Construction of a New Hypersurface Family Using the Spherical Product in Minkowski Geometry

Abstract

The spherical product of two curves, composed of a total of n components, gives rise to spherical product surfaces in Euclidean space ${\mathbb{E}}^n$, frequently resulting in surfaces of revolution, including superquadrics, which often exhibit inherent symmetry. When $(n-1)$-planar curves are considered, this construction enables the generation of hypersurfaces in n-dimensional spaces. Building upon this geometric framework, we conduct the first-ever investigation of spherical product hypersurfaces in the context of Minkowski geometry. We define these hypersurfaces in four-dimensional Minkowski space ${\mathbb{E}}_1^4$ and derive explicit expressions for their Gaussian and mean curvatures. We also determine the conditions under which such hypersurfaces are flat or minimal. Furthermore, we reinterpret certain hyperquadrics as specific instances of spherical product hypersurfaces in ${\mathbb{E}}_1^4$, supported by visual illustrations. Finally, we extend the construction to arbitrary-dimensional Minkowski spaces, providing a unified formulation for spherical product hypersurfaces across higher-dimensional Lorentzian geometries.

1. Introduction

Embeddings of product spaces were first defined by Kuiper between Euclidean spaces, as in the following:

$$\begin{array}{c}F=\alpha \otimes \beta :M\times {\mathbb{S}}^n\to {\mathbb{E}}^{m+n+d}\hfill \\ F(u,v)=({\alpha }_1\left(u\right),{\alpha }_2\left(u\right),\dots ,{\alpha }_{m+d-1}\left(u\right),{\alpha }_{m+d}\left(u\right)\beta \left(v\right)),\hfill \end{array}$$

(1)

where $\alpha :M\to {\mathbb{E}}^{m+d}$ is an embedding defined from an m-dimensional manifold M into ${\mathbb{E}}^{m+d}$ and $\beta :{\mathbb{S}}^n\to {\mathbb{E}}^{n+1}$ is the standard embedding from an n-sphere into ${\mathbb{E}}^{n+1}$. The product $\alpha \left(u\right)\otimes \beta \left(v\right)$ results in a higher-dimensional embedding, which can be interpreted as a surface in a higher-dimensional Euclidean space, following the work of Kuiper [1].

A special case of these types of embeddings was given for two Euclidean plane curves $\alpha ,$ $\beta :I\subset \mathbb{R}\to {\mathbb{E}}^2$, $\alpha \left(u\right)=({\alpha }_1\left(u\right),{\alpha }_2\left(u\right)), \beta \left(v\right)=({\beta }_1\left(v\right),{\beta }_2\left(v\right))$ in [2]. They defined the spherical product $\alpha \left(u\right)\otimes \beta \left(v\right)$ of two planar curves as follows:

$$\alpha \left(u\right)\otimes \beta \left(v\right)=F(u,v)=({\alpha }_1\left(u\right),{\alpha }_2\left(u\right){\beta }_1\left(v\right),{\alpha }_2\left(u\right){\beta }_2\left(v\right)),$$

(2)

which is a surface in ${\mathbb{E}}^3$. In another study [3], taking $\alpha \left(u\right)=({\alpha }_1\left(u\right),{\alpha }_2\left(u\right),{\alpha }_3\left(u\right))$, $\beta \left(v\right)=({\beta }_1\left(v\right),{\beta }_2\left(v\right))$, they give the parametrization

$$F(u,v)=({\alpha }_1\left(u\right),{\alpha }_2\left(u\right),{\alpha }_3\left(u\right){\beta }_1\left(v\right),{\alpha }_3\left(u\right){\beta }_2\left(v\right))$$

in Euclidean 4-space.

Hsu derived both implicit and parametric representations for asymmetric and symmetric spherical product surfaces and applied these results to geometric modeling [4]. The most well-known special instances of spherical product surfaces are superquadrics and rotational surfaces, concepts that encompass a wide range of geometries—including helicoidal and ruled surfaces—with accompanying illustrations [5,6,7,8,9]. In particular, superquadrics and hyperquadrics can be obtained as spherical products of superhyperbolas and superellipses, whose simple forms are familiar from analytic geometry; accordingly, shapes such as the superhyperboloid with one leaf, the superhyperboloid with two leaves, the superellipsoid, and the toroid can be generated [5,10]. Such superquadrics have applications in computer vision and computer graphics [11,12,13].

Furthermore, surfaces are also known as a subcategory of hypersurfaces in the field of geometry. In $n-$dimensional spaces, hypersurfaces are considered as $(n-1)$-dimensional surfaces and can be locally represented as level sets of smooth functions. For instance, let $\Psi :D\subset {\mathbb{E}}_1^n\to \mathbb{R}$ be a smooth real-valued function defined on an open subset D of the Minkowski space. Then the hypersurface M can be expressed as

$$M=\left\{x\in D:\Psi \left(x\right)=k\right\},$$

where $k\in \mathbb{R}$ is a constant. A number of related studies can be found in [14,15,16,17].

A 2-surface is created in ${\mathbb{E}}^4$ using the spherical product [3]. A hypersurface is obtained in the same space [18]. Furthermore, a hypersurface can be constructed in Minkowski space ${\mathbb{E}}_1^4$ by the spherical product of three planar curves, and various investigations can be performed. The construction of hypersurfaces via spherical products inherently involves combining geometrical objects with structured behavior, often resulting in parametrizations that exhibit various types of symmetry, such as rotational or reflective patterns. In particular, superquadrics derived from such constructions are well-known for their symmetrical forms.

In this paper, the spherical product is considered on hypersurfaces for the first time in Minkowski 4-space. Initially, spherical product hypersurfaces are defined and it is shown under which conditions they are timelike in ${\mathbb{E}}_1^4$. The Gaussian curvature and the mean curvature are obtained. A necessary and sufficient condition for a hypersurface to be flat is established, which asserts that at least one curve constituting the hypersurface must be a straight line. Moreover, the condition for minimality is examined. The subsequent part focuses on defining superquadrics in hypersurface form in ${\mathbb{E}}_1^4$, providing illustrative examples, and plotting their projections onto ${\mathbb{E}}_1^3$. Finally, Section 5 generalizes spherical product hypersurfaces and presents their related parameterization in n-dimensional Minkowski space, denoted as ${\mathbb{E}}_1^n$. Throughout, the rotational and reflection symmetries inherited from the generating curves persist and inform the curvature-based criteria across dimensions.

