Geometrical Analysis on Submanifolds in Riemannian Manifolds Attached with Silver Structure

Abstract

In this paper, we analyze a silver Riemannian structure on a Riemannian manifold. We compute some fundamental properties of the induced structure on submanifolds immersed in a silver Riemannian manifold and also obtain some results for induced structures on submanifolds of codimension 2. Moreover, we explore the conditions for totally geodesic and minimal submanifolds in a silver Riemannian manifold. Finally, we also give an example of submanifolds in a silver Riemannian manifold.

1. Introduction

Irrational numbers are as fascinating as rational numbers. A remarkable example is the golden ratio $\phi =(1+\sqrt{5})/2$, which arises as the positive solution of the quadratic equation $x^2-x-1=0$. Like the golden ratio, another interesting irrational number is $\varphi =(1+\sqrt{2})$, which is the positive root of the equation $x^2-2x-1=0$ and is called the silver ratio or silver mean. Hretcanu and Crasmareanu [1] studied the generalized Fibonacci sequences on Riemannian manifolds and investigated some applications of the metallic means family in 2013. In [2], de Spinadel defined the metallic mean family as the set of positive eigenvalues of the material equation given by (1). This consists of some generalizations of the golden mean, silver mean, bronze mean, copper mean, and nickel mean, as well as many other means.

The golden and silver structures are distinct types of metallic structures on manifolds, described algebraically by different constant ratios. The essential distinction is the numerical value of their respective metallic mean or ratio. Recent research has focused on several kinds of polynomial structures, including almost product, almost complex, almost tangent, and f-structure [3,4]. The silver number, also known as the silver ratio or silver mean, is a positive solution to the equation $x^2-x-1=0$. It is an irrational number similar to the golden ratio. The precise mathematical behavior and specific geometric properties of the manifold are determined by the $\widehat{\theta }$ and $\theta$ values, with the silver structure having a different set of eigenvalues and associated sequences (Pell instead of Fibonacci) compared to the golden structure. It has applications in design, architecture, and physics. Additionally, Chandra and Rani [5] employed the silver mean to characterize fractal geometry.

Submanifolds in metallic Riemannian manifolds were explored by Hretcanu and Crasmareanu [6]. Yano and Goldberg [7] defined a polynomial structure on a manifold in 1970. In 1971, A. F. Horadam [8] defined the Pell numbers by the Pell sequence $\left\{P_n\right\}$. Primo and Reyes [9] studied some of the silver number’s algebraic and geometric features in 2007. It was the first time, to the best of our knowledge, that the silver structure on a manifold was introduced in the literature and the silver structure on a differential manifold was studied by Ozkan and Peltek [10].

Crasmareanu and Hretcanu [11] studied golden differential geometry and Gherici [12] studied the induced structure on golden Riemannian manifolds in 2008 and 2018, respectively. The authors examined submanifolds of metallic manifolds in [13,14,15] and dealt with submanifolds with a golden structure in [16,17,18,19,20]. Additionally, numerous writers have examined various structures, such as submanifolds [21,22], slant submanifolds [17], quasi-hemi-slant submanifolds [23,24], invariant submanifolds, anti-invariant and semi-invariant submanifolds [19,25,26], and some generic submanifolds in Riemannian manifolds with a golden structure [27].

The geometric impact of a silver structure on a submanifold is determined by how the structure in the ambient manifold interacts with the submanifold’s induced geometry. A silver structure on the manifold decomposes the tangent space into two complementary, orthogonal distributions (subspaces) associated with its eigenvalues, the silver ratio $(\theta =1+\sqrt{2})$ and $(2-\theta =1-\sqrt{2})$. The structure imposes a framework similar to an almost product structure, dividing the tangent bundle. In fact, the silver structure establishes a decomposable framework on the underlying manifold, defining how vector fields and geometric attributes behave in the space and its embedded submanifolds.

The structure of this paper is as follows: Section 2 introduces the silver structure on a differential manifold, followed by the definition of Pell sequences and an exploration of the connections between the silver ratio, tangent silver ratio, and complex silver ratio. A number of important findings are established in Section 3, which focuses on the characteristics of the induced structure on submanifolds within silver Riemannian manifolds.

2. Silver Riemannian Manifolds

Definition 1

([28]). Consider a Riemannian manifold $(M,g)$. A silver structure θ on M is defined as a non-vanishing tensor field of type (1, 1) that satisfies the relation

$${\theta }^2=2\theta +I,$$

(1)

where I denotes the identity transformation on the tangent bundle $TM$. The metric g is said to be θ-compatiple if it satisfies

$$g(\theta X,Y)=g(X,\theta Y)$$

(2)

for all $X,Y$ vector fields on M. If we use (1) and change $\theta X$ to X in (2), we get

$$g(\theta X,\theta Y)=2g(X,\theta Y)+g(X,Y).$$

(3)

The Riemannian metric given in (2) is termed as θ-compatible, and the structure $(M,\theta ,g)$ is called a silver Riemannian manifold.

Proposition 1 

([8,9]). Let M be a silver structure θ. With any integer n

$${\theta }^n=P_n\theta +P_{n-1}I,$$

(4)

where $\left(P_n\right)$ is the Pell sequence.

Now, applying Binet’s formula [7], $P_n={\displaystyle \frac{\varphi -{(2-\varphi )}^n}{2\sqrt{2}}}$ in the Pell sequence (4), we gain

$${\theta }_n={\displaystyle \frac{\varphi -{(2-\varphi )}^n}{2\sqrt{2}}}\theta +{\displaystyle \frac{{\varphi }^{n-1}-{(2-\varphi )}^{n-1}}{2\sqrt{2}}}I.$$

Proposition 2 

([10]). 1.  The silver ratio $\varphi$ and $(2-\varphi )$ are the eigenvalues of the silver structure θ.

2. 

A silver structure θ on M is an isomorphism on $T_{x}M$, where $T_{x}M$ is a tangent space defined on M for every $x\in M$.

3. 

Structure θ is invertible and its inverse $\widehat{\theta }={\theta }^{-1}$ holds:

$${\widehat{\theta }}^2=-2\widehat{\theta }+I.$$

If $T,J$, and P represent an almost tangent structure, an almost complex structure, and an almost product structure, respectively, then $-T,-P$, and $-J$ are also almost tangent, almost complex, and almost product structures, respectively [11].

Proposition 3

([10]). If θ represents a silver structure, then $\widehat{\theta }=2I-\theta$ is likewise a silver structure.

The statement that follows exhibits how it is simple to construct a relationship between a silver structure and an almost product structure.

Theorem 1

([10]). Consider a silver structure θ on manifold M; then

$$P={\displaystyle \frac{1}{\sqrt{2}}}(\theta -I)$$

(5)

defines an almost product structure on M.

By Equation (5), we can give the following definitions:

Definition 2

([10]). 1.  On an almost tangent manifold $(M,T)$, tensor field ${\theta }_t$, given by

$${\theta }_t=I+\sqrt{2}T$$

and satisfying the equation ${{\theta }_t}^2-2{\theta }_t+I=0$, is called the tangent silver structure on $(M,T)$.

Considering the relation $x^2-2x+1=0$ over the real field $\mathbb{R}$, we get the tangent real silver ratio ${\varphi }_t=1$.

2. 

On an almost complex manifold $(M,T)$, tensor field ${\theta }_c$, given by

$${\theta }_c=I+\sqrt{2}J$$

and satisfying the relation ${{\theta }_c}^2-2{\theta }_c+3I=0$, is known as the complex silver structure on $(M,J)$.

Over the real field $\mathbb{R}$, we have $x^2-2x+3=0$ and we get complex silver ratio ${\varphi }_c=1+i\sqrt{2}$.

We have

(i). Tangent silver ratio: ${\varphi }_t=1$;

(ii). Silver ratio: $\varphi =1+\sqrt{2}$;

(iii). Complex silver ratio: $\begin{array}{cc}{\varphi }_c\hfill & =1+i\sqrt{2}\hfill \\ & ={\varphi }_t+i(\varphi -{\varphi }_t).\hfill \end{array}$

3. Properties of Induced Structure on Submanifolds in Silver Riemannian Manifold

In this section, we consider an n-dimensional submanifold M of codimension r, which is immersed in an $(n+r)$-dimensional silver Riemannian manifold $(\widehat{M},\widehat{g},\theta )$, where $r,n\in N$.

We use $T_{x}M$ to represent the tangent space and $T_x^{\perp }M$ to represent the normal space on M, where $x\in M$. Let i be the differential of the immersion $i:M\to \widehat{M}$ and g is the induced Riemannian metric on M, given by

$$g(X,Y)=\widehat{g}(X,Y),$$

for any $X,Y\in \chi \left(M\right).$

The normal space $T_x^{\perp }M$ has a local orthonormal basis $N_1,N_2,\dots ,N_r$. Assume that the indices $\alpha ,\beta ,\gamma$ are run across the range $1,2,\dots ,r$.

The vector fields $\theta \left(iX\right)$ and $\theta \left(N_{\alpha }\right)$ can be split down into their tangent and normal components as follows for each $X\in T_{x}M$:

$$\left\{\begin{array}{c}\theta \left(iX\right)=i\left(RX\right)+\sum _{\alpha =1}^r{u}_{\alpha }\left(X\right)N_{\alpha },\hfill \\ \theta \left(N_{\alpha }\right)=i\left({\xi }_{\alpha }\right)+\sum _{\beta =1}^r{a}_{\alpha \beta }{N}_{\beta }\hfill \end{array}\right.$$

(6)

where R indicates a (1, 1) tensor field on M, $u_{\alpha }$ are 1-forms on M, $\xi \in \chi \left(M\right)$, and ${\left(a_{\alpha \beta }\right)}_r$ signifies an $r\times r$ matrix with smooth real valued functions defined on M as components.

