Towards a Differential Geometry of Superpolar Manifolds: Curvature, Singularities, and Geometric Structures

1. Introduction

If mathematics is truly to become the language of nature, also of the living, it might retrace the steps taken by Kepler, Galileo and Newton, beginning with a generalization of the conic sections—a program initiated by Gabriel Lamé in 1818 [1]. Lamé introduced supercircles (more generally superellipses) defined by the equation

$${\left|{\displaystyle \frac{x}{a}}\right|}^n+{\left|{\displaystyle \frac{y}{b}}\right|}^n=1,$$

(1)

of which the classical ellipses, circles, squares, and diamonds arise as the special cases. Originally intended to model the shape of crystals [1], superellipses have recently been found in many natural forms and phenomena [2,3,4,5,6,7]. It allows modeling, for example, tree rings of softwoods in two numbers only. Direct quantification of area through integration is straigthforward, but recently, also methods for computing the perimeter have been found [8,9].

The Gielis superformula [10] constitutes a still broader generalization of the conic sections: it extends Lamé’s superellipses to arbitrary rotational symmetries by allowing independent exponents and a symmetry parameter m, generating supershapes. In the natural sciences, this superformula has proved a remarkably versatile model for a wide range of biological forms. It is a generic geometric continuous transformation between both abstract and natural forms, their development and evolution.

Inspired by this, the notions of a superelliptic inner product [11,12] and of a star derivative [13] have been introduced. The superelliptic product allowed to define orthogonality adapted to the shape. For example, two arms of a starfish, spaced $2\pi /5$ apart, are orthogonal in the superelliptic sense, and this extends to any two adjacent radii in regular polygons. The star derivative allowed for the computation of the curvature in a superelliptic sense. As an immediate consequence, every superelliptic circle possesses constant superelliptic curvature. In this precise sense, all supershapes are reduced to circles. The apparent anisotropy of the shapes in Figure 1 and Figure 2 arises solely from their Euclidean embedding, but intrinsically they are circles and spheres, respectively. In addition, irregularities—regions of vanishing or infinite classical curvature—are absorbed into the underlying metric. The stretchable radius has constant length everywhere, so that the curves (and surfaces) generated by the superformula are simultaneously unit circles and totally osculating shapes [14]. The same geometric framework yields a natural link with quaternion algebras. By reinterpreting rotation in the superelliptic setting one recovers, for example, the classical logarithmic spiral as a circle also when the spiral is modified by the superformula [15].

The star derivative itself may be viewed as the composition of the ordinary differential operator with a radial scaling function; the separation of variables is therefore built into the calculus from the outset. The resulting differential geometry provides a coherent description of curvature, singularities, and geometric structures, and opens the way to a unified treatment of higher-dimensional analogs of the classical theory of curves and surfaces.

In the present paper, we show how both constructions extend to general manifolds. Although earlier work concentrated on the Gielis formula, the superelliptic inner product and the star derivative admit far wider application: any sufficiently smooth and periodic function can serve as the radial function. Because the adjective “superelliptic” has become too restrictive, we henceforth adopt the term superpolar. The term “polar” emphasizes that one begins from a distinguished center, and “superpolar” then allows this position vector to stretch according to a well-defined intrinsic or prescribed geometry.

In this article, we first construct the superpolar geometry itself, with examples of superpolar lines, planes, and spheres. Then we examine manifolds equipped with this structure, the associated Riemann tensor fields, and the geometry of superpolar submanifolds. In the final section, examples of conic submanifolds and spheres, rectifying submanifolds, and rectifying hypersurfaces are provided.

2. Foundations of Superpolar Geometry

Let

$$\mathbf{u}=(u_1,\dots ,u_n),\mathbf{v}=(v_1,\dots ,v_n)\in {\mathbb{R}}^n.$$

We define the superpolar inner product

$$g_{{\mathbb{S}}_{\mathcal{E}}}:{\mathbb{R}}^n\times {\mathbb{R}}^n⟶\mathbb{R}$$

by

$$\begin{array}{c}\hfill g_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{u},\mathbf{v})={\displaystyle \frac{1}{r{\left(\phi \right)}^2}}{\displaystyle \sum _{i=1}^n}{\eta }_i{u}_i{v}_i,\end{array}$$

(2)

where ${\eta }_i>0$ for $i=1,\dots ,n$.

Particular instances of $r\left(\phi \right)$ are superellipses, the superformula and the general superformula [16]

$$\begin{array}{c}\hfill r\left(\phi \right)=c\left(\phi \right){\left({\displaystyle \frac{1}{\alpha }}{\left|cos{\displaystyle \frac{m_1{f}_1\left(\phi \right)}{4}}\right|}^{n_2}+{\displaystyle \frac{1}{\beta }}{\left|sin{\displaystyle \frac{m_2{f}_2\left(\phi \right)}{4}}\right|}^{n_3}\right)}^{-1/n_1}\end{array}$$

(3)

where $\phi \in [-\pi ,\pi ],f_1\left(\phi \right),f_2\left(\phi \right),c\left(\phi \right)$ are continuous functions; $m_1,m_2,\alpha ,\beta ,n_1,n_2,n_3$ are real numbers, with $\alpha ,\beta ,n_1\ne 0$. The standard Gielis Superformula [10] is obtained for $f_1\left(\phi \right)=f_2\left(\phi \right)=\phi$ , moderating the function $c\left(\phi \right)$. If, in addition, $m_1=m_2=4$ and $n_1=n_2=n_3$, Lamé’s superellipses result. The denominator 4 is strictly unnecessary, since the same results can be simply obtained by changing the values $m_1$ and $m_2$, but it is used to preserve the connection to the original form of the Gielis Superformula and the special case of Lamé curves for $m=4$, in a rectangular coordinate system.

To ensure the metric coefficient is well-defined and non-degenerate, we require $c\left(\phi \right)>0$ and assume that the parameters $\alpha ,\beta ,m_1,m_2$ and functions $f_1\left(\phi \right),f_2\left(\phi \right)$ are chosen such that the trigonometric terms do not vanish simultaneously, thereby avoiding any singularities:

$${\displaystyle \frac{1}{\alpha }}{\left|cos{\displaystyle \frac{m_1{f}_1\left(\phi \right)}{4}}\right|}^{n_2}+{\displaystyle \frac{1}{\beta }}{\left|sin{\displaystyle \frac{m_2{f}_2\left(\phi \right)}{4}}\right|}^{n_3}>0,\forall \phi .$$

(4)

Since ${\eta }_i>0$ and $r\left(\phi \right)>0$, the bilinear form $g_{{\mathbb{S}}_{\mathcal{E}}}$ is symmetric and positive definite:

To show that the bilinear form $g_{{\mathbb{S}}_{\mathcal{E}}}$ determines a valid Riemannian metric, we need to prove that it is symmetric, bilinear, and positive definite on ${\mathbb{R}}^n$.

1.

Bilinearity: Let $\mathbf{u},\mathbf{v},\mathbf{w}\in {\mathbb{R}}^n$ and let $c_1,c_2\in \mathbb{R}$. By definition, we have:

$$\begin{array}{cc}\hfill g_{{\mathbb{S}}_{\mathcal{E}}}(c_1\mathbf{u}+c_2\mathbf{v},\mathbf{w})& ={\displaystyle \frac{1}{r{\left(\phi \right)}^2}}{\displaystyle \sum _{i=1}^n}{\eta }_i(c_1{u}_i+c_2{v}_i)w_i\hfill \\ \hfill & ={\displaystyle \frac{1}{r{\left(\phi \right)}^2}}\left[c_1{\displaystyle \sum _{i=1}^n}{\eta }_i{u}_i{w}_i+c_2{\displaystyle \sum _{i=1}^n}{\eta }_i{v}_i{w}_i\right]\hfill \\ \hfill & =c_1{g}_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{u},\mathbf{w})+c_2{g}_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{v},\mathbf{w}).\hfill \end{array}$$

(5)

Due to the linearity of the summation and scalar multiplication, linearity in the second argument is proven analogously. Thus, $g_{{\mathbb{S}}_{\mathcal{E}}}$ is bilinear.

2.

Symmetry: For any $\mathbf{u},\mathbf{v}\in {\mathbb{R}}^n$, since standard multiplication in $\mathbb{R}$ is commutative ($u_i{v}_i=v_i{u}_i$), we obtain:

$$\begin{array}{cc}\hfill g_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{u},\mathbf{v})& ={\displaystyle \frac{1}{r{\left(\phi \right)}^2}}{\displaystyle \sum _{i=1}^n}{\eta }_i{u}_i{v}_i={\displaystyle \frac{1}{r{\left(\phi \right)}^2}}{\displaystyle \sum _{i=1}^n}{\eta }_i{v}_i{u}_i=g_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{v},\mathbf{u}).\hfill \end{array}$$

(6)

Therefore, $g_{{\mathbb{S}}_{\mathcal{E}}}$ is a symmetric form.

3.

Positive Definiteness: For any $\mathbf{u}\in {\mathbb{R}}^n$, consider the evaluation of $g_{{\mathbb{S}}_{\mathcal{E}}}$ with itself:

$$\begin{array}{c}\hfill g_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{u},\mathbf{u})={\displaystyle \frac{1}{r{\left(\phi \right)}^2}}{\displaystyle \sum _{i=1}^n}{\eta }_i{\left(u_i\right)}^2.\end{array}$$

(7)

Since $c\left(\phi \right)>0$ and the trigonometric terms do not vanish simultaneously ($\sum >0$), the radius function satisfies $r\left(\phi \right)\ne 0$, which implies that ${\displaystyle \frac{1}{r{\left(\phi \right)}^2}}>0$. Furthermore, given that ${\eta }_i>0$ for all $i=1,\dots ,n$ and ${\left(u_i\right)}^2\ge 0$, every term in the sum is non-negative.

Thus, $g_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{u},\mathbf{u})\ge 0$, and equality holds if and only if $u_i=0$ for all $i=1,\dots ,n$ (i.e., $\mathbf{u}=\mathbf{0}$). Consequently, $g_{{\mathbb{S}}_{\mathcal{E}}}$ is positive definite.

Hence, it determines a Riemannian metric on ${\mathbb{R}}^n$. The resulting metric space is called the n-dimensional superpolar space and is denoted by

$${\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n=\left({\mathbb{R}}^n,g_{{\mathbb{S}}_{\mathcal{E}}}\right).$$

The metric tensor associated with $g_{{\mathbb{S}}_{\mathcal{E}}}$ is

$$G={\displaystyle \frac{1}{r{\left(\phi \right)}^2}}diag({\eta }_1,{\eta }_2,\dots ,{\eta }_n),$$

and

$$g_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{u},\mathbf{v})={\mathbf{u}}^{T}G\mathbf{v}.$$

The induced norm is given by

$${\parallel \mathbf{u}\parallel }_{{\mathbb{S}}_{\mathcal{E}}}=\sqrt{g_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{u},\mathbf{u})}.$$

Two vectors $\mathbf{u},\mathbf{v}\in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n$ are said to be superpolar orthogonal if

$$g_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{u},\mathbf{v})=0.$$

Let $\{e_1,\dots ,e_n\}$ be the standard basis of ${\mathbb{R}}^n$. The corresponding superpolar orthonormal basis

$$\{{\widehat{e}}_1,\dots ,{\widehat{e}}_n\}$$

is defined by

$${\widehat{e}}_i={\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_i}}}{e}_i,i=1,\dots ,n.$$

It follows immediately that

$$g_{{\mathbb{S}}_{\mathcal{E}}}({\widehat{e}}_i,{\widehat{e}}_j)={\delta }_{ij},1\le i,j\le n.$$

Therefore, every vector

$$\mathbf{u}\in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n$$

admits the representation;

$$\mathbf{u}={\displaystyle \sum _{i=1}^n}{u}_i{\widehat{e}}_i={\displaystyle \sum _{i=1}^n}{u}_i{\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_i}}}{e}_i.$$

This metric construction furnishes a natural geometric setting in which classical Euclidean geometry is generalized through a superpolar deformation governed by the parameters $r\left(\phi \right)$ and ${\left\{{\eta }_i\right\}}_{i=1}^n$. It is defined as the superpolar cross product of two vectors $\{\mathbf{u},\mathbf{v}\}$ in ${\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^3$, given by

$$\mathbf{u}{\times }_{{\mathbb{S}}_{\mathcal{E}}}\mathbf{v}={\Lambda }^{*}\left|\begin{array}{ccc}{\displaystyle \frac{r{\left(\phi \right)}^2{e}_1}{{\eta }_1}}& {\displaystyle \frac{r{\left(\phi \right)}^2{e}_2}{{\eta }_2}}& {\displaystyle \frac{r{\left(\phi \right)}^2{e}_3}{{\eta }_3}}\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right|,$$

(8)

where

$${\Lambda }^{*}={\displaystyle \frac{1}{r{\left(\phi \right)}^3}}\sqrt{{\eta }_1{\eta }_2{\eta }_3},{\eta }_1,{\eta }_2,{\eta }_3\in {\mathbb{R}}^{+}.$$

Determining the cosine of the angle between two non-zero superpolar vectors $\mathbf{u}$ and $\mathbf{v}$ is given by

$$cos\theta ={\displaystyle \frac{g_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{u},\mathbf{v})}{{∥\mathbf{u}∥}_{{\mathbb{S}}_{\mathcal{E}}}{∥\mathbf{v}∥}_{{\mathbb{S}}_{\mathcal{E}}}}}.$$

Consequently, the angular parameterization of a superellipse or a superellipsoid can be obtained through the parameter $\theta$, which serves as the intrinsic angular coordinate induced by the superpolar metric structure. In particular, the parameter $\theta$ generalizes the classical Euclidean polar and spherical angular variables, allowing the geometry of superpolar manifolds to be expressed in a smooth parametric form compatible with the underlying inner product structure. This formulation preserves the geometric interpretation of angles while extending the classical trigonometric framework to the superpolar setting [12].