2. Basic Notations

The four-dimensional Minkowski space ${\mathbb{E}}_1^4$ is a real vector space ${\mathbb{R}}^4$ equipped with the metric tensor $〈.,.〉=d{x}_1^2+d{x}_2^2+d{x}_3^2-d{x}_4^2$, where $x_i$ are the pseudo-Euclidean coordinates of type $(3,1)$ [19].

Let $x=\left(x_1,x_2,x_3,x_4\right)$, $y=\left(y_1,y_2,y_3,y_4\right)$, and $z=\left(z_1,z_2,z_3,z_4\right)$ be three vectors in ${\mathbb{E}}_1^4$. The scalar product $〈x,y〉$, the vector product $x\times y\times z$, and the norm of the vector x are given by

$$〈x,y〉=x_1{y}_1+x_2{y}_2+x_3{y}_3-x_4{y}_4,$$

$$x\times y\times z=\left|\begin{array}{cccc}{e}_1& e_2& e_3& -e_4\\ x_1& x_2& x_3& x_4\\ y_1& y_2& y_3& y_4\\ z_1& z_2& z_3& z_4\end{array}\right|,$$

and

$$∥x∥=\sqrt{\left|〈x,x〉\right|}.$$

A vector x in ${\mathbb{E}}_1^4$ is spacelike if $〈x,x〉>0$ or $x=0$, timelike if $〈x,x〉<0$, and lightlike (null) if $〈x,x〉=0$ and $x\ne 0$ [20].

If $M:F(u,v,w)$ is a hypersurface in ${\mathbb{E}}_1^4$, then M is represented as

$$\begin{array}{ccc}\hfill F& :& {\mathbb{E}}^3\to {\mathbb{E}}_1^4\hfill \\ \hfill (u,v,w)& \to & F(u,v,w)\hfill \\ \hfill F(u,v,w)& =& (F_1(u,v,w),F_2(u,v,w),F_3(u,v,w),F_4(u,v,w))\hfill \end{array}$$

and principle normal of M is given by

$$N={\displaystyle \frac{F_u\times F_v\times F_w}{\left|\left|F_u\times F_v\times F_w\right|\right|}}.$$

(3)

Note that a hypersurface in ${\mathbb{E}}_1^4$ is spacelike (timelike) if the normal vector is timelike (spacelike).

The matrices corresponding to the first fundamental form I and the second fundamental form $II$ are, respectively, given by

$$I=\left[\begin{array}{ccc}e& f& a\\ f& g& b\\ a& b& c\end{array}\right]\mathrm{and}II=\left[\begin{array}{ccc}l& m& p\\ m& n& t\\ p& t& v\end{array}\right],$$

(4)

where

$$\begin{array}{ccc}\hfill e& =& 〈F_u,F_u〉,f=〈F_u,F_v〉,g=〈F_v,F_v〉,\hfill \\ \hfill a& =& 〈F_u,F_w〉,b=〈F_v,F_w〉,c=〈F_w,F_w〉\hfill \end{array}$$

(5)

are the coefficients of the first fundamental form and

$$\begin{array}{ccc}\hfill l& =& 〈F_{uu},N〉,m=〈F_{uv},N〉,n=〈F_{vv},N〉,\hfill \\ \hfill p& =& 〈F_{uw},N〉,t=〈F_{vw},N〉,v=〈F_{ww},N〉\hfill \end{array}$$

(6)

are the coefficients of the second fundamental form.

For a hypersurface M in ${\mathbb{E}}_1^4$, the shape operator matrix is obtained as $S=\left(I^{-1}\right)II$. Also, the Gaussian curvature K and the mean curvature H are given by

$$K=\det \left(S\right)={\displaystyle \frac{\det \left(II\right)}{\det \left(I\right)}}\mathrm{and}H={\displaystyle \frac{\mathrm{tr}\left(S\right)}{3}},$$

(7)

respectively. From Equations (4) and (7), we have

$$K={\displaystyle \frac{(ln-m^2)v+2mpt-p^{2}n-t^{2}l}{(eg-f^2)c+2abf-a^{2}g-b^{2}e}}$$

(8)

and

$$H={\displaystyle \frac{(en+gl-2fm)c+(eg-f^2)v-a^{2}n-b^{2}l-2(apg+bte-abm-atf-bpf)}{3((eg-f^2)c+2abf-a^{2}g-b^{2}e)}}.$$

(9)

Note that a hypersurface M is flat (minimal) if $K=0$ ($H=0$). [16].

3. Spherical Product Hypersurfaces in Minkowski 4-Space ${\mathbb{E}}_{\mathbf{1}}^{\mathbf{4}}$

In this section, the concept of rotational embedding is considered. Upon closer examination, the spherical product refers to the multiplication of the last component of the first parameterization by each component of the other parameterization, resulting in the generation of new components. Consequently, the resulting surface is defined in a higher-dimensional space. This spherical product can then be applied twice to construct a hypersurface that incorporates three curves. The following formal definition is given.

Definition 1. 

Suppose the functions ${\alpha }_i,$${\beta }_i,$${\gamma }_i$$(i=1,2)$ are differentiable and the curves $\alpha ,$ $\beta ,$ $\gamma :I\subset \mathbb{R}\to {\mathbb{E}}_1^2$ are given by $\alpha \left(u\right)=\left({\alpha }_1\left(u\right),{\alpha }_2\left(u\right)\right),$ $\beta \left(v\right)=\left({\beta }_1\left(v\right),{\beta }_2\left(v\right)\right),$ $\gamma \left(w\right)=\left({\gamma }_1\left(w\right),{\gamma }_2\left(w\right)\right).$ Then, a spherical product hypersurface in ${\mathbb{E}}_1^4$ is defined by ($\alpha \left(u\right)\otimes \beta \left(v\right)\otimes \gamma \left(w\right)$) as

$$\begin{array}{ccc}\hfill F\left(u,v,w\right)=\alpha \left(u\right)\otimes \beta \left(v\right)\otimes \gamma \left(w\right)& =& \left[\begin{array}{c}{\alpha }_1\left(u\right)\\ {\alpha }_2\left(u\right)\end{array}\right]\otimes \left[\begin{array}{c}{\beta }_1\left(v\right)\\ {\beta }_2\left(v\right)\end{array}\right]\otimes \left[\begin{array}{c}{\gamma }_1\left(w\right)\\ {\gamma }_2\left(w\right)\end{array}\right]\hfill \\ & =& \left[\begin{array}{c}{\alpha }_1\left(u\right)\\ {\alpha }_2\left(u\right)\end{array}\right]\otimes \left[\begin{array}{c}{\beta }_1\left(v\right)\\ {\beta }_2\left(v\right){\gamma }_1\left(w\right)\\ {\beta }_2\left(v\right){\gamma }_2\left(w\right)\end{array}\right].\hfill \end{array}$$