Proposition 4

([6]). Let $(\widehat{M},\widehat{g},\theta )$ be a silver Riemannian manifold and M a submanifold on $\widehat{M}$; then the induced structure $\sum =(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ on $\widehat{M}$ holds the below equalities:

$$\begin{array}{cc}\hfill R^2\left(X\right)& =X+2R\left(X\right)-\sum _{\alpha }{u}_{\alpha }\left(X\right){\xi }_{\alpha },\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill u_{\alpha }\left(RX\right)& =2u_{\alpha }\left(X\right)-\sum _{\beta }{a}_{\alpha \beta }{u}_{\beta }\left(X\right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill a_{\alpha \beta }& =a_{\beta \alpha },\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill u_{\beta }\left({\xi }_{\alpha }\right)& ={\delta }_{\alpha \beta }+2a_{\alpha \beta }-\sum _{\gamma }{a}_{\alpha \gamma }{a}_{\gamma \beta },\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill R{\xi }_{\alpha }& =2{\xi }_{\alpha }-\sum _{\beta }{a}_{\alpha \beta }{\xi }_{\beta },\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill u_{\alpha }\left(X\right)& =g(X,{\xi }_{\alpha }),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill g(RX,Y)& =g(X,RY),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill g(RX,RY)& =2g(X,RY)+g(X,Y)+\sum _{\alpha }{u}_{\alpha }\left(X\right)u_{\alpha }\left(Y\right),\hfill \end{array}$$

(14)

for every $X,Y\in \chi \left(M\right)$, where ${\delta }_{\alpha \beta }$ is the Kronecker delta.

Definition 3

([6]). A submanifold M of a manifold $\widehat{M}$, equipped with a structure tensor field θ, is said to be invariant with respect to θ if $\theta \left(T_{x}M\right)\subset T_x\left(M\right)$ for all $x\in M$.

Remark 1.

The structure $\sum =(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ induced on the submanifold M by the silver Riemannian structure $(g,\theta )$ is invariant if $u_{\alpha }=0$ (${\xi }_{\alpha }=0$) for every $\alpha \in (1,2,\dots ,r)$.

The Gauss and Weingarten formula are given by

$$\begin{array}{cc}\hfill {\widehat{\nabla }}_{X}Y& ={\nabla }_{X}Y+\sum _{\alpha }{h}_{\alpha }(X,Y)N_{\alpha },\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\widehat{\nabla }}_X{N}_{\alpha }& =-A_{\alpha }X+{\nabla }_X^{\perp }{N}_{\alpha },\hfill \end{array}$$

(16)

where the shape operator $A_{\alpha }=A_{N_{\alpha }}$ and the second fundamental form is

$$h(X,Y)=\sum _{\alpha }{h}_{\alpha }(X,Y)N_{\alpha }.$$

Also,

$$h_{\alpha }(X,Y)=g(A_{\alpha }X,Y)=g(h(X,Y),N_{\alpha })$$

(17)

for any $X,Y\in \chi \left(M\right)$.

Consider two local orthogonal bases, $\{N_1,N_2,\dots ,N_r\}$ and $\{N_1^{\prime },N_2^{\prime },\dots ,N_r^{\prime }\}$; on normal space $T_x^{\perp }M$, and for the frame $\{N_1,N_2,\dots ,N_r\}$, the decomposition of $N_{\alpha }^{\prime }$ is given by

$$N_{\alpha }^{\prime }=\sum _{\alpha }{k}_{\alpha }^{\prime }{N}_{\gamma }.$$

If $\left(k_{\alpha }^{\prime }\right)$ is an $r\times r$ orthogonal matrix, then for any $\alpha \in \{1,2,\dots ,r\}$, we have

$$u_{\alpha }^{\prime }=\sum _{\gamma }{k}_{\alpha }^{\gamma }{u}_{\gamma }{\xi }_{\gamma }^{\prime }=\sum _{\gamma }{k}_{\alpha }^{\prime }{\xi }_{\gamma }$$

and

$$a_{\alpha \beta }^{\prime }=\sum _{\gamma }{k}_{\alpha }^{\gamma }{a}_{\gamma \delta }{k}_{\beta }^{\delta }$$

Hence, ${\xi }_1,{\xi }_2,\dots ,{\xi }_r$ are vector fields that are linearly independent, which suggests that ${\xi }_1^{\prime },{\xi }_2^{\prime },\dots ,{\xi }_r^{\prime }$ are linearly independent as well.

The decomposition for a normal connection ${\nabla }_X^{\perp }{N}_{\alpha }$ is as follows:

$${\nabla }_X^{\perp }N=\sum _{\beta }{l}_{\alpha \beta }\left(X\right)N_{\beta },$$

(18)

for any $X\in \chi \left(M\right)$. We have obtained an $r\times r$ matrix ${\left(l_{\alpha \beta }\left(X\right)\right)}_r$ of 1-form on submanifold M. Using

$$\widehat{g}(N_{\alpha },N_{\beta })={\delta }_{\alpha \beta },$$

we turn up

$$\widehat{g}({\nabla }_X^{\perp }{N}_{\alpha },N_{\beta })+\widehat{g}(N_{\alpha },{\nabla }_X^{\perp }{N}_{\beta })=0$$

which is equivalent to

$$\widehat{g}(\sum _{\gamma }{l}_{\alpha \gamma }\left(X\right)N_{\gamma },N_{\beta })+\widehat{g}(N_{\alpha },\sum _{\gamma }{l}_{\beta \gamma }\left(X\right)N_{\gamma })=0$$

for any $X\in \chi \left(M\right)$. Thus, we obtain

$$l_{\alpha \gamma }=-l_{\gamma \alpha }$$

(19)

for any $\alpha ,\gamma \in \{1,2,\dots ,r\}$.

Remark 2.

For the upcoming results, we then use the acronym Θ to denote “M is an n-dimensional submanifold of codimension r in a silver Riemannian manifold” for convenience. Consequently, we get the following result:

Theorem 2.

Let Θ be equipped with structure $(\widehat{M},\widehat{g},\theta )$. If θ is parallel with respect to the Levi–Civita connection $\widehat{\nabla }$ associated with the metric $\widehat{g}$, then the structure $\sum =(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ on M induced by the structure θ exhibits the following properties (see [16]):

$$\begin{array}{cc}\hfill \left({\nabla }_{X}R\right)Y& =\sum _{\alpha }{h}_{\alpha }(X,Y){\xi }_{\alpha }+\sum _{\alpha }{u}_{\alpha }\left(Y\right)A_{\alpha }X,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \left({\nabla }_X{u}_{\alpha }\right)Y& =\sum _{\beta }{a}_{\alpha \beta }{h}_{\beta }(X,Y)-h_{\alpha }(X,RY)-\sum _{\beta }{u}_{\beta }\left(Y\right)l_{\alpha \beta }\left(X\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\nabla }_X{\xi }_{\alpha }+R\left(A_{\alpha }X\right)& =\sum _{\beta }{a}_{\alpha \beta }{A}_{\beta }X+\sum _{\beta }{l}_{\alpha \beta }\left(X\right){\xi }_{\beta },\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill X\left(a_{\alpha \beta }\right)+u_{\alpha }\left(A_{\beta }X\right)+u_{\beta }\left(A_{\alpha }X\right)& =\sum _{\gamma }[l_{\alpha \gamma }\left(X\right)a_{\gamma \beta }+l_{\beta \gamma }\left(X\right)a_{\alpha \gamma }],\hfill \end{array}$$

(23)

for any $X,Y\in \chi \left(M\right)$, where $h_{\alpha }(X,Y)=g(A_{\alpha }X,Y)$.

Proof. 

Using condition $\widehat{\nabla }\theta =0,$ we obtain

$$\theta \left({\widehat{\nabla }}_{X}Y\right)={\widehat{\nabla }}_X\theta Y.$$

Using Equation (6), we get from the above equation

$$\theta \left({\widehat{\nabla }}_{X}Y\right)={\widehat{\nabla }}_{X}RY+\sum _{\alpha }\{u_{\alpha }\left(Y\right){\widehat{\nabla }}_X{N}_{\alpha }+N_{\alpha }{\widehat{\nabla }}_X\left(u_{\alpha }\left(Y\right)\right)\}.$$

Using Equations (15) and (16) above, we have

$$\begin{array}{c}\theta \left({\nabla }_{X}Y\right)+\sum _{\alpha }{h}_{\alpha }(X,Y){\theta N}_{\alpha }=\left({\nabla }_{X}R\right)Y+R\left({\nabla }_{X}Y\right)+\sum _{\alpha }{h}_{\alpha }(X,RY)N_{\alpha }\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\sum _{\alpha }{u}_{\alpha }\left(Y\right)A_{\alpha }X+\sum _{\alpha }{u}_{\alpha }\left(Y\right){\nabla }_X^{\perp }{N}_{\alpha }+\left\{\sum _{\alpha }\left({\nabla }_X{u}_{\alpha }\right)Y+\sum _{\alpha }{u}_{\alpha }\left({\nabla }_{X}Y\right)\right\}{N}_{\alpha }.\end{array}$$

