Gradient Expanding Ricci Solitons Type Inequalities on Submanifolds in Quaternion Kaehler Manifolds with Bi-Slant Factor

Abstract

In this article, we study the Ricci soliton on quaternion bi-slant submanifolds of quaternion Kaehler manifolds. We obtain a lower-bound-type inequality in terms of expanding gradient Ricci solitons with a gradient-type vector field for the quaternion bi-slant submanifold of quaternion Kaehler manifolds. Additionally, we derive a series of lower-bound-type inequalities for semi-slant submanifolds, $CR$-submanifolds, hemi-slant submanifolds, slant submanifolds, invariant anti-invaraint submanifolds and totally real submanifolds in the same quaternion Kaehler manifolds. Finally, we discuss a double inequality for expanding gradient Ricci solitons on submanifolds in quaternion Kaehler manifolds and extend the same double inequality in terms of gradient Ricci solitons with a scalar concircular field on semi-slant, quaternion $CR$-submanifolds of quaternion Kaehler manifolds.

1. Introduction

An important area of research that offers deep insights into the geometric and analytic properties of Riemannian manifolds is inequalities in terms of gradient solitons. Soliton related inequalities are profound mathematical statements in differential geometry and analysis that connect geometric properties like curvatures, which relate extrinsic to intrinsic invariants for submanifolds in different ambient manifolds with soliton behavior. They lead to insights on manifold structure, volume growth, and sharp analytical bounds, with important applications in physics (fluid mechanics, optics, and gravity) where stable, localized waves (solitons) appear, as well as mathematical analysis for PDE theory [1]. Moreover, in fields like computer science (algorithm efficiency, proving bounds), physics (quantum chaos), and cryptography (securing data with prime number features), inequality and number theory work together to create limits, estimate values, and represent constraints.

In 1988, Hamilton [2] first presented the ideas behind Ricci flow. It demonstrates that the limit of the solution for the Ricci flow is the Ricci soliton. Furthermore, geometric flow theory, in particular the Ricci flow, has attracted the attention of numerous mathematicians within the last 20 years.

The family of metrics $g\left(t\right)$ on a Riemannian manifold $\mathrm{\Xi }$ evolve into the Ricci flow [2], if

$${\displaystyle \frac{\partial }{\partial t}}g\left(t\right)=-2S_{ric}\left(t\right)g\left(t\right),g_0=g\left(0\right).$$

Definition 1 

([2]). On the Riemannian manifold, $(\mathrm{\Xi },g)$, a Ricci soliton ($RS$) is a triplet $(g,\mathcal{F},\lambda )$ obeying

$$S_{ric}+\lambda g+{\displaystyle \frac{1}{2}}{\mathfrak{L}}_{\mathcal{F}}g=0,$$

(1)

where the Ricci tensor is $S_{Ric}$, and $\mathfrak{L}$ is the Lie-derivative across the vector field $\mathcal{F}$. Depending on the constant λ, the manifold $(\mathrm{\Xi },g,\mathcal{F},\lambda )$ is called a Ricci shrinker, expander, or stable soliton, whether $\lambda <0$, $\lambda >0$, or $\lambda =0$.

Equation (1) becomes

$$S_{ric}+\mathcal{H}ess\left(f\right)=\lambda g,$$

(2)

where $\mathcal{H}ess$ denotes the Hessian operator with respect to metric g. It shows that $(\mathrm{\Xi },g,\mathcal{F}=\nabla f,\lambda )$ is a gradient Ricci soliton $\left(GRs\right)$.

A lower bound type inequality on the geometry of metric g in terms of $GRs$ for a smooth function f on ambient space $\mathrm{\Xi }$ was established in 2020 by Blaga and Carasmareanu [3].

$$\left|\right|S_{ric}{\left|\right|}_g^2\ge {\left|\right|\mathcal{H}ess\left(f\right)\left|\right|}_g^2-{\displaystyle \frac{1}{n}}{\left(\mathrm{\Delta }f\right)}^2,$$

(3)

where $S_{ric}$ and $\mathcal{H}ess$ indicate the Ricci tensor and the Hessian of a smooth function f on $\mathrm{\Xi }$, respectively.