(10)

Hence, $F\left(u,v,w\right)$ is represented by

$$F\left(u,v,w\right)=\left({\alpha }_1\left(u\right),{\alpha }_2\left(u\right){\beta }_1\left(v\right),{\alpha }_2\left(u\right){\beta }_2\left(v\right){\gamma }_1\left(w\right),{\alpha }_2\left(u\right){\beta }_2\left(v\right){\gamma }_2\left(w\right)\right).$$

(11)

As seen from (10), the spherical product $\beta \left(v\right)\otimes \gamma \left(w\right)$ corresponds to the spherical product surface

$$G\left(v,w\right)=\left({\beta }_1\left(v\right),{\beta }_2\left(v\right){\gamma }_1\left(w\right),{\beta }_2\left(y\right){\gamma }_2\left(w\right)\right)$$

(12)

in 3-dimensional Minkowski space.

Example 1. 

Let the three curves $\alpha ,\beta$, and γ be chosen as $\alpha \left(u\right)=\left({\alpha }_1\left(u\right),u\right),$ $\beta \left(v\right)=\left(sinhv,coshv\right)$, and $\gamma \left(w\right)=\left(sinhw,coshw\right)$. Then, the spherical product of $\alpha \left(u\right),\beta \left(v\right),\gamma \left(w\right)$ is congruent to a rotational hypersurface and represented by

$$M:F\left(u,v,w\right)=\left({\alpha }_1\left(u\right),usinhv,ucoshvsinhw,ucoshvcoshw\right).$$

Assume that a spherical product hypersurface is denoted by (11). The first partial derivatives are

$$\begin{array}{ccc}\hfill F_u& =& {\displaystyle \frac{\partial F\left(u,v,w\right)}{\partial u}}=\left({\alpha }_1^{\prime },{\alpha }_2^{\prime }{\beta }_1,{\alpha }_2^{\prime }{\beta }_2{\gamma }_1,{\alpha }_2^{\prime }{\beta }_2{\gamma }_2\right),\hfill \\ \hfill F_v& =& {\displaystyle \frac{\partial F\left(u,v,w\right)}{\partial v}}=\left(0,{\alpha }_2{\beta }_1^{\prime },{\alpha }_2{\beta }_2^{\prime }{\gamma }_1,{\alpha }_2{\beta }_2^{\prime }{\gamma }_2\right),\hfill \\ \hfill F_w& =& {\displaystyle \frac{\partial F\left(u,v,w\right)}{\partial w}}=\left(0,0,{\alpha }_2{\beta }_2{\gamma }_1^{\prime },{\alpha }_2{\beta }_2{\gamma }_2^{\prime }\right).\hfill \end{array}$$

(13)

Using (3) and (13), we get the unit normal vector of the hypersurface M as

$$N={\displaystyle \frac{{\alpha }_2^2{\beta }_2}{\sqrt{\left|W\right|}}}\left({\alpha }_2^{\prime }\left({\beta }_1{\beta }_2^{\prime }-{\beta }_1^{\prime }{\beta }_2\right)\left({\gamma }_1{\gamma }_2^{\prime }-{\gamma }_1^{\prime }{\gamma }_2\right),-{\alpha }_1^{\prime }{\beta }_2^{\prime }\left({\gamma }_1{\gamma }_2^{\prime }-{\gamma }_1^{\prime }{\gamma }_2\right),{\alpha }_1^{\prime }{\beta }_1^{\prime }{\gamma }_2^{\prime },{\alpha }_1^{\prime }{\beta }_1^{\prime }{\gamma }_1^{\prime }\right),$$

(14)

where

$W={\alpha }_2^4{\beta }_2^2\left({\left({\gamma }_1{\gamma }_2^{\prime }-{\gamma }_1^{\prime }{\gamma }_2\right)}^2\left({\alpha }_2^{\prime 2}{\left({\beta }_1{\beta }_2^{\prime }-{\beta }_1^{\prime }{\beta }_2\right)}^2+{\alpha }_1^{\prime 2}{\beta }_2^{\prime 2}\right)+{\alpha }_1^{\prime 2}{\beta }_1^{\prime 2}\left({\gamma }_2^{\prime 2}-{\gamma }_1^{\prime 2}\right)\right)$.

It is known that if the unit normal vector is spacelike (that is, $\langle N,N\rangle >0$), then the hypersurface is timelike; if the unit normal vector is timelike, then the hypersurface is spacelike. If $\langle N,N\rangle =0$, the hypersurface is called lightlike. Therefore, we obtain the following corollary.

Corollary 1. 

Let M be a spherical product hypersurface in ${\mathbb{E}}_1^4$ given by the parameterization (11).

(i) 

In case

$$|{\gamma }_2^{\prime }|>|{\gamma }_1^{\prime }|,$$

the hypersurface M is timelike.

(ii) 

In case

$$|{\gamma }_2^{\prime }|<|{\gamma }_1^{\prime }|,$$

we have the following subcases:

(a) 

If

$${\left({\gamma }_1{\gamma }_2^{\prime }-{\gamma }_1^{\prime }{\gamma }_2\right)}^2\left({\alpha }_2^{\prime 2}{({\beta }_1{\beta }_2^{\prime }-{\beta }_1^{\prime }{\beta }_2)}^2+{\alpha }_1^{\prime 2}{\beta }_2^{\prime 2}\right)<{\alpha }_1^{\prime 2}{\beta }_1^{\prime 2}\left(|{\gamma }_2^{\prime 2}-{\gamma }_1^{\prime 2}|\right),$$

then the hypersurface M is spacelike.

(b) 

If

$${\left({\gamma }_1{\gamma }_2^{\prime }-{\gamma }_1^{\prime }{\gamma }_2\right)}^2\left({\alpha }_2^{\prime 2}{({\beta }_1{\beta }_2^{\prime }-{\beta }_1^{\prime }{\beta }_2)}^2+{\alpha }_1^{\prime 2}{\beta }_2^{\prime 2}\right)>{\alpha }_1^{\prime 2}{\beta }_1^{\prime 2}\left(|{\gamma }_2^{\prime 2}-{\gamma }_1^{\prime 2}|\right),$$

then the hypersurface M is timelike.