Using Equations (6) and (18) above, we get

$$\begin{array}{c}\sum _{\alpha }{h}_{\alpha }(X,Y){\xi }_{\alpha }+\sum _{\alpha }{h}_{\alpha }(X,Y)\sum _{\beta }{a}_{\alpha \beta }{N}_{\beta }=\left({\nabla }_{X}R\right)Y+\sum _{\alpha }{h}_{\alpha }(X,RY)N_{\alpha }\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\sum _{\alpha }{u}_{\alpha }\left(Y\right)A_{\alpha }X+\sum _{\alpha }{u}_{\alpha }\left(Y\right)\left\{\sum _{\beta }{l}_{\alpha \beta }\left(X\right)N_{\beta }\right\}+\sum _{\alpha }\left({\nabla }_X{u}_{\alpha }\right)\left(Y\right)N_{\alpha }.\end{array}$$

After calculating the normal and tangential components, we have

$$\left({\nabla }_{X}R\right)Y=\sum _{\alpha }{h}_{\alpha }(X,Y){\xi }_{\alpha }+\sum _{\alpha }{u}_{\alpha }\left(Y\right)A_{\alpha }X$$

(24)

where $h_{\alpha }(X,Y)=g(A_{\alpha }X,Y)$ and

$$\left({\nabla }_X{u}_{\alpha }\right)Y=\sum _{\beta }{h}_{\alpha }(X,Y)a_{\alpha \beta }-\sum _{\alpha }{u}_{\alpha }\left(Y\right)l_{\alpha \beta }\left(X\right)-h_{\alpha }(X,RY).$$

(25)

Again using (6) and $\widehat{\nabla }\theta =0,$ we get

$$\theta \left({\widehat{\nabla }}_X{N}_{\alpha }\right)={\widehat{\nabla }}_X{\xi }_{\alpha }+\sum _{\beta }\{\left({\widehat{\nabla }}_X{a}_{\alpha \beta }\right)N_{\beta }+a_{\alpha \beta }\left({\widehat{\nabla }}_X{N}_{\beta }\right)\}.$$

By using Equations (15) and (16) in the above equation, we have

$$\begin{array}{c}-\theta \left(A_{\alpha }X\right)+\theta \left({\nabla }_X^{\perp }{N}_{\alpha }\right)={\nabla }_X{\xi }_{\alpha }+\sum _{\alpha }{h}_{\alpha }(X,{\xi }_{\alpha })N_{\alpha }+\sum _{\beta }\left(X{a}_{\alpha \beta }\right)N_{\beta }\hfill \\ \hfill -\sum _{\beta }{a}_{\alpha \beta }{A}_{\beta }X+\sum _{\beta }{a}_{\alpha \beta }{\nabla }_X^{\perp }{N}_{\beta }.\end{array}$$

Using Equations (6) and (18) above, we get

$$\begin{array}{c}-R\left(A_{\alpha }X\right)-\sum _{\alpha }{u}_{\alpha }\left(A_{\alpha }X\right)N_{\alpha }+\sum _{\beta }{l}_{\alpha \beta }\left(X\right){\xi }_{\beta }={\nabla }_X{\xi }_{\alpha }+\sum _{\alpha }{h}_{\alpha }(X,{\xi }_{\alpha })N_{\alpha }\hfill \\ \hfill +\sum _{\beta }\left(X{a}_{\alpha \beta }\right)N_{\beta }-\sum _{\beta }{a}_{\alpha \beta }{A}_{\beta }X+\sum _{\beta }{a}_{\alpha \beta }\left(\sum _{\gamma }{l}_{\alpha \gamma }\left(X\right)N_{\gamma }\right).\end{array}$$

When we contrast the normal and tangential components on both sides, we obtain

$${\nabla }_X{\xi }_{\alpha }+R\left(A_{\alpha }X\right)=\sum _{\beta }{l}_{\alpha \beta }\left(X\right){\xi }_{\beta }+\sum _{\beta }{a}_{\alpha \beta }{A}_{\beta }X$$

and

$$X\left(a_{\alpha \beta }\right)+u_{\alpha }\left(A_{\beta }X\right)+u_{\beta }\left(A_{\alpha }X\right)=\sum _{\gamma }[l_{\alpha \gamma }\left(X\right)a_{\gamma \beta }+l_{\beta \gamma }\left(X\right)a_{\alpha \gamma }].$$

(26)

□

Definition 4

([14]). $(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ is a induced structure on the submanifold M of a silver Riemannian manifold $\widehat{M}$, which is normal if the equality

$$N_R(X,Y)=2\sum _{\alpha }d{u}_{\alpha }(X,Y){\xi }_{\alpha }$$

holds for all vector fields $X,Y\in \chi \left(M\right)$.

Remark 3.

For a Levi=-Civita connection $\widehat{\nabla }$ associated with metric $\widehat{g}$, $\widehat{\nabla }\theta =0$ implies that the structure θ is integrable, i.e., its Nijenhuis torsion tensor vanishes

$$\begin{array}{cc}\hfill N_{\theta }(X,Y)& =[\theta X,\theta Y]+{\theta }^2[X,Y]-\theta [\theta X,Y]-\theta [X,\theta Y],\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill N_{\theta }(X,Y)& =\left(D_{\theta X}\theta \right)Y-\left(D_{\theta Y}\theta \right)X-\theta \{\left(D_X\theta \right)Y-\left(D_Y\theta \right)X\}.\hfill \end{array}$$

(28)

for any $X,Y\in \chi \left(M\right)$.

Proposition 5.

Let Θ with structure $(\widehat{M},\widehat{g},\theta )$. If the induced structure $(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ on M is normal and the normal connection ${\nabla }^{\perp }$ on M identically vanishes (i.e., $l_{\alpha \beta }=0$), then we have (see [16])

$$\sum _{\alpha }g(Y,{\xi }_{\alpha })(R{A}_{\alpha }-A_{\alpha }R)X-\sum _{\alpha }g(X,{\xi }_{\alpha })(R{A}_{\alpha }-A_{\alpha }R)Y=0$$

(29)

Proof. 

Definition 4 gives us

$$N_R(X,Y)=2\sum _{\alpha }d{u}_{\alpha }(X,Y){\xi }_{\alpha }$$

(30)

Using (20) and (21), we get

$$\begin{array}{cc}\hfill \left({\nabla }_{RX}R\right)Y& =\sum _{\alpha }{h}_{\alpha }(RX,Y){\xi }_{\alpha }+\sum _{\alpha }{u}_{\alpha }\left(Y\right)A_{\alpha }RX,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \left({\nabla }_{RY}R\right)X& =\sum _{\alpha }{h}_{\alpha }(RY,X){\xi }_{\alpha }+\sum _{\alpha }{u}_{\alpha }\left(X\right)A_{\alpha }RY,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill R\left({\nabla }_{X}R\right)Y& =\sum _{\alpha }{h}_{\alpha }(X,Y)R{\xi }_{\alpha }+\sum _{\alpha }{u}_{\alpha }\left(Y\right)R{A}_{\alpha }X\hfill \end{array}$$

Using (11) in the above equation, we have

$$R\left({\nabla }_{X}R\right)Y=2\sum _{\alpha }{h}_{\alpha }(X,Y){\xi }_{\alpha }-\sum _{\alpha }{h}_{\alpha }(X,Y)\sum _{\beta }{a}_{\alpha \beta }{\xi }_{\beta }+\sum _{\alpha }{u}_{\alpha }\left(Y\right)R{A}_{\alpha }X$$

(33)

$$R\left({\nabla }_{Y}R\right)X=2\sum _{\alpha }{h}_{\alpha }(Y,X){\xi }_{\alpha }-\sum _{\alpha }{h}_{\alpha }(Y,X)\sum _{\beta }{a}_{\alpha \beta }{\xi }_{\beta }+\sum _{\alpha }{u}_{\alpha }\left(X\right)R{A}_{\alpha }Y$$

(34)

Using Equations (28), (31), (32), (33) and (34) we have

$$\begin{array}{c}{N}_R(X,Y)=\sum _{\alpha }g(A_{\alpha }RX,Y){\xi }_{\alpha }+\sum _{\alpha }g(Y,{\xi }_{\alpha })A_{\alpha }RX-\sum _{\alpha }g(A_{\alpha }RY,X){\xi }_{\alpha }-\sum _{\alpha }g(X,{\xi }_{\alpha })A_{\alpha }RY\hfill \\ -2\sum _{\alpha }g(A_{\alpha }X,Y){\xi }_{\alpha }+\sum _{\alpha }g(A_{\alpha }X,Y)\sum _{\beta }{a}_{\alpha \beta }{\xi }_{\beta }-\sum _{\alpha }g(Y,{\xi }_{\alpha })R{A}_{\alpha }X\\ \hfill +2\sum _{\alpha }g(A_{\alpha }Y,X){\xi }_{\alpha }-\sum _{\alpha }g(A_{\alpha }Y,X)\sum _{\beta }{a}_{\alpha \beta }{\xi }_{\beta }+\sum _{\alpha }g(X,{\xi }_{\alpha })R{A}_{\alpha }Y\end{array}$$

As we know that $g(A_{\alpha }RX,Y)=g\left(R(A_{\alpha }X,Y)\right)$, we have from above

$$\begin{array}{c}{N}_R(X,Y)=-\sum _{\alpha }g((R{A}_{\alpha }-A_{\alpha }R,X),Y){\xi }_{\alpha }-\sum _{\alpha }g(Y,{\xi }_{\alpha })(R{A}_{\alpha }-A_{\alpha }R)X\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\sum _{\alpha }g(X,{\xi }_{\alpha })(R{A}_{\alpha }-A_{\alpha }R)Y+\sum _{\beta }[g(A_{\alpha }X,Y)-g(A_{\alpha }Y,X)](a_{\alpha \beta }{\xi }_{\beta }-2)\end{array}$$

As $g(A_{\alpha }X,Y)=g(A_{\alpha }Y,X)$, we have

$$\begin{array}{c}{N}_R(X,Y)=-\sum _{\alpha }g((R{A}_{\alpha }-A_{\alpha }R,X),Y){\xi }_{\alpha }-\sum _{\alpha }g(Y,{\xi }_{\alpha })(R{A}_{\alpha }-A_{\alpha }R)X\hfill \\ \hfill +\sum _{\alpha }g(X,{\xi }_{\alpha })(R{A}_{\alpha }-A_{\alpha }R)Y,\end{array}$$

(35)

for any $X,Y\in \chi \left(M\right)$.