Blaga and Carasmareanu [3] also extend (3) and derive a double inequality in terms of gradient Ricci solitons such as

$${\left|\right|\mathcal{H}ess\left(f\right)\left|\right|}_g^2-{\displaystyle \frac{1}{n}}{\left(\mathrm{\Delta }f\right)}^2\le \left|\right|S_{ric}{\left|\right|}_g^2\le {\left|\right|\mathcal{H}ess\left(f\right)\left|\right|}_g^2+{\displaystyle \frac{{\delta }^2}{n}},$$

(4)

where the simultaneous equalities for $n\ge 3$ hold if and only if the scalar curvature $\delta =\mathrm{\Delta }f=0=\lambda$ and $\mathcal{H}ess\left(f\right)=-S_{ric}.$ In [4], the authors derived the solitonic inequality for submanifolds of trans-Sasakian manifolds with a slant factor. Furthermore, some more articles on gradient Ricci solitons and geometric inequalities have been published, which are relevant to this current topic (for more details, see [5,6,7,8]).

Moreover, the combination of scalar concircular fields [9] and Ricci solitons is a topic of significant study in differential geometry and mathematical physics, primarily because it imposes strong geometric constraints on the manifold, frequently leading to rigidity conclusions and specific categories. Scalar concircular fields are advanced ideas in differential geometry that describe scalar functions $\left(\phi \right)$ on Riemannian manifolds. Their corresponding vector fields (gradients of type $\mathcal{F}=\nabla f$) have particular properties related to concircular transformations, which maintain geodesic circles [9]. Essentially, they are scalar fields whose gradients operate as concircular vector fields that leave the manifold’s curvature structure consistent. Therefore, we provide the following definition:

Definition 2 

([9]). If the scalar field $f\in C^{\infty }\left(\mathbb{Q}\right)$ holds the equation, it is known as a scalar concircular field.

$$\mathcal{H}ess\left(f\right)(G,K)=\mho g(G,K),$$

(5)

for vector fields $G,K$ and where $\mathcal{H}ess$ is the Hessian, a scalar field is ℧ and the Riemannian metric is g.

In view of (2) and (5), the manifold becomes an Einstein manifold, which means it holds the trivial soliton (where the metric is Einstein). Incorporating scalar concircular fields into Ricci soliton research creates a framework for classifying manifolds and establishing geometric inequalities. For example, a connected Ricci soliton is trivial if $\mathrm{\Delta }\left(f\right)\le {\phi }^2-S_{ric},$ where $\phi$ is a scalar function.

A key result in this domain, proven by writers such as Chen and Deshmukh [10,11], was that a compact almost Ricci soliton with a concircular potential vector field is isometric to the Euclidean sphere ${\mathbb{S}}^3$. This is an effective characterization result that can be expressed in terms of inequalities, for example, lower bounds on curvature integrals. Chen exhibits how Ricci solitons and concircular fields can be defined individually and integrated to yield classification theorems for inequality analysis used to establish rigidity [12].

On the other hand, in 1974, S. Ishihara [13] proposed a quaternion Kaehler manifold, sometimes known as a quaternion manifold, as a Riemannian manifold whose holonomy group is a subgroup of $Sp\left(1\right).Sp\left(m\right)=Sp\left(1\right)\times Sp\left(m\right)/\{\pm 1\}$. The existence of a 3-dimensional vector bundle V of tensor of type $(1,1)$ on a quaternion manifold with local cross-sections of basically Hermitian structures satisfying specific axioms is well known [14]. If the quaternion sectional curvature c of a quaternion manifold is constant, it is a quaternion space form. A quaternionic submanifold $\mathbb{B}$ in a quaternion manifold $\tilde{\mathbb{Q}}$ is a submanifold for which the tangent space is invariant under each section of V [15].

The establishment of a straightforward relationship between the essential intrinsic invariants and the important extrinsic invariants of the submanifolds is a central problem in the general theory of Riemannian submanifolds. Certain forms of inequalities can provide such a correlation.

Quaternion Kähler geometry and soliton analysis are linked by the investigation of certain metric structures on manifolds, with geometric features frequently influencing the behavior and presence of solitons. Certain Ricci-flat quaternion Kähler metrics, for example, can be solutions to gravitational instanton equations, which have direct applications in theoretical physics by characterizing the structure of certain physical solitons [16].