(c) 

If

$${\left({\gamma }_1{\gamma }_2^{\prime }-{\gamma }_1^{\prime }{\gamma }_2\right)}^2\left({\alpha }_2^{\prime 2}{({\beta }_1{\beta }_2^{\prime }-{\beta }_1^{\prime }{\beta }_2)}^2+{\alpha }_1^{\prime 2}{\beta }_2^{\prime 2}\right)={\alpha }_1^{\prime 2}{\beta }_1^{\prime 2}\left(|{\gamma }_2^{\prime 2}-{\gamma }_1^{\prime 2}|\right),$$

then the hypersurface M is lightlike.

(iii) 

In case

$$\left|{\gamma }_2^{\prime }\right|=\left|{\gamma }_1^{\prime }\right|,$$

the hypersurface M is lightlike if

$${\left({\gamma }_1{\gamma }_2^{\prime }-{\gamma }_1^{\prime }{\gamma }_2\right)}^2\left({\alpha }_2^{\prime 2}{({\beta }_1{\beta }_2^{\prime }-{\beta }_1^{\prime }{\beta }_2)}^2+{\alpha }_1^{\prime 2}{\beta }_2^{\prime 2}\right)=0,$$

otherwise timelike.

The coefficients of the first fundamental form are obtained as

$$\begin{array}{ccc}\hfill e& =& {\alpha }_1^{\prime 2}+{\alpha }_2^{\prime 2}\left[{\beta }_1^2+{\beta }_2^2\left({\gamma }_1^2-{\gamma }_2^2\right)\right]={\alpha }_1^{\prime 2}+{\alpha }_2^{\prime 2}〈G\left(v,w\right),G\left(v,w\right)〉,\hfill \\ \hfill f& =& {\alpha }_2^{\prime }{\alpha }_2\left[{\beta }_1^{\prime }{\beta }_1+{\beta }_2^{\prime }{\beta }_2\left({\gamma }_1^2-{\gamma }_2^2\right)\right]={\alpha }_2^{\prime }{\alpha }_2〈G\left(v,w\right),G_v\left(v,w\right)〉,\hfill \\ \hfill a& =& {\alpha }_2^{\prime }{\alpha }_2{\beta }_2^2\left({\gamma }_1^{\prime }{\gamma }_1-{\gamma }_2^{\prime }{\gamma }_2\right)={\alpha }_2^{\prime }{\alpha }_2〈G\left(v,w\right),G_w\left(v,w\right)〉,\hfill \\ \hfill g& =& {\alpha }_2^2\left[{\beta }_1^{\prime 2}+{\beta }_2^{\prime 2}\left({\gamma }_1^2-{\gamma }_2^2\right)\right]={\alpha }_2^2〈G_v\left(v,w\right),G_v\left(v,w\right)〉,\hfill \\ \hfill b& =& {\alpha }_2^2{\beta }_2^{\prime }{\beta }_2\left({\gamma }_1^{\prime }{\gamma }_1-{\gamma }_2^{\prime }{\gamma }_2\right)={\alpha }_2^2〈G_v\left(v,w\right),G_w\left(v,w\right)〉,\hfill \\ \hfill c& =& {\alpha }_2^2{\beta }_2^2\left({\gamma }_1^{\prime 2}-{\gamma }_2^{\prime 2}\right)={\alpha }_2^2〈G_w\left(v,w\right),G_w\left(v,w\right)〉.\hfill \end{array}$$

(15)

By using (4) and (15), one can find that $det\left(I\right)=-W$.

Now, we get the second partial derivatives as

$$\begin{array}{ccc}\hfill F_{uu}& =& \left({\alpha }_1^{″},{\alpha }_2^{″}{\beta }_1,{\alpha }_2^{″}{\beta }_2{\gamma }_1,{\alpha }_2^{″}{\beta }_2{\gamma }_2\right),\hfill \\ \hfill F_{uv}& =& \left(0,{\alpha }_2^{\prime }{\beta }_1^{\prime },{\alpha }_2^{\prime }{\beta }_2^{\prime }{\gamma }_1,{\alpha }_2^{\prime }{\beta }_2^{\prime }{\gamma }_2\right),\hfill \\ \hfill F_{uw}& =& \left(0,0,{\alpha }_2^{\prime }{\beta }_2{\gamma }_1^{\prime },{\alpha }_2^{\prime }{\beta }_2{\gamma }_2^{\prime }\right),\hfill \\ \hfill F_{vv}& =& \left(0,{\alpha }_2{\beta }_1^{″},{\alpha }_2{\beta }_2^{″}{\gamma }_1,{\alpha }_2{\beta }_2^{″}{\gamma }_2\right),\hfill \\ \hfill F_{vw}& =& \left(0,0,{\alpha }_2{\beta }_2^{\prime }{\gamma }_1^{\prime },{\alpha }_2{\beta }_2^{\prime }{\gamma }_2^{\prime }\right),\hfill \\ \hfill F_{ww}& =& \left(0,0,{\alpha }_2{\beta }_2{\gamma }_1^{″},{\alpha }_2{\beta }_2{\gamma }_2^{″}\right).\hfill \end{array}$$

(16)

Taking

$$\begin{array}{ccc}\hfill {\lambda }_1\left(u\right)& =& {\alpha }_1^{″}{\alpha }_2^{\prime }-{\alpha }_2^{″}{\alpha }_1^{\prime },\hfill \\ \hfill {\lambda }_2\left(v\right)& =& {\beta }_1{\beta }_2^{\prime }-{\beta }_1^{\prime }{\beta }_2,\hfill \\ \hfill {\lambda }_3\left(v\right)& =& {\beta }_1^{\prime }{\beta }_2^{″}-{\beta }_1^{″}{\beta }_2^{\prime },\hfill \\ \hfill {\lambda }_4\left(w\right)& =& {\gamma }_1{\gamma }_2^{\prime }-{\gamma }_1^{\prime }{\gamma }_2,\hfill \\ \hfill {\lambda }_5\left(w\right)& =& {\gamma }_1^{″}{\gamma }_2^{\prime }-{\gamma }_1^{\prime }{\gamma }_2^{″}\hfill \end{array}$$

(17)

and with the help of (6), (14), and (17), we write the coefficients of the second fundamental form as

$$\begin{array}{ccc}\hfill l& =& {\displaystyle \frac{{\alpha }_2^2{\beta }_2}{\sqrt{\left|W\right|}}}{\lambda }_1\left(u\right){\lambda }_2\left(v\right){\lambda }_4\left(w\right),\hfill \\ \hfill n& =& {\displaystyle \frac{{\alpha }_1^{\prime }{\alpha }_2^3{\beta }_2}{\sqrt{\left|W\right|}}}{\lambda }_3\left(v\right){\lambda }_4\left(w\right),\hfill \\ \hfill v& =& {\displaystyle \frac{{\alpha }_1^{\prime }{\beta }_1^{\prime }{\alpha }_2^3{\beta }_2^2}{\sqrt{\left|W\right|}}}{\lambda }_5\left(w\right),\hfill \\ \hfill m& =& p=t=0.\hfill \end{array}$$

(18)

Theorem 1. 

