Solitons of η-Ricci–Bourguignon Type on Submanifolds in (LCS)_m Manifolds

Abstract

In this research article, we concentrate on the exploration of submanifolds in an ${\left(LCS\right)}_m$-manifold $\tilde{B}$. We examine these submanifolds in the context of two distinct vector fields, namely, the characteristic vector field and the concurrent vector field. Initially, we consider some classifications of $\eta$-Ricci–Bourguignon (in short, $\eta$-RB) solitons on both invariant and anti-invariant submanifolds of $\tilde{B}$ employing the characteristic vector field. We establish several significant findings through this process. Furthermore, we investigate additional results by using $\eta$-RB solitons on invariant submanifolds of $\tilde{B}$ with concurrent vector fields, and discuss a supporting example.

1. Overview

It is well known that a Riemannian manifold (in short, $RM$) $(\tilde{B},\mathfrak{g})$ is said to be a Ricci soliton [1] if it satisfies the following condition:

$$\begin{array}{c}\hfill {\pounds }_{\mathcal{U}}\mathfrak{g}+2R\widetilde{ic}ci+2\varrho \mathfrak{g}=0\end{array}$$

(1)

where $\mathfrak{g}$, ${\pounds }_{\mathcal{U}}$, $R\widetilde{ic}ci$, and $\varrho$ respectively represent the Riemannian metric, the Lie derivative operator in the direction of vector field $\mathcal{U}$, the Ricci tensor, and some constant. In addition, a Ricci soliton $(\tilde{B},\mathcal{U},\varrho ,\mathfrak{g})$ is said to be shrinking, steady, or expanding according to whether $\varrho$ is negative, zero, or positive, respectively.

Over the past two decades, many researchers have dedicated their efforts to the exploration of Ricci solitons and the expansion of their concepts. Among these scholars, J.P. Bourguignon stands out for introducing a fresh extension, referred to as RB solitons, in the work documented in [2]. Drawing inspiration from unpublished works by Lichnerowicz and a paper authored by Aubin [3], in 1981 J.P. Bourguignon pioneered a novel progression known as the RB flow [2], extending the original Ricci flow.

On $RM$, an RB flow is an intrinsic geometric flow of which the fixed points are solitons. RB flow [2] is shown by

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathfrak{g}}{\partial t}}=-2(R\widetilde{ic}ci-\delta \tilde{r}\mathfrak{g}),\mathfrak{g}\left(0\right)=g_0.\end{array}$$

(2)

Here, the scalar curvature concerning the metric $\mathfrak{g}$ is denoted by $\tilde{r}$ and a non-zero real constant is symbolized as $\delta$.

For special values of the constant $\delta$ in (2), certain situations arise for the tensor $R\widetilde{ic}ci-\delta \tilde{r}\mathfrak{g}$, as described below [2]:

• When $\delta$ equals ${\displaystyle \frac{1}{2}}$, the Einstein tensor takes the form $R\widetilde{ic}c-{\displaystyle \frac{\tilde{r}}{2}}\mathfrak{g}$, as seen in the Einstein soliton [4].
• In the case where $\delta$ is ${\displaystyle \frac{1}{m}}$, the traceless Ricci tensor is provided by $R\widetilde{ic}c-{\displaystyle \frac{\tilde{r}}{m}}\mathfrak{g}$, illustrating the behavior of the traceless Ricci tensor in relation to $R\widetilde{ic}c$ and $\tilde{r}$.
• For $\delta$ equal to ${\displaystyle \frac{1}{2(m-1)}}$, the Schouten tensor can be expressed as $R\widetilde{ic}c-{\displaystyle \frac{\tilde{r}}{2(m-1)}}\mathfrak{g}$, as observed in the Schouten soliton [2], showcasing the dependence of the Schouten tensor on $R\widetilde{ic}c$ and $\tilde{r}$.
• When $\delta =0$, the Ricci tensor $R\widetilde{ic}c$ plays a pivotal role, as found in Ricci solitons [1].

The flow achieves a static state when, in the case of two dimensions, the Einstein tensor, traceless Ricci tensor, and Schouten tensor all assume a value of zero. Moreover, the value of $\delta$ is consistently arranged in a descending order in higher dimensions. The existence and uniqueness of a solution for this geometric flow have been demonstrated over a short time interval. Specifically, for sufficiently small values of t, the Equation (2) possesses a singular solution that is uniquely defined when $\delta$ is less than ${\displaystyle \frac{1}{2(m-1)}}$.

On the other hand, the solution to the Ricci flow equation can be interpreted as a quasi-Einstein metric or a Ricci solitons, as discussed in [2,5]. Within a comprehensive $RM$, the concept of the RB flow was introduced by Aubin [3]. Furthermore, many scholars have explored the characteristics of RB solitons, as evidenced in works such as [6,7,8].

An $RM$$(\tilde{B},\mathfrak{g})$ with a dimension of at least $m\ge 3$ is called an RB soliton if it satisfies the following condition:

$$\begin{array}{c}\hfill \frac{1}{2}{\pounds }_{\mathcal{U}}\mathfrak{g}+R\widetilde{ic}c+(\varrho +\delta \tilde{r})\mathfrak{g}=0.\end{array}$$

(3)

Analogous to the characterization of a Ricci soliton, an RB soliton is categorized as expanding when $\varrho >0$, steady when $\varrho =0$, and shrinking when $\varrho <0$. By perturbing the equation that defines RB solitons in (3) by multiples of a certain $(0,2)$-tensor field $\eta \otimes \eta$, we can obtain more general notion, namely, $\eta$-RB solitons [9] such as

$$\begin{array}{c}\hfill \frac{1}{2}{\pounds }_{\mathcal{U}}\mathfrak{g}+R\widetilde{ic}c+(\varrho +\delta \tilde{r})g+\gamma \eta \otimes \eta =0,\end{array}$$

(4)

where $\gamma$ stands for a real constant and $\eta$ represents a 1-form. The soliton known as $\eta$-RB can be simplified to an $\eta$-Ricci soliton by setting $\delta$ equal to zero in Equation (4). Recent works involve soliton types [10,11,12,13], k-almost Yamabe solitons [14], soliton theory [15,16,17,18], singularity theory [19], submanifold theory [20,21], tangent bundle problems [22,23,24,25,26], and classical differential geometry [27,28], all of which have been studied by many mathematicians in recent decades. The main methods, techniques, and results in these papers inspired us to carry out the present research.

The concept of Lorentzian concircular structure manifolds, often abbreviated as ${\left(LCS\right)}_m$-manifolds, was originally introduced by Shaikh in [29]. This notion extends the concept of LP-Sasakian manifolds, which was initially put forward by Matsumoto [30] and later by Mihai and Rosca [31]. Moreover, the implications of ${\left(LCS\right)}_m$-manifolds within the broader domains of general relativity and cosmology have been explored by Shaikh and Baishya in their works [32,33]. Extensive investigations into ${\left(LCS\right)}_m$-manifolds have been conducted by researchers such as Hui [34], Hui and Atceken [35], and Shaikh et al. [36,37,38], among others.

This article focuses on the investigation of invariant and anti-invariant submanifolds endowed with $\eta$-RB solitons in ${\left(LCS\right)}_m$-manifolds, admitting both the characteristic vector field and concurrent vector fields. The paper’s structure is as follows. Section 2 introduces fundamental concepts and definitions utilized throughout the subsequent discussion. Section 3 is dedicated to examining $\eta$-RB solitons on invariant submanifolds of ${\left(LCS\right)}_m$-manifolds with both the characteristic vector field and concurrent vector fields. In Section 4, we extend our study to anti-invariant submanifolds of ${\left(LCS\right)}_m$-manifolds exhibiting $\eta$-RB solitons with a characteristic vector field. Moving on to Section 5, we delve into additional results pertaining to contact CR-submanifolds within ${\left(LCS\right)}_m$-manifolds, where $\eta$-RB solitons are present alongside concurrent vector fields. Finally, Section 6 offers concluding remarks and observations.

