Hypersurfaces of revolution family supplying ∆ r = A r in pseudo-Euclidean space E 73

: In this study, we introduce a family of hypersurfaces of revolution characterized by six parameters in the seven-dimensional pseudo-Euclidean space E 73 . These hypersurfaces exhibit intriguing geometric properties, and our aim is to analyze them in detail. To begin, we calculate the matrices corresponding to the fundamental form, Gauss map, and shape operator associated with this hypersurface family. These matrices provide essential information about the local geometry of the hypersurfaces, including their curvatures and tangent spaces. Using the Cayley-Hamilton theorem, we employ matrix algebra techniques to determine the curvatures of the hypersurfaces. This theorem allows us to express the characteristic polynomial of a matrix in terms of the matrix itself, enabling us to compute the curvatures e ﬀ ectively. In addition, we establish equations that describe the interrelation between the mean curvature and the Gauss-Kronecker curvature of the hypersurface family. These equations provide insights into the geometric behavior of the surfaces and o ﬀ er a deeper understanding of their intrinsic properties. Furthermore, we investigate the relationship between the Laplace-Beltrami operator, a di ﬀ erential operator that characterizes the geometry of the hypersurfaces, and a speciﬁc 7 × 7 matrix denoted as A . By studying this relation, we gain further insights into the geometric structure and di ﬀ erential properties of the hypersurface family. Overall, our study contributes to the understanding of hypersurfaces of revolution in E 73 , o ﬀ ering mathematical insights and establishing connections between various geometric quantities and operators associated with this family


Introduction
Chen [11][12][13][14] originally proposed the notion of submanifolds of finite order immersed in m-space E m or pseudo-Euclidean m-space E m ν 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 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 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 E m . Dursun [22] introduced hypersurfaces with a pointwise 1-type Gauss map in 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 E 3 , Takahashi [43] explored the concept of minimal surfaces, where spheres and minimal surfaces with ∆r = λr, λ ∈ R are the exclusive types of surfaces. Ferrandez et al. [23] identified that surfaces ∆H = A 3×3 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 ∆r = A 3×3 r + B 3×1 , which include minimal surfaces, spheres, and circular cylinders. Stamatakis and Zoubi [42] established the properties of surfaces of revolution defined by ∆ III x = A 3×3 x. Kim et al. [36] focused on the Cheng-Yau operator and the Gauss map of surfaces of revolution.
In 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 ∆ I R = A 4×4 R. Furthermore, Güler [27] obtained the fundamental form IV and curvature formulas of the hypersphere.
In Minkowski 4-space E 4 1 , 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 ∆H = αH with a constant α. 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 λ-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 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 E 7 3 . This family, denoted as r, is characterized by six parameters. The paper focuses on computing various matrices associated with r, including the fundamental form, Gauss map, and shape operator. The Cayley-Hamilton theorem is employed to determine the curvatures of r.
Furthermore, the paper establishes equations that describe the relationship between the mean curvature and Gauss-Kronecker curvature of r. Additionally, the paper explores the connection between the Laplace-Beltrami operator of r and a 7 × 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 E 7 3 . 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 E 7 3 and utilize the previously discussed family to compute it.
Finally, we serve a conclusion in the last section.

Preliminaries
In this paper, we use the following notations, formulas, eqations, etc.
For clarity, E m v represents a pseudo-Euclidean m-space with coordinates denoted as (x 1 , x 2 , . . . , x m ) with index v. The canonical pseudo-Euclidean metric tensor on E m v is represented by g and defined Let M be an m-dimensional semi-Riemannian submanifold, and is embedded in E m v , and the Levi-Civita connections [38] associated with M are denoted as ∇, ∇, respectively. We utilize X, Y, Z, and W to denote vector fields tangent to M, and ξ, ς 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 ξ is a symmetric endomorphism of the tangent space T p M at each point p ∈ M for each ξ ∈ T ⊥ p 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 ∇, and h denotes the second fundamental form of M. The Codazzi equation is given by where∇h denotes the covariant derivative of h w.r.t. the Levi-Civita connection ∇, 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 E n+1 with its shape operator S, and position vector x. Consider a local orthonormal frame field {e 1 , e 2 , . . . , e n } consisting of principal directions of M coinciding with the principal curvature k i for i = 1, 2, . . . , n. Let the dual basis of this frame field be { f 1 , f 2 , . . . , f n }. Then, the first structural equation of Cartan is determined by where ω i j indicates the connection forms coinciding with the chosen frame field. By the Codazzi equation, we derive the equations.
We let s j = σ j (k 1 , k 2 , . . . , k n ), where σ j denotes the j-th elementary symmetric function defined by We consider the notation According to the given definition, we have r 0 i = 1 and s n+1 = s n+2 = · · · = 0. The function s k is referred to as the k-th mean curvature of the oriented hypersurface M. The mean curvature H = 1 n s 1 is also defined, and the Gauss-Kronecker curvature of M is K = s n . If s j ≡ 0, the hypersurface M is known as j-minimal.