Let a spherical product hypersurface be parameterized by (11) in ${\mathbb{E}}_1^4.$ Then, the Gaussian curvature is yielded as

$$K={\displaystyle \frac{-{\alpha }_1^{\prime 2}{\alpha }_2^8{\beta }_2^4{\beta }_1^{\prime }{\lambda }_1\left(u\right){\lambda }_2\left(v\right){\lambda }_3\left(v\right){\lambda }_4^2\left(w\right){\lambda }_5\left(w\right)}{W\sqrt{{\left|W\right|}^3}}}.$$

(19)

Here, ${\lambda }_1\left(u\right),$ ${\lambda }_2\left(v\right),$ ${\lambda }_3\left(v\right),$ ${\lambda }_4\left(w\right),$ and ${\lambda }_5\left(w\right)$ are given in (17).

Proof. 

By using (8), (15), and (18), we obtain the result. □

Theorem 2. 

Suppose that a spherical product hypersurface M in Minkowski 4-space is parameterized by (11). Then, the necessary and sufficient condition for M to be flat is that one of the generating curves of the hypersurface is a straight line.

Proof. 

Assume a hypersurface obtained by spherical product is given by (11). If its Gaussian curvature vanishes (flatness), then by the use of (19), we yield that any of the following equations hold:

$$\begin{array}{ccc}\hfill {\lambda }_1\left(u\right)& =& 0,\hfill \\ \hfill {\lambda }_2\left(v\right)& =& 0,\hfill \\ \hfill {\lambda }_3\left(v\right)& =& 0,\hfill \\ \hfill {\lambda }_4\left(w\right)& =& 0,\hfill \\ \hfill {\lambda }_5\left(w\right)& =& 0,\hfill \end{array}$$

which means that at least one of the curves $\alpha \left(u\right)$, $\beta \left(v\right)$, or $\gamma \left(w\right)$ is a straight line. The converse implication is straightforward. □

Theorem 3. 

Suppose that a spherical product hypersurface M in Minkowski 4-space is parameterized by (11). Then, the mean curvature function is

$$H=-{\displaystyle \frac{{\alpha }_2^5{\beta }_2}{3W\sqrt{\left|W\right|}}}\left\{\begin{array}{c}{\lambda }_4\left(w\right)\left[\begin{array}{c}{\alpha }_2{\lambda }_1\left(u\right){\lambda }_2\left(v\right)\left(〈G_v,G_v〉〈G_w,G_w〉-{〈G_v,G_w〉}^2\right)\\ +{\alpha }_1^{\prime }{\lambda }_3\left(v\right)\left({\alpha }_1^{\prime 2}〈G_w,G_w〉+{\alpha }_2^{\prime 2}\left(〈G,G〉〈G_w,G_w〉-{〈G,G_w〉}^2\right)\right)\end{array}\right]\\ +{\beta }_2{\alpha }_1^{\prime }{\beta }_1^{\prime }{\lambda }_5\left(w\right)\left({\alpha }_1^{\prime 2}〈G_v,G_v〉+{\alpha }_2^{\prime 2}\left(〈G,G〉〈G_v,G_v〉-{〈G,G_v〉}^2\right)\right)\end{array}\right\}.$$

(20)

Here, $G=G(v,w)$ is spherical product surface in ${\mathbb{E}}_1^3$ parameterized by (12).

Proof. 

Assume that a spherical product hypersurface is given by (11) in ${\mathbb{E}}_1^4.$ Using (5), (6), and (9), we obtain the result. □

Corollary 2. 

Suppose that a spherical product hypersurface is parameterized by (11) in ${\mathbb{E}}_1^4$. By using (20), the following statements are satisfied:

(a) 

If γ passes through the origin and is congruent to a straight line, then the spherical product hypersurface is minimal.

(b) 

If all three curves α, β, and γ are straight lines, then the spherical product hypersurface is minimal.

Example 2. 

Let M be a spherical product hypersurface given with the parametrization (10) in ${\mathbb{E}}_1^4$. For the functions $\alpha \left(u\right)=\left(cosu,u\right)$, $\beta \left(v\right)=\left(sinv,v\right)$, and $\gamma \left(w\right)=\left(w,2w\right)$, we get the parametrization of M as

$$M:F\left(u,v,w\right)=\left(cosu,usinv,uvw,2uvw\right).$$

(21)

The unit normal vector field of M is given by

$$N={\displaystyle \frac{1}{\sqrt{3}}}(0,0,-2,-1),$$

from which it follows that the spherical product hypersurface is timelike. Moreover, since all coefficients of the second fundamental form vanish, the spherical product hypersurface is both flat and minimal. Choosing $w=1$, the corresponding $3D$ projection can be plotted by using Julia (v1.12.1) with the GLMakie library (v0.13.6) (see Figure 1).

• The projection onto three-dimensional space is obtained by eliminating the fourth component of the parametrization; specifically, the third and fourth components are added together and represented as a new third component in ${\mathbb{R}}^3$.

4. Hyperquadrics in Minkowski 4-Space ${\mathbb{E}}_{\mathbf{1}}^{\mathbf{4}}$

In this section, the relationship between hyperquadrics and the concept of the spherical product will be presented.

Superellipses are special cases of Lame curves, named after Gabriel Lame, and they have the equation

$${\left({\displaystyle \frac{Y_1}{c_1}}\right)}^{\frac{2}{ϵ}}+{\left({\displaystyle \frac{Y_2}{c_2}}\right)}^{\frac{2}{ϵ}}=1,$$

where $ϵ$ is a positive real constant and $c_1$, $c_2$ are nonzero real constants. Thus, the parametric equation can be given by

$$\alpha \left(u\right)=\left(c_1{cos}^{ϵ}u,c_2{sin}^{ϵ}u\right).$$

(22)

In (22), when $ϵ=1$, obviously it corresponds to an ellipse, which is a subject of analytic geometry [11].