The superpolar hypersphere of radius R and center $\mathbf{C}$ is the subset

$${\mathbb{S}}_{{\mathbb{S}}_{\mathcal{E}}}^{n-1}=\{\mathbf{x}\in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n{:\Vert \mathbf{x}-\mathbf{C}\Vert }_{{\mathbb{S}}_{\mathcal{E}}}=R\}$$

(9)

The following figures illustrate a sequence of superpolar circles and spheres obtained from the standard Gielis superformula. Figure 1 and Figure 2 illustrate the superpolar circles and spheres generated using different parameter sets of the standard Gielis Superformula. It can be observed that when $n_1=n_2=n_3=2$ the resulting superpolar circle and spheres is identical to the classical Euclidean circle and sphere, respectively.

Let $\mathbf{p}\in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n$ be a fixed point and let $\mathbf{v}\in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n$ be a non-zero direction vector. The superpolar line passing through $\mathbf{p}$ in the direction of $\mathbf{v}$ is defined as the set

$$\ell =\{\mathbf{x}\in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n:\mathbf{x}=\mathbf{p}+t\mathbf{v},t\in \mathbb{R}\}.$$

(10)

Here, t is a real parameter, and the expression $\mathbf{x}\left(t\right)=\mathbf{p}+t\mathbf{v}$ is called the superpolar parametric equation of the line with superpolar direction $\mathbf{v}$.

Figure 3 illustrates the superpolar lines generated using different parameter sets of the Gielis Superformula. It can be observed that when $n_1=n_2=n_3=2$ the resulting superpolar line is identical to the classical Euclidean straight line.

The set

$$\mathcal{P}=\{\mathbf{x}\in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n:g_{{\mathbb{S}}_{\mathcal{E}}}(\mathbf{x}-\mathbf{p},\mathbf{v})=0\}$$

(11)

is defined as superpolar hyperplane that is superpolar orthogonal to $\mathbf{v}$ and passes through $\mathbf{p}$.

Figure 4 illustrates the superpolar planes generated using different parameter sets of the Gielis Superformula. It can be observed that when $n_1=n_2=n_3=2$ the resulting superpolar plane is identical to the classical Euclidean plane.

For each point $p\in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n$, the superpolar tangent space at p is denoted by

$$T_p{\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n.$$

A superpolar smooth vector field on ${\mathbb{R}}^n$ is a smooth map

$$X:{\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n\to T{\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n$$

satisfying

$$X\left(p\right)\in T_p{\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n,\forall p\in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n.$$

The set of all smooth vector fields on ${\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n$ is denoted by

$${\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left({\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n\right).$$

With respect to the orthonormal frame

$$\{{\widehat{e}}_1={\displaystyle \frac{{\partial }_{*}}{{\partial }_{*}{x}_1}},{\widehat{e}}_2{\displaystyle \frac{{\partial }_{*}}{{\partial }_{*}{x}_2}},\dots ,{\widehat{e}}_n={\displaystyle \frac{{\partial }_{*}}{{\partial }_{*}{x}_n}}\},$$

every vector field

$$X=({\widehat{X}}_1,{\widehat{X}}_2,...,{\widehat{X}}_n)\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left({\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n\right)$$

can be uniquely represented as

$$X={\displaystyle \sum _{i=1}^n}{\widehat{X}}_i{\widehat{e}}_i={\displaystyle \sum _{i=1}^n}{X}_i{e}_i,$$

where $e_i={\displaystyle \frac{\partial }{{\partial }_{x_i}}}\in \chi \left({\mathbb{R}}^n\right)$, then we have

$$X_i={\widehat{X}}_i{\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_i}}}\in C^{\infty }({\mathbb{R}}^n,\mathbb{R})$$

are smooth functions.

Definition 1 

(Component Representation Operator). Let

$$E=\{{\widehat{e}}_1,{\widehat{e}}_2,\dots ,{\widehat{e}}_n\}$$

be a fixed ordered basis of ${\mathbb{R}}^n$. Define the linear operator

$$ϝ:{\mathbb{R}}^n⟶span\left(E\right)$$

by

$$ϝ(v_1,\dots ,v_n)={\displaystyle \sum _{i=1}^n}{v}_i{\widehat{e}}_i.$$

Since the basis E is fixed, the vectors ${\widehat{e}}_i$ are regarded constants; consequently, differentiation acts only on the scalar coordinate functions.

Definition 2 

(Star Derivative). Let $U\subseteq {\mathbb{R}}^m$ be open and let

$$f:U⟶span\left(E\right),f\left(x\right)={\displaystyle \sum _{i=1}^n}{f}_i\left(x\right){\widehat{e}}_i,$$

where $E=\{{\widehat{e}}_1,\dots ,{\widehat{e}}_n\}$ is a fixed basis. Thestar derivativeof f is defined componentwise by

$$f^{*}\left(x\right)={\displaystyle \sum _{i=1}^n}{f}_i^{\prime }\left(x\right){\widehat{e}}_i,$$

or equivalently,

$$f^{*}\left(x\right)=\left(f_1^{\prime }\left(x\right){\widehat{e}}_1,f_2^{\prime }\left(x\right){\widehat{e}}_2,\dots ,f_n^{\prime }\left(x\right){\widehat{e}}_n\right).$$

Equivalently,

$$f^{*}\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f\left(x\right)}{h}}.$$

Definition 3 

(Star Gradient). Let $U\subseteq {\mathbb{R}}^n$ be an open set and let

$$f:U⟶\mathbb{R}$$

be differentiable. Let

$$E=\{{\widehat{e}}_1,{\widehat{e}}_2,\dots ,{\widehat{e}}_n\}$$

be a fixed ordered basis of ${\mathbb{R}}^n$. Thestar gradientof f is defined by

$${\nabla }^{*}f=ϝ(\nabla f),$$

that is,

$${\nabla }^{*}f\left(x\right)={\displaystyle \sum _{i=1}^n}{\displaystyle \frac{\partial f}{\partial x_i}}\left(x\right){\widehat{e}}_i.$$

The framework of superpolar geometry provides a systematic mathematical approach for analyzing figures characterized by irregularities, sharp corners, asymmetry, and other non-smooth geometric features. This framework is particularly useful for extending conventional geometric analysis to shapes that cannot be adequately represented by standard smooth curves or regular geometric structures. As example of continuous functions $r\left(\phi \right)$, Figure 1 and Figure 2 present superpolar circles generated using the standard Gielis superformula, providing classical examples characterized by regularity, symmetry, and smoothly varying radial structures. In the same sense, the graphs in Figure 3 and Figure 4 are lines and planes in the superpolar sense. In contrast, Figure 5 and Figure 6 present geometries generated using the generalized formulation. The resulting geometries exhibit a gradual transition from regular and symmetric configurations to increasingly irregular, asymmetric, and highly spiked forms. They can all be considered as circles and spheres within the superpolar framework.

Table 1. Parameters used for the cases illustrated in Figure 5.

Case	c	d	${\mathit{m}}_1$	${\mathit{m}}_2$	${\mathit{n}}_1$	${\mathit{n}}_2$	${\mathit{n}}_3$	$\mathit{\alpha }$	$\mathit{\beta }$	${\mathit{f}}_1\left(\mathit{\phi }\right)$	${\mathit{f}}_2\left(\mathit{\phi }\right)$	$\mathit{c}\left(\mathit{\phi }\right)$
(a)	$-\pi$	$\pi$	30	30	$-21$	3	3	2	7	$3sin{\left[2\left|2\phi -{\displaystyle \frac{\pi }{3}}\right|\right]}^3$	$2cos{\left[3\left(\phi +{\displaystyle \frac{\pi }{2}}\right)\right]}^2$	1
(b)	$-\pi$	$\pi$	30	30	$-31$	3	3	2	7	$3sin{\left[2\left|\phi -{\displaystyle \frac{\pi }{3}}\right|\right]}^3$	$2cos{\left[3\left(\phi +{\displaystyle \frac{\pi }{2}}\right)\right]}^2$	1
(c)	$-\pi$	$\pi$	30	30	$-5$	3	3	2	7	$3sin{\left[2\left|\phi -{\displaystyle \frac{\pi }{3}}\right|\right]}^3$	$2cos{\left[3\left(\phi +{\displaystyle \frac{\pi }{2}}\right)\right]}^2$	1
(d)	$-\pi$	$\pi$	30	30	31	3	3	2	7	$3sin{\left[2\left|\phi -{\displaystyle \frac{\pi }{3}}\right|\right]}^3$	$2cos{\left[3\left(\phi +{\displaystyle \frac{\pi }{2}}\right)\right]}^2$	1
(e)	$-\pi$	$\pi$	30	30	5	3	3	2	7	$3sin{\left[2\left|\phi -{\displaystyle \frac{\pi }{3}}\right|\right]}^3$	$2cos{\left[3\left(\phi +{\displaystyle \frac{\pi }{2}}\right)\right]}^2$	1

Table 1. Parameters used for the cases illustrated in Figure 5.

Case	c	d	${\mathit{m}}_1$	${\mathit{m}}_2$	${\mathit{n}}_1$	${\mathit{n}}_2$	${\mathit{n}}_3$	$\mathit{\alpha }$	$\mathit{\beta }$	${\mathit{f}}_1\left(\mathit{\phi }\right)$	${\mathit{f}}_2\left(\mathit{\phi }\right)$	$\mathit{c}\left(\mathit{\phi }\right)$
(a)	$-\pi$	$\pi$	30	30	$-21$	3	3	2	7	$3sin{\left[2\left|2\phi -{\displaystyle \frac{\pi }{3}}\right|\right]}^3$	$2cos{\left[3\left(\phi +{\displaystyle \frac{\pi }{2}}\right)\right]}^2$	1
(b)	$-\pi$	$\pi$	30	30	$-31$	3	3	2	7	$3sin{\left[2\left|\phi -{\displaystyle \frac{\pi }{3}}\right|\right]}^3$	$2cos{\left[3\left(\phi +{\displaystyle \frac{\pi }{2}}\right)\right]}^2$	1
(c)	$-\pi$	$\pi$	30	30	$-5$	3	3	2	7	$3sin{\left[2\left|\phi -{\displaystyle \frac{\pi }{3}}\right|\right]}^3$	$2cos{\left[3\left(\phi +{\displaystyle \frac{\pi }{2}}\right)\right]}^2$	1
(d)	$-\pi$	$\pi$	30	30	31	3	3	2	7	$3sin{\left[2\left|\phi -{\displaystyle \frac{\pi }{3}}\right|\right]}^3$	$2cos{\left[3\left(\phi +{\displaystyle \frac{\pi }{2}}\right)\right]}^2$	1
(e)	$-\pi$	$\pi$	30	30	5	3	3	2	7	$3sin{\left[2\left|\phi -{\displaystyle \frac{\pi }{3}}\right|\right]}^3$	$2cos{\left[3\left(\phi +{\displaystyle \frac{\pi }{2}}\right)\right]}^2$	1

3. Superpolar Manifolds

In this section, we present a framework for manifolds constructed within the setting of superpolar geometry.

A superpolar Riemannian metric $g_{{\mathbb{S}}_{\mathcal{E}}}$ on superpolar manifold M is defined as a mapping $p\mapsto g_{{\mathbb{S}}_{\mathcal{E}}}\in \Omega \left(T_{{\mathbb{S}}_{\mathcal{E}}}M\right)$ for $p\in M$. Here, ${\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$ denotes the superpolar tangent space, and $\Omega \left({\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)\right)$ represents the superpolar dual space of ${\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$. The mapping $g_{{\mathbb{S}}_{\mathcal{E}}}$ is said to be a superpolar Riemannian metric if, for all $X,Y\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$, the following conditions hold:

$$g_{{\mathbb{S}}_{\mathcal{E}}}(X,Y)=g_{{\mathbb{S}}_{\mathcal{E}}}(Y,X),\mathrm{and}{g}_{{\mathbb{S}}_{\mathcal{E}}}(X,X)>0.$$

Under these conditions, the pair $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$ is called a superpolar Riemannian manifold.