Now, from

$$2d{u}_{\alpha }(X,Y)=X\left(u_{\alpha }\left(Y\right)\right)-Y\left(u_{\alpha }\left(X\right)\right)-u_{\alpha }[X,Y]$$

we have

$$2d{u}_{\alpha }(X,Y)=\left({\nabla }_X{u}_{\alpha }\right)Y-\left({\nabla }_Y{u}_{\alpha }\right)X$$

(36)

for any $X,Y\in \chi \left(M\right)$, and using (21), we obtain

$$\begin{array}{c}2d{u}_{\alpha }(X,Y)=g(A_{\alpha }Y,RX)-g(A_{\alpha }X,RY)+\sum _{\beta }g(A_{\beta }X,Y)a_{\alpha \beta }\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\sum _{\beta }g(A_{\beta }Y,X)a_{\alpha \beta }-\sum _{\beta }{u}_{\beta }\left(Y\right)l_{\alpha \beta }\left(X\right)+\sum _{\beta }{u}_{\beta }\left(X\right)l_{\alpha \beta }\left(Y\right)\end{array}$$

As $g(A_{\beta }X,Y)=g(A_{\beta }Y,X)$, we have from above

$$2d{u}_{\alpha }(X,Y)=-g((R{A}_{\alpha }-A_{\alpha }R)X,Y)+\sum _{\beta }[u_{\beta }\left(X\right)l_{\alpha \beta }\left(Y\right)-u_{\beta }\left(Y\right)l_{\alpha \beta }\left(X\right)]$$

(37)

Using Definition 4 and Equations (35) and (37), we have

$$\sum _{\alpha }g(Y,{\xi }_{\alpha })(R{A}_{\alpha }-A_{\alpha }R)X-\sum _{\alpha }g(X,{\xi }_{\alpha })(R{A}_{\alpha }-A_{\alpha }R)Y=\sum _{\beta }[g(X,{\xi }_{\alpha })l_{\alpha \beta }\left(Y\right)-g(Y,{\xi }_{\alpha })l_{\alpha \beta }\left(X\right)]{\xi }_{\alpha }$$

for every $X,Y\in \chi \left(M\right)$.

Considering, $l_{\alpha \beta }=0$, i.e., the normal connection ${\nabla }^{\perp }$ on M vanishes, we get the result. □

Proposition 6.

Assuming the above result, Proposition 6 is independent of the choice of basis in the normal space $T_x^{\perp }\left(M\right)$ for any $x\in M$.

Proof. 

Consider another basis $\left\{N_{\alpha }^{\prime }\right\}$ in $T_x^{\perp }\left(M\right)$; then we know that

$$N_{\alpha }^{\prime }=\sum _{\beta }{O}_{\alpha \beta }{N}_{\alpha }$$

(38)

where $O_{\alpha \beta }$ is an orthogonal matrix.

We know the condition

$${\overline{\nabla }}_X{N}_{\alpha }^{\prime }=0,$$

then using Equation (38) above, we get

$${\overline{\nabla }}_X\sum _{\beta }{O}_{\alpha \beta }{N}_{\alpha }=0$$

$$\sum _{\beta }{\overline{\nabla }}_X\left(O_{\alpha \beta }\right)N_{\alpha }+\sum _{\beta }{O}_{\alpha \beta }{{\overline{\nabla }}_{X}N}_{\alpha }=0,$$

$$\sum _{\beta }{\overline{\nabla }}_X\left(O_{\alpha \beta }\right)N_{\alpha }=0$$

(39)

$$\sum _{\beta }X\left(O_{\alpha \beta }\right)N_{\alpha }=0$$

(40)

for $x\in M$.

This gives $O_{\alpha \beta }=$ constant, because $\left\{N_{\beta }\right\}$ is a linearly independent set other than above. From (16)

$${\overline{\nabla }}_X{N}_{\alpha }^{\prime }=-A_{\alpha }^{\prime }X$$

and

$${\overline{\nabla }}_X{N}_{\alpha }^{\prime }=\sum _{\beta }{\overline{\nabla }}_X\left(O_{\alpha \beta }\right)N_{\beta }+\sum _{\beta }{O}_{\alpha \beta }{{\overline{\nabla }}_{X}N}_{\beta }$$

This implies that

$${\overline{\nabla }}_X{N}_{\alpha }^{\prime }=\sum _{\beta }{\overline{\nabla }}_X\left(O_{\alpha \beta }\right)N_{\beta }-\sum _{\beta }{O}_{\alpha \beta }{A}_{\beta }X$$

(41)

Using (39) in the above, we get

$$A_{\alpha }^{\prime }X=\sum _{\beta }{O}_{\alpha \beta }{A}_{\beta }X$$

(42)

Now, from (6) and using (38), we have

$$\theta \left(N_{\alpha }^{\prime }\right)=i\left({\xi }_{\alpha }^{\prime }\right)+\sum _{\beta }\sum _{\gamma }{a}_{\alpha \beta }^{\prime }{O}_{\alpha \beta }{N}_{\gamma }$$

(43)

From (38), we can write

$$\theta \left(N_{\alpha }^{\prime }\right)=\sum _{\beta }{O}_{\alpha \beta }\theta \left(N_{\alpha }\right)$$

(44)

Using again (6) in the above, we get

$$\theta \left(N_{\alpha }^{\prime }\right)=i(\sum _{\beta }{O}_{\alpha \beta }{\xi }_{\alpha })+\sum _{\beta }\sum _{\gamma }\left(O_{\alpha \beta }{a}_{\alpha \gamma }{N}_{\gamma }\right)$$

(45)

We derive from (43) and (45)

$${\xi }_{\alpha }^{\prime }=\sum _{\beta }{O}_{\alpha \beta }{\xi }_{\alpha }$$

(46)

and

$$\sum _{\beta }\sum _{\gamma }{a}_{\alpha \beta }^{\prime }{O}_{\alpha \beta }{N}_{\gamma }=\sum _{\beta }\sum _{\gamma }\left(O_{\alpha \beta }{a}_{\alpha \gamma }{N}_{\gamma }\right).$$

(47)

With $\alpha =1,2,\dots ,r$, based on $\left\{N_{\alpha }^{\prime }\right\}$, the criterion for Proposition 7 now becomes

$$\sum _{\alpha }g(Y,{\xi }_{\alpha }^{\prime })(R{A}_{\alpha }^{\prime }-A_{\alpha }^{\prime }R)X-\sum _{\alpha }g(X,{\xi }_{\alpha }^{\prime })(R{A}_{\alpha }^{\prime }-A_{\alpha }^{\prime }R)Y=0$$

Using (42) and (46) in the above gives

$$\sum _{\alpha }{O}_{\alpha \beta }{O}_{\beta \gamma }[g(Y,{\xi }_{\beta })(R{A}_{\gamma }-A_{\gamma }R)X-g(X,{\xi }_{\beta })(R{A}_{\gamma }-A_{\gamma }R)Y]=0$$

Due to orthogonality of $O_{\alpha \beta }$, the above equation can be written as

$$\sum _{\alpha }[g(Y,{\xi }_{\beta })(R{A}_{\gamma }-A_{\gamma }R)X-g(X,{\xi }_{\beta })(R{A}_{\gamma }-A_{\gamma }R)Y]=0$$

This implies that Proposition 6 is independent of the choice of basis in the normal space $T_x^{\perp }\left(M\right)$ for any $x\in M$. □

Lemma 1.

Let Θ. Consider an induced structure $(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ on M; then for all $X,Y\in \mathfrak{X}\left(M\right)$, the expression $g((R{A}_{\alpha }-A_{\alpha }R)X,Y)$ on M is skew-symmetric.

Proof. 

$$\begin{array}{cc}\hfill g\left(\left(R{A}_{\alpha }-A_{\alpha }R\right)X,Y\right)& =g(R{A}_{\alpha }X,Y)-g(A_{\alpha }RX,Y)\hfill \\ \hfill & =g\left(X,R{A}_{\alpha }Y\right)-g(X,A_{\alpha }RY)\hfill \\ \hfill & =-\left[g\left(X,A_{\alpha }RY\right)-g\left(X,R{A}_{\alpha }Y\right)\right]\hfill \\ \hfill & =-g\left((A_{\alpha }R-R{A}_{\alpha })Y,X\right).\hfill \end{array}$$

⟹$g\left(\left(R{A}_{\alpha }-A_{\alpha }R\right)X,Y\right)$ is skew-symmetric for any $X,Y\in \chi \left(M\right)$. □

Proposition 7.

