.1\right)$, and let the generating hypersurface $\gamma$ of $\mathfrak{r}$ be parametrized by the Riemann hypersphere defined by Example 3. An HRF $\mathfrak{r}$ has the same results denoted by Example 5.

6.   Conclusions

This research has presented a detailed analysis of a family of hypersurfaces of revolution $\mathfrak{r}$ is characterized by six parameters in the seven-dimensional pseudo-Euclidean space ${\mathbb{E}}_{3}^{7}$, and its geometric properties have been thoroughly explored.

The main focus of the paper was on computing and investigating various matrices associated with $\mathfrak{r}$. The fundamental form, Gauss map, and shape operator matrices were derived, providing essential information about the local geometry of the hypersurfaces. By utilizing the Cayley-Hamilton theorem, the curvatures of $\mathfrak{r}$ were determined, facilitating a comprehensive understanding of their intrinsic curvature properties. Moreover, the paper established equations that describe the relationship between the mean curvature and Gauss-Kronecker curvature of $\mathfrak{r}$. These equations shed light on the geometric behavior of the hypersurfaces and offer valuable insights into their intrinsic properties. Additionally, the paper investigated the connection between the Laplace-Beltrami operator of $\mathfrak{r}$ and a specific $7\times 7$ matrix. This exploration further deepened our understanding of the geometric structure and differential properties of the hypersurface family.

In summary, this research contributes to the understanding of hypersurfaces of revolution in ${\mathbb{E}}_{3}^{7}$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that they have no conflicts of interest to report regarding the present study.