2. Fundamental Concepts

In a Lorentzian manifold denoted by $(\tilde{B},\mathfrak{g})$, according to reference [39], a vector field ⋁ can be defined through the equation $\mathfrak{g}(\mathcal{U},\bigvee )=\Omega \left(\mathcal{U}\right)$ for all $\mathcal{U}\in \Gamma \left(T\tilde{B}\right)$ and $\bigvee \in \Gamma \left(T\tilde{B}\right)$. This vector field ⋁ is called a concircular vector field under the condition that the equation

$$\begin{array}{c}\hfill ({\tilde{\nabla }}_{\mathcal{U}}\Omega )\left(\mathcal{V}\right)=\alpha \{\mathfrak{g}(\mathcal{U},\mathcal{V})+\omega \left(\mathcal{U}\right)\Omega \left(\mathcal{V}\right)\}\end{array}$$

(5)

holds for all $\mathcal{U},\mathcal{V}\in \Gamma \left(T\tilde{B}\right)$. Here, $\alpha$ represents a nonzero scalar, $\omega$ denotes a closed 1-form, and $\tilde{\nabla }$ signifies the operator responsible for the covariant derivative of the Lorentzian metric $\mathfrak{g}$.

Consider an m-dimensional Lorentzian manifold, denoted by $\tilde{B}$$(m>2)$, which possesses a unit timelike concircular vector field, referred to as $\zeta$ (a significant feature of $\tilde{B}$). This vector field $\zeta$ is recognized as the characteristic vector field of $\tilde{B}$. In this context, it holds that

$$\begin{array}{c}\hfill \mathfrak{g}(\zeta ,\zeta )=-1.\end{array}$$

(6)

Because $\zeta$ is a unit concircular vector field, it implies the existence of a nonzero 1-form, denoted by $\eta$, such that

$$\begin{array}{c}\hfill \mathfrak{g}(\mathcal{U},\zeta )=\eta \left(\mathcal{U}\right),\end{array}$$

(7)

the equation of form

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_{\mathcal{U}}\eta \right)\left(\mathcal{V}\right)=\alpha \{\mathfrak{g}(\mathcal{U},\mathcal{V})+\eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right)\},\end{array}$$

(8)

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{\mathcal{U}}\zeta =\alpha \{\mathcal{U}+\eta \left(\mathcal{U}\right)\zeta \},\end{array}$$

(9)

holds for all $\mathcal{U},\mathcal{V}\in \Gamma \left(T\tilde{B}\right)$, and $\alpha$ is a non-zero scalar function that satisfies the following:

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{\mathcal{U}}\alpha =\left(\mathcal{U}\alpha \right)=\delta \eta \left(\mathcal{U}\right),\end{array}$$

(10)

where $\delta$ is determined by a specific scalar function such that $\delta =-\left(\zeta \alpha \right)$.

Now, consider the case where we choose

$$\begin{array}{c}\hfill \psi \mathcal{U}={\displaystyle \frac{1}{\alpha }}{\tilde{\nabla }}_{\mathcal{U}}\zeta .\end{array}$$

(11)

Then, from (9) and (11), we have

$$\begin{array}{c}\hfill \psi \mathcal{U}=\mathcal{U}+\eta \left(\mathcal{U}\right)\zeta \end{array}$$

(12)

and

$$\begin{array}{c}\hfill \mathfrak{g}(\psi \mathcal{U},\mathcal{V})=\mathfrak{g}(\mathcal{U},\psi \mathcal{V}),\end{array}$$

(13)

where $\psi$ takes on the role of a symmetric $(1,1)$-tensor, recognized as the manifold’s structure tensor. The Lorentzian manifold denoted as $\tilde{B}$, is combined with a unit timelike concircular vector field denoted by $\zeta$, its corresponding 1-form $\eta$, and a $(1,1)$ tensor field named $\psi$. This combination leads to the characterization of a Lorentzian concircular structure manifold, which is often referred to as an ${\left(LCS\right)}_m$-manifold in a shorter form, as discussed in [29]. Notably, in the special case where we set $\alpha$ to be equal to 1, we arrive at the LP-Sasakian structure initially introduced in [30].

Several relationships are established on $\tilde{B}$, as follows [29]:

$$\begin{array}{c}\hfill \eta \left(\zeta \right)=-1,\psi \zeta =0,\eta \left(\psi \mathcal{U}\right)=0,\mathfrak{g}(\psi \mathcal{U},\psi \mathcal{V})=\mathfrak{g}(\mathcal{U},\mathcal{V})+\eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right),\end{array}$$

(14)

$$\begin{array}{c}\hfill {\psi }^2\mathcal{U}=\mathcal{U}+\eta \left(\mathcal{U}\right)\zeta ,\end{array}$$

(15)

$$\begin{array}{c}\hfill R\widetilde{ic}c(\mathcal{U},\zeta )=(m-1)({\alpha }^2-\delta )\eta \left(\mathcal{U}\right),\end{array}$$

(16)

$$\widetilde{RC}(\mathcal{U},\mathcal{V})\zeta =({\alpha }^2-\delta )\{\eta \left(\mathcal{V}\right)\mathcal{U}-\eta \left(\mathcal{U}\right)\mathcal{V}\},$$

(17)

$$\begin{array}{c}\hfill \widetilde{RC}(\zeta ,\mathcal{V})\mathcal{W}=({\alpha }^2-\delta )\{\mathfrak{g}(\mathcal{V},\mathcal{W})\zeta -\eta \left(\mathcal{W}\right)\mathcal{V}\},\end{array}$$

(18)

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_{\mathcal{U}}\psi \right)\mathcal{V}=\alpha \{\mathfrak{g}(\mathcal{U},\mathcal{V})\zeta +2\eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right)\zeta +\eta \left(\mathcal{V}\right)\mathcal{U}\},\end{array}$$

(19)

$$\begin{array}{c}\hfill \left(\mathcal{U}\delta \right)=\beta \eta \left(\mathcal{U}\right),\end{array}$$

(20)

for all $\mathcal{U},\mathcal{V},\mathcal{W}\in \Gamma \left(T\tilde{B}\right)$, with $\beta$ as a scalar function equal to $-\left(\zeta \delta \right)$ and with $\widetilde{RC}$, $R\widetilde{ic}c$ representing the Riemannian curvature tensor and Ricci tensor of $\tilde{B}$, respectively.

A vector field $\nu$ on a (semi-)$RM$$(\tilde{B},\mathfrak{g})$ [40] is identified as a concircular vector field if it obeys the following condition:

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{\mathcal{W}}\nu =\kappa \mathcal{W}.\end{array}$$

(21)

Here, $\kappa$ signifies a nontrivial smooth function defined on $\tilde{B}$. This concircular vector field $\nu$ earns the title of a concurrent vector field when the specific choice $\kappa =1$ is made within the context of Equation (21). Several mathematicians and researchers have dedicated investigations to exploring manifolds endowed with specific types of vector fields, including [41,42,43].

Consider a submanifold S of dimension d in $\tilde{B}$ such that d is smaller than m. This submanifold inherits an induced metric denoted by $\mathfrak{g}$. Additionally, assume the use of the induced metric connection on both the tangent bundle $TS$ and the normal bundle $T^{\perp }S$ of S, represented by ∇ and ${\nabla }^{\perp }$, respectively. In this context, the Gauss and Weingarten formulas can be stated as follows [40]:

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{\mathcal{U}}\mathcal{V}={\nabla }_{\mathcal{U}}\mathcal{V}+\mathfrak{h}(\mathcal{U},\mathcal{V})\end{array}$$

(22)

and

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{\mathcal{U}}\mathrm{N}=-A_{\mathrm{N}}\mathcal{U}+{\nabla }_{\mathcal{U}}^{\perp }\mathrm{N}\end{array}$$

(23)

for all $\mathcal{U},\mathcal{V}\in \Gamma \left(TS\right)$ and $\mathrm{N}\in \Gamma \left(T^{\perp }S\right)$, where:

• $\mathfrak{h}$ denotes the second fundamental form associated with S in $\tilde{B}$
• $A_{\mathrm{N}}$ represents the shape operator.