Let $f\in C^{\infty }({\mathbb{R}}^n,\mathbb{R})$, then, for $X=(X_1,X_2,...,X_n)\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$, the star directional derivative on a superellitic Riemannian manifold is defined as follows:

$$X\left[f\right]={\displaystyle \sum _{i=1}^n}{\displaystyle \frac{{\eta }_i}{r{\left(\phi \right)}^2}}{X}_i{\displaystyle \frac{{\partial }_{*}f}{{\partial }_{*}{x}_i}}=g_{{\mathbb{S}}_{\mathcal{E}}}(X,{\nabla }^{*}f).$$

Example 1. 

Consider the scalar-valued function $f(x,y)=x^{2}y+y^3$ and $X=(x^2{\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_1}}},xy{\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_2}}})\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$. The superpolar gradient of the function is

$${\nabla }^{*}f(x,y)={\displaystyle \frac{\partial f}{\partial x}}{\widehat{e}}_1+{\displaystyle \frac{\partial f}{\partial y}}{\widehat{e}}_2.=(2xy{\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_1}}},(x^2+3y^2){\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_2}}}).$$

Thus the star directional derivative computed as

$$X\left[f\right]=g_{{\mathbb{S}}_{\mathcal{E}}}(X,{\nabla }^{*}f)=3x^{3}y+3x{y}^3.$$

The superpolar Lie bracket is defined by

$${[·,·]}_{{\mathbb{S}}_{\mathcal{E}}}:{\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left({\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n\right)\times {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left({\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n\right)\to {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left({\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n\right),$$

then the superpolar Lie bracket of the superpolar vector fields X and Y can be written explicitly as

$${[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}={\displaystyle \sum _{i,j=1}^n}(X_i\left[{\widehat{Y}}_j\right]-Y_j\left[{\widehat{X}}_i\right]){\widehat{e}}_i.$$

(12)

Moreover, the superpolar Lie bracket on the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$ is defined for a differentiable function f and vector fields $X,Y\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$ by

$${[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}\left(f\right)=(X\left[Y\left[f\right]\right]-Y\left[X\left[f\right]\right]).$$

(13)

With this bracket operation, the space ${\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$ acquires the structure of an infinite-dimensional Lie algebra over $\mathbb{R}$. In particular, the Lie bracket satisfies the superpolar Jacobi identity:

$${[{[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}},Z]}_{{\mathbb{S}}_{\mathcal{E}}}+{[{[Y,Z]}_{{\mathbb{S}}_{\mathcal{E}}},X]}_{{\mathbb{S}}_{\mathcal{E}}}+{[{[Z,X]}_{{\mathbb{S}}_{\mathcal{E}}},Y]}_{{\mathbb{S}}_{\mathcal{E}}}=0,$$

(14)

for all $X,Y,Z\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$. Furthermore, ${\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$ may be regarded as a module over the ring $\mathcal{F}\left(M\right)$ of smooth functions on M. Specifically, for $f\in \mathcal{F}\left(M\right)$ and $X\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$, the vector field $fX$ is defined pointwise by

$${\left(fX\right)}_p=f\left(p\right)X_p,p\in M.$$

With this module structure, the superpolar Lie bracket satisfies the following compatibility relation:

$${[fX,gY]}_{{\mathbb{S}}_{\mathcal{E}}}=fg{[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}+f\left(Xg\right)Y-g\left(Yf\right)X,$$

for all $f,g\in \mathcal{F}\left(M\right)$ and $X,Y\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$. A superpolar Lie algebra over $\mathbb{R}$ is a real vector space $\mathfrak{g}$ together with a bilinear map

$${[·,·]}_{{\mathbb{S}}_{\mathcal{E}}}:{\mathfrak{g}}_{{\mathbb{S}}_{\mathcal{E}}}\times {\mathfrak{g}}_{{\mathbb{S}}_{\mathcal{E}}}⟶{\mathfrak{g}}_{{\mathbb{S}}_{\mathcal{E}}},$$

called the superpolar Lie bracket, satisfying the following axioms for all $X,Y,Z\in {\mathfrak{g}}_{{\mathbb{S}}_{\mathcal{E}}}$:

1.

Skew-symmetry:

$${[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}=-{[Y,X]}_{{\mathbb{S}}_{\mathcal{E}}}.$$

2.

Jacobi identity:

$${[{[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}},Z]}_{{\mathbb{S}}_{\mathcal{E}}}+{[{[Y,Z]}_{{\mathbb{S}}_{\mathcal{E}}},X]}_{{\mathbb{S}}_{\mathcal{E}}}+{[[Z,X],Y]}_{{\mathbb{S}}_{\mathcal{E}}}=0.$$

The superpolar connection is a map

$$\mathfrak{D}:{\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left({\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n\right)\times {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left({\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n\right)\to {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left({\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n\right),(X,Y)\mapsto {\mathfrak{D}}_{X}Y.$$

The superpolar covariant derivative of the vector field $Y={\sum }_{i=1}^n{\widehat{Y}}_i{\widehat{e}}_i$ with respect to $X={\sum }_{i=1}^n{\widehat{X}}_i{\widehat{e}}_i$ is defined as follows:

$${\mathfrak{D}}_{X}Y={\displaystyle \sum _{i=1}^n}{g}_{{\mathbb{S}}_{\mathcal{E}}}({\widehat{X}}_i{\widehat{e}}_i,{\nabla }^{*}{\widehat{Y}}_i){\widehat{e}}_i.$$

Example 2. 

Consider the vector fields

$$X(x,y)=(x^2+y){\widehat{e}}_1+xy{\widehat{e}}_2=\left((x^2+y){\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_1}}},xy{\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_2}}}\right),Y(x,y)=y{\widehat{e}}_1+x{\widehat{e}}_2=\left(y{\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_1}}},x{\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_2}}}\right),$$

defined on ${\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^2$. The directional derivative of the vector field Y along the vector field X is defined by

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{X}Y& =X[Y_1,Y_2]=(X\left[Y_1\right],X\left[Y_2\right])=(g_{{\mathbb{S}}_{\mathcal{E}}}(X,{\nabla }^{*}{\widehat{Y}}_1){\widehat{e}}_1+\left(g_{{\mathbb{S}}_{\mathcal{E}}}(X,{\nabla }^{*}{\widehat{Y}}_2){\widehat{e}}_2\right)\hfill \\ \hfill & =xy{\widehat{e}}_1+(x^2+y){\widehat{e}}_2\hfill \\ \hfill & =(xy{\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_1}}},(x^2+y){\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_2}}})\hfill \end{array}$$

Definition 4. 

The mapping $g_{{\mathbb{S}}_{\mathcal{E}}}:M\to {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}$, $(X,Y)\mapsto g_{{\mathbb{S}}_{\mathcal{E}}}(X,Y)$ defined on the superpolar manifold M is called a superpolar metric tensor if it satisfies the following conditions for each $p\in M$:

1.

$g_{{\mathbb{S}}_{\mathcal{E}}}(X,Y)=g_{{\mathbb{S}}_{\mathcal{E}}}(Y,X)$ for all $X,Y\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$,

2.

$g_{{\mathbb{S}}_{\mathcal{E}}}(X,X)>0$ for all $X\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right),X\ne 0$,

3.

$g_{{\mathbb{S}}_{\mathcal{E}}}$ is a bilinear mapping,

where ${\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$ denotes the superpolar tangent space of the superpolar manifold M. Moreover, the following relation holds for the components of the superpolar metric tensor:

$$g_{{\mathbb{S}}_{\mathcal{E}}}(X,Y)={\displaystyle \sum _{k,l=1}^n}{g}_{{\mathbb{S}}_{\mathcal{E}}kl}{X}_k{Y}_l,$$

where $g_{{\mathbb{S}}_{\mathcal{E}}}({\widehat{e}}_k,{\widehat{e}}_l)=g_{{\mathbb{S}}_{\mathcal{E}}kl}$ represent the components of the superpolar metric tensor. Since $g_{{\mathbb{S}}_{\mathcal{E}}}$ is bilinear, it can be represented by a superpolar matrix, whose inverse is denoted by $g_{{\mathbb{S}}_{\mathcal{E}}}^{ij}$.

Definition 5. 

Let $\mathfrak{D}$ denote the superpolar covariant derivative on the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$. The operator $\mathfrak{D}$ is called a superpolar affine connection if it satisfies the following properties:

1.

${\mathfrak{D}}_{\lambda X+\mu Y}Z=\lambda {\mathfrak{D}}_{X}Z+\mu {\mathfrak{D}}_{Y}Z$,

2.

${\mathfrak{D}}_X(\lambda Y+\mu Z)=\lambda {\mathfrak{D}}_{X}Y+\mu {\mathfrak{D}}_{X}Z$,

3.

${\mathfrak{D}}_{fX}Y=f{\mathfrak{D}}_{X}Y$,

4.

${\mathfrak{D}}_{X}fY=f{\mathfrak{D}}_{X}Y+X\left[f\right]Y$,

where $\lambda ,\mu \in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}$, $X,Y,Z\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$, and f is a superpolar differentiable function.

Proposition 1. 

Let $\mathfrak{D}$ denote the superpolar affine connection on the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$. Then

(1)

${\displaystyle {\mathfrak{D}}_{{\widehat{e}}_i}Y={\displaystyle \sum _{k=1}^n}\left\{\frac{\partial {\widehat{Y}}_k}{\partial u_i}+{\displaystyle \sum _{j=1}^n}{\Gamma }_{ij}^k{\widehat{Y}}_j\right\}{\widehat{e}}_k}$,

(2)

${\displaystyle {\Gamma }_{kl}^r={\displaystyle \sum _{h=1}^n}\frac{g_{{\mathbb{S}}_{\mathcal{E}}}^{hr}}{2}\left\{\frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}lh}}{\partial u_k}+\frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}kh}}{\partial u_l}-\frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}kl}}{\partial u_h}\right\}}$,

where $Y={\sum }_{k=1}^n{\widehat{Y}}_k{\widehat{e}}_k$ and $\left(g_{{\mathbb{S}}_{\mathcal{E}}}^{hr}\right)$ is the inverse matrix of $\left(g_{{\mathbb{S}}_{\mathcal{E}}kh}\right)$.

Proof. 

Statement (1) is an immediate consequence of properties given in Definition 5. To prove (2), let us put $X={\widehat{e}}_h,Y={\widehat{e}}_k,Z={\widehat{e}}_l$ in the superpolar Koszul formula characterized by

$$\begin{array}{cc}\hfill 2g_{{\mathbb{S}}_{\mathcal{E}}}{({\mathfrak{D}}_{Y}Z,X)}_{{\mathbb{S}}_{\mathcal{E}}}=& Y{g}_{{\mathbb{S}}_{\mathcal{E}}}{(Z,X)}_{{\mathbb{S}}_{\mathcal{E}}}+Z{g}_{{\mathbb{S}}_{\mathcal{E}}}{(X,Y)}_{{\mathbb{S}}_{\mathcal{E}}}-X{g}_{{\mathbb{S}}_{\mathcal{E}}}{(Y,Z)}_{{\mathbb{S}}_{\mathcal{E}}}\hfill \\ \hfill & -g_{{\mathbb{S}}_{\mathcal{E}}}(Y,[Z,X])+g_{{\mathbb{S}}_{\mathcal{E}}}{(Z,{[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}})}_{{\mathbb{S}}_{\mathcal{E}}}+g_{{\mathbb{S}}_{\mathcal{E}}}(X,{[Y,Z]}_{{\mathbb{S}}_{\mathcal{E}}}).\hfill \end{array}$$

(15)

Since the brackets are zero, it leaves

$$2g_{{\mathbb{S}}_{\mathcal{E}}}({\mathfrak{D}}_{{\widehat{e}}_k}{\widehat{e}}_l,{\widehat{e}}_h)={\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}lh}}{\partial u_k}}+{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}kh}}{\partial u_l}}-{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}kl}}{\partial u_h}}.$$

But from the definition of Christoffel symbols we have

$$2g_{{\mathbb{S}}_{\mathcal{E}}}({\mathfrak{D}}_{{\widehat{e}}_k}{\widehat{e}}_l,{\widehat{e}}_h)=2{\displaystyle \sum _{k=1}^n}{\Gamma }_{ij}^k{g}_{kt}.$$

Attacking both equations with ${\sum }_h{g}_{{\mathbb{S}}_{\mathcal{E}}}^{hr}$ yields the required formula. □

Definition 6. 