Let Θ$(r\ge 2)$ in a silver Riemannian manifold with structure $(\widehat{M},\widehat{g},\theta )$, where the structure θ is parallel with respect to the Levi–Civita connection $\widehat{\nabla }$ on $\widehat{M}$ with an induced structure $(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ defined on M. If the normal connection ${\nabla }^{\perp }$ vanishes identically on $T^{\perp }\left(M\right)$ (i.e., $l_{\alpha \beta }=0$), then the tangent vector fields $\left\{{\xi }_{\alpha }\right\}$ for any $\alpha \in \{1,2,\dots ,r\}$ are linearly independent if and only if $det(I_r+2A-A^2)$ does not vanish for all $x\in M$ ($I_r$ being an $(r\times r)$ identity matrix) (see [16]).

Proof. 

Let $k_1,k_2,\dots ,k_r$ be real numbers such that

$$k_1{\xi }_1+k_2{\xi }_2+\cdots +k_r{\xi }_r=0$$

(48)

at any point $x\in M$.

From Equations (10) and (12), we obtain

$$g({\xi }_{\alpha },{\xi }_{\beta })=u_{\alpha }\left({\xi }_{\alpha }\right)={\delta }_{\alpha \beta }+2a_{\alpha \beta }-\sum _{\gamma }{a}_{\alpha \gamma }{a}_{\gamma \beta }.$$

(49)

Taking the inner product of (48) with ${\xi }_{\alpha }$ for any $\alpha \in \{1,2,\dots ,r\}$, we obtain

$$k_{1}g({\xi }_1,{\xi }_{\alpha })+k_{2}g({\xi }_2,{\xi }_{\alpha })+\cdots +k_{r}g({\xi }_r,{\xi }_{\alpha })=0.$$

Using (3.42) in the above equation, we obtain the following system:

$$\left\{\begin{array}{c}{k}_1\left(1+2a_{11}-\sum _{\gamma }{a}_{1\gamma }{a}_{\gamma 1}\right)+k_2\left(2a_{12}-\sum _{\gamma }{a}_{1\gamma }{a}_{\gamma 2}\right)+\cdots +k_r\left(2a_{1r}-\sum _{\gamma }{a}_{1\gamma }{a}_{\gamma r}\right)=0\hfill \\ k_1\left(2a_{21}-\sum _{\gamma }{a}_{2\gamma }{a}_{\gamma 1}\right)+k_2\left(1+2a_{22}-\sum _{\gamma }{a}_{2\gamma }{a}_{\gamma 2}\right)+\cdots +k_r\left(2a_{2r}-\sum _{\gamma }{a}_{2\gamma }{a}_{\gamma r}\right)=0\hfill \\ ⋮\hfill \\ k_1\left(2a_{r1}-\sum _{\gamma }{a}_{r\gamma }{a}_{\gamma 1}\right)+k_2\left(2a_{r2}-\sum _{\gamma }{a}_{r\gamma }{a}_{\gamma 2}\right)+\cdots +k_r\left(1+2a_{rr}-\sum _{\gamma }{a}_{r\gamma }{a}_{\gamma r}\right)=0.\hfill \end{array}\right.$$

This system has the unique solution $k_1=k_2=\cdots =k_r=0$ if and only if its determinant is non-zero. The determinant is that of the matrix $I_r+2A-A^2$. □

We study some characterizations of any submanifold of a locally decomposable silver Riemannian manifold in which the codimension of the submanifold is greater than or equal to the rank of the set of tangent vector fields of the induced structure on it by the metallic or silver Riemannian structure of the ambient manifold.

The choice of codimension 2 for submanifolds is principally motivated by the algebraic nature of the structures and the geometric quality of the associated attributes. An almost product structure divides tangent space into two complementary distributions. In many geometric applications, a codimension 2 submanifold permits a natural interaction between the manifold’s normal space and these two distributions. Specifically, silver structures have bear resemblance to almost complex structures ($J^2=-I$). Just as complex geometry frequently highlights codimension 2 (complex hypersurfaces), golden geometry uses this codimension to investigate “silver-like” phenomena in a comparable structural context.

Remark 4.

Now, we use the acronym Ξ to denote n-dimensional submanifold M with codimension 2 over a silver Riemannian structure $(\widehat{M},\widehat{g},\theta )$ for convenience. Consequently, we get the following result:

Lemma 2.

Consider an Ξ with induced normal structure $(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ and structure θ being parallel with a respective Levi–Civita connection $\overline{\nabla }$. If the normal connection ${\nabla }^{\perp }$ on M vanishes identically (i.e., $l_{\alpha \beta }=0$), then

$$\begin{array}{c}g(Y,{\xi }_1)(R{A}_1-A_{1}R)X+g(Y,{\xi }_2)(R{A}_2-A_{2}R)X\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+g((R{A}_1-A_{1}R)X,Y){\xi }_1+g((R{A}_2-A_{2}R)X,Y){\xi }_2=0\end{array}$$

(50)

for all $X,Y\in \chi \left(M\right)$.

Proof. 

From Proposition 5 it follows that

$$g(Y,{\xi }_1)(R{A}_1-A_{1}R)X+g(Y,{\xi }_2)(R{A}_2-A_{2}R)X$$

$$=g(X,{\xi }_1)(R{A}_1-A_{1}R)Y+g(X,{\xi }_2)(R{A}_2-A_{2}R)Y$$

for all $X,Y\in \chi \left(M\right)$.

Taking the inner product with $Z\in \chi \left(M\right)$:

$$\begin{array}{c}g(Y,{\xi }_1)g((R{A}_1-A_{1}R)X,Z)+g(Y,{\xi }_2)g((R{A}_2-A_{2}R)X,Z)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=g(X,{\xi }_1)g((R{A}_1-A_{1}R)Y,Z)+g(X,{\xi }_2)g((R{A}_2-A_{2}R)Y,Z).\end{array}$$

(51)

Interchanging Z and Y gives

$$\begin{array}{c}g(Z,{\xi }_1)g((R{A}_1-A_{1}R)X,Y)+g(Z,{\xi }_2)g((R{A}_2-A_{2}R)X,Y)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=g(X,{\xi }_1)g((R{A}_1-A_{1}R)Z,Y)+g(X,{\xi }_2)g((R{A}_2-A_{2}R)Z,Y).\end{array}$$

(52)

Adding (50) and (51), we obtain

$$\begin{array}{c}g\left(\right[g(Y,{\xi }_1)(R{A}_1-A_{1}R)X+g(Y,{\xi }_2)(R{A}_2-A_{2}R)X\hfill \\ \hfill +g((R{A}_1-A_{1}R)X,Y){\xi }_1\hfill \\ \hfill +g((R{A}_2-A_{2}R)X,Y){\xi }_2],Z)=0\end{array}$$

for all $Z\in \chi \left(M\right)$. Thus,

$$g(Y,{\xi }_1)(R{A}_1-A_{1}R)X+g(Y,{\xi }_2)(R{A}_2-A_{2}R)X$$

$$+g((R{A}_1-A_{1}R)X,Y){\xi }_1+g((R{A}_2-A_{2}R)X,Y){\xi }_2=0.$$

□

Lemma 3.

Consider an Ξ with induced normal structure $(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ and structure θ being parallel with respect to Levi–Civita connection $\overline{\nabla }$. If the normal connection ${\nabla }^{\perp }$ on M vanishes identically (i.e., $l_{\alpha \beta }=0$) and $\sigma \ne 0$, then

$$\left.\begin{array}{cc}\hfill (R{A}_1-A_{1}R){\xi }_1& =0\hfill \\ \hfill (R{A}_2-A_{2}R){\xi }_2& =0\hfill \\ \hfill (R{A}_1-A_{1}R){\xi }_2& =0\hfill \\ \hfill (R{A}_2-A_{2}R){\xi }_1& =0\hfill \end{array}\right\}$$

(53)

Proof. 

By Equation (48) and using a similar process as in Lemma 2 (for more details see [16]), we obtain the results. □

Lemma 4.

Consider Ξ with an induced normal structure $(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ and structure θ being parallel with respect to Levi–Civita connection $\overline{\nabla }$. If the normal connection ${\nabla }^{\perp }$ on M vanishes identically (i.e., $l_{\alpha \beta }=0$), $\sigma \ne 0$, and $traceA=0$, then the tensor field R commutes with the Weingarten operator $A_{\alpha }$, $\alpha \in \{1,2\}$ if we have

$$\begin{array}{cc}\hfill (R{A}_1-A_{1}R)X& =0\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill (R{A}_2-A_{2}R)X& =0\hfill \end{array}$$

(55)

for any $X\in \chi \left(M\right)$.

Proof. 

We can write

$$g((R{A}_{\alpha }-A_{\alpha }R)X,{\xi }_{\beta })=-g((R{A}_{\alpha }-A_{\alpha }R){\xi }_{\beta },X)$$

(56)

where $\alpha ,\beta \in \{1, 2\}$.

From Lemma 3 we have

$$(R{A}_{\alpha }-A_{\alpha }R){\xi }_{\beta }=0$$

for any $\alpha ,\beta \in \{1, 2\}$.