The relationship between the second fundamental form $\mathfrak{h}$ and the shape operator $A_{\mathrm{N}}$ is provided by [40]

$$\begin{array}{c}\hfill \mathfrak{g}(\mathfrak{h}(\mathcal{U},\mathcal{V}),\mathrm{N})=\mathfrak{g}(A_{\mathrm{N}}\mathcal{U},\mathcal{V}).\end{array}$$

(24)

Combining both the Gauss and Weingarten formulas, we obtain the Gauss equation

$$\begin{array}{ccc}\hfill RC(\mathcal{U},\mathcal{V})\mathcal{W}& =& \widetilde{RC}(\mathcal{U},\mathcal{V})\mathcal{W}+A_{\mathfrak{h}(\mathcal{U},\mathcal{W})}\mathcal{V}\hfill \\ & & -A_{\mathfrak{h}(\mathcal{V},\mathcal{W})}\mathcal{U},\hfill \end{array}$$

(25)

for all $\mathcal{U},\mathcal{V},\mathcal{W}\in \Gamma \left(TS\right)$. Here, $RC$ represents the Riemannian curvature tensor of S.

A submanifold S in $\tilde{B}$ is classified as totally umbilical when the following relation holds:

$$\begin{array}{c}\hfill \mathfrak{h}(\mathcal{U},\mathcal{V})=\mathfrak{g}(\mathcal{U},\mathcal{V})\mathcal{H}\end{array}$$

(26)

where $\mathcal{H}$ signifies the mean curvature vector of S. Furthermore, if the tensor $\mathfrak{h}$ is zero, then the submanifold S is called totally geodesic, while if the vector $\mathcal{H}$ is zero, then S is considered minimal within the context of $\tilde{B}$ [40].

A submanifold S in an ${\left(LCS\right)}_m$-manifold $\tilde{B}$ is termed “invariant” if the structure vector field $\zeta$ lies tangent to S at every point on S. Additionally, for any vector field $\mathcal{U}$ tangent to S at each point on S, the condition of $\psi \mathcal{U}$ being tangent to S is satisfied, meaning that $\psi \left(TS\right)$ is a subset of $TS$ at each point p on S. The submanifold S is referred to as “anti-invariant” if the vector $\psi \mathcal{U}$ is normal to S for any $\mathcal{U}$ tangent to S, implying that $\psi \left(TS\right)$ lies within the normal bundle $T^{\perp }S$ at each point p on S, where $T^{\perp }S$ represents the bundle of vectors normal to S. It was established in [40] that an invariant submanifold of $\tilde{B}$ can itself be identified as an ${\left(LCS\right)}_m$-manifold [38].

3. Solitons Demonstrating $\mathit{\eta }$-RB Characteristics on Invariant Submanifolds

Let us take an $\eta$-RB soliton $(\mathfrak{g},\zeta ,\varrho ,\gamma ,\delta )$ on a submanifold S of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$. Then, we have

$$\begin{array}{c}\hfill \left({\pounds }_{\zeta }\mathfrak{g}\right)(\mathcal{U},\mathcal{V})+2Ricci(\mathcal{U},\mathcal{V})+2(\varrho +\delta r)\mathfrak{g}(\mathcal{U},\mathcal{V})+2\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right)=0.\end{array}$$

(27)

From (11) and (22), we can write

$$\begin{array}{c}\hfill \alpha \psi \mathcal{U}={\tilde{\nabla }}_{\mathcal{U}}\zeta ={\nabla }_{\mathcal{U}}\zeta +\mathfrak{h}(\mathcal{U},\zeta ).\end{array}$$

(28)

If S is invariant in $\tilde{B}$, upon equating tangential and normal components of (28) we obtain

$$\begin{array}{c}\hfill \alpha \psi \mathcal{U}={\nabla }_{\mathcal{U}}\zeta \mathrm{and}\mathfrak{h}(\mathcal{U},\zeta )=0.\end{array}$$

(29)

Again, from (7), (12), and (29) we find

$$\begin{array}{cc}\hfill \left({\pounds }_{\zeta }\mathfrak{g}\right)(\mathcal{U},\mathcal{V})& =\mathfrak{g}({\nabla }_{\mathcal{U}}\zeta ,\mathcal{V})+\mathfrak{g}(\mathcal{U},{\nabla }_{\mathcal{V}}\zeta )\hfill \\ \hfill & =2\alpha \left[\mathfrak{g}(\mathcal{U},\mathcal{V})+\eta \left(\mathcal{V}\right)\eta \left(\mathcal{U}\right)\right].\hfill \end{array}$$

(30)

Now, by combining (30) and (27), we arrive at

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{V})=-\left(\alpha +\varrho +\delta r\right)\mathfrak{g}(\mathcal{U},\mathcal{V})-(\gamma +\alpha )\eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right),\end{array}$$

(31)

which implies that S is an $\eta$-Einstein manifold. In addition, from (26) and (29) we obtain $\eta \left(\mathcal{U}\right)\mathcal{H}=0$, that is, $\mathcal{H}=0$. Because $\eta \left(\mathcal{U}\right)$ is not equal to 0, this shows that S is minimal in $\tilde{B}$. Thus, we can provide the following result.

Theorem 1. 

Let $(\mathfrak{g},\zeta ,\varrho ,\gamma ,\delta )$ be an η-RB soliton on an invariant submanifold S of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$. Then, S is:

1.

An η-Einstein manifold

2.

Minimal in $\tilde{B}$, provided that S is totally umbilical.

A direct result stemming from Theorem 1 follows:

• For $\delta =\frac{1}{2}$, (31) provides

  $$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{V})=-\left(\alpha +\varrho +\frac{r}{2}\right)\mathfrak{g}(\mathcal{U},\mathcal{V})-(\gamma +\alpha )\eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right),\end{array}$$

  (32)

  that is, S is an $\eta$-Einstein soliton.
• For $\delta =\frac{1}{2(d-1)}$, (31) becomes

  $$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{V})=-\left(\alpha +\varrho +\frac{r}{2(d-1)}\right)\mathfrak{g}(\mathcal{U},\mathcal{V})-(\gamma +\alpha )\eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right),\end{array}$$

  (33)

  that is, S is an $\eta$-Schouten soliton.
• For $\delta =0$, (31) reduces to

  $$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{V})=-(\alpha +\varrho )\mathfrak{g}(\mathcal{U},\mathcal{V})-(\gamma +\alpha )\eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right),\end{array}$$

  (34)

  that is, S is an $\eta$-Ricci soliton.

Thus, we can conclude the following result.

Corollary 1. 

Let $(\mathfrak{g},\zeta ,\varrho ,\gamma ,\delta )$ be an η-Einstein soliton, η-Schouten soliton, or η-Ricci soliton on an invariant submanifold S of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$. Then, S is:

1.

An η-Einstein manifold

2.

Minimal in $\tilde{B}$, provided that S is totally umbilical.

Because S is a d-dimensional invariant submanifold of an ${\left(LCS\right)}_m$-manifold, and using the relations (25) and (16) along with the fact that $\mathfrak{h}(\mathcal{U},\zeta )=0$, we can derive the following:

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\zeta )=(d-1)({\alpha }^2-\delta )\eta \left(\mathcal{U}\right).\end{array}$$

(35)

On taking $\mathcal{V}=\zeta$ in (31) along with (35), (6), and (7), we obtain

$$\begin{array}{c}\hfill \varrho +\gamma =-\left[(d-1)({\alpha }^2-\delta )+2\alpha +\delta r\right],\end{array}$$

(36)

$\eta \left(\mathcal{U}\right)$ not being equal to 0. Thus, we can state the following theorem.

Theorem 2. 

If ($\mathfrak{g},\zeta ,\varrho ,\gamma ,\delta$) is an η-RB soliton on an invariant submanifold S of an ${\left(LCS\right)}_m$ manifold, then ϱ and γ are related by (36).