The components of the superpolar affine connection $\mathfrak{D}$ on the superpolar Riemannian manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$ are given by

$${\Gamma }_{kl}^r={\displaystyle \frac{1}{2}}{\displaystyle \sum _{h=1}^n}{g}_{{\mathbb{S}}_{\mathcal{E}}}^{hr}[{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}lh}}{\partial u_k}}+{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}kh}}{\partial u_l}}-{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}kl}}{\partial u_h}}].$$

Here, ${\Gamma }_{kl}^r$ are called to as the superpolar Christoffel symbols. Moreover, if ${\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}ij}}{\partial u_t}}=g_{{\mathbb{S}}_{\mathcal{E}}ij,t}$, the expression can be rewritten as:

$${\Gamma }_{kl}^r={\displaystyle \frac{1}{2}}{\displaystyle \sum _{h=1}^n}{g}_{{\mathbb{S}}_{\mathcal{E}}}^{hr}(g_{{\mathbb{S}}_{\mathcal{E}}lh,k}+g_{{\mathbb{S}}_{\mathcal{E}}hk,l}-g_{{\mathbb{S}}_{\mathcal{E}}kl,h}).$$

(16)

In particular, if one selects $X={\widehat{e}}_k$ and $Y={\widehat{e}}_l$, then the following relation holds:

$${\mathfrak{D}}_{{\widehat{e}}_k}{\widehat{e}}_l={\displaystyle \sum _{m=1}^n}{\Gamma }_{kl}^m{\widehat{e}}_m.$$

Theorem 1. 

Let $\mathfrak{D}$ be a superpolar affine connection on the superpolar Riemannian manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$. Then, for $X,Y\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$, the following expression holds:

$${\mathfrak{D}}_{X}Y={\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k=1}^n}{\widehat{X}}_k{\widehat{e}}_k\left[{\widehat{Y}}_m\right]+{\displaystyle \sum _{k,l=1}^n}{\widehat{X}}_k{\widehat{Y}}_l{\Gamma }_{kl}^m\right){\widehat{e}}_m.$$

(17)

Proof. Let $X={\sum }_{k=1}^n{\widehat{X}}_k{\widehat{e}}_k$ and $Y={\sum }_{l=1}^n{\widehat{Y}}_l{\widehat{e}}_l$. The superpolar covariant derivative of the superpolar vector field Y in the direction of X is computed as follows:

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{X}Y& ={\mathfrak{D}}_{{\sum }_{k=1}^n{\widehat{X}}_k{\widehat{e}}_k}Y\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^n}{\widehat{X}}_k{\mathfrak{D}}_{{\widehat{e}}_k}\left({\displaystyle \sum _{l=1}^n}{\widehat{Y}}_l{\widehat{e}}_l\right)\hfill \\ \hfill & ={\displaystyle \sum _{k,l=1}^n}{\widehat{X}}_k({\widehat{e}}_k\left[{\widehat{Y}}_l\right]{\widehat{e}}_l+{\widehat{Y}}_l{\mathfrak{D}}_{{\widehat{e}}_k}{\widehat{e}}_l)\hfill \\ \hfill & ={\displaystyle \sum _{k,l=1}^n}{\widehat{X}}_k\left({\widehat{e}}_k\left[{\widehat{Y}}_l\right]{\widehat{e}}_l+{\widehat{Y}}_l{\displaystyle \sum _{m=1}^n}{\Gamma }_{kl}^m{\widehat{e}}_m\right)\hfill \\ \hfill & ={\displaystyle \sum _{l=1}^n}\left({\displaystyle \sum _{k=1}^n}{\widehat{X}}_k{\widehat{e}}_k\left[{\widehat{Y}}_l\right]\right){\widehat{e}}_l+{\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k,l=1}^n}{\widehat{X}}_k{\widehat{Y}}_l{\Gamma }_{kl}^m\right){\widehat{e}}_m.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k=1}^n}{\widehat{X}}_k{\widehat{e}}_k\left[{\widehat{Y}}_m\right]\right){\widehat{e}}_m+{\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k,l=1}^n}{\widehat{X}}_k{\widehat{Y}}_l{\Gamma }_{kl}^m\right){\widehat{e}}_m\hfill \\ \hfill & ={\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k=1}^n}{\widehat{X}}_k{\widehat{e}}_k\left[{\widehat{Y}}_m\right]+{\displaystyle \sum _{k,l=1}^n}{\widehat{X}}_k{\widehat{Y}}_l{\Gamma }_{kl}^m\right){\widehat{e}}_m.\hfill \end{array}$$

Theorem 2. 

The superpolar affine connection $\mathfrak{D}$ on the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$ is symmetric if and only if ${\Gamma }_{kl}^m={\Gamma }_{lk}^m$.

Proof. Let

$$X={\displaystyle \sum _{k=1}^n}{X}_k{\widehat{e}}_k,Y={\displaystyle \sum _{l=1}^n}{Y}_l{\widehat{e}}_l.$$

Using Equation (17), we obtain

$${\mathfrak{D}}_{X}Y={\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k=1}^n}{X}_k{\widehat{e}}_k\left[Y_m\right]+{\displaystyle \sum _{k,l=1}^n}{X}_k{Y}_l{\Gamma }_{kl}^m\right){\widehat{e}}_m$$

(18)

and similarly,

$${\mathfrak{D}}_{Y}X={\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k=1}^n}{Y}_k{\widehat{e}}_k\left[X_m\right]+{\displaystyle \sum _{k,l=1}^n}{Y}_k{X}_l{\Gamma }_{kl}^m\right){\widehat{e}}_m.$$

(19)

By combining Equations (18) and (19), we derive

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{X}Y-{\mathfrak{D}}_{Y}X& ={\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k=1}^n}{X}_k{\widehat{e}}_k\left[Y_m\right]-{\displaystyle \sum _{l=1}^n}{Y}_l{\widehat{e}}_l\left[X_m\right]\right){\widehat{e}}_m+{\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k,l=1}^n}{X}_k{Y}_l({\Gamma }_{kl}^m-{\Gamma }_{lk}^m)\right){\widehat{e}}_m.\hfill \end{array}$$

Taking Equation (12) into account, this expression can be rewritten as

$${\mathfrak{D}}_{X}Y-{\mathfrak{D}}_{Y}X={[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}+{\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k,l=1}^n}{X}_k{Y}_l({\Gamma }_{kl}^m-{\Gamma }_{lk}^m)\right){\widehat{e}}_m.$$

Suppose that the superpolar affine connection $\mathfrak{D}$ is symmetric, that is, ${[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}={\mathfrak{D}}_{X}Y-{\mathfrak{D}}_{Y}X$. Then we obtain

$${\displaystyle \sum _{m=1}^n}\left({\displaystyle \sum _{k,l=1}^n}{X}_k{Y}_l({\Gamma }_{kl}^m-{\Gamma }_{lk}^m)\right){\widehat{e}}_m=0.$$

From this, it immediately follows that

$${\Gamma }_{kl}^m={\Gamma }_{lk}^m.$$

Conversely, if we assume ${\Gamma }_{kl}^m={\Gamma }_{lk}^m$, then we arrive at

$${[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}={\mathfrak{D}}_{X}Y-{\mathfrak{D}}_{Y}X.$$

4. Superpolar Riemann Tensor Fields

Definition 7. 

Let $\mathfrak{D}$ be a superpolar affine connection on the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$. The mapping $\mathfrak{T}$, defined for $X,Y\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$ by

$$\begin{array}{cc}\hfill \mathfrak{T}:{\varphi }_{*}\left(M\right)\times {\varphi }_{*}\left(M\right)& \to {\varphi }_{*}\left(M\right)\hfill \\ \hfill (X,Y)& \to \mathfrak{T}(X,Y)={\mathfrak{D}}_{X}Y-{\mathfrak{D}}_{Y}X-{[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}},\hfill \end{array}$$

(20)

is called the superpolar torsion of $\mathfrak{D}$, or the superpolar torsion tensor field.

Theorem 3. 

Let $\mathfrak{T}$be a superpolar tensor field on the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$. Then $\mathfrak{T}$ satisfies the following properties:

1.

$\mathfrak{T}(X,Y)=-\mathfrak{T}(Y,X)$,

2.

$\mathfrak{T}(fX,Y)=\mathfrak{T}(X,fY)=f\mathfrak{T}(X,Y)$.

Proof. 

From the definition of the superpolar torsion tensor field $\mathfrak{T}$ given in Equation (20), we obtain

$$\begin{array}{cc}\hfill \mathfrak{T}(X,Y)& =({\mathfrak{D}}_{X}Y-{\mathfrak{D}}_{Y}X-{[Y,X]}_{{\mathbb{S}}_{\mathcal{E}}})\hfill \\ \hfill & =-({\mathfrak{D}}_{X}Y-{[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}-{\mathfrak{D}}_{Y}X)\hfill \\ \hfill & =-\mathfrak{T}(Y,X).\hfill \end{array}$$

This establishes the first assertion of the theorem. For the second part, by using the defining properties of the superpolar affine connection given in Definition 5 together with Equation (12), we obtain

$$\begin{array}{cc}\hfill \mathfrak{T}(fX,Y)& ={\mathfrak{D}}_{fX}Y-{\mathfrak{D}}_Y\left(fX\right)-[fX,Y]\hfill \\ \hfill & =f{\mathfrak{D}}_{X}Y-f{\mathfrak{D}}_{Y}X-Y\left[f\right]X+Y\left[f\right]X\hfill \\ \hfill & -f{[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}\hfill \\ \hfill & =f({\mathfrak{D}}_{X}Y-{\mathfrak{D}}_{Y}X-{[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}})\hfill \\ \hfill & =f\mathfrak{T}(X,Y).\hfill \end{array}$$

Moreover, by invoking the first part of the theorem, we derive

$$\begin{array}{cc}\hfill \mathfrak{T}(X,fY)& =-\mathfrak{T}(fY,X)\hfill \\ \hfill & =-f\mathfrak{T}(Y,X)\hfill \\ \hfill & =f\mathfrak{T}(X,Y).\hfill \end{array}$$

□

Definition 8. 

Let $\mathfrak{D}$ be a superpolar affine connection on the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$. If $\mathfrak{T}=0$, then $\mathfrak{D}$ is said to be torsion-free (or symmetric), and the following identity holds:

$${\mathfrak{D}}_{X}Y-{\mathfrak{D}}_{Y}X={[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}.$$

The unique torsion-free connection compatible with $g_{{\mathbb{S}}_{\mathcal{E}}}$ is called the superpolar Levi–Civita connection and is characterized by:

$${\mathfrak{D}}_{X}Y-{\mathfrak{D}}_{Y}X={[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}},$$

and satisfies

$$X\left(g_{{\mathbb{S}}_{\mathcal{E}}}(Y,Z)\right)=g_{{\mathbb{S}}_{\mathcal{E}}}({\mathfrak{D}}_{X}Y,Z)+g_{{\mathbb{S}}_{\mathcal{E}}}(Y,{\mathfrak{D}}_{X}Z).$$

Definition 9. 

Let $\mathfrak{D}$ be a superpolar affine connection on the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$. If the following condition is satisfied, namely ${\mathfrak{D}}_X{g}_{{\mathbb{S}}_{\mathcal{E}}}=0$, then $\mathfrak{D}$ is said to be compatible with the metric $g_{{\mathbb{S}}_{\mathcal{E}}}$. For $X,Y,Z\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$, this condition is expressed as

$${\mathfrak{D}}_X{g}_{{\mathbb{S}}_{\mathcal{E}}}(X,Y)=X\left[g_{{\mathbb{S}}_{\mathcal{E}}}(Y,Z)\right]-g_{{\mathbb{S}}_{\mathcal{E}}}({\mathfrak{D}}_{X}Y,Z)-g_{{\mathbb{S}}_{\mathcal{E}}}(Y,{\mathfrak{D}}_{X}Z).$$

Definition 10. 

Let $\mathfrak{D}$ be a superpolar affine connection on the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$. If $\mathfrak{D}$ is both symmetric and compatible with $g_{{\mathbb{S}}_{\mathcal{E}}}$, then it is called the superpolar Levi-Civita connection, or equivalently, the superpolar Riemannian connection.

Example 3. 