By analogy, the equation of a superhyperbola in Minkowski plane can be given as

$${\left({\displaystyle \frac{Y_1}{c_1}}\right)}^{\frac{2}{ϵ}}-{\left({\displaystyle \frac{Y_2}{c_2}}\right)}^{\frac{2}{ϵ}}=1$$

which has the following parametrization:

$$\alpha \left(u\right)=\left(c_1{cosh}^{ϵ}u,c_2{sinh}^{ϵ}u\right).$$

Definition 2. 

Suppose three curves denoted by α, β, and γ are chosen in the form of superellipses or superhyperbola in the Minkowski plane. The spherical product of α, β, and γ is congruent to a hypersurface, called a hyperquadric in ${\mathbb{E}}_1^4$.

With the help of the spherical product of three superellipses or superhyperbola, we obtain some different types of superellipsoids, superhyperboloids, or supertoroids in hypersurface form.

Proposition 1. 

Let the three superellipses α, β, and γ be given by

$$\begin{array}{ccc}\hfill \alpha \left(u\right)& =& \left(c_1{cos}^{{ϵ}_1}u,{sin}^{{ϵ}_1}u\right),\hfill \\ \hfill \beta \left(v\right)& =& \left(c_2{cos}^{{ϵ}_2}v,{sin}^{{ϵ}_2}v\right),\hfill \\ \hfill \gamma \left(w\right)& =& \left(c_3{cos}^{{ϵ}_3}w,c_4{sin}^{{ϵ}_3}w\right).\hfill \end{array}$$

The spherical product of them is obtained as

$$F(u,v,w)=\left(c_1{cos}^{{ϵ}_1}u,c_2{sin}^{{ϵ}_1}u{cos}^{{ϵ}_2}v,c_3{sin}^{{ϵ}_1}u{sin}^{{ϵ}_2}v{cos}^{{ϵ}_3}w,c_4{sin}^{{ϵ}_1}u{sin}^{{ϵ}_2}v{sin}^{{ϵ}_3}w\right).$$

(23)

This corresponds to a hyperellipsoid, which has the equation in Cartesian coordinates

$${\left({\displaystyle \frac{Y_1}{c_1}}\right)}^{\frac{2}{{\epsilon }_1}}+{\left\{{\left({\displaystyle \frac{Y_2}{c_2}}\right)}^{\frac{2}{{\epsilon }_2}}+{\left\{{\left({\displaystyle \frac{Y_3}{c_3}}\right)}^{\frac{2}{{\epsilon }_3}}+{\left({\displaystyle \frac{Y_4}{c_4}}\right)}^{\frac{2}{{\epsilon }_3}}\right\}}^{{\displaystyle \frac{{\epsilon }_3}{{\epsilon }_2}}}\right\}}^{\frac{{\epsilon }_2}{{\epsilon }_1}}=1.$$

Example 3. 

Let M be a superellipsoid given with the parametrization (23) in ${\mathbb{E}}_1^4$. Choosing $c_1=3$, $c_2=2$, $c_3=c_4=1$, ${ϵ}_1=2$, ${ϵ}_2=3$, ${ϵ}_3=1$, and $w={\displaystyle \frac{\pi }{2}}$, the corresponding $3D$ projection can be plotted by using Julia (v1.12.1) with the GLMakie library (v0.13.6), (see Figure 2).

• The projection onto three-dimensional space is obtained by eliminating the fourth component of the parametrization; specifically, the third and fourth components are added together and represented as a new third component in ${\mathbb{R}}^3$.

Proposition 2. 

Let the three curves (one superhyperbola and two superellipses) α, β, and γ be given by

$$\begin{array}{ccc}\hfill \alpha \left(u\right)& =& \left(c_1{sinh}^{{ϵ}_1}u,{cosh}^{{ϵ}_1}u\right),\hfill \\ \hfill \beta \left(v\right)& =& \left(c_2{cos}^{{ϵ}_2}v,{sin}^{{ϵ}_2}v\right),\hfill \\ \hfill \gamma \left(w\right)& =& \left(c_3{cos}^{{ϵ}_3}w,c_4{sin}^{{ϵ}_3}w\right).\hfill \end{array}$$

The spherical product of them is obtained as

$$F(u,v,w)=\left(c_1{sinh}^{{ϵ}_1}u,c_2{cosh}^{{ϵ}_1}u{cos}^{{ϵ}_2}v,c_3{cosh}^{{ϵ}_1}u{sin}^{{ϵ}_2}v{cos}^{{ϵ}_3}w,c_4{cosh}^{{ϵ}_1}u{sin}^{{ϵ}_2}v{sin}^{{ϵ}_3}w\right).$$

(24)

This parametric indication corresponds to an $hyper-$hyperboloid with one leaf in ${\mathbb{E}}_1^4$, and the following Cartesian equation is satisfied:

$$-{\left({\displaystyle \frac{Y_1}{c_1}}\right)}^{\frac{2}{{\epsilon }_1}}+{\left\{{\left({\displaystyle \frac{Y_2}{c_2}}\right)}^{\frac{2}{{\epsilon }_2}}+{\left\{{\left({\displaystyle \frac{Y_3}{c_3}}\right)}^{\frac{2}{{\epsilon }_3}}+{\left({\displaystyle \frac{Y_4}{c_4}}\right)}^{{\displaystyle \frac{2}{{\epsilon }_3}}}\right\}}^{\frac{{\epsilon }_3}{{\epsilon }_2}}\right\}}^{\frac{{\epsilon }_2}{{\epsilon }_1}}=1.$$

Example 4. 

Let M be a $hyper-$hyperboloid given with the parametrization (24) in ${\mathbb{E}}_1^4$. Choosing $c_1=c_2=1$, $c_3=c_4=2$, ${ϵ}_1=2$, ${ϵ}_2=2$, ${ϵ}_3=3$, and $w={\displaystyle \frac{\pi }{2}}$, the related hypersurfaces’ projection in 3-dimensional space can be plotted by using Julia (see Figure 3).

Proposition 3. 