In particular, if we take $\gamma =0$, then (36) becomes

$$\begin{array}{c}\hfill \varrho =-\left[(d-1)({\alpha }^2-\delta )+2\alpha +\delta r\right].\end{array}$$

(37)

From this, we can state the following result.

Theorem 3. 

If ($\mathfrak{g},\zeta ,\varrho ,\delta$) is an RB soliton on an invariant submanifold S of an ${\left(LCS\right)}_m$ manifold $\tilde{B}$, then S is expanding, shrinking, or steady according to the nature of ϱ in (37), that is, $\varrho >0$, $\varrho <0$, or $\varrho =0$, respectively.

Next, we study $\eta$-RB solitons with a concurrent vector field on invariant submanifolds. For an isometric immersion $\pi :S\to \tilde{B}$, we have

$$\begin{array}{c}\hfill \nu ={\nu }^T+{\nu }^{\perp },\end{array}$$

(38)

where $\nu \in \Gamma \left(TS\right)$ and where ${\nu }^T$ and ${\nu }^{\perp }$ denote the tangential and normal components of $\nu$ on S, respectively.

Because $\nu$ is a concurrent vector field on $\tilde{B}$, from the definition of concurrent vector field and (38) we obtain

$$\begin{array}{c}\hfill \mathcal{U}={\tilde{\nabla }}_{\mathcal{U}}{\nu }^T+{\tilde{\nabla }}_{\mathcal{U}}{\nu }^{\perp }.\end{array}$$

(39)

Per (22) and (23), (39) takes the following form:

$$\begin{array}{c}\hfill \mathcal{U}={\nabla }_{\mathcal{U}}{\nu }^T+\mathfrak{h}(\mathcal{U},{\nu }^T)-A_{{\nu }^{\perp }}\mathcal{U}+{\nabla }_{\mathcal{U}}^{\perp }{\nu }^{\perp }.\end{array}$$

(40)

By comparing the tangential and normal components of (40), we can conclude that

$$\begin{array}{c}\hfill {\nabla }_{\mathcal{U}}{\nu }^T=\mathcal{U}+A_{{\nu }^{\perp }}\mathcal{U}\end{array}$$

(41)

and

$$\begin{array}{c}\hfill \mathfrak{h}(\mathcal{U},{\nu }^T)=-{\nabla }_{\mathcal{U}}^{\perp }{\nu }^{\perp }.\end{array}$$

(42)

From the definition of a Lie derivative, we immediately have

$$\begin{array}{c}\hfill \left({\pounds }_{{\nu }^T}\mathfrak{g}\right)(\mathcal{U},\mathcal{W})=\mathfrak{g}({\nabla }_{\mathcal{U}}{\nu }^T,\mathcal{W})+\mathfrak{g}(\mathcal{U},{\nabla }_{\mathcal{W}}{\nu }^T).\end{array}$$

(43)

Using (24) and (41) in (43), we achieve

$$\begin{array}{c}\hfill \left({\pounds }_{{\nu }^T}\mathfrak{g}\right)(\mathcal{U},\mathcal{W})=2\mathfrak{g}(\mathcal{U},\mathcal{W})+2\mathfrak{g}(\mathfrak{h}(\mathcal{U},\mathcal{W}),{\nu }^{\perp }).\end{array}$$

(44)

Now, assuming that the invariant submanifold S admits an $\eta$-RB soliton, we can write

$$\begin{array}{c}\hfill \left({\pounds }_{{\nu }^T}\mathfrak{g}\right)(\mathcal{U},\mathcal{W})+2Ricci(\mathcal{U},\mathcal{W})+2(\varrho -\delta r)\mathfrak{g}(\mathcal{U},\mathcal{W})+2\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{W}\right)=0.\end{array}$$

(45)

By applying (44) and (45), we can conclude that

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{W})=-\left\{(\varrho +\delta r+1)\mathfrak{g}(\mathcal{U},\mathcal{W})+\mathfrak{g}(\mathfrak{h}(\mathcal{U},\mathcal{W}),{\nu }^{\perp })+\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{W}\right)\right\}.\end{array}$$

(46)

Thus, the following theorem is derived.

Theorem 4. 

If S is an invariant submanifold of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$ admitting an η-RB soliton with concurrent vector field ν, then the Ricci tensor $Ricci$ of S is provided by (46).

Now, if S is also a totally geodesic submanifold, then (46) turns into

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{W})=-\left\{(\varrho +\delta r+1)\mathfrak{g}(\mathcal{U},\mathcal{W})+\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{W}\right)\right\}.\end{array}$$

(47)

Then, we have the following theorem.

Theorem 5. 

If S is a totally geodesic invariant submanifold of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$ admitting an η-RB soliton with a concurrent vector field ν, then S is an η-Einstein manifold.

Now, putting $\mathcal{W}=\zeta$ in (47) and using (7), (14), and (35), we have

$$\begin{array}{c}\hfill \varrho +\gamma =-[1+(d-1)({\alpha }^2-\delta )+\delta r].\end{array}$$

(48)

Based on the information provided earlier, we can deduce the following theorem.

Theorem 6. 

If $(S,\mathfrak{g},\zeta ,\varrho ,\gamma ,\delta )$ is an η-RB soliton as a totally geodesic invariant submanifold S of an ${\left(LCS\right)}_m$ manifold $\tilde{B}$ with a concurrent vector field, then ϱ and γ are related by (48); hence, the scalar curvature is non-zero scalar.

Corollary 2. 

If $(S,\mathfrak{g},\nu ,\varrho ,\gamma ,\frac{1}{2})$ is an η-Einstein soliton with a concurrent vector field ν as invariant submanifold S of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$, then:

1.

The Ricci tensor of S is provided by

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{W})=-\left\{\left(\varrho +\frac{r}{2}+1\right)\mathfrak{g}(\mathcal{U},\mathcal{W})+\mathfrak{g}(\mathfrak{h}(\mathcal{U},\mathcal{W}),{\nu }^{\perp })+\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{W}\right)\right\}.\end{array}$$

(49)

2.

If S is totally geodesic, S is an η-Einstein manifold and ϱ and γ are related by $\varrho +\gamma =-[1+\frac{(d-1)}{2}(2{\alpha }^2-1)+\frac{r}{2}]$.

Corollary 3. 

If $(S,\mathfrak{g},\nu ,\varrho ,\gamma ,\frac{1}{2(d-1)})$ is an η-Schouten soliton with a concurrent vector field ν as the invariant submanifold S of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$, then:

1.

The Ricci tensor of S is provided by

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{W})=-\left\{\left(\varrho +\frac{r}{2(d-1)}+1\right)\mathfrak{g}(\mathcal{U},\mathcal{W})+\mathfrak{g}(\mathfrak{h}(\mathcal{U},\mathcal{W}),{\nu }^{\perp })+\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{W}\right)\right\}.\end{array}$$

(50)

2.

S is an η-Einstein manifold and ϱ and γ are related by $\varrho +\gamma =-[1+\frac{(2(d-1){\alpha }^2-1)}{2}+\frac{r}{2(d-1)}]$ when S is totally geodesic.

Corollary 4. 

If $(S,\mathfrak{g},\nu ,\varrho ,\gamma ,0)$ is an η-Ricci soliton with a concurrent vector field ν as the invariant submanifold S of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$, then:

1.

The Ricci tensor of S is provided by

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{W})=-\left\{\left(\varrho +1\right)\mathfrak{g}(\mathcal{U},\mathcal{W})+\mathfrak{g}(\mathfrak{h}(\mathcal{U},\mathcal{W}),{\nu }^{\perp })+\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{W}\right)\right\}.\end{array}$$

(51)

2.

If S is totally geodesic, then S is an η-Einstein manifold and ϱ and γ are related as $\varrho +\gamma =-[1+{\alpha }^2(d-1)]$.

By providing the subsequent example, we can demonstrate the presence of this soliton on an invariant submanifold $S^3$ of an ${\left(LCS\right)}_5$-manifold denoted by $\tilde{B}$.

Example 1. 