Consider a two-dimensional superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$ with the parametrization φ, where $R\in {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}$ and ϕ represents a superpolar angle:

$$\phi (R,\varphi )=Rcos\varphi {\widehat{e}}_1+Rsin\varphi {\widehat{e}}_2.$$

We now determine the components of the superpolar metric tensor $g_{{\mathbb{S}}_{\mathcal{E}}}$ and the associated superpolar Christoffel symbols ${\Gamma }_{kl}^m$. The superpolar partial derivatives of φ with respect to R and ϕ are computed as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \phi }{\partial R}}& ={\widehat{e}}_R=cos\varphi {\widehat{e}}_1+sin\varphi {\widehat{e}}_2,\hfill \\ \hfill {\displaystyle \frac{\partial \phi }{\partial \varphi }}& ={\widehat{e}}_{\varphi }=-Rsin\varphi {\widehat{e}}_1+Rcos\varphi {\widehat{e}}_2.\hfill \end{array}$$

Hence, the components of the superpolar metric tensor are obtained as

$$\begin{array}{cc}\hfill g_{{\mathbb{S}}_{\mathcal{E}}11}& =g_{{\mathbb{S}}_{\mathcal{E}}}{({\widehat{e}}_R,{\widehat{e}}_R)}_{{\mathbb{S}}_{\mathcal{E}}}=1\hfill \\ \hfill g_{{\mathbb{S}}_{\mathcal{E}}12}& =g_{{\mathbb{S}}_{\mathcal{E}}}{({\widehat{e}}_R,{\widehat{e}}_{\varphi })}_{{\mathbb{S}}_{\mathcal{E}}}=0\hfill \\ \hfill g_{{\mathbb{S}}_{\mathcal{E}}22}& =g_{{\mathbb{S}}_{\mathcal{E}}}{({\widehat{e}}_{\varphi },{\widehat{e}}_{\varphi })}_{{\mathbb{S}}_{\mathcal{E}}}=R^2.\hfill \end{array}$$

Since $g_{{\mathbb{S}}_{\mathcal{E}}}$ is symmetric, we also have $g_{{\mathbb{S}}_{\mathcal{E}}12}=g_{{\mathbb{S}}_{\mathcal{E}}21}=0$. Accordingly, the superpolar matrix corresponding to the components of $g_{{\mathbb{S}}_{\mathcal{E}}}$ is given by

$$\left[g_{{\mathbb{S}}_{\mathcal{E}}R\varphi }\right]=\left[\begin{array}{cc}1& 0\\ 0& R^2\end{array}\right].$$

Furthermore, the superpolar determinant of this matrix is computed as follows:

$$\begin{array}{c}\hfill det\left[g_{{\mathbb{S}}_{\mathcal{E}}R\varphi }\right]=R^2.\end{array}$$

Therefore, the inverse of the superpolar matrix $g_{{\mathbb{S}}_{\mathcal{E}}r\varphi }$ is obtained as

$$\begin{array}{c}\hfill \left[g_{{\mathbb{S}}_{\mathcal{E}}}^{R\varphi }\right]=\left[\begin{array}{cc}1& 0\\ 0& 1/R^2\end{array}\right].\end{array}$$

We now compute the superpolar Christoffel symbols using the metric coefficients:

$$\begin{array}{cc}\hfill {\Gamma }_{11}^1& ={\displaystyle \frac{1}{2}}{\displaystyle \sum _{h=1}^2}{g}_{{\mathbb{S}}_{\mathcal{E}}}^{h1}(g_{{\mathbb{S}}_{\mathcal{E}}1h,1}+g_{{\mathbb{S}}_{\mathcal{E}}h1,1}-g_{{\mathbb{S}}_{\mathcal{E}}11,h})\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}[g_{{\mathbb{S}}_{\mathcal{E}}}^{11}\left({\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}11}}{\partial R}}\right)+g_{{\mathbb{S}}_{\mathcal{E}}}^{21}({\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}12}}{\partial R}}+{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}21}}{\partial R}}-{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}11}}{\partial \varphi }})]\hfill \\ \hfill & =0\hfill \\ \hfill {\Gamma }_{12}^1& ={\displaystyle \frac{1}{2}}{\displaystyle \sum _{h=1}^2}{g}_{{\mathbb{S}}_{\mathcal{E}}}^{h1}(g_{{\mathbb{S}}_{\mathcal{E}}2h,1}+g_{{\mathbb{S}}_{\mathcal{E}}h1,2}-g_{{\mathbb{S}}_{\mathcal{E}}12,h})\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}[g_{{\mathbb{S}}_{\mathcal{E}}}^{11}({\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}21}}{\partial R}}+{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}21}}{\partial \varphi }}-{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}12}}{\partial R}})+g_{{\mathbb{S}}_{\mathcal{E}}}^{12}({\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}22}}{\partial R}}+{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}21}}{\partial \varphi }}-{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}12}}{\partial \varphi }})]\hfill \\ \hfill & =0\hfill \\ \hfill {\Gamma }_{22}^1& ={\displaystyle \frac{1}{2}}{\displaystyle \sum _{h=1}^2}{g}_{{\mathbb{S}}_{\mathcal{E}}}^{h1}(g_{{\mathbb{S}}_{\mathcal{E}}2h,2}+g_{{\mathbb{S}}_{\mathcal{E}}h2,2}-g_{{\mathbb{S}}_{\mathcal{E}}22,h})\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}[g_{{\mathbb{S}}_{\mathcal{E}}}^{11}({\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}21}}{\partial \varphi }}+{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}12}}{\partial \varphi }}-{\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}22}}{\partial R}})+g_{{\mathbb{S}}_{\mathcal{E}}}^{21}\left({\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}22}}{\partial \varphi }}\right)]\hfill \\ \hfill & =-R.\hfill \end{array}$$

In a similar manner, one obtains

$${\Gamma }_{11}^2={\Gamma }_{22}^2=0\mathrm{and}{\Gamma }_{21}^2={\Gamma }_{12}^2=1/R.$$

We now compute the values of the superpolar Levi-Civita connection with respect to the superpolar basis vectors:

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{{\widehat{e}}_R}{\widehat{e}}_R& ={\displaystyle \sum _{k=1}^2}{\Gamma }_{11}^k{\widehat{e}}_k={\Gamma }_{11}^1{\widehat{e}}_1+{\Gamma }_{11}^2{\widehat{e}}_2\hfill \\ \hfill & =0,\hfill \\ \hfill {\mathfrak{D}}_{{\widehat{e}}_{\varphi }}{\widehat{e}}_{\varphi }& ={\displaystyle \sum _{k=1}^2}{\Gamma }_{22}^k{\widehat{e}}_k={\Gamma }_{22}^1{\widehat{e}}_1+{\Gamma }_{22}^2{\widehat{e}}_2\hfill \\ \hfill & =-R·{\widehat{e}}_R.\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{{\widehat{e}}_{\varphi }}{\widehat{e}}_R& ={\displaystyle \sum _{k=1}^2}{\Gamma }_{21}^k{\widehat{e}}_k={\Gamma }_{21}^1{\widehat{e}}_1+{\Gamma }_{21}^2{\widehat{e}}_2\hfill \\ \hfill & ={\displaystyle \frac{1}{R}}{\widehat{e}}_{\varphi }.\hfill \end{array}$$

where ${\widehat{e}}_1={\widehat{e}}_R$ and ${\widehat{e}}_2={\widehat{e}}_{\varphi }$.

Definition 11. 

Let $\mathfrak{D}$ be the superpolar Levi-Civita connection on the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$. The superpolar curvature tensor is defined as the mapping for all $X,Y,Z\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$ by

$$R_{{\mathbb{S}}_{\mathcal{E}}}(X,Y)Z={\mathfrak{D}}_X{\mathfrak{D}}_{Y}Z-{\mathfrak{D}}_Y{\mathfrak{D}}_{X}Z-{\mathfrak{D}}_{{[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}}Z.$$

Here, $\mathfrak{D}$ denotes the superpolar Riemannian connection on the superpolar Riemann manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$, and it defines a mapping $(X,Y)\mapsto {\mathfrak{D}}_{X}Y$ for all $X,Y\in {\chi }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$. In addition, the components $R_{{\mathbb{S}}_{\mathcal{E}}}{jkl}^i:U\subset M\to {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}$ are defined by choosing $X={\widehat{e}}_k$, $Y={\widehat{e}}_l$, and $Z={\widehat{e}}_j$ as

$$R_{{\mathbb{S}}_{\mathcal{E}}}({\widehat{e}}_k,{\widehat{e}}_l){\widehat{e}}_j={\displaystyle \sum _{i=1}^n}{R}_{{\mathbb{S}}_{\mathcal{E}}}{jkl}^i{\widehat{e}}_i.$$

(21)

Theorem 4. 

The superpolar Riemann curvature tensor of the superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$ has the following component expression:

$$R_{{\mathbb{S}}_{\mathcal{E}}}{jkl}^i={\displaystyle \frac{\partial {\Gamma }_{lj}^i}{\partial u_k}}-{\displaystyle \frac{\partial {\Gamma }_{kj}^i}{\partial u_l}}+{\displaystyle \sum _{m=1}^n}({\Gamma }_{km}^i{\Gamma }_{lj}^m-{\Gamma }_{kj}^m{\Gamma }_{lm}^i).$$

(22)

Proof. 

Using Equation (17), we compute

$$\begin{array}{cc}\hfill R_{{\mathbb{S}}_{\mathcal{E}}}{jkl}^i& =R_{{\mathbb{S}}_{\mathcal{E}}}({\widehat{e}}_k,{\widehat{e}}_l){\widehat{e}}_j\hfill \\ \hfill & ={\mathfrak{D}}_{{\widehat{e}}_k}\left({\mathfrak{D}}_{{\widehat{e}}_l}{\widehat{e}}_j\right)-{\mathfrak{D}}_{{\widehat{e}}_l}\left({\mathfrak{D}}_{{\widehat{e}}_k}{\widehat{e}}_j\right)-{\mathfrak{D}}_{[}{\widehat{e}}_k,{\widehat{e}}_l{]}_{{\mathbb{S}}_{\mathcal{E}}}{\widehat{e}}_j.\hfill \end{array}$$

Since ${[{\widehat{e}}_k,{\widehat{e}}_l]}_{{\mathbb{S}}_{\mathcal{E}}}=0$, this reduces to

$$\begin{array}{cc}\hfill R_{{\mathbb{S}}_{\mathcal{E}}}{jkl}^i& ={\mathfrak{D}}_{{\widehat{e}}_k}\left({\displaystyle \sum _{m=1}^n}{\Gamma }_{lj}^m{\widehat{e}}_m\right)-{\mathfrak{D}}_{{\widehat{e}}_l}\left({\displaystyle \sum _{m=1}^n}{\Gamma }_{kj}^m{\widehat{e}}_m\right).\hfill \end{array}$$

Expanding the covariant derivatives, we obtain

$$\begin{array}{cc}\hfill R_{{\mathbb{S}}_{\mathcal{E}}}{jkl}^i& ={\displaystyle \sum _{m=1}^n}\left(\left({\displaystyle \frac{\partial {\Gamma }_{lj}^m}{\partial u_k}}\right){\widehat{e}}_m+{\Gamma }_{lj}^m{\mathfrak{D}}_{{\widehat{e}}_k}{\widehat{e}}_m\right)-{\displaystyle \sum _{m=1}^n}\left(\left({\displaystyle \frac{\partial {\Gamma }_{kj}^m}{\partial u_l}}\right){\widehat{e}}_m+{\Gamma }_{kj}^m{\mathfrak{D}}_{{\widehat{e}}_l}{\widehat{e}}_m\right).\hfill \end{array}$$

Substituting ${\mathfrak{D}}_{{\widehat{e}}_k}{\widehat{e}}_m={\sum }_{r=1}^n{\Gamma }_{km}^r{\widehat{e}}_r$ and ${\mathfrak{D}}_{{\widehat{e}}_l}{\widehat{e}}_m={\sum }_{r=1}^n{\Gamma }_{lm}^r{\widehat{e}}_r$ yields

$$\begin{array}{cc}\hfill R_{{\mathbb{S}}_{\mathcal{E}}}{jkl}^i& ={\displaystyle \sum _{m=1}^n}\left({\displaystyle \frac{\partial {\Gamma }_{lj}^m}{\partial u_k}}\right){\widehat{e}}_m-{\displaystyle \sum _{m=1}^n}\left({\displaystyle \frac{\partial {\Gamma }_{kj}^m}{\partial u_l}}\right){\widehat{e}}_m+{\displaystyle \sum _{m=1}^n}{\displaystyle \sum _{r=1}^n}{\Gamma }_{lj}^m{\Gamma }_{km}^r{\widehat{e}}_r-{\displaystyle \sum _{m=1}^n}{\displaystyle \sum _{i=1}^n}{\Gamma }_{kj}^m{\Gamma }_{lm}^i{\widehat{e}}_i.\hfill \end{array}$$

Collecting coefficients of ${\widehat{e}}_i$ completes the derivation.

$$\begin{array}{cc}\hfill & ={\displaystyle \sum _{i=1}^n}\left({\displaystyle \frac{\partial {\Gamma }_{lj}^i}{\partial u_k}}-{\displaystyle \frac{\partial {\Gamma }_{kj}^i}{\partial u_l}}+{\displaystyle \sum _{m=1}^n}({\Gamma }_{km}^i{\Gamma }_{lj}^m-{\Gamma }_{kj}^m{\Gamma }_{lm}^i)\right){\widehat{e}}_i\hfill \\ \hfill & ={\displaystyle \frac{\partial {\Gamma }_{lj}^i}{\partial u_k}}-{\displaystyle \frac{\partial {\Gamma }_{kj}^i}{\partial u_l}}+{\displaystyle \sum _{m=1}^n}({\Gamma }_{km}^i{\Gamma }_{lj}^m-{\Gamma }_{kj}^m{\Gamma }_{lm}^i).\hfill \end{array}$$

□

Definition 12. 