Quaternionic Kähler manifolds are ideal for soliton analysis, especially the study of solvsolitons like algebraic Ricci solitons on solvable Lie groups, which emerge as main orbits or hypersurfaces in non-compact cases. Furthermore, recent advances in quaternion-Kähler harmonic Ricci flow have introduced the concept of geometric solitons as self-similar solutions to these flows, giving researchers a tool to examine singularity creation and the potential classification of compact structures [17].

We are aware that the normalized normal scalar curvature and mean curvature of manifolds are extrinsic invariants, while the normalized scalar curvature of the submanifold is referred to as an intrinsic invariant of manifolds. Therefore, the purpose of this study is to deduce various inequalities in terms of gradient Ricci solitons for QR-submanifolds in quaternion space forms and for quaternion bi-slant submanifolds [18]. We present a double inequality for gradient Ricci solitons on submanifolds in quaternion Kaehler manifolds and infer the same double inequality in terms of a scalar concircular field on quaternion bi-slant submanifolds of quaternion Kaehler manifolds.

Furthermore, for specific examples like semi-slant, hemi-slant, CR, slant, invariant, and fully real submanifolds in the same ambient space form, we find the same solitonic inequality.

2. Quaternion Kaehler Manifolds

In this section, we will discuss some basic definitions and notions for Quaternion Kaehler Manifolds.

Assume that $\tilde{\mathbb{Q}}$ is a Riemannian manifold of $4n$ dimensions with metric tensor g. If there is a 3-dimensional vector bundle V formed by tensors of type $(1,1)$ with local basis of almost Hermitian structures ${\mathbb{J}}_1$, ${\mathbb{J}}_2$, and ${\mathbb{J}}_3$ such that $\tilde{\mathbb{Q}}$ is a quaternion Kaehler manifold $\left(QKM\right)$

(a)

$${\mathbb{J}}_1^2={\mathbb{J}}_2^2={\mathbb{J}}_3^2=-I_d,$$

(6)

$${\mathbb{J}}_1{\mathbb{J}}_2=-{\mathbb{J}}_2{\mathbb{J}}_1={\mathbb{J}}_3,{\mathbb{J}}_2{\mathbb{J}}_3=-{\mathbb{J}}_3{\mathbb{J}}_2={\mathbb{J}}_1,{\mathbb{J}}_3{\mathbb{J}}_1=-{\mathbb{J}}_1{\mathbb{J}}_3={\mathbb{J}}_2,$$

(7)

$$g({\mathbb{J}}_{1}G,{\mathbb{J}}_{1}K)=g({\mathbb{J}}_{2}G,{\mathbb{J}}_{2}K)=g({\mathbb{J}}_{3}G,{\mathbb{J}}_{3}K)=g(G,K),$$

(8)

where the identity tensor field of type $(1,1)$ on $\tilde{\mathbb{Q}}$ is denoted by $I_d$.

(b)

${\overline{\nabla }}_G\mathbb{J}$ is a local cross-section of V for any local cross-section $\mathbb{J}$ of V and any vector G tangent to $\tilde{\mathbb{Q}}$, where $\overline{\nabla }$ indicates the Riemannian connection on $\tilde{\mathbb{Q}}$.

(c)

Local 1-forms $\gamma$, $\sigma$, and $\tau$ exit there in a way that

$${\overline{\nabla }}_G{\mathbb{J}}_1=\tau \left(G\right){\mathbb{J}}_2-\sigma \left(G\right){\mathbb{J}}_3,$$

(9)

$${\overline{\nabla }}_G{\mathbb{J}}_2=-\tau \left(G\right){\mathbb{J}}_1+\gamma \left(G\right){\mathbb{J}}_3,$$

(10)

$${\overline{\nabla }}_G{\mathbb{J}}_3=\sigma \left(G\right){\mathbb{J}}_1-\gamma \left(G\right){\mathbb{J}}_2.$$

(11)

Assuming that G is a unit vector tangent to the quaternion manifold $\tilde{\mathbb{Q}}$, an orthonormal frame is formed by $\{G,{\mathbb{J}}_{1}G,$${\mathbb{J}}_{2}G,{\mathbb{J}}_{3}G\}$. For any orthonormal vectors $G,H$ tangent to $\tilde{\mathbb{Q}}$, the plane spanned by $G,K$ is said to be totally real if $\mathbb{Q}\left(G\right)$ and $\mathbb{Q}\left(K\right)$ are orthogonal. A quaternion plane is any plane that is part of a quaternion section. A quaternion sectional curvature is the sectional curvature of a quaternion plane [19].