Recall the example of a five-dimensional ${\left(LCS\right)}_5$-manifold in [44], that is,

$$\tilde{B}=\{(X_1,Y_1,Z,X_2,Y_2)\in {\mathbb{R}}^5,\psi ,\zeta ,\eta ,\mathfrak{g}\},$$

where $(X_1,Y_1,Z,X_2,Y_2)$ are the standard coordinates in ${\mathbb{R}}^5$ with $\alpha ={exp}^{-2Z}$ not equal to 0 and $\delta =2{exp}^{-4Z}$.

The linearly independent vector fields on $\tilde{B}$ are denoted by $v_i$ for $i=\{1,2,3,4,5\}$ such that

$$\begin{array}{ccc}& & v_1={exp}^{-Z}(X_1\partial X_1+Y_1\partial Y_1),v_2={exp}^{-Z}\partial Y_1,\hfill \\ & & v_3={exp}^{-2Z}\partial Z,v_4={exp}^{-Z}(X_2\partial X_2+Y_1\partial Y_2),\hfill \\ & & v_5={exp}^{-Z}\partial Y_2.\hfill \end{array}$$

Thus, $\mathfrak{g}$ and ψ are respectively defined by

$$\mathfrak{g}(v_i,v_i)=1,\mathfrak{g}(v_i,v_j)=0,i\ne j=\{1,2,4,5\},$$

$$\mathfrak{g}(v_3,v_3)=-1,\mathfrak{g}(v_i,v_3)=0,i=\{1,2,4,5\},$$

and

$$\psi v_i=v_i,\psi v_3=0,i=\{1,2,4,5\}.$$

Thanks to the linearity of these tensors, it is quite easy to compute (6), (14), and (15).

For any vector field $\mathcal{U},\mathcal{V}\in \Gamma \left(T\tilde{B}\right)$, we can write

$$\mathcal{U}=a_1{v}_1+b_1{v}_2+c_1{v}_3+d_1{v}_4+e_1{v}_5,$$

$$\mathcal{V}=a_2{v}_1+b_2{v}_2+c_2{v}_3+d_2{v}_4+e_2{v}_5,$$

where $a_i,b_i,c_i,d_i,e_i\in \mathbb{R}$, $i=1,2$ such that $a_1{a}_2+b_1{b}_2-c_1{c}_2+d_1{d}_2+e_1{e}_2\ne 0$. Hence,

$$\mathfrak{g}(\mathcal{U},\mathcal{V})=a_1{a}_2+b_1{b}_2-c_1{c}_2+d_1{d}_2+e_1{e}_2.$$

In addition, we obtain

$$\begin{array}{ccc}\hfill {\tilde{\nabla }}_{\mathcal{U}}\mathcal{V}& =& {exp}^{-2Z}[a_1{c}_2{v}_1+b_1{c}_2{v}_2+(a_1{a}_2+b_1{b}_2+d_1{d}_2+e_1{e}_2)v_3+d_1{c}_2{v}_4+e_1{c}_2{v}_5]\hfill \\ & & +{exp}^{-Z}[-b_1{a}_2{v}_1+b_1{a}_2{v}_2-e_1{e}_2{v}_4+e_1{d}_2{v}_5].\hfill \end{array}$$

We can consider an invariant submanifold S of three dimensions in the five-dimensional ${\left(LCS\right)}_5$-manifold, that is,

$$S=\{(X,Y,Z)\ne (0,0,0)\in {\mathbb{R}}^3\},$$

where $(X,Y,Z)$ are the standard coordinates in ${\mathbb{R}}^3$ (shown in [44]). The linearly independent vector fields at each point of S are denoted by ${\mathcal{E}}_i$ for $i=\{1,2,3\}$ such that

$$\begin{array}{c}\hfill {\mathcal{E}}_1={exp}^{-Z}(X\partial X+Y\partial Y),{\mathcal{E}}_2={exp}^{-Z}\partial Y,{\mathcal{E}}_3={exp}^{-2Z}\partial Z.\end{array}$$

By applying Koszul’s formula, the $RC$ of S (see [44]) can be obtained easily; hence, the components $Ricci$ of the Ricci tensor of S are as follows:

$$Ricci({\mathcal{E}}_1,{\mathcal{E}}_1)=2{exp}^{-4Z}-{exp}^{-2Z},Ricci({\mathcal{E}}_2,{\mathcal{E}}_2)=2{exp}^{-4Z}-{exp}^{-2Z},$$

and

$$Ricci({\mathcal{E}}_3,{\mathcal{E}}_3)=-2{exp}^{-4Z}.$$

Because

$$r=\sum _{i=1}^{3}Ricci({\mathcal{E}}_i,{\mathcal{E}}_i),$$

we can compute

$$r=2({exp}^{-4Z}-{exp}^{-2Z}).$$

Now, we can use (31) to find

$$Ricci({\mathcal{E}}_3,{\mathcal{E}}_3)=-(2{exp}^{-2Z}-4{exp}^{-6Z}+4{exp}^{-8Z}+\varrho +\gamma ).$$

By equating the values of $Ricci({\mathcal{E}}_3,{\mathcal{E}}_3)$, we arrive at the following relation:

$$\varrho +\gamma =-2{exp}^{-2Z}+2{exp}^{-4Z}+4{exp}^{-6Z}-4{exp}^{-8Z}.$$

We can also verify this relation for $d=3$ using (36). Thus, $\mathfrak{g}$ provides an η-RB soliton on S of dimension 3.

4. Solitons Exhibiting $\mathit{\eta }$-RB Properties on Anti-Invariant Submanifolds

If S is an anti-invariant submanifold in $\tilde{B}$, then $\psi \left(\mathcal{U}\right)\in \Gamma \left(T^{\perp }S\right)$, ∀$\mathcal{U}\in \Gamma \left(TS\right)$; hence, from (29), we obtain

$${\nabla }_{\mathcal{U}}\zeta =0\mathrm{and}\mathfrak{h}(\mathcal{U},\zeta )=-\alpha \psi \left(\mathcal{U}\right).$$

Then, we have

$$\begin{array}{c}\hfill \left({\pounds }_{\zeta }\mathfrak{g}\right)(\mathcal{U},\mathcal{V})=\mathfrak{g}({\nabla }_{\mathcal{U}}\zeta ,\mathcal{V})+\mathfrak{g}(\mathcal{U},{\nabla }_{\mathcal{V}}\zeta )=0.\end{array}$$

(52)

By means of (52) and (27), we have

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{V})=-(\varrho +\delta r)\mathfrak{g}(\mathcal{U},\mathcal{V})-\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right),\end{array}$$

(53)

which implies that S is an $\eta$-Einstein manifold.

Theorem 7. 

If $(S,\mathfrak{g},\zeta ,\varrho ,\gamma ,\delta )$ is an anti-invariant submanifold of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$ admitting an η-RB soliton, then we have the following:

1.

The Ricci tensor $Ricci$ of S is provided by (53).

2.

S is an η-Einstein manifold.

For $\delta =\frac{1}{2}$, $\delta =\frac{1}{2(d-1)}$, and $\delta =0$, (53) provides the same result as shown in (32), (33), and (34).

Corollary 5. 

If an η-Einstein soliton, η-Schouten soliton, or η-Ricci soliton on an anti-invariant submanifold S of an ${\left(LCS\right)}_m$ manifold $\tilde{B}$ is η-Einstein, then their Ricci tensors are provided by

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{V})=-(\varrho +\frac{r}{2})\mathfrak{g}(\mathcal{U},\mathcal{V})-\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right),\end{array}$$

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{V})=-(\varrho +\frac{r}{2(d-1)})\mathfrak{g}(\mathcal{U},\mathcal{V})-\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right),\end{array}$$

and

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{V})=-\varrho \mathfrak{g}(\mathcal{U},\mathcal{V})-\gamma \eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right),\end{array}$$

respectively; hence, S is an η-Einstein manifold in each case.