A superpolar manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}})$ is said to be superpolar flat if $R_{{\mathbb{S}}_{\mathcal{E}}}=0$, where $R_{{\mathbb{S}}_{\mathcal{E}}}$ denotes the superpolar Riemann curvature tensor of the manifold.

Theorem 5. 

The superpolar Euclidean space ${\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n$ is a superpolar flat manifold; that is, its superpolar Riemann curvature tensor satisfies $R_{{\mathbb{S}}_{\mathcal{E}}}=0$.

Proof. 

The superpolar Euclidean space ${\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^n$ is equipped with a constant superpolar Euclidean metric, which is invariant at every point of the space. Considering Equation (16), we have

$${\Gamma }_{kl}^r={\displaystyle \frac{1}{2}}{\displaystyle \sum _{h=1}^n}{g}_{{\mathbb{S}}_{\mathcal{E}}}^{hr}(g_{{\mathbb{S}}_{\mathcal{E}}lh,k}+g_{{\mathbb{S}}_{\mathcal{E}}hk,l}-g_{{\mathbb{S}}_{\mathcal{E}}kl,h}),$$

where ${\displaystyle \frac{\partial g_{{\mathbb{S}}_{\mathcal{E}}ij}}{\partial u_t}}=g_{{\mathbb{S}}_{\mathcal{E}}ij,t}$. Since the components of the superpolar Euclidean metric are constant in the superpolar sense, their corresponding superpolar derivatives vanish, i.e., 0. Therefore, from Equation (16), we obtain

$${\Gamma }_{kl}^r=0.$$

Substituting this result into Equation (22), it follows immediately that

$$R_{{\mathbb{S}}_{\mathcal{E}}}{jkl}^i=0.$$

□

5. Superpolar Submanifolds

Let $(M,g_{{\mathbb{S}}_{\mathcal{E}}}{(·,·)}_m)$ be a superpolar Riemannian manifold, and let $T_{{\mathbb{S}}_{\mathcal{E}}p}M$ denote the superpolar tangent space of M at a point p. Furthermore, suppose that $F:M\to {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ is a superpolar differentiable mapping of class up to order k. The superpolar differential of F at the point $p\in M$ is defined as the linear map

$$d_{{\mathbb{S}}_{\mathcal{E}}}{F}_p:T_{{\mathbb{S}}_{\mathcal{E}}p}M\to T_{{\mathbb{S}}_{\mathcal{E}}F\left(p\right)}{\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m,d_{{\mathbb{S}}_{\mathcal{E}}}{F}_p\left(X_p\right)\left(g\right)={\displaystyle \frac{\partial (g\circ F)}{\partial X_p}},$$

where $g\in C_{{\mathbb{S}}_{\mathcal{E}}}^k\left({\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m\right)$.

The mapping F is called a superpolar immersion whenever the differential $d_{{\mathbb{S}}_{\mathcal{E}}}{F}_p$ is injective for every point $p\in M$. Moreover, if the metric is preserved under the differential, namely,

$$g_{{\mathbb{S}}_{\mathcal{E}}}{(X_p,Y_p)}_m=g_{{\mathbb{S}}_{\mathcal{E}}}(d_{{\mathbb{S}}_{\mathcal{E}}}{F}_p\left(X_p\right),d_{{\mathbb{S}}_{\mathcal{E}}}{F}_p\left(Y_p\right)),\forall X_p,Y_p\in T_{{\mathbb{S}}_{\mathcal{E}}}^{p}M,$$

then the superpolar immersion F is referred to as a superpolar isometric immersion.

It follows immediately that $T_{{\mathbb{S}}_{\mathcal{E}}p}M$ can be regarded as a subspace of $T_{{\mathbb{S}}_{\mathcal{E}}F\left(p\right)}{\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Consequently, the ambient superpolar tangent space admits the following superpolar orthogonal decomposition:

$$T_{{\mathbb{S}}_{\mathcal{E}}p}{\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m=T_{{\mathbb{S}}_{\mathcal{E}}p}M\oplus N_{{\mathbb{S}}_{\mathcal{E}}p}M,$$

where $N_{{\mathbb{S}}_{\mathcal{E}}p}M$ denotes the superpolar normal space of M at p, and its elements are referred to as superpolar normal vectors to M.

Let

$$D:{\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m\to \bigcup T_{{\mathbb{S}}_{\mathcal{E}}q}{\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m,q\mapsto D\left(q\right)\subset T_{{\mathbb{S}}_{\mathcal{E}}q}{\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$$

be a mapping satisfying $dim\left(D\right(q\left)\right)=r$. Such an assignment is called an r-dimensional superpolar distribution. The distribution D is said to be superpolar differentiable of class $C_{{\mathbb{S}}_{\mathcal{E}}}^k$ provided that, for every $q\in {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$, there exist r superpolar linearly independent vector fields of class $C_{{\mathbb{S}}_{\mathcal{E}}}^k$ spanning $D\left(q\right)$. We denote by ${\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(D\right)$ the collection of allsuperpolar sections of the distribution D. Furthermore, D is called superpolar involutive if

$${[X,Y]}_{{\mathbb{S}}_{\mathcal{E}}}\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(D\right),$$

for every $X,Y\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(D\right)$, where ${[·,·]}_{{\mathbb{S}}_{\mathcal{E}}}$ denotes the superpolar Lie bracket.

Let M be a superpolar submanifold of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. M is called an superpolar integral manifold of D when $T_{{\mathbb{S}}_{\mathcal{E}}}^{p}M=D\left(p\right)$ for every $p\in M$. If there is no other superpolar submanifold containing M, then it is said to be maximal. If in addition there is a maximal superpolar integral manifold of $D\left(p\right)$ for every $p\in M$, then D is called superpolar integrable.

The following theorem establishes the superpolar extension of the classical Frobenius theorem.

Theorem 6.(superpolar Frobenius theorem). Let D be a superpolar involutive distribution on ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Then, for each $q\in {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$, there exists a unique maximal superpolar integral manifold M associated with $D\left(q\right)$. Furthermore, every superpolar integral manifold passing through q is an open superpolar submanifold of M.

Proof. 

Let X and Y be superpolar tangent vector fields on M, and let $\tilde{X}$ and $\tilde{Y}$ denote arbitrary extensions of these vector fields to the ambient space ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. The covariant derivative of $\tilde{Y}$ along $\tilde{X}$ admits the following superpolar orthogonal decomposition:

$${\tilde{\mathfrak{D}}}_{\tilde{X}}\tilde{Y}={\mathfrak{D}}_{X}Y+h_{{\mathbb{S}}_{\mathcal{E}}}(X,Y),$$

(23)

where ${\mathfrak{D}}_{X}Y$ and $h_{{\mathbb{S}}_{\mathcal{E}}}(X,Y)$ denote the superpolar tangential and superpolar normal components of ${\tilde{\mathfrak{D}}}_{\tilde{X}}\tilde{Y}$ with respect to M, respectively. Equation (23) is referred to as the superpolar Gauss formula, while the tensor $h_{{\mathbb{S}}_{\mathcal{E}}}$ is called the superpolar second fundamental form. In particular, the submanifold M is said to be superpolar totally geodesic whenever $h_{{\mathbb{S}}_{\mathcal{E}}}$ vanishes identically. □

Proposition 2. 

Let $F:M\to {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ be a superpolar immersion of the superpolar Riemannian manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}}{(·,·)}_m)$ into ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Then the following statements hold:

1.

$\mathfrak{D}$ defines a superpolar Riemannian connection on M.

2.

h is a superpolar bilinear and symmetric tensor.

Proof. 

Fix an arbitrary point $p\in M$. Around the image point $F\left(p\right)$, extend all vector fields and functions defined on M to a neighborhood in ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Let X, U, and V be superpolar tangent vector fields on M, where X is chosen arbitrarily, and denote their corresponding extensions by $\tilde{X}$, $\tilde{U}$, and $\tilde{V}$.

1. By means of the superpolar Koszul formula, define

$$K(\tilde{U},\tilde{V},\tilde{X})=2g_{{\mathbb{S}}_{\mathcal{E}}}({\tilde{\mathfrak{D}}}_{\tilde{U}}\tilde{V},\tilde{X}).$$

Let $\tilde{f}\in C_{{\mathbb{S}}_{\mathcal{E}}}^1\left({\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m\right)$. Restricting $\tilde{f}$ to M yields

$$\tilde{f}{{|}_M=f\in C_{{\mathbb{S}}_{\mathcal{E}}}^1\left(M\right),{\displaystyle \frac{\partial \tilde{f}}{\partial \tilde{U}}}|}_M={\displaystyle \frac{\partial f}{\partial U}},$$

together with

$$\left(g_{{\mathbb{S}}_{\mathcal{E}}}(\tilde{U},\tilde{V})\right){|}_M=g_{{\mathbb{S}}_{\mathcal{E}}}{(U,V)}_m,\left({[\tilde{U},\tilde{V}]}_{{\mathbb{S}}_{\mathcal{E}}}\right){{|}_M={[U,V]}_{{\mathbb{S}}_{\mathcal{E}}},\left({\tilde{\mathfrak{D}}}_{\tilde{U}}\tilde{V}\right)|}_M={\tilde{\mathfrak{D}}}_{U}V.$$

Consequently,

$$K(\tilde{U},\tilde{V},\tilde{X})=K(U,V,X).$$

Since X is tangent to M, the decomposition given by the superpolar Gauss formula implies

$$g_{{\mathbb{S}}_{\mathcal{E}}}({\tilde{\mathfrak{D}}}_{U}V,X)=g_{{\mathbb{S}}_{\mathcal{E}}}({\mathfrak{D}}_{U}V+h_{{\mathbb{S}}_{\mathcal{E}}}(U,V),X)=g_{{\mathbb{S}}_{\mathcal{E}}}{({\mathfrak{D}}_{U}V,X)}_m,$$

because the second fundamental form is superpolar normal to M. Therefore, $\mathfrak{D}$ satisfies the defining properties of a superpolar Riemannian connection.

2. The superpolar bilinearity of h follows immediately from its definition. To establish symmetry, observe that

$$\begin{array}{cc}\hfill h_{{\mathbb{S}}_{\mathcal{E}}}(U,V)-h_{{\mathbb{S}}_{\mathcal{E}}}(V,U)& =({\tilde{\mathfrak{D}}}_{U}V-{\tilde{\mathfrak{D}}}_{V}U)+({\mathfrak{D}}_{V}U-{\mathfrak{D}}_{U}V)\hfill \\ \hfill & ={[U,V]}_{{\mathbb{S}}_{\mathcal{E}}}-{[U,V]}_{{\mathbb{S}}_{\mathcal{E}}}=0.\hfill \end{array}$$

Hence, h is symmetric. □

Let $\{w_1,\dots ,w_n\}$ be a superpolar orthonormal basis of $T_{{\mathbb{S}}_{\mathcal{E}}p}M$. The superpolar mean curvature is defined by

$$H_{{\mathbb{S}}_{\mathcal{E}}}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{h}_{{\mathbb{S}}_{\mathcal{E}}}(w_i,w_i).$$

We call Msuperpolar minimal if $H_{{\mathbb{S}}_{\mathcal{E}}}=0$ and superpolar totally umbilical if

$$h_{{\mathbb{S}}_{\mathcal{E}}}(X,Y)=g_{{\mathbb{S}}_{\mathcal{E}}}{(X,Y)}_m{H}_{{\mathbb{S}}_{\mathcal{E}}}$$

for every $X,Y$ superpolar tangent to M.

Example 4. 