Consider a $4n$-dimensional quaternion space form of constant quaternion sectional curvature c, denoted by $\tilde{\mathbb{Q}}\left(c\right)$. The following is the form of the curvature tensor field ${\tilde{\mathbb{R}}}_{em}$ of $\tilde{\mathbb{Q}}$ [13]:

$$\begin{array}{cc}\hfill {\tilde{\mathbb{R}}}_{em}(G,K)H& ={\displaystyle \frac{c}{4}}[g(K,H)E-g(G,H)K\hfill \\ \hfill & +g({\mathbb{J}}_{1}K,H){\mathbb{J}}_{1}G-g({\mathbb{J}}_{1}G,H){\mathbb{J}}_{1}K+2g(G,{\mathbb{J}}_{1}K){\mathbb{J}}_{1}H\hfill \\ \hfill & +g({\mathbb{J}}_{2}K,H){\mathbb{J}}_{2}G-g({\mathbb{J}}_{2}G,H){\mathbb{J}}_{2}K+2g(G,{\mathbb{J}}_{2}K){\mathbb{J}}_{2}H\hfill \\ \hfill & +g({\mathbb{J}}_{3}K,H){\mathbb{J}}_{3}G-g({\mathbb{J}}_{3}G,H){\mathbb{J}}_{3}K+2g(G,{\mathbb{J}}_{3}K){\mathbb{J}}_{3}H],\hfill \end{array}$$

(12)

for any $G,K,H\in \mathrm{\Gamma }\left(T\tilde{\mathbb{Q}}\right)$.

One way to express the Equation (12) is as

$$\begin{array}{ccc}\hfill {\tilde{\mathbb{R}}}_{em}(G,K)H& =& {\displaystyle \frac{c}{4}}[g(K,H)G-g(G,H)K+\sum _{\alpha =1}^{3}g({\mathbb{J}}_{\alpha }K,H){\mathbb{J}}_{\alpha }G\hfill \\ & -& g({\mathbb{J}}_{\alpha }G,H){\mathbb{J}}_{\alpha }K+2g(G,{\mathbb{J}}_{\alpha }K){\mathbb{J}}_{\alpha }H].\hfill \end{array}$$

(13)

3. Submanifolds of Quaternion Kaehler Manifolds

Consider a $4n$-dimensional $QKM$. $\tilde{\mathcal{Q}}$ has an m-dimensional submanifold $\mathbb{B}$, and the induced Levi-Civita connection on $\mathbb{B}$ is represented by ${\nabla }^{\mathbb{B}}$. The induced Riemannian metric on $\mathbb{B}$ is expressed by the same notation g. Assume that the curvature tensor of the induced connection ${\nabla }^{\mathbb{B}}$ is ${\mathbb{R}}_{em}$.

The formula for Gauss is provided by

$$\begin{array}{cc}\hfill {\tilde{\mathbb{R}}}_{em}(G,K,H,L)& ={\mathbb{R}}_{em}(G,K,H,L)-g(\hslash (G,L),\hslash (K,H))\hfill \\ \hfill & +g(\hslash (K,L),g(G,H\left)\right),\hfill \end{array}$$

(14)

where in $\tilde{\mathbb{Q}}$, ℏ represents the second fundamental form of $\mathbb{B}$.

The action of the local basis serves as the primary framework for the geometry of submanifold $\mathbb{B}$ of $QKM$$\tilde{\mathbb{Q}}$. On each tangent space to $\mathbb{B}$, $\{{\mathbb{J}}_1,{\mathbb{J}}_2,{\mathbb{J}}_3\}$.

Let ${\mathbb{B}}^m$ be a submanifold of a $QKM$${\mathbb{Q}}^{4n}$. For any $G\in \mathrm{\Gamma }\left(T\mathbb{B}\right)$, we are able to compose

$${\mathbb{J}}_{\alpha }G={\mathcal{T}}_{\alpha }G+{\mathcal{N}}_{\alpha }G,$$

where ${\mathcal{T}}_{\alpha }G$ (resp. ${\mathcal{N}}_{\alpha }G$) is the tangential component (resp. normal component) of ${\mathbb{J}}_{\alpha }G$.