Because ${\nabla }_{\mathcal{U}}\zeta =0$, from (17) and (35) we obtain

$$\begin{array}{c}\hfill RC(\mathcal{U},\mathcal{V})\zeta =0\mathrm{and}Ricci(\mathcal{U},\zeta )=0.\end{array}$$

(54)

Putting $\mathcal{V}=\zeta$ in (53) and using (54) yields

$$\begin{array}{c}\hfill \gamma +\varrho =-\delta r.\end{array}$$

(55)

As $\gamma$, $\varrho$, and $\delta$ represent certain constants, we can deduce the following theorem.

Theorem 8. 

If $(S,\mathfrak{g},\zeta ,\varrho ,\gamma ,\delta )$ is an anti-invariant submanifold S of an ${\left(LCS\right)}_m$ manifold $\tilde{B}$ admitting an η-RB soliton, then:

1.

ϱ and γ are related by (55).

2.

The scalar curvature is a constant.

Furthermore, the following immediate implication of Theorem 8 can be observed.

Corollary 6. 

Let $(S,\mathfrak{g},\zeta ,\varrho ,\gamma ,0)$ be an anti-invariant submanifold S of an ${\left(LCS\right)}_m$ manifold $\tilde{B}$ admitting an η-Ricci soliton. If S is steady, then $(\mathfrak{g},\zeta ,\varrho ,\gamma ,0)$ is always a Ricci soliton on S.

5. Solitons with $\mathit{\eta }$-RB Structure on Contact CR-Submanifolds

An isometrically immersed submanifold S of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$ is said to be a contact CR-submanifold of $\tilde{B}$ if:

• $\zeta$ is tangent to S.
• The tangent bundle is divided into two differentiable distributions $\mathrm{D}$ and ${\mathrm{D}}^{\perp }$ such that $TS=\mathrm{D}\oplus {\mathrm{D}}^{\perp }$.
• The distribution $\mathrm{D}:u\to {\mathrm{D}}_u\subset T_{u}S$ is invariant to $\psi$, that is, $\psi {\mathrm{D}}_{\mathrm{u}}\subset {\mathrm{D}}_{\mathrm{u}}$ for each $u\in S$.
• The distribution ${\mathrm{D}}^{\perp }:u\to {\mathrm{D}}_u^{\perp }\subset T_{u}S$ is anti-invariant to $\psi$, that is, $\psi {\mathrm{D}}_{\mathrm{u}}^{\perp }\subset T_u^{\perp }S$ for each $u\in S$.

Note that the horizontal and vertical distributions are represented by $\mathrm{D}$ and ${\mathrm{D}}^{\perp }$, respectively.

Consider the projection operators $\mathrm{E}$ and $\mathrm{F}$ associated with $\mathrm{D}$ and ${\mathrm{D}}^{\perp }$, defined in such a way that they satisfy the following conditions:

$${\mathrm{E}}^2=\mathrm{E},{\mathrm{F}}^2=\mathrm{F},\mathrm{EF}=\mathrm{FE}=0,\mathrm{E}+\mathrm{F}=\mathrm{I}$$

(56)

where $\mathrm{I}$ stands for the identity transformation S.

For any vector field $\mathcal{U}\in \Gamma \left(TS\right)$ and $\mathcal{V}\in \Gamma \left(T^{\perp }S\right)$, we have

$$\psi \mathcal{U}=\mathrm{EU}+\mathrm{FU}$$

(57)

and

$$\psi \mathcal{V}=\mathrm{BV}+\mathrm{CV},$$

(58)

where $\mathrm{EU}$ and $\mathrm{FU}$ represent the tangential and normal parts of $\psi \mathcal{U}$ while $\mathrm{BV}$ and $\mathrm{CV}$ denote the tangential and normal components of $\psi \mathcal{V}$. Additionally, $\mathrm{B}$ operates as an endomorphism on the normal bundle $T^{\perp }S$ associated with $TS$, while $\mathrm{C}$ serves as an endomorphism acting on the sub-bundle of the normal bundle $T^{\perp }S$.

It follows from the definition of a contact CR-submanifold S within $\tilde{B}$ that the decompositions of the tangent and normal spaces at any point $u\in S$ can be expressed as follows:

$$T_{u}S={\mathcal{D}}_u\left(S\right)\oplus \left\{\zeta \right\}\oplus {\mathcal{D}}^{\perp }u\left(S\right)\left\{\zeta \right\},T_u^{\perp }S=\psi {\mathcal{D}}_u^{\perp }\left(S\right)\oplus N_u\left(S\right)$$

(59)

where $N_u\left(S\right)$ represents the orthogonal complement of $\psi {\mathcal{D}}^{\perp }u\left(S\right)$ within $T^{\perp }uS$. It is noteworthy that $\mathcal{D}u\left(S\right)=\psi \mathcal{D}u\left(S\right)$, $\psi {\mathcal{D}}^{\perp }u\left(S\right)=\mathrm{F}{\mathcal{D}}^{\perp }u\left(S\right)$, $\psi N_u\left(S\right)=\mathrm{C}{N}_u\left(S\right)=N_u\left(S\right)$. Consequently, the following representation is valid:

$$\nu ={\nu }^T+{\nu }^{\perp }+\psi \left({\nu }^{\perp }\right)+\mathfrak{f}\zeta +\mu ,$$

(60)

where $\mathfrak{f}$ signifies a constant function, $\nu \in \Gamma \left(T\tilde{B}\right)$, ${\nu }^T\in \mathcal{D}\left(S\right)$, ${\nu }^{\perp }\in {\mathcal{D}}^{\perp }\left(S\right)$, and $\mu \in N\left(S\right)$.

On the other hand, it is well known that the covariant derivative of the tensor field $\psi$ is provided by

$$\left({\tilde{\nabla }}_{\mathcal{U}}\psi \right)\mathcal{V}={\tilde{\nabla }}_{\mathcal{U}}\psi \mathcal{V}-\psi {\tilde{\nabla }}_{\mathcal{U}}\mathcal{V}.$$

Considering that $\nu$ is a concurrent vector field, the following relationship holds:

$$\psi \mathcal{U}=\psi {\tilde{\nabla }}_{\mathcal{U}}\nu ={\tilde{\nabla }}_{\mathcal{U}}\psi \nu -\left({\tilde{\nabla }}_{\mathcal{U}}\psi \right)\nu .$$

(61)

Using the expression (60), we find that $\psi \nu =\mathrm{E}{\nu }^T+\mathrm{F}{\nu }^{\perp }+{\nu }^{\perp }+\mathrm{C}\mu$. Substituting this with (19) into (61), we obtain

$$\begin{array}{ccc}\hfill \psi \mathcal{U}& =& {\tilde{\nabla }}_{\mathcal{U}}\psi {°}^{\mathrm{T}}+{\tilde{\nabla }}_{\mathcal{U}}\psi {°}^{\perp }+{\tilde{\nabla }}_{\mathcal{U}}{\psi }^2{°}^{\perp }+{\tilde{\nabla }}_{\mathcal{U}}\psi \mathfrak{f}\zeta +{\tilde{\nabla }}_{\mathcal{U}}\psi \mu -\alpha \{\mathfrak{g}(\mathcal{U},{\nu }^T)\zeta +\mathfrak{g}(\mathcal{U},{\nu }^{\perp })\zeta \hfill \\ & & +\mathfrak{g}(\mathcal{U},\psi \left({\nu }^{\perp }\right))\zeta +\mathfrak{g}(\mathcal{U},\mathfrak{f}\zeta )\zeta +\mathfrak{g}(\mathcal{U},\mu )\zeta +2\eta \left(\mathcal{U}\right)\eta \left({\nu }^T\right)\zeta +2\eta \left(\mathcal{U}\right)\eta \left({\nu }^{\perp }\right)\zeta \hfill \\ & & +2\eta \left(\mathcal{U}\right)\eta \left(\psi {\nu }^{\perp }\right)\zeta +2\eta \left(\mathcal{U}\right)\eta \left(\mathfrak{f}\zeta \right)\zeta +2\eta \left(\mathcal{U}\right)\eta \left(\mu \right)\zeta +\eta \left({\nu }^T\right)\mathcal{U}+\eta \left({\nu }^{\perp }\right)\mathcal{U}\hfill \\ & & +\eta \left(\psi \left({\nu }^{\perp }\right)\right)\mathcal{U}+\eta \left(\mathfrak{f}\zeta \right)\mathcal{U}+\eta \left(\mu \right)\mathcal{U}\}.\hfill \end{array}$$