In the three-dimensional superpolar space ${\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^3$, the superpolar sphere illustrated in Figure 6 is an example of a totally umbilical surface. Indeed, for every tangent vector fields $X,Y\in T{S}_{{\mathbb{S}}_{\mathcal{E}}}^2$, the second fundamental form satisfies

$$h_{{\mathbb{S}}_{\mathcal{E}}}(X,Y)=g_{{\mathbb{S}}_{\mathcal{E}}}(X,Y)H_{{\mathbb{S}}_{\mathcal{E}}}.$$

Thus, the principal curvatures of the superpolar sphere are equal at every point, and hence superpolar sphere is totally umbilical.

On the other hand, the catenoid shown in Figure 7 is an example of a superpolar minimal surface in ${\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}}^3$. Its mean curvature vanishes identically, that is,

$$H_{{\mathbb{S}}_{\mathcal{E}}}=0.$$

Therefore, the superpolar catenoid

$$\begin{array}{cc}\hfill x& ={\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_1}}}cosh\theta cos\phi \hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill y& ={\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_1}}}cosh\theta sin\phi \hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill z& ={\displaystyle \frac{r\left(\phi \right)}{\sqrt{{\eta }_1}}}\theta \hfill \end{array}$$

(26)

is a superpolar minimal surface.

The superpolar first normal space at $p\in M$ in ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ is a superpolar subspace of the superpolar normal space $N_{{\mathbb{S}}_{\mathcal{E}}p}M$ defined by

$$\mathrm{Im}{h}_p={\mathrm{Span}}_{*}\{h_{{\mathbb{S}}_{\mathcal{E}}}(X,Y):X,Y\in T_{{\mathbb{S}}_{\mathcal{E}}p}M\},$$

where ${\mathrm{Span}}_{*}$ is used in the superpolar sense.

Let $\xi$ be a superpolar normal vector field along M in ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. The covariant derivative of $\xi$ with respect to a superpolar tangent vector field $\tilde{X}$ admits the unique superpolar orthogonal decomposition

$${\tilde{\mathfrak{D}}}_{\tilde{X}}\xi =-A_{\xi }\left(X\right)+D_{{\mathbb{S}}_{\mathcal{E}}X}\xi ,$$

(27)

where $A_{\xi }\left(X\right)$ and $D_{{\mathbb{S}}_{\mathcal{E}}X}\xi$ denote the superpolar tangential and superpolar normal components of ${\tilde{\mathfrak{D}}}_{\tilde{X}}\xi$ relative to M, respectively. Equation (27) is referred to as the superpolar Weingarten formula, while the operator A is called the superpolar shape operator.

Proposition 3. 

Let $F:M\to {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ be a superpolar immersion of the superpolar Riemannian manifold $(M,g_{{\mathbb{S}}_{\mathcal{E}}}{(·,·)}_m)$ into ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$, and let ξ be a superpolar normal vector field along M. Then the following assertions hold:

1.

The superpolar shape operator $A_{\xi }$ is superpolar linear and satisfies

$$g_{{\mathbb{S}}_{\mathcal{E}}}(h_{{\mathbb{S}}_{\mathcal{E}}}(X,Y),\xi )=g_{{\mathbb{S}}_{\mathcal{E}}}{(A_{\xi }\left(X\right),Y)}_m,$$

for every superpolar tangent vector fields X, Y and every superpolar normal vector field ξ along M.

2.

$D_{*}$ defines a superpolar natura connection satisfying

$$D_{{\mathbb{S}}_{\mathcal{E}}X}\left(g_{{\mathbb{S}}_{\mathcal{E}}}(\xi ,\eta )\right)=g_{{\mathbb{S}}_{\mathcal{E}}}(D_{{\mathbb{S}}_{\mathcal{E}}X}\xi ,\eta )+g_{{\mathbb{S}}_{\mathcal{E}}}(\xi ,D_{{\mathbb{S}}_{\mathcal{E}}X}\eta ),$$

for every superpolar tangent vector field X and every pair of superpolar normal vector fields ξ, η along M.

Proof. 

1.

Let $f,g\in C_{{\mathbb{S}}_{\mathcal{E}}}^1\left(M\right)$ and let $p\in M$. By the linearity of the ambient superpolar Riemannian connection,

$${\tilde{\mathfrak{D}}}_{\left(f{X}_p\right)}\left(g\xi \right)=f\left({\displaystyle \frac{\partial g}{\partial X_p}}\xi +g{\tilde{\mathfrak{D}}}_{X_p}\xi \right),$$

where $\tilde{\mathfrak{D}}$ denotes the superpolar Riemannian connection on ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Applying the superpolar Weingarten formula (27) and comparing the superpolar tangential components, we obtain

$$-A_{\left(g\xi \right)}\left(f{X}_p\right)=-\left(fg\right)A_{\xi }\left(X_p\right),$$

which proves the superpolar linearity of the shape operator.

To establish the stated identity, let Y be a superpolar tangent vector field and let $\xi$ be a superpolar normal vector field along M. Since $g_{{\mathbb{S}}_{\mathcal{E}}}(Y,\xi )=0$, differentiation along $X_p$ yields

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \left(g_{{\mathbb{S}}_{\mathcal{E}}}(Y,\xi )\right)}{\partial X_p}}=g_{{\mathbb{S}}_{\mathcal{E}}}({\tilde{\mathfrak{D}}}_{X_p}Y,\xi )+g_{{\mathbb{S}}_{\mathcal{E}}}(Y,{\tilde{\mathfrak{D}}}_{X_p}\xi )=0.\end{array}$$

Using the superpolar Gauss and superpolar Weingarten formulas (23) and (27), we conclude that

$$\begin{array}{c}\hfill g_{{\mathbb{S}}_{\mathcal{E}}}(h_{{\mathbb{S}}_{\mathcal{E}}}(X_p,Y),\xi )-g_{{\mathbb{S}}_{\mathcal{E}}}{(Y,A_{\xi }\left(X_p\right))}_m=0,\end{array}$$

which is equivalent to

$$g_{{\mathbb{S}}_{\mathcal{E}}}(h_{{\mathbb{S}}_{\mathcal{E}}}(X_p,Y),\xi )=g_{{\mathbb{S}}_{\mathcal{E}}}{(A_{\xi }\left(X_p\right),Y)}_m.$$

2.

The proof follows the same argument as in the first part. Replacing the tangent vector field Y by a superpolar normal vector field $\eta$ and applying the superpolar Weingarten formula immediately gives

$$D_{{\mathbb{S}}_{\mathcal{E}}X}\left(g_{{\mathbb{S}}_{\mathcal{E}}}(\xi ,\eta )\right)=g_{{\mathbb{S}}_{\mathcal{E}}}(D_{{\mathbb{S}}_{\mathcal{E}}X}\xi ,\eta )+g_{{\mathbb{S}}_{\mathcal{E}}}(\xi ,D_{{\mathbb{S}}_{\mathcal{E}}X}\eta ),$$

thereby proving that $D_{{\mathbb{S}}_{\mathcal{E}}}$ is a superpolar natural connection.

□

5.1. Superpolar Conic Submanifolds and Spheres

A superpolar cone in ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ with vertex q is defined as a superpolar submanifold formed by a family of superpolar lines emanating from the point q. A local parametrization of such a superpolar cone is given by

$$\Lambda (x_1,x_2,\dots ,x_n)=q+x_1\Lambda (x_2,\dots ,x_n),x_i\in I_i\subset {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}},i=1,\dots ,n,$$

(28)

where $\Lambda (x_2,\dots ,x_n)$ is a local coordinate chart of an $(n-1)$-dimensional superpolar Riemannian manifold. An superpolar conic submanifold of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ with vertex q is defined to be any open subset of a superpolar cone having vertex q, where $q\in {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ is constant.

A superpolar submanifold of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ is called superpolar spherical if it is contained in a superpolar hypersphere of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$.

A nonzero vector field $\tilde{U}\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left({\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m\right)$ is said to be superpolar concurrent provided that

$${\tilde{\mathfrak{D}}}_{\tilde{Y}}\tilde{U}=\tilde{Y},\forall \tilde{Y}\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left({\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m\right).$$

(29)

As an illustration, consider the radial vector field $\tilde{U}$ and let

$$\tilde{Y}={\displaystyle \sum _{i=1}^m}{y}_i\widehat{e_i}.$$

Then,

$$\begin{array}{cc}\hfill {\tilde{\mathfrak{D}}}_{{\tilde{Y}}_p}\tilde{U}& ={\displaystyle \sum _{i=1}^m}{\displaystyle \frac{\partial x_i}{\partial {\tilde{Y}}_p}}\widehat{e_i}\hfill \\ \hfill & ={\displaystyle \sum _{i,j=1}^m}{\delta }_{ij}{y}_j\left(p\right)\widehat{e_i}\hfill \\ \hfill & =\tilde{Y}\left(p\right),\forall p\in {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m,\hfill \end{array}$$

which shows that the radial vector field is a superpolar concurrent vector field. Let $F:M\to {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ be a superpolar isometric immersion. For every point $p\in M$, denote by $F^{T_{{\mathbb{S}}_{\mathcal{E}}}}$ and $F^{N_{{\mathbb{S}}_{\mathcal{E}}}}$ the superpolar orthogonal projections onto $T_{{\mathbb{S}}_{\mathcal{E}}}^{p}M$ and $N_{{\mathbb{S}}_{\mathcal{E}}p}M$, respectively. Then, we write as follows:

$$F=F^{T_{{\mathbb{S}}_{\mathcal{E}}}}+F^{N_{{\mathbb{S}}_{\mathcal{E}}}}.$$

In what follows, we characterize superpolar submanifods in ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ in terms of the superpolar tangential and superpolar normal components of their position vectors.

Proposition 4. 

Let $F:M\to {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ be a superpolar isometric immersion. Then

1.

$F^{N_{{\mathbb{S}}_{\mathcal{E}}}}=0$ if and only if M is a superpolar conic submanifold with vertex at 0.

2.

$F^{T_{{\mathbb{S}}_{\mathcal{E}}}}=0$ if and only if M is contained in a superpolar spherical manifold centered at 0.

Proof. 

1.

Assume first that $F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)=0$ for every $p\in M$. Define

$$w_1={\displaystyle \frac{F}{{\parallel F\parallel }_{{\mathbb{S}}_{\mathcal{E}}}}}\mathrm{and}f={\parallel F\parallel }_{{\mathbb{S}}_{\mathcal{E}}},$$

so that $F=f{w}_1$ and $w_1\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$. Since

$$g_{{\mathbb{S}}_{\mathcal{E}}}(w_1,w_1)=1,$$

taking the superpolar derivative with respect to $w_1$ yields

$${\displaystyle \frac{\partial \left(g_{{\mathbb{S}}_{\mathcal{E}}}(w_1,w_1)\right)}{\partial w_1}}=2g_{{\mathbb{S}}_{\mathcal{E}}}({\tilde{\mathfrak{D}}}_{w_1}{w}_1,w_1)=0,$$

where $\tilde{\mathfrak{D}}$ denotes the superpolar Riemannian connection on ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Furthermore, since F is superpolar concurrent, we have

$${\tilde{\mathfrak{D}}}_{w_1}F=w_1.$$

On the other hand,

$${\tilde{\mathfrak{D}}}_{w_1}F={\tilde{\mathfrak{D}}}_{w_1}\left(f{w}_1\right)={\displaystyle \frac{\partial f}{\partial w_1}}{w}_1+f{\tilde{\mathfrak{D}}}_{w_1}{w}_1.$$

Since the left-hand side is equal to $w_1$ and $w_1$ is superpolar orthogonal to ${\tilde{\mathfrak{D}}}_{w_1}{w}_1$, it follows immediately that

$${\tilde{\mathfrak{D}}}_{w_1}{w}_1=0.$$

Therefore, the superpolar integral curves of $w_1$ are superpolar geodesics in ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Moreover, because F is the position vector field and $F^{N_{{\mathbb{S}}_{\mathcal{E}}}}=0$, these geodesics pass through the origin. Consequently, M is a superpolar conic submanifold with vertex at 0.

Conversely, suppose that M is a superpolar conic submanifold with vertex at 0. Then M admits the given parametrization:

$$\Lambda (x_1,x_2,\dots ,x_n)=x_1\Lambda (x_2,\dots ,x_n),x_i\in I_i\subset {\mathbb{R}}_{{\mathbb{S}}_{\mathcal{E}}},i=1,\dots ,n.$$

From this parametrization, it is evident that $\Lambda (x_1,\dots ,x_n)$ is superpolar parallel to one of the basis vectors of the superpolar tangent space. Hence,

$$F^{N_{{\mathbb{S}}_{\mathcal{E}}}}=0.$$

This establishes the first assertion.

2.