Using this relation in (56), we have

$$(R{A}_1-A_{1}R)X=0.$$

Similarly,

$$(R{A}_2-A_{2}R)X=0.$$

□

Now, for an n-dimensional submanifold M of codimension 2 over a silver Riemannian structure $(\widehat{M},\widehat{g},\theta )$ with an induced normal structure $(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_2)$ for $\alpha ,\beta \in \{1, 2\}$, suppose that the normal connection ${\nabla }^{\perp }$ on M vanishes identically (i.e., $l_{\alpha \beta }=0$); then relations in Proposition 4 take the following form:

$$\begin{array}{cc}\hfill R^2\left(X\right)& =2R\left(X\right)+X-u_1\left(X\right){\xi }_1-u_2\left(X\right){\xi }_2\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill u_1\left(RX\right)& =2u_1\left(X\right)-a_{11}{u}_1\left(X\right)-a_{12}{u}_2\left(X\right)\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill u_2\left(RX\right)& =2u_2\left(X\right)-a_{21}{u}_1\left(X\right)-a_{22}{u}_2\left(X\right)\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill u_1\left({\xi }_1\right)& =1+2a_{11}-a_{11}^2-a_{12}^2\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill u_2\left({\xi }_2\right)& =1+2a_{22}-a_{12}^2-a_{22}^2\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill R{\xi }_1& =2{\xi }_1-a_{11}{\xi }_1-a_{12}{\xi }_2\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill R{\xi }_2& =2{\xi }_2-a_{21}{\xi }_1-a_{22}{\xi }_2\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill u_1\left({\xi }_2\right)=u_2\left({\xi }_1\right)& =a_{12}(2-a_{11}-a_{22})\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill g(RX,RY)& =2g(X,RY)+g(X,Y)+u_1\left(X\right)u_1\left(Y\right)+u_2\left(X\right)u_2\left(Y\right)\hfill \end{array}$$

(65)

for every $X,Y\in \chi \left(M\right)$. We denote $A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)$.

Moreover, under the condition that the normal connection ${\nabla }^{\perp }$ on M vanishes identically (i.e., $l_{\alpha \beta }=0$), Theorem 2 leads to

$$\begin{array}{cc}\hfill \left({\nabla }_{X}R\right)Y& =h_1(X,Y){\xi }_1+h_2(X,Y){\xi }_2+u_1\left(Y\right)A_{1}X+u_2\left(Y\right)A_{2}X\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill \left({\nabla }_X{u}_1\right)Y& =a_{11}{h}_1(X,Y)+a_{12}{h}_2(X,Y)-h_1(X,RY)\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill \left({\nabla }_X{u}_2\right)Y& =a_{21}{h}_1(X,Y)+a_{22}{h}_2(X,Y)-h_2(X,RY)\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill {\nabla }_X{\xi }_1+R\left(A_{1}X\right)& =a_{11}{A}_{1}X+a_{12}{A}_{2}X\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill {\nabla }_X{\xi }_2+R\left(A_{2}X\right)& =a_{21}{A}_{1}X+a_{22}{A}_{2}X\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill X\left(a_{11}\right)& =-2u_1\left(A_{1}X\right)\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill X\left(a_{22}\right)& =-2u_2\left(A_{2}X\right)\hfill \end{array}$$

(72)

for any $X,Y\in \chi \left(M\right)$, where $h_1(X,Y)=g(A_{1}X,Y)$ and $h_2(X,Y)=g(A_{2}X,Y)$.

Remark 5.

A simpler assumption for these above relations is $a_{11}+a_{22}=0$. Thus, $traceA=0$. Under this assumption, if we denote $a_{11}=-a_{22}=a$, $a_{12}=-a_{21}=b$, and $1-a^2-b^2=\sigma$ from (57) to (65), we can write

$$\begin{array}{cc}\hfill u_1\left(RX\right)& =(2-a)u_1\left(X\right)-b{u}_2\left(X\right),\hfill \\ \hfill u_2\left(RX\right)& =(2-a)u_2\left(X\right)-b{u}_1\left(X\right),\hfill \\ \hfill u_1\left({\xi }_2\right)=u_2\left({\xi }_1\right)& =2b,\hfill \\ \hfill u_1\left({\xi }_1\right)=u_2\left({\xi }_2\right)& =a+\sigma ⟺g({\xi }_1,{\xi }_1)=g({\xi }_2,{\xi }_2)=a+\sigma ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill R{\xi }_1& =(2-a){\xi }_1-b{\xi }_2\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill R{\xi }_2& =(2-a){\xi }_2-b{\xi }_1\hfill \end{array}$$

(74)

Proposition 8.

Consider an Ξ with an induced normal structure $(R,g,u_{\alpha },{\xi }_{\alpha },{\left(a_{\alpha \beta }\right)}_r)$ and structure θ being parallel with respect to Levi–Civita connection $\overline{\nabla }$. If the normal connection ${\nabla }^{\perp }$ on M vanishes identically (i.e., $l_{\alpha \beta }=0$), $traceA=0$, and $\sigma \ne 0$, then we have

$$\begin{array}{cc}\hfill (2a+\sigma )A_1{\xi }_1+2b{A}_1{\xi }_2& =h_1({\xi }_1,{\xi }_1){\xi }_1+h_1({\xi }_1,{\xi }_2){\xi }_2\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill (2a+\sigma )A_1{\xi }_2+2b{A}_1{\xi }_2& =h_1({\xi }_1,{\xi }_2){\xi }_1+h_1({\xi }_2,{\xi }_2){\xi }_2\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill (2a+\sigma )A_2{\xi }_1+2b{A}_2{\xi }_2& =h_2({\xi }_1,{\xi }_1){\xi }_1+h_2({\xi }_1,{\xi }_2){\xi }_2\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill (2a+\sigma )A_2{\xi }_2+2b{A}_2{\xi }_2& =h_2({\xi }_1,{\xi }_2){\xi }_1+h_2({\xi }_2,{\xi }_2){\xi }_2\hfill \end{array}$$

(78)

Proof. 

Applying R to both sides of (54):

$$R^2{A}_{1}X-R{A}_{1}RX=0$$

for any $X\in \chi \left(M\right)$.

Putting $X={\xi }_1$ and using (57):

$$2R\left(A_1{\xi }_1\right)+A_1{\xi }_1-u_1\left(A_1{\xi }_1\right){\xi }_1-u_2\left(A_1{\xi }_1\right){\xi }_2=R{A}_{1}R{\xi }_1.$$

Using (73):

$$\begin{array}{c}\hfill (5-2R)A_1{\xi }_1+(R-2)a{A}_1{\xi }_1+(R-2)b{A}_1{\xi }_2=h_1({\xi }_1,{\xi }_1){\xi }_1+h_1({\xi }_1,{\xi }_2){\xi }_2.\end{array}$$

(79)

Putting $X={\xi }_2$ and using (57):

$$2R\left(A_1{\xi }_2\right)+A_1{\xi }_2-u_1\left(A_1{\xi }_2\right){\xi }_1-u_2\left(A_1{\xi }_2\right){\xi }_2=R{A}_{1}R{\xi }_2.$$

Using (74):

$$\begin{array}{c}\hfill (5-2R)A_1{\xi }_2+(R-2)a{A}_1{\xi }_2+(R-2)b{A}_1{\xi }_1=h_1({\xi }_1,{\xi }_2){\xi }_1+h_1({\xi }_2,{\xi }_2){\xi }_2.\end{array}$$

(80)

Replacing X by $RX$ in (42):

$$R{A}_{1}RX-A_1{R}^{2}X=0.$$

Putting $X={\xi }_1$ and using (57) and (41):

$$[2R-5+2a+u_1\left({\xi }_1\right)]A_1{\xi }_1+[2-R]a{A}_1{\xi }_1+[2b-Rb+u_2\left({\xi }_1\right)]A_1{\xi }_2=0.$$

As $u_2\left({\xi }_1\right)=2b$ and $u_1\left({\xi }_1\right)=a+\sigma$:

$$(2R-5+2a+\sigma )A_1{\xi }_1+(2-R)a{A}_1{\xi }_1+(4-R)b{A}_1{\xi }_2=0.$$

(81)

Putting $X={\xi }_2$ and using (57) and (74):

$$(2R-5+2a+\sigma )A_1{\xi }_2+(2-R)a{A}_1{\xi }_2+(4-R)b{A}_1{\xi }_2=0.$$

(82)

Adding (79) and (81):

$$(2a+\sigma )A_1{\xi }_1+2b{A}_1{\xi }_2=h_1({\xi }_1,{\xi }_1){\xi }_1+h_1({\xi }_1,{\xi }_2){\xi }_2.$$

(83)

Adding (80) and (82):

$$(2a+\sigma )A_1{\xi }_2+2b{A}_1{\xi }_2=h_1({\xi }_1,{\xi }_1){\xi }_1+h_1({\xi }_1,{\xi }_2){\xi }_2.$$

(84)

Similarly, applying R to (55) and following the same process yields (77) and (78). □

Theorem 3.

Consider Θ being parallel with respect to Levi–Civita connection $\widehat{\nabla }$ on $\widehat{M}$ (i.e., $\widehat{\nabla }\theta =0$). If the vector fields ${\xi }_{\alpha }$ ($\alpha =1,2,3,\dots ,r$) are linearly independent, the tensor $T_r\left(R\right)$ is constant and the submanifold M is totally umbilical; thus, M must be totally geodesic (see [16]).

Proof. 