(62)

Because $\zeta$ is tangent to S, by using (15), (22), (23), and (57) in (62) we obtain

$$\begin{array}{ccc}\hfill \mathrm{EU}+\mathrm{FU}& =& {\nabla }_{\mathcal{U}}\mathrm{E}{\nu }^T+\mathfrak{h}(\mathcal{U},\mathrm{E}{\nu }^T)+{\nabla }_{\mathcal{U}}^{\perp }\mathrm{F}{\nu }^{\perp }-A_{\mathrm{F}{\nu }^{\perp }}\mathcal{U}+{\nabla }_{\mathcal{U}}{\nu }^{\perp }\hfill \\ & & +\mathfrak{h}(\mathcal{U},{\nu }^{\perp })+{\nabla }_{\mathcal{U}}^{\perp }\mathrm{C}\mu -A_{\mathrm{C}\mu }\mathcal{U}-\alpha \{\mathfrak{g}(\mathcal{U},{\nu }^T)\zeta +\mathfrak{g}(\mathcal{U},{\nu }^{\perp })\zeta \hfill \\ & & -\mathfrak{f}\eta \left(\mathcal{U}\right)\zeta -\mathfrak{f}\mathcal{U}\}.\hfill \end{array}$$

(63)

On contrasting the tangential and normal elements, we can deduce the subsequent relationships

$$\begin{array}{ccc}\hfill \mathrm{EU}& =& {\nabla }_{\mathcal{U}}\mathrm{E}{\nu }^T-A_{\mathrm{F}{\nu }^{\perp }}\mathcal{U}+{\nabla }_{\mathcal{U}}{\nu }^{\perp }-A_{\mathrm{C}\mu }\mathcal{U}\hfill \\ & & -\alpha \{\mathfrak{g}(\mathcal{U},{\nu }^T+{\nu }^{\perp })\zeta -\mathfrak{f}{\psi }^2\mathcal{U}\}\hfill \end{array}$$

(64)

and

$$\begin{array}{c}\hfill \mathrm{FU}=\mathfrak{h}(\mathcal{U},\mathrm{E}{\nu }^T)+{\nabla }_{\mathcal{U}}^{\perp }\mathrm{F}{\nu }^{\perp }+{\nabla }_{\mathcal{U}}^{\perp }\mathrm{C}\mu +\mathfrak{h}(\mathcal{U},{\nu }^{\perp }).\end{array}$$

(65)

It follows from the definition of the Lie derivative that

$$\begin{array}{c}\hfill \left({\pounds }_{{\nu }^{\perp }}\mathfrak{g}\right)(\mathcal{U},\mathcal{V})=\mathfrak{g}({\nabla }_{\mathcal{U}}{\nu }^{\perp },\mathcal{V})+\mathfrak{g}({\nabla }_{\mathcal{V}}{\nu }^{\perp },\mathcal{U})\end{array}$$

(66)

for all $\mathcal{U},\mathcal{V}\in {\mathcal{D}}^{\perp }\left(S\right)$.

Using (64) in (66) and $\mathrm{EU}=\mathrm{EV}=0$, ∀$\mathcal{U},\mathcal{V}\in {\mathcal{D}}^{\perp }\left(S\right)$, we arrive at

$$\begin{array}{ccc}\hfill \left({\pounds }_{{\nu }^{\perp }}\mathfrak{g}\right)(\mathcal{U},\mathcal{V})& =& -\mathfrak{g}({\nabla }_{\mathcal{U}}\mathrm{E}{\nu }^T,\mathcal{V})+\mathfrak{g}(A_{\mathrm{F}{\nu }^{\perp }}\mathcal{U},\mathcal{V})+\mathfrak{g}(A_{\mathrm{C}\mu }\mathcal{U},\mathcal{V})\hfill \\ & & +2\alpha g(\mathcal{U},\mathcal{V})+\mathfrak{g}(\mathcal{U},\mathrm{EV})-\mathfrak{g}(\mathcal{U},{\nabla }_{\mathcal{V}}\mathrm{E}{\nu }^T)\hfill \\ & & +\mathfrak{g}(A_{\mathrm{F}{\nu }^{\perp }}\mathcal{V},\mathcal{U})+\mathfrak{g}(A_{\mathrm{C}\mu }\mathcal{U},\mathcal{V})\hfill \\ & =& -\mathfrak{g}({\nabla }_{\mathcal{U}}\mathrm{E}{\nu }^T,\mathcal{V})-\mathfrak{g}(\mathcal{U},{\nabla }_{\mathcal{V}}\mathrm{E}{\nu }^T)\hfill \\ & & +2\mathfrak{g}(\mathfrak{h}(\mathcal{U},\mathcal{V}),\mathrm{F}{\nu }^{\perp })+2\mathfrak{g}(\mathfrak{h}(\mathcal{U},\mathcal{V}),\mathrm{C}\mu )\hfill \\ & & +2\alpha g(\mathcal{U},\mathcal{V}).\hfill \end{array}$$

Moreover, it is possible to rephrase this as

$$\begin{array}{ccc}\hfill \left({\pounds }_{{\nu }^{\perp }}\mathfrak{g}\right)(\mathcal{U},\mathcal{V})& =& 2\mathfrak{g}(\mathfrak{h}(\mathcal{U},\mathcal{V}),\mathrm{F}{\nu }^{\perp })+2\mathfrak{g}(\mathfrak{h}(\mathcal{U},\mathcal{V}),\mathrm{C}\mu )\hfill \\ & & +2\alpha g(\mathcal{U},\mathcal{V})\hfill \end{array}$$

(67)

because the induced structure $\mathrm{E}$ is an $\left(LCS\right)$-structure on S. It follows that $\left({\nabla }_{\mathcal{U}}\mathrm{E}\right)\mathcal{V}=\alpha \{\mathfrak{g}(\mathcal{U},\mathcal{V})\zeta +2\eta \left(\mathcal{U}\right)\eta \left(\mathcal{V}\right)\zeta +\eta \left(\mathcal{V}\right)\mathcal{U}\}$.

By employing the definition of $\eta$-RB soliton, the relation (67), and with the fact that ${\mathcal{D}}^{\perp }u\left(S\right)$ is the orthogonal complement of $\mathcal{D}u\left(S\right)\oplus \zeta$ while considering the significance of $\mathrm{B}h=0$ in the geometry of submanifolds, the resulting Ricci tensor can be expressed as

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{V})=-(\alpha +\varrho +\delta r)\mathfrak{g}(\mathcal{U},\mathcal{V})-\mathfrak{g}\left(\mathfrak{h}\right(\mathcal{U},\mathcal{V}),\mathrm{C}\mu ).\end{array}$$

(68)

Based on the above considerations, we can draw the following conclusion.

Theorem 9. 

If $(S,\mathfrak{g},\nu ,\varrho ,\gamma ,\delta )$ is an η-RB soliton on a contact CR-submanifold of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$ with a concurrent vector field, then the Ricci tensor $Ricc{i}_{{\mathrm{D}}^{\perp }}$ for ${\mathrm{D}}^{\perp }$ is provided by (68).