Let the three superhyperbolas α, β, and γ be given by

$$\begin{array}{ccc}\hfill \alpha \left(u\right)& =& \left(c_1{sinh}^{{ϵ}_1}u,{cosh}^{{ϵ}_1}u\right),\hfill \\ \hfill \beta \left(v\right)& =& \left(c_2{sinh}^{{ϵ}_2}v,{cosh}^{{ϵ}_2}v\right),\hfill \\ \hfill \gamma \left(w\right)& =& \left(c_3{sinh}^{{ϵ}_3}w,c_4{cosh}^{{ϵ}_3}w\right).\hfill \end{array}$$

The spherical product corresponds to a hyper-hyperboloid with two leaves in ${\mathbb{E}}_1^4$ and has the parametrization

$$F(u,v,w)=\left(c_1{sinh}^{{ϵ}_1}u,c_2{cosh}^{{ϵ}_1}u{sinh}^{{ϵ}_2}v,c_3{cosh}^{{ϵ}_1}u{cosh}^{{ϵ}_2}v{sinh}^{{ϵ}_3}w,c_4{cosh}^{{ϵ}_1}u{cosh}^{{ϵ}_2}v{cosh}^{{ϵ}_3}w\right)$$

(25)

Moreover, this satisfies the following equation:

$$-{\left({\displaystyle \frac{Y_1}{c_1}}\right)}^{\frac{2}{{\epsilon }_1}}+{\left\{-{\left({\displaystyle \frac{Y_2}{c_2}}\right)}^{\frac{2}{{\epsilon }_2}}+{\left\{-{\left({\displaystyle \frac{Y_3}{c_3}}\right)}^{\frac{2}{{\epsilon }_3}}+{\left({\displaystyle \frac{Y_4}{c_4}}\right)}^{\frac{2}{{\epsilon }_3}}\right\}}^{{\displaystyle \frac{{\epsilon }_3}{{\epsilon }_2}}}\right\}}^{\frac{{\epsilon }_2}{{\epsilon }_1}}=1.$$

Example 5. 

Let M be a superhyperboloid given with the parametrization (25) in ${\mathbb{E}}_1^4$. Taking $c_3=c_4$ and ${ϵ}_1={ϵ}_2={ϵ}_3=1$, the Gaussian curvature K and the mean curvature H are obtained by

$$K={\displaystyle \frac{c_1^3{c}_2^3{c}_3^6{cosh}^{10}ucoshv}{W\sqrt{{\left|W\right|}^3}}}$$

and

$$H={\displaystyle \frac{c_1{c}_2{c}_3^2{cosh}^{6}u{cosh}^{3}v}{3W\sqrt{\left|W\right|}}}\left\{{cosh}^{2}u{cosh}^{2}v\left(c_1^2{c}_2^2-c_3^4\right)+c_3^2\left(c_1^2+c_2^2\left({cosh}^{2}u{sinh}^{2}v-{sinh}^{2}u\right)\right)\right\},$$

respectively. In particular, choosing $c_1=2$, $c_2=-1$, $c_3=c_4=3$, and $w={\displaystyle \frac{\pi }{2}}$, the corresponding hypersurface’s projection in 3-dimensional space can be plotted by using Julia (see Figure 4).

Proposition 4. 

Let the curves α, β, and γ be defined as

$$\begin{array}{ccc}\hfill \alpha \left(u\right)& =& \left(c_1{sin}^{{ϵ}_1}u,R+{cos}^{{ϵ}_1}u\right),u\in \mathbb{R},\hfill \\ \hfill \beta \left(v\right)& =& \left(c_2{cos}^{{ϵ}_2}v,{sin}^{{ϵ}_2}v\right),\hfill \\ \hfill \gamma \left(w\right)& =& \left(c_3{cos}^{{ϵ}_3}w,c_4{sin}^{{ϵ}_3}w\right),\hfill \end{array}$$

where $c_1,c_2,c_3,c_4,R,{ϵ}_1,{ϵ}_2,{ϵ}_3$ are real constants. The spherical product of these three curves leads to the parametrization

$$F(u,v,w)=\left(c_1{sin}^{{ϵ}_1}u,c_2(R+{cos}^{{ϵ}_1}u){cos}^{{ϵ}_2}v,c_3(R+{cos}^{{ϵ}_1}u){sin}^{{ϵ}_2}v{cos}^{{ϵ}_3}w,c_4(R+{cos}^{{ϵ}_1}u){sin}^{{ϵ}_2}v{sin}^{{ϵ}_3}w\right).$$

This parametrization corresponds to a toroidal hypersurface in ${\mathbb{E}}^4$ and satisfies the Cartesian equation

$$\left[{\left({\displaystyle \frac{y_1}{c_1}}\right)}^{\frac{2}{{ϵ}_1}}+{\left[\left({\left({\displaystyle \frac{y_2}{c_2}}\right)}^{\frac{2}{{ϵ}_2}}+{\left({\left({\left({\displaystyle \frac{y_3}{c_3}}\right)}^{\frac{2}{{ϵ}_3}}+{\left({\displaystyle \frac{y_4}{c_4}}\right)}^{\frac{2}{{ϵ}_3}}\right)}^{{\displaystyle \frac{{ϵ}_3}{{ϵ}_2}}}\right)}^{\frac{{ϵ}_2}{2}}-R\right)\right]}^{\frac{2}{{ϵ}_1}}\right]=1$$

Example 6. 

Consider the toroidal hypersurface described by the parametrization $F(u,v,w)$ introduced above in ${\mathbb{E}}^4$. Let us choose the constants as $c_1=2, c_2=c_3=c_4=1$, ${ϵ}_1=3, {ϵ}_2={ϵ}_3=1$, $R=2$, and fix the parameter $w={\displaystyle \frac{\pi }{2}}$. Under these selections, a 3-dimensional projection of the corresponding hypersurface can be visualized using Julia (see Figure 5).

5. Generalized Spherical Product Hypersurfaces in Minkowski $\mathit{n}$-Space

In this part, the representation of spherical product hypersurfaces are generalized to the n-dimension.