From equality (23),

$${\nabla }_X\left(a_{\alpha \beta }\right)+u_{\alpha }\left(A_{\beta }X\right)+u_{\beta }\left(A_{\alpha }X\right)=\sum _{\gamma }[l_{\alpha \gamma }\left(X\right)a_{\gamma \beta }+l_{\beta \gamma }\left(X\right)a_{\alpha \gamma }].$$

(85)

Taking $\alpha =\beta$,

$${\nabla }_X\left(a_{\beta \beta }\right)+2u_{\beta }\left(A_{\beta }X\right)=\sum _{\gamma }[l_{\beta \gamma }\left(X\right)a_{\gamma \beta }+l_{\beta \gamma }\left(X\right)a_{\beta \gamma }].$$

(86)

Since $l_{\alpha \beta }$ is skew-symmetric and $a_{\alpha \beta }$ is symmetric, ${\sum }_{\alpha \gamma }\left[l_{\alpha \gamma }\left(X\right)a_{\alpha \gamma }\right]=0$. Given $T_r\left(R\right)=$ constant, we have ${\sum }_{\alpha }{a}_{\alpha \alpha }=$ constant.

$$\sum _{\beta }{A}_{\beta }{\xi }_{\beta }=0.$$

Since ${\xi }_{\beta }$ are linearly independent, we have $A_{\beta }=0$. Hence M is totally geodesic. □

Theorem 4.

Consider Θ being parallel with respect to Levi–Civita connection $\widehat{\nabla }$ on $\widehat{M}$ (i.e., $\widehat{\nabla }\theta =0$). If the vector fields ${\xi }_{\alpha }$ ($\alpha =1,2,3,\dots ,r$) are linearly independent, the tensor $T_r\left(R\right)$ is constant and ${\sum }_{\phi }\left({\nabla }_{e_{\phi }}R\right)e_{\phi }=0$; then M is minimal (see [16]).

Proof. 

From equality (20),

$$\left({\nabla }_{X}R\right)Y=\sum _{\alpha }[h_{\alpha }(X,Y){\xi }_{\alpha }+u_{\alpha }\left(Y\right)A_{\alpha }X].$$

(87)

Taking $X=Y=e_{\phi }$,

$$\sum _{\phi }\left({\nabla }_{e_{\phi }}R\right)e_{\phi }=\sum _{\alpha }\left[A_{\alpha }\sum _{\phi }{u}_{\alpha }\left(e_{\phi }\right)e_{\phi }+\sum _{\phi }{h}_{\alpha }(e_{\phi },e_{\phi }){\xi }_{\alpha }\right].$$

Using (22),

$$\sum _{\phi }\left({\nabla }_{e_{\phi }}R\right)e_{\phi }=\sum _{\alpha }[A_{\alpha }{\xi }_{\alpha }+\sum _{\phi }{h}_{\alpha }(e_{\phi },e_{\phi }){\xi }_{\alpha }].$$

Given $T_r\left(R\right)=$ constant, from Theorem 3

$$\sum _{\alpha }{A}_{\alpha }{\xi }_{\alpha }=0.$$

Thus,

$$\sum _{\alpha }\sum _{\phi }{h}_{\alpha }(e_{\phi },e_{\phi }){\xi }_{\alpha }=0.$$

Since ${\xi }_{\alpha }$ are linearly independent, we have $h_{\alpha }(e_{\phi },e_{\phi })=0$. It follows that M is minimal. □

Lemma 5.

Let Θ be endowed with structure θ. If ${\xi }_{\alpha }$ ($\alpha =1,2,3,\dots ,r$) are linearly independent, then

$$T_r\left(R\right)=-T_r\left(a_{\alpha \beta }\right),$$

where $(r=n)$.

Lemma 6.

Let Θ endowed with structure θ. If ${\xi }_{\alpha }$ ($\alpha =1,2,3,\dots ,r$) are linearly independent and ${\nabla }_{X}R=0$, then $T_r\left(a_{\alpha \beta }\right)$ is constant.

Example 1

(See [8]). Consider ambient space to be a $(2p+q)$-dimensional Euclidian space $E^{2p+q},(p,q\in N)$. Let $\theta :E^{2p+q}\to E^{2p+q}$ be an (1, 1) tensor field defined by

$$\theta (x^i,y^i,z^j)=(\phi x^i,\phi y^i,(2-\phi )z^j),$$

where $i\in \{1,\dots ,p\}$ and $j\in \{1,\dots ,q\}$.

For every point $(x^i,y^i,z^j)\in E^{2p+q}$, $\phi =1+\sqrt{2}$ and $(2-\phi )=1-\sqrt{2}$ are the two roots of the equality $x^2=2x+1$.

On the other hand, for $(x^i,y^i,z^j)\in E^{2p+q},$ where $i\in \{1,\dots ,p\}$ and $j\in \{1,\dots ,q\}$, we have

$$\begin{array}{cc}\hfill {\theta }^2(x^i,y^i,z^j)& =({\phi }^2{x}^i,{{\phi }^{2}y}^i,{(2-\phi )}^2{z}^j)\hfill \\ \hfill {\theta }^2(x^i,y^i,z^j)& =(x^i,y^i,z^j)+2(\phi x^i,\phi y^i,(2-\phi )z^j)\hfill \\ \hfill {\theta }^2& =2\theta +I.\hfill \end{array}$$

For $(x^i,y^i,z^j),(u^i,v^i,w^j)\in E^{2p+q}$, we have

$$\langle \theta (x^i,y^i,z^j),(u^i,v^i,w^j)\rangle =\langle (x^i,y^i,z^j),\theta (u^i,v^i,w^j)\rangle$$

This shows that product $\langle ,\rangle$ on $E^{2p+q}$ is $\theta$-compatible.

Thus, silver structure $\theta ,$ defined on $(E^{2p+q},\langle ,\rangle )$ and $(E^{2p+q},\langle .\rangle ,\theta )$ is a silver Riemannian manifold.

In the above point, we recognize $iX$ with X (where $X\in \mathfrak{X}\left(E^{2p+q}\right))$.

Evidently, $E^{2p+q}=E^p\times E^p\times E^q$, and we have a hypersurface

$$\begin{array}{cc}\hfill S^{p-1}\left(r_1\right)& =\left\{\left(x^i\right),\sum _{i=1}^p{\left(x^i\right)}^2={r_1}^2\right\},\hfill \\ \hfill S^{p-1}\left(r_2\right)& =\left\{\left(y^i\right),\sum _{i=1}^p{\left(x^i\right)}^2={r_2}^2\right\},\hfill \\ \hfill S^{q-1}\left(r_3\right)& =\left\{\left(z^j\right),\sum _{j=1}^q{\left(x^j\right)}^2={r_3}^2\right\},\hfill \end{array}$$

for each space $E^p$ and $E^q$, respectively, where ${r_1}^2+{r_2}^2+{r_3}^2=R^2$, $i\in \{1,\dots ,p\}$ and $j\in \{1,\dots ,q\}$.

We construct product manifold $S^{p-1}\left(r_1\right)\times S^{p-1}\left(r_2\right)\times S^{q-1}\left(r_3\right)$. Every point of $S^{p-1}\left(r_1\right)\times S^{p-1}\left(r_2\right)\times S^{q-1}\left(r_3\right)$ has the coordinate $(x^i,y^i,z^j)$ for $(i\in \{1,\dots ,p\}$ and $j\in \{1,\dots ,q\})$ such that

$$\sum _{i=1}^p{\left(x^i\right)}^2+\sum _{i=1}^p{\left(x^i\right)}^2+\sum _{j=1}^q{\left(x^j\right)}^2={R_s}^2.$$

Thus, $S^{p-1}\left(r_1\right)\times S^{p-1}\left(r_2\right)\times S^{q-1}\left(r_3\right)$ is a submanifold of codimension 3 in $E^{2p+q}$, a submanifold of codimension 2 in $S^{2p+q-1}\left(R_s\right)$, and a hypersurface in $S^{2p+q-2}\left(r\right).$

Therefore, we have

$$S^{p-1}\left(r_1\right)\times S^{p-1}\left(r_2\right)\times S^{q-1}\left(r_3\right)\to S^{2p+q-2}\left(r\right)\to S^{2p+q-1}\left(R_s\right)\to E^{2p+q}$$

Also, its tangent space $T_{(x^i,y^i,z^j)}M$ at a point $(x^i,y^i,z^j)$ is as given below:

$$T_{(x^i,0^i,0^j)}{S}^{p-1}\left(r_1\right)\oplus T_{(0^i,y^i,0^j)}{S}^{p-1}\left(r_2\right)\oplus T_{(0^i,0^i,z^j)}{S}^{q-1}\left(r_3\right).$$

We can see that any tangent vector $(X^i,Y^i,Z^j)\in T_{(x^i,y^i,z^j)}{E}^{2p+q}$ belongs to $T_{(x^i,y^i,z^j)}M$ for every point $(x^i,y^i,z^j)\in M$ if and only if

$$\sum _{i=1}^p{x}^i,X^i=\sum _{i=1}^p{y}^i,Y^i=\sum _{i=1}^q{z}^i,Z^i=0.$$

In addition, since $(X^i,Y^i,Z^j)$ is a tangent vector on the sphere $S^{2p+q-1}\left(R_s\right)$,

$$T_{(x^i,y^i,z^j)}M\subset T_{(x^i,y^i,z^j)}{S}^{2p+q-1}\left(R_s\right)$$

for every point $(x^i,y^i,z^j)\in M.$

Let us consider a local orthogonal basis $\left(N_{\gamma }\right),\gamma =1,2,3$ of $T_{(x^i,y^i,z^j)}^{\perp }(S^{p-1}\left(r_1\right)\times S^{p-1}\left(r_2\right)\times S^{q-1}\left(r_3\right))$ in every point $(x^i,y^i,z^j)\in S^{p-1}\left(r_1\right)\times S^{p-1}\left(r_2\right)\times S^{q-1}\left(r_3\right)$, given by

$$\begin{array}{cc}\hfill N_1& ={\displaystyle \frac{1}{R_s}}(x^i,y^i,z^j),\hfill \\ \hfill N_2& ={\displaystyle \frac{1}{R_s}}(x^i,y^i,{-z}^j),\hfill \\ \hfill N_3& ={\displaystyle \frac{1}{r_3}}\left({\displaystyle \frac{r_2}{r_1}}{x}^i,{-{\displaystyle \frac{r_1}{r_2}}y}^i,0\right).\hfill \end{array}$$

From (6), decomposing $\theta \left(N_{\alpha }\right)$ into its tangential and normal components at $S^{p-1}\left(r_1\right)\times S^{p-1}\left(r_2\right)\times S^{q-1}\left(r_3\right)$, we get

$$\theta \left(N_{\alpha }\right)={\xi }_1+a_{\alpha 1}{N}_1+a_{\alpha 2}{N}_2+a_{\alpha 3}{N}_3$$

(88)

where $\alpha \in \{1, 2, 3\}$.