Curvatures in
In this section, we reveal the curvature formulas of any hypersurface r = r(u, v, w, α, β, γ) in E 7 3 . Theorem 1. A hypersurface r in E 7 3 has the following curvature formulas, K 0 = 1 by definition, where a 6 λ 6 + a 5 λ 5 + a 4 λ 4 + a 3 λ 3 + a 2 λ 2 + a 1 λ + a 0 = 0 denotes the characteristic polynomial equation P S (λ) = 0 of the shape operator matrix S, a 6 = det g i j , a 0 = det h i j , and g i j , h i j are the first, and the second fundamental form matrices, respectively. Proof. The solution matrix g i j −1 · h i j supplies the shape operator matrix S of hypersurface r in pseudo-Euclidean 7-space E 7 3 . In E 7 3 , computing the curvature formula K i , where i = 0, 1, . . . , 6, we reveal the characteristic polynomial equation det(S − λI 6 ) = 0 of S. Then, we obtain 6 0 K 0 = 1, Definition 4. A space-like hypersurface r is called j-maximal if K j = 0, where j = 1, ..., 6. Theorem 2. A hypersurface r = r(u, v, w, α, β, γ) in E 7 3 has the following relation where I, II, . . . , VII determines the fundamental form matrices, O 6 represents the zero matrix having order 6 × 6 of the hypersurface. Proof. Regarding n = 6 in (2.1), it works.

Hypersurfaces of revolution family in
In this section, we define the hypersurfaces of revolution family (HRF), then find its differential geometric properties in pseudo-Euclidean 7-space E 7 3 . An HR in Riemannian space forms were given in [21].
An HRF M of Euclidean (n + 1)-space constructed by a hypersurface around rotating axis does not meet is acquired by taking the orbit of under the orthogonal transformations of (n + 1)-space.

5.
Hypersurfaces of revolution family satisfying ∆r = Ar in E 7 3 In this section, our focus is on the Laplace-Beltrami operator of a smooth function in E 7 3 . We will proceed to compute it utilizing the HRF, which is defined by Eq (4.1). Definition 6. The Laplace-Beltrami operator of a smooth function f = f (x 1 , x 2 , ..., x 6 ) | D (D ⊂ R 6 ) of class C 6 depends on the first fundamental form g i j of a hypersurface r, and is the operator defined by where g i j = (g kl ) −1 andĝ = det g i j .
By using the inverse matrix of the first fundamental form matrix g i j 6×6 , we have the following. For an HRF given by Eq (4.1) , g i j = 0 when i j except for i, j < 4. Therefore, the Laplace-Beltrami operator of the HRF r = r(u, v, w, α, β, γ) is given by By using the derivatives of the functions in (5.2), w.r.t. u, v, w, α, β, γ, resp., we obtain the following. Theorem 4. The Laplace-Beltrami operator of the HRF r denoted by Eq (4.1) is given by ∆r = 6K 1 G, where K 1 denotes the mean curvature, G represents the Gauss map of r.

Conclusions
This research has presented a detailed analysis of a family of hypersurfaces of revolution r is characterized by six parameters in the seven-dimensional pseudo-Euclidean space E 7 3 , and its geometric properties have been thoroughly explored.
The main focus of the paper was on computing and investigating various matrices associated with 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 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 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 r and a specific 7 × 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 E 7 3 .

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