Next, we apply Equation (64) to determine the Ricci tensor $Ricc{i}_{\mathrm{D}}$ for $\mathrm{D}$. During this process, we substitute ${\nu }^T$ with $\mathrm{E}{\nu }^T$, leading to the following expression:

$$\begin{array}{ccc}\hfill {\nabla }_{\mathcal{U}}{\nu }^T& =& -\mathrm{EU}-A_{\mathrm{F}{\nu }^{\perp }}\mathcal{U}+{\nabla }_{\mathcal{U}}{\nu }^{\perp }-A_{\mathrm{C}\mu }\mathcal{U}\hfill \\ & & -\alpha \{\mathfrak{g}(\mathcal{U},{\nu }^T+{\nu }^{\perp })\zeta -\mathfrak{f}{\psi }^2\mathcal{U}\}.\hfill \end{array}$$

(69)

For all $\mathcal{U},\mathcal{V}\in \mathcal{D}\left(S\right)$, the Lie derivative along ${\nu }^T$ is provided by

$$\begin{array}{ccc}\hfill \left({\pounds }_{{\nu }^T}\mathfrak{g}\right)(\mathcal{U},\mathcal{V})& =& \mathfrak{g}({\nabla }_{\mathcal{U}}{\nu }^T,\mathcal{V})+\mathfrak{g}({\nabla }_{\mathcal{V}}{\nu }^T,\mathcal{U})\hfill \\ & =& -\mathfrak{g}(\mathrm{EU},\mathcal{V})-\mathfrak{g}(\mathrm{EV},\mathcal{U})+\mathfrak{g}({\nabla }_{\mathcal{U}}{\nu }^{\perp },\mathcal{V})+\mathfrak{g}({\nabla }_{\mathcal{V}}{\nu }^{\perp },\mathcal{U})\hfill \\ & & -2\mathfrak{g}\left(\mathfrak{h}\right(\mathcal{U},\mathcal{V}),\mathrm{C}\mu )+2\alpha g(\mathcal{U},\mathcal{V}),\hfill \end{array}$$

which can be rewritten as

$$\begin{array}{ccc}\hfill \left({\pounds }_{{\nu }^T}\mathfrak{g}\right)(\mathcal{U},\mathcal{V})-\left({\pounds }_{{\nu }^{\perp }}\mathfrak{g}\right)(\mathcal{U},\mathcal{V})& =& -\mathfrak{g}(\mathrm{EU},\mathcal{V})-\mathfrak{g}(\mathrm{EV},\mathcal{U})\hfill \\ & & -2\mathfrak{g}\left(\mathfrak{h}\right(\mathcal{U},\mathcal{V}),\mathrm{C}\mu )+2\alpha g(\mathcal{U},\mathcal{V}).\hfill \end{array}$$

(70)

However, for all $\mathcal{U},\mathcal{V}\in \mathcal{D}\left(S\right)$, the Lie derivative along ${\nu }^{\perp }$ is found below:

$$\begin{array}{ccc}\hfill \left({\pounds }_{{\nu }^{\perp }}\mathfrak{g}\right)(\mathcal{U},\mathcal{V})& =& \mathfrak{g}(\mathrm{EU},\mathcal{V})+\mathfrak{g}(\mathrm{EV},\mathcal{U})+2\mathfrak{g}\left(\mathfrak{h}\right(\mathcal{U},\mathcal{V}),\mathrm{C}\mu )+2\alpha g(\mathcal{U},\mathcal{V})\hfill \\ & & -\mathfrak{g}({\nabla }_{\mathcal{U}}\mathrm{E}{\nu }^T,\mathcal{V})-\mathfrak{g}({\nabla }_{\mathcal{V}}\mathrm{E}{\nu }^T,\mathcal{U}).\hfill \end{array}$$

(71)

On combining (70) and (71), we find that

$$\begin{array}{c}\hfill \left({\pounds }_{{\nu }^T}\mathfrak{g}\right)(\mathcal{U},\mathcal{V})=4\alpha g(\mathcal{U},\mathcal{V})-\mathfrak{g}({\nabla }_{\mathcal{U}}\mathrm{E}{\nu }^T,\mathcal{V})-\mathfrak{g}({\nabla }_{\mathcal{V}}\mathrm{E}{\nu }^T,\mathcal{U}).\end{array}$$

We can rewrite this as

$$\begin{array}{c}\hfill \left({\pounds }_{{\nu }^T}\mathfrak{g}\right)(\mathcal{U},\mathcal{V})=2\alpha g(\mathcal{U},\mathcal{V}),\end{array}$$

where we have used $\mathfrak{g}({\nabla }_{\mathcal{V}}\mathrm{E}{\nu }^T,\mathcal{U})=\mathfrak{g}(\left({\nabla }_{\mathcal{V}}\mathrm{E}\right){\nu }^T,\mathcal{U})+\mathfrak{g}(\mathrm{E}{\nabla }_{\mathcal{V}}{\nu }^T,\mathcal{U})$ ⇒$\mathfrak{g}({\nabla }_{\mathcal{V}}\mathrm{E}{\nu }^T,\mathcal{U})=\mathfrak{g}(\mathrm{E}{\nabla }_{\mathcal{V}}{\nu }^T,\mathcal{U})=\mathfrak{g}({\nabla }_{\mathcal{V}}{\nu }^T,\mathrm{E}\mathcal{U})$.

As a result, we obtain

$$\begin{array}{c}\hfill Ricci(\mathcal{U},\mathcal{V})=-(\alpha +\varrho +\delta r)\mathfrak{g}(\mathcal{U},\mathcal{V}).\end{array}$$

(72)

Therefore, we can formulate the following theorem.

Theorem 10. 

If $(S,\mathfrak{g},\nu ,\varrho ,\gamma ,\delta )$ is an η-RB soliton on a contact CR-submanifold of an ${\left(LCS\right)}_m$-manifold $\tilde{B}$ with a concurrent vector field, then the Ricci tensor $Ricc{i}_{\mathrm{D}}$ for $\mathrm{D}$ is provided by (72).

It is worth noting that Theorems 9 and 10 are applicable to specific scenarios of $\delta$, particularly when S embodies an $\eta$-Einstein soliton, an $\eta$-Schouten soliton, or an $\eta$-Ricci soliton. Additionally, we emphasize that $Ricc{i}_{\mathrm{D}}=Ricc{i}_{{\mathrm{D}}^{\perp }}$ holds true when S is considered ${\mathrm{D}}^{\perp }$-totally geodesic.

6. Concluding Remarks

It is a widely acknowledged fact that Ricci solitons were initially introduced by R.S. Hamilton [1]. Subsequently, many researchers have dedicated their efforts to investigating and extending the concept of Ricci solitons. Among these extensions, the most recent addition is the RB soliton, pioneered by J.P. Bourguignon [3].

In the present article, we have delved into the exploration of $\eta$-RB solitons, an extension of RB solitons, specifically focusing on their presence within invariant and anti-invariant submanifolds denoted by S in an ${\left(LCS\right)}_m$-manifold $\tilde{B}$. Drawing insights from the discourse presented in Section 3 and Section 4, we can derive the following conclusions.

Nature of submanifold	vector field	nature of solitons	submanifold
invariant	characteristic	$\eta$-RB	$\eta$-Einstein
anti-invariant	characteristic	$\eta$-RB	$\eta$-Einstein
invariant	characteristic	$\eta$-Einstein, $\eta$-Schouten, $\eta$-Ricci	$\eta$-Einstein
anti-invariant	characteristic	$\eta$-Einstein, $\eta$-Schouten, $\eta$-Ricci	$\eta$-Einstein
invariant	concurrent	$\eta$-RB	$\eta$-Einstein
invariant	concurrent	$\eta$-Einstein, $\eta$-Schouten, $\eta$-Ricci	$\eta$-Einstein

The preceding discussion prompts several inquiries that merit consideration for future research:

• Are the findings presented in this paper applicable to vector fields other than characteristic and concurrent vector fields?
• Do the results of this paper extend to semi-generic submanifolds?
• If different connections than those in this article were employed, what novel outcomes might be derived?
• In light of the existence of warped product CR-submanifolds $S=S_{\perp }{\times }_l{S}_T$ originating from ${\left(LCS\right)}_n$-manifolds $\tilde{B}$, where $S_{\perp }$ represents an anti-invariant submanifold tangent to $\zeta$ and $S_T$ signifies an invariant submanifold of $\tilde{B}$ [45], what are the necessary conditions on S within $\tilde{B}$ to render it an Einstein manifold under the influence of the gradient $\eta$-RB soliton?