From $a_{\alpha \beta }=\langle \theta \left(N_{\alpha }\right),N_{\beta }\rangle$, $\alpha ,\beta \in \{1, 2, 3\}$ is given by

$$\begin{array}{cc}\hfill a_{11}& =a_{22}={\displaystyle \frac{1}{{R_s}^2}}(\phi r_1^2+\phi r_2^2+(2-\phi )r_3^2)=1+{\displaystyle \frac{\sqrt{2}}{{R_s}^2}}(r_1^2+r_2^2-r_3^2),\hfill \\ \hfill a_{12}& =a_{21}={\displaystyle \frac{1}{{R_s}^2}}(\phi r_1^2+\phi r_2^2-(2-\phi )r_3^2)=\sqrt{2}+{\displaystyle \frac{1}{{R_s}^2}}(r_1^2+r_2^2-r_3^2),\hfill \\ \hfill a_{33}& ={\displaystyle \frac{1}{r_3^2}}(\phi r_1^2+\phi r_2^2)={\displaystyle \frac{1}{r_3^2}}(1+\sqrt{2})(r_1^2+r_2^2),\hfill \\ \hfill a_{13}& =a_{31}=a_{23}=a_{32}=0.\hfill \end{array}$$

(i)

Thus, the matrix $\mathcal{A}={\left(a_{\alpha \beta }\right)}_3$ is given by

$$\mathcal{A}=\left(\begin{array}{ccc}1+{\displaystyle \frac{\sqrt{2}}{{R_s}^2}}(r_1^2+r_2^2-r_3^2)& \sqrt{2}+{\displaystyle \frac{1}{{R_s}^2}}(r_1^2+r_2^2-r_3^2)& 0\\ \sqrt{2}+{\displaystyle \frac{1}{{R_s}^2}}(r_1^2+r_2^2-r_3^2)& 1+{\displaystyle \frac{\sqrt{2}}{{R_s}^2}}(r_1^2+r_2^2-r_3^2)& 0\\ 0& 0& {\displaystyle \frac{1}{r_3^2}}(1+\sqrt{2})(r_1^2+r_2^2)\end{array}\right)$$

(89)

(ii)

From (88), we can write for $\alpha =1$

$${\xi }_1=\theta \left(N_1\right)-a_{11}{N}_1-a_{12}{N}_2-a_{13}{N}_3$$

Putting the required values in the above equation, we obtain

$${\xi }_1={\displaystyle \frac{(2r_3^2-{R_s}^2)}{{R_s}^3}}({\phi x}^i,{\phi y}^i,{-(2-\phi )z}^j)={\displaystyle \frac{(2r_3^2-{R_s}^2)}{{R_s}^3}}\theta (x^i,y^i,-z^j)$$

(90)

Similarly, we have

$${\xi }_2={\displaystyle \frac{(2r_3^2-{R_s}^2)}{{R_s}^3}}({\phi x}^i,{\phi y}^i,{(2-\phi )z}^j)={\displaystyle \frac{(2r_3^2-{R_s}^2)}{{R_s}^3}}\theta (x^i,y^i,z^j)$$

(91)

$${\xi }_3={\displaystyle \frac{(2r_3^2-{R_s}^2)}{r_3^2}}\left({\displaystyle \frac{r_2}{r_1{r}_3}}{\phi x}^i,{-{\displaystyle \frac{r_1}{r_2{r}_3}}\phi y}^i,0\right)$$

(92)

(iii)

From equality (12), we have

$$u_{\alpha }\left(X\right)=u(X^i,Y^i,Z^j)=\langle (X^i,Y^i,Z^j),{\xi }_{\alpha }\rangle$$

(93)

Using (93), we obtain

$$\begin{array}{cc}\hfill u_1& ={\displaystyle \frac{1}{R_s}}(\phi {X^{i}x}^i+{\phi Y}^i{y}^i+(2-\phi )Z^j{z}^j)\hfill \end{array}$$

(94)

$$\begin{array}{cc}\hfill u_2& ={\displaystyle \frac{1}{R_s}}(\phi {X^{i}x}^i+{\phi Y}^i{y}^i-(2-\phi )Z^j{z}^j)\hfill \end{array}$$

(95)

$$\begin{array}{cc}\hfill u_3& ={\displaystyle \frac{1}{r_3}}\left({\displaystyle \frac{r_2}{r_1}}\phi {X^{i}x}^i-{\displaystyle \frac{r_1}{r_2}}{\phi Y}^i{y}^i\right)\hfill \end{array}$$

(96)

(i)

By decomposing $\theta \left(N\right)$ into its tangential and normal components at sphere $S^{2p+q-1}\left(R_s\right)$, we get

$$\theta \left(N\right)=RX+u\left(X\right)N$$

$$RX=\theta \left(N\right)-u\left(X\right)N$$

(97)

Since $RX=R(X^i,Y^i,Z^j)$; then from above, we have

$$\begin{array}{c}RX=\theta (X^i,Y^i,Z^j)-u_1\left(X\right)N_1-u_2\left(X\right)N_2-u_3\left(X\right)N_3\hfill \\ RX=\left(\phi X^i-\left[{\displaystyle \frac{2}{{R_s}^2}}(\phi {X^{i}x}^i+{\phi Y}^i{y}^i)+{\displaystyle \frac{r_2}{{r_{1}r}_3^2}}\left({\displaystyle \frac{r_2}{r_1}}\phi {X^{i}x}^i-{\displaystyle \frac{r_1}{r_2}}{\phi Y}^i{y}^i\right)\right]x^i,\right.\\ \phi Y^i-\left[{\displaystyle \frac{2}{{R_s}^2}}(\phi {X^{i}x}^i+{\phi Y}^i{y}^i)+{\displaystyle \frac{r_{12}}{{r_{2}r}_3^2}}\left({\displaystyle \frac{r_2}{r_1}}\phi {X^{i}x}^i-{\displaystyle \frac{r_1}{r_2}}{\phi Y}^i{y}^i\right)\right]y^i,\\ \hfill \left.{(2-\phi )Z}^j-\left[{\displaystyle \frac{2}{R^2}}(2-\phi )Z^j{z}^j\right]z^j\right)\end{array}$$

(98)

Thus, we have $\theta \left(T_{(x^i,y^i,z^j)}(S^{p-1}\left(r_1\right)\times S^{p-1}\left(r_2\right)\times S^{q-1}\left(r_3\right))\right)\subseteq \left(T_{(x^i,y^i,z^j)}(S^{p-1}\left(r_1\right)\times S^{p-1}\left(r_2\right)\times S^{q-1}\left(r_3\right))\right)$, and we deduce the $(R,u_{\alpha },{\xi }_{\alpha },\left(a_{\alpha \beta }\right))$-induced structure $(S^{p-1}\left(r_1\right)\times S^{p-1}\left(r_2\right)\times S^{q-1}\left(r_3\right))$ by the silver Riemannian structure $(\theta ,\langle \rangle )$ on $E^{2p+q}$, and this is effectively determined by the equalities (88)–(98).

4. Conclusions

We provide a brief description of the outstanding results of this article. In the beginning, we constructed some fundamental properties of induced structures on submanifolds immersed in silver Riemannian manifolds. After, we obtained some results for induced normal structures on submanifolds of codimension 2. The main purpose of this article is characterize the geometrical analysis of submanifolds in Riemannian manifolds attached with silver structures. Moreover, we provide a non-trivial example of submanifolds of a silver Riemannian manifold.

Mathematical or Geometrical viewpoint: The silver structure refers to a concept in pure mathematics, specifically differential geometry, having implications for algebra and number theory. The silver structure’s fundamental role in differential geometry is to establish precise geometric conditions and relationships between distinct structures on a manifold or submanifolds, the properties of which are then studied using standard differential geometric tools like the second fundamental form and curvature inequalities. The compatibility of silver Riemannian manifolds enables the investigation of specific metric characteristics and curvature relationships on such manifolds. The study of these silver structures focuses on polynomial structures on manifolds and their accompanying numerical sequences, for example, Pell sequences for the silver ratio.

Physical viewpoint: The silver ratio structure exists in architecture, design, and even fractal geometry, but the exact “silver structure” on a differentiable manifold is still mostly theoretical. It is mentioned in research articles as an example of Clifford algebras and linkages in primary fiber bundles, which have applications in theoretical physics such as gauge theories.