The spherical product of two curves in ${\mathbb{E}}_{\mathbf{1}}^{\mathbf{3}}$:

Firstly, let us consider the plane curves $C_1$ and $C_2$ with the parametrizations $C_1=\left({\alpha }_1,{\alpha }_2\right)$ and $C_2=\left({\alpha }_3,{\alpha }_4\right)$. The spherical product of these two curves is

$$C_1\otimes C_2=\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right]\otimes \left[\begin{array}{c}{\alpha }_3\\ {\alpha }_4\end{array}\right].$$

Thus, the spherical product hypersurface generated by these curves is parametrized by

$$C_1\otimes C_2=\left({\alpha }_1,{\alpha }_2{\alpha }_3,{\alpha }_2{\alpha }_4\right).$$

The spherical product of three curves in ${\mathbb{E}}_{\mathbf{1}}^{\mathbf{4}}$:

Secondly, suppose that $C_1$, $C_2$, and $C_3$ are plane curves with the parametrizations $C_1=\left({\alpha }_1,{\alpha }_2\right)$, $C_2=\left({\alpha }_3,{\alpha }_4\right)$, and $C_3=\left({\alpha }_5,{\alpha }_6\right)$. The spherical product of these curves is

$$C_1\otimes C_2\otimes C_3=\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right]\otimes \left[\begin{array}{c}{\alpha }_3\\ {\alpha }_4\end{array}\right]\otimes \left[\begin{array}{c}{\alpha }_5\\ {\alpha }_6\end{array}\right].$$

Thus, the spherical product hypersurface generated by these curves has the following parametrization:

$$C_1\otimes C_2\otimes C_3=\left({\alpha }_1,{\alpha }_2{\alpha }_3,{\alpha }_2{\alpha }_4{\alpha }_5,{\alpha }_2{\alpha }_4{\alpha }_6\right).$$

The spherical product of four curves in ${\mathbb{E}}_{\mathbf{1}}^{\mathbf{5}}$:

Let $C_1$, $C_2$, $C_3$, and $C_4$ be plane curves with the parametrizations $C_1=\left({\alpha }_1,{\alpha }_2\right)$, $C_2=\left({\alpha }_3,{\alpha }_4\right)$, $C_3=\left({\alpha }_5,{\alpha }_6\right)$, and $C_4=\left({\alpha }_7,{\alpha }_8\right)$. The spherical product of these curves is

$$C_1\otimes C_2\otimes C_3\otimes C_4=\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right]\otimes \left[\begin{array}{c}{\alpha }_3\\ {\alpha }_4\end{array}\right]\otimes \left[\begin{array}{c}{\alpha }_5\\ {\alpha }_6\end{array}\right]\otimes \left[\begin{array}{c}{\alpha }_7\\ {\alpha }_8\end{array}\right].$$

Thus, the spherical product hypersurface generated by these curves has the following parametrization:

$$C_1\otimes C_2\otimes C_3\otimes C_4=\left({\alpha }_1,{\alpha }_2{\alpha }_3,{\alpha }_2{\alpha }_4{\alpha }_5,{\alpha }_2{\alpha }_4{\alpha }_6{\alpha }_7,{\alpha }_2{\alpha }_4{\alpha }_6{\alpha }_8\right).$$

By following the steps outlined above, we get the equation of a spherical product hypersurface in Minkowski n-space.

The spherical product of $\mathit{n}-\mathbf{1}$ curves in ${\mathbb{E}}_{\mathbf{1}}^{\mathit{n}}$:

Now, let us consider the curves $C_1,C_2,\dots ,C_{n-1}$ in the Minkowski plane, which are given by $C_1=\left({\alpha }_1,{\alpha }_2\right), C_2=\left({\alpha }_3,{\alpha }_4\right)$, $C_3=\left({\alpha }_5,{\alpha }_6\right)$, …$,C_{n-1}=\left({\alpha }_{2n-3},{\alpha }_{2n-2}\right)$. Then, a spherical product hypersurface in ${\mathbb{E}}_1^n$ is defined as $C_1\otimes C_2\otimes C_3\otimes \dots \otimes C_{n-1}$ and represented as

$$\begin{array}{ccc}& & C_1\otimes C_2\otimes C_3\otimes \dots \otimes C_{n-1}\hfill \\ & =& \left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right]\otimes \left[\begin{array}{c}{\alpha }_3\\ {\alpha }_4\end{array}\right]\otimes \left[\begin{array}{c}{\alpha }_5\\ {\alpha }_6\end{array}\right]\otimes \dots \otimes \left[\begin{array}{c}{\alpha }_{2n-3}\\ {\alpha }_{2n-2}\end{array}\right]\hfill \\ & =& {\alpha }_1{x}^1\left(p\right)+\sum _{i=2}^{n-1}\left(\prod _{j=1}^{i-1}{\alpha }_{2j}\right){\alpha }_{2i-1}{x}^i\left(p\right)+\left(\prod _{j=1}^{n-2}{\alpha }_{2j}\right){\alpha }_{2n-2}{x}^n\left(p\right),\hfill \end{array}$$

where $x^i\left(p\right)=e_i$, $i=1,\dots ,n$ are coordinate functions in ${\mathbb{E}}_1^n.$

6. Conclusions

The concept of embeddings of product spaces was first defined by Kuiper between Euclidean spaces. Then spherical product immersion, a special case of embeddings of product spaces, has been studied by many authors so far.

In this work, we have considered the concept of the spherical product to construct hypersurfaces in Minkowski space ${\mathbb{E}}_1^4$ and aimed to generalize such hypersurfaces to higher dimensions. Particularly, some classifications of spherical product hypersurfaces in ${\mathbb{E}}_1^4$ have been provided, focusing on flat and minimal cases. In addition, some hyperquadric examples have been presented, as well as their projections to 3-dimensional space. We have seen that there is a close relationship between hyperquadrics and the concept of the spherical product.

These results place spherical product hypersurfaces as a flexible and analyzable class within Minkowski geometry, suggesting applications in geometric modeling and mathematical physics where symmetry, causal character, and curvature constraints are paramount. Promising directions include classification by causal type, rigidity under prescribed curvature, and variational problems for spherical product data in higher dimensions. In addition, the spherical product contributes rotational or Lorentzian symmetries, while hyperquadrics obtained through spherical products inherit all symmetries of the associated quadratic form. Consequently, the curvature conditions derived in this paper—such as flatness, minimality, and causal character—are compatible with and, in some cases, dictated by these underlying symmetry groups.

From the viewpoint of general relativity, our timelike spherical product hypersurfaces in ${\mathbb{E}}_1^4$ may be interpreted as spacelike slices whose mean curvature H encodes the extrinsic geometry. In this setting, the minimal ($H=0$) and constant mean curvature cases are related to vacuum and constant mean curvature hypersurfaces used in spacetime models, while the flat condition (vanishing intrinsic curvature) corresponds to spatial slices with locally Euclidean geometry, compatible with embeddings inspired by Minkowski, Schwarzschild, or de Sitter spacetimes.