Assume that

$$F^{T_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)=0,\forall p\in M.$$

For every $U\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$ and every $p\in M$, we have

$${\displaystyle \frac{\partial \left(g_{{\mathbb{S}}_{\mathcal{E}}}(F,F)\right)}{\partial U_p}}=2g_{{\mathbb{S}}_{\mathcal{E}}}({\tilde{\mathfrak{D}}}_{U_p}F,F).$$

Since

$${\tilde{\mathfrak{D}}}_{U}F=U$$

and

$$F^{T_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)=0,$$

it follows that

$${\displaystyle \frac{\partial \left(g_{{\mathbb{S}}_{\mathcal{E}}}(F,F)\right)}{\partial U_p}}=0.$$

Hence, $g_{{\mathbb{S}}_{\mathcal{E}}}(F,F)$ is constant on M. Therefore, M is contained in a superpolar hypersphere of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ centered at the origin.

Conversely, if M is contained in a superpolar hypersphere centered at 0, then the position vector field is everywhere superpolar normal to M, which immediately implies

$$F^{T_{{\mathbb{S}}_{\mathcal{E}}}}=0.$$

This completes the proof.

□

5.2. Superpolar Rectifying Submanifolds

In this section, we introduce the notion of superpolar rectifying submanifolds.

Definition 13. 

Let $F:M\to {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ be a superpolar immersion of a superpolar Riemannian manifold into ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. The submanifold M is called an superpolar rectifying submanifold of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ if

$$g_{{\mathbb{S}}_{\mathcal{E}}}(F\left(p\right),Im{h}_p)=0,\forall p\in M.$$

Furthermore, if

$$F^{T_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)\ne 0\mathrm{and}{F}^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)\ne 0,\forall p\in M,$$

then M is called a proper superpolar rectifying submanifold.

Remark 1. 

Let M be a proper superpolar rectifying submanifold of dimension n in ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Then

$$m>n+dim(Im{h}_p),\forall p\in M.$$

The above inequality follows by contradiction. Suppose that

$$m=n+dim(Im{h}_p).$$

Observe that the inequality

$$m<n+dim(Im{h}_p)$$

cannot occur since

$$dim(Im{h}_p)\le dim\left(N_{{\mathbb{S}}_{\mathcal{E}}p}M\right),$$

for every $p\in M$. Consequently,

$$g_{{\mathbb{S}}_{\mathcal{E}}}(F\left(p\right),Im{h}_p)=g_{{\mathbb{S}}_{\mathcal{E}}}(F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right),Im{h}_p)=0,\forall p\in M.$$

Since

$$dim(Im{h}_p)=m-n,$$

it follows that

$$F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)\in Im{h}_p.$$

Hence,

$$g_{{\mathbb{S}}_{\mathcal{E}}}\left(F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right),F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)\right)=0,\forall p\in M,$$

which implies

$$F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)=0,\forall p\in M.$$

This contradicts the assumption that M is proper, thereby completing the proof.

Theorem 7. 

Let $F:M\to {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ be a superpolar isometric immersion of a superpolar Riemannian manifold into ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ with $F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\ne 0$. Then M is a proper superpolar rectifying submanifold if and only if $F^{T_{{\mathbb{S}}_{\mathcal{E}}}}$ is a superpolar concurrent vector field on M. The converse is also true.

Proof. 

Let $\tilde{\mathfrak{D}}$ and $\mathfrak{D}$ denote the superpolar Riemannian connection on ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ and the induced superpolar Riemannian connection on M, respectively. Let $X,Y\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$. Since F is a concurrent vector field, we have

$$X={\tilde{\mathfrak{D}}}_{X}F={\tilde{\mathfrak{D}}}_X\left(F^{T_{{\mathbb{S}}_{\mathcal{E}}}}+F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\right)={\tilde{\mathfrak{D}}}_X{F}^{T_{{\mathbb{S}}_{\mathcal{E}}}}+{\tilde{\mathfrak{D}}}_X{F}^{N_{{\mathbb{S}}_{\mathcal{E}}}}.$$

Applying the superpolar Gauss and superpolar Weingarten formulas yields

$$X={\mathfrak{D}}_X{F}^{T_{{\mathbb{S}}_{\mathcal{E}}}}+h_{{\mathbb{S}}_{\mathcal{E}}}(X,F^{T_{{\mathbb{S}}_{\mathcal{E}}}})-A_{F^{N_{{\mathbb{S}}_{\mathcal{E}}}}}\left(X\right)+D_{{\mathbb{S}}_{\mathcal{E}}X}{F}^{N_{{\mathbb{S}}_{\mathcal{E}}}}.$$

Comparing the superpolar tangent components of both sides, we obtain

$$X={\mathfrak{D}}_X{F}^{T_{{\mathbb{S}}_{\mathcal{E}}}}-A_{F^{N_{{\mathbb{S}}_{\mathcal{E}}}}}\left(X\right).$$

(30)

Assume first that M is a proper superpolar rectifying submanifold. By Definition 13, for every $p\in M$,

$$g_{{\mathbb{S}}_{\mathcal{E}}}\left(F\left(p\right),h_{{\mathbb{S}}_{\mathcal{E}}}(X_p,Y_p)\right)=g_{{\mathbb{S}}_{\mathcal{E}}}\left(F^{T_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)+F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right),h_{{\mathbb{S}}_{\mathcal{E}}}(X_p,Y_p)\right)=g_{{\mathbb{S}}_{\mathcal{E}}}\left(F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right),h_{{\mathbb{S}}_{\mathcal{E}}}(X_p,Y_p)\right)=0.$$

Therefore, by the first assertion of Proposition 3,

$$A_{F^{N_{{\mathbb{S}}_{\mathcal{E}}}}}\left(X\right)=0,\forall X\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right).$$

This immediately gives

$${\mathfrak{D}}_X{F}^{T_{{\mathbb{S}}_{\mathcal{E}}}}=X,$$

showing that $F^{T_{{\mathbb{S}}_{\mathcal{E}}}}$ is a superpolar concurrent vector field on M.

Conversely, suppose that

$${\mathfrak{D}}_X{F}^{T_{{\mathbb{S}}_{\mathcal{E}}}}=X,\forall X\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right).$$

Then we obtain

$$A_{F^{N_{{\mathbb{S}}_{\mathcal{E}}}}}\left(X\right)=0,\forall X\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right).$$

By the first assertion of Proposition 3, it follows that

$$g_{{\mathbb{S}}_{\mathcal{E}}}\left(F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right),h_{{\mathbb{S}}_{\mathcal{E}}}(X_p,Y_p)\right)=0,\forall p\in M.$$

Since $F^{T_{{\mathbb{S}}_{\mathcal{E}}}}$ is tangent to M, we also have

$$g_{{\mathbb{S}}_{\mathcal{E}}}\left(F^{T_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right),h_{{\mathbb{S}}_{\mathcal{E}}}(X_p,Y_p)\right)=0.$$

Consequently,

$$g_{{\mathbb{S}}_{\mathcal{E}}}\left(F\left(p\right),h_{{\mathbb{S}}_{\mathcal{E}}}(X_p,Y_p)\right)=0,\forall p\in M,$$

which, by Definition 13, shows that M is a proper superpolar rectifying submanifold. This completes the proof. □

5.3. Superpolar Rectifying Hypersurfaces

We begin by characterizing the superpolar totally geodesic submanifolds of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$.

Proposition 5. 

A superpolar n-dimensional superpolar totally geodesic submanifold of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ is an open subset of a superpolar linear subspace ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^n$ of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Conversely, every open subset of a superpolar linear subspace ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^n$ is a superpolar totally geodesic submanifold.

Proof. 

Let $F:M\to {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ be a superpolar immersion of an n-dimensional superpolar totally geodesic submanifold. Since M is superpolar totally geodesic, its second fundamental form vanishes identically. Therefore, the superpolar Gauss formula reduces to

$${\tilde{\mathfrak{D}}}_{X}Y={\mathfrak{D}}_{X}Y,X,Y\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right).$$

Hence, every superpolar geodesic of M is also a superpolar geodesic of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$, and conversely. Suppose that M contains a point p satisfying

$$F\left(p\right)=0,$$

Then $T_{{\mathbb{S}}_{\mathcal{E}}}^{p}M$ is a superpolar linear subspace of $T_{{\mathbb{S}}_{\mathcal{E}}0}{\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Since the superpolar geodesics of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ are precisely the superpolar straight lines, it follows that M is an open subset of the superpolar linear subspace ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^n$ whose superpolar tangent space at the origin is $T_{{\mathbb{S}}_{\mathcal{E}}}^{p}M$.

The converse follows immediately, since every superpolar linear subspace has vanishing second fundamental form and is therefore superpolar totally geodesic. □

Now let M be a superpolar hypersurface of ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ with superpolar unit normal vector field Z. Assume further that M is a superpolar rectifying submanifold. In this case, the superpolar first normal space at each point is the superpolar line spanned by Z, and the defining condition in Definition 13 becomes

$$g_{{\mathbb{S}}_{\mathcal{E}}}(F\left(p\right),Z\left(p\right))=0,\forall p\in M.$$

Since

$$F\left(p\right)=F^{T_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)+F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)$$

and

$$F^{N_{{\mathbb{S}}_{\mathcal{E}}}}=\lambda Z$$

for some smooth function $\lambda$ on M, we obtain

$$F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)=0,\forall p\in M.$$

Therefore, a superpolar rectifying hypersurface cannot be proper when its dimension is $m-1$.

The following theorem provides a complete classification of superpolar rectifying hypersurfaces in ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$.

Theorem 8. 

Let $F:M\to {\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$ be a superpolar immersion of an $(m-1)$-dimensional superpolar Riemannian manifold into ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$. Then M is a superpolar rectifying hypersurface if and only if it is an open subset of a superpolar hyperplane.

Proof. 

Let $X\in {\Gamma }_{{\mathbb{S}}_{\mathcal{E}}}\left(M\right)$ be arbitrary. From the previous discussion,

$$F^{N_{{\mathbb{S}}_{\mathcal{E}}}}\left(p\right)=0,\forall p\in M.$$

Hence,

$$F=F^{T_{{\mathbb{S}}_{\mathcal{E}}}},$$

and since the position vector field is superpolar concurrent in ${\mathbb{E}}_{{\mathbb{S}}_{\mathcal{E}}}^m$, we have

$${\tilde{\mathfrak{D}}}_X{F}^{T_{{\mathbb{S}}_{\mathcal{E}}}}=X.$$

Applying the superpolar Gauss formula (23), we obtain

$$X={\tilde{\mathfrak{D}}}_X{F}^{T_{{\mathbb{S}}_{\mathcal{E}}}}={\mathfrak{D}}_X{F}^{T_{{\mathbb{S}}_{\mathcal{E}}}}+h_{{\mathbb{S}}_{\mathcal{E}}}(X,F^{T_{{\mathbb{S}}_{\mathcal{E}}}}).$$

Since

$$h_{{\mathbb{S}}_{\mathcal{E}}}(X,F^{T_{{\mathbb{S}}_{\mathcal{E}}}})=0,$$

the shape operator satisfies

$$A_Z\left(X\right)=0$$

by the first assertion of Proposition 3. Consequently, M is superpolar totally geodesic. The desired conclusion now follows directly from Proposition 5. □

6. Conclusions

This paper extends earlier results [11,12,13,14] in a Riemannian framework. By replacing the Euclidean metric with a superpolar metric, a broad class of natural and abstract geometric objects can be investigated within a unified intrinsic setting. In particular, all Gielis curves, including non-smooth and polygonal shapes, are interpreted as superpolar circles with respect to the proposed metric. Consequently, geometric objects that are extrinsically distinct in Euclidean space become intrinsically equivalent in superpolar geometry.

This intrinsic equivalence considerably simplifies the mathematical treatment of manifolds and submanifolds. Objects such as superpolar spheres with sharp corners or polygonal boundaries possess the same intrinsic geometric structure as the classical Euclidean sphere under the proposed metric. Therefore, many differential geometric constructions can be extended beyond the smooth Euclidean setting without introducing additional analytical complexity. The proposed framework thus provides a natural generalization of manifold and submanifold theory while preserving the essential geometric properties of the underlying space.

Another important consequence of this approach is the reinterpretation of singularities. Features that are regarded as geometric singularities from the Euclidean viewpoint, such as corners or non-smooth boundaries, need not represent intrinsic singularities in superpolar geometry. Instead, they arise as artifacts of the Euclidean metric rather than of the geometric object itself. This observation offers a new perspective on the notion of regularity and suggests that geometric singularities should be understood relative to the metric structure under consideration.

Overall, the proposed superpolar geometry establishes an alternative geometric paradigm in which shape is determined by intrinsic metric properties rather than by Euclidean appearance. From this perspective, geometrically different objects may share an identical intrinsic structure, allowing polygonal and smooth geometries to be studied within the same differential geometric framework. It is expected that this viewpoint will stimulate further developments in differential geometry and find applications in all of the natural sciences, certain human sciences, and in various fields of technology.