In case, if ${\mathbb{J}}_{\alpha }\left(T\mathbb{B}\right)\subset T^{\perp }\mathbb{B}$, then $\mathbb{B}$ is totally real (${\mathcal{T}}_{\alpha }=0$) and if ${\mathbb{J}}_{\alpha }\left(T\mathbb{B}\right)\subset T\mathbb{B}$, then $\mathbb{B}$ is quaternion (${\mathcal{N}}_{\alpha }=0$).

An n-dimensional totally real submanifold of a $4m$-dimensional $\tilde{\mathbb{Q}}$ is said to be a Lagrangian submanifold if $n=m$.

At point $p\in \mathbb{B}$, the squared norm of ${\mathcal{T}}_{\alpha }$ is provided as

$$\begin{array}{c}\hfill \left|\right|{\mathcal{T}}_{\alpha }{\left|\right|}^2=\sum _{\alpha =1}^3\sum _{i,j=1}^m{g}^2({\mathbb{J}}_{\alpha }{\nu }_i,{\nu }_j),\end{array}$$

(15)

wherein $\{{\nu }_1,\dots ,{\nu }_m\}$ is any orthonormal frame basis of the tangent space $T_p\mathbb{B}$.

Definition 3 

([20]). A submanifold $\mathbb{B}$ of a $QKM$$\tilde{\mathbb{Q}}$ is referred to as slant if the angle $\mathrm{\Theta }\left(G\right)$ between $\mathbb{J}G$ and $T_x\mathbb{B}$ is constant for each non-zero vector G tangent to $\mathbb{B}$ at $x\in \mathbb{B}$, linearly independent on ξ for each non-zero vector G tangent to $\mathbb{B}$ at $x\in \mathbb{B}$.

This angle Θ is known as the slant angle of the submanifold in this sense. If neither $\mathrm{\Theta }=0$ nor $\mathrm{\Theta }={\displaystyle \frac{\pi }{2}}$, a slant submanifold $\mathbb{B}$ is regarded as valid slant submanifold.

In addition, we can also see the following scenarios [20]:

(i)

If $\mathrm{\Theta }=0$, a slant submanifold $\mathbb{B}$ is an invariant submanifold.

(ii)

If $\mathrm{\Theta }={\displaystyle \frac{\pi }{2}}$, it is an anti-invariant submanifold.

Furthermore, invariant and anti-invariant submanifolds are extended by the slant submanifold [21].

Carriazo et al. proposed the concept of bi-slant submanifolds as a logical extension of CR, semi-slant, slant, and hemi-slant submanifolds (see [22]). Furthermore, in an essentially Hermitian manifold, Papaghiuc [23] created a different classification of submanifolds known as the semi-slant submanifolds, which includes proper CR-submanifolds and proper slant-submanifolds as special examples. Bi-slant submanifolds in particular can yield semi-slant submanifolds [22], hemi-slant submanifolds, CR submanifolds, and slant submanifolds [21,24]. Quaternion CR-submanifolds of $QKM$ were discussed by Barros et al. in [25] (some results are also presented in [26]). Slant submanifolds in $QKM$ were further studied by Sahin [27]. Numerous writers have since examined some sharp inequalities with these submanifolds in $QKM$ and other relevant ambient manifolds [28,29,30,31,32].

A submanifold $\mathbb{B}$ of a $QKM$$\tilde{\mathbb{Q}}$ is said to be a quaternion bi-slant submanifold $\left(QBSS\right)$, if we have

1.

$T\mathbb{B}={\mathbb{D}}_{{\mathrm{\Theta }}_1}\oplus {\mathbb{D}}_{{\mathrm{\Theta }}_2}$, for any $i=1,2$, the distribution ${\mathbb{D}}_i$ is slant distribution with slant angle ${\mathrm{\Theta }}_i$.

2.

${\mathbb{J}}_{\alpha }{\mathbb{D}}_{{\mathrm{\Theta }}_1}\perp {\mathbb{D}}_{{\mathrm{\Theta }}_2}$ and ${\mathbb{J}}_{\alpha }{\mathbb{D}}_{{\mathrm{\Theta }}_2}\perp {\mathbb{D}}_{{\mathrm{\Theta }}_1}$, for $\alpha =1,2,3$,

wherein ${\mathbb{D}}_{{\mathrm{\Theta }}_1}$ and ${\mathbb{D}}_{{\mathrm{\Theta }}_2}$ are two orthogonal distributions of $\mathbb{B}$ with slant angle ${\mathrm{\Theta }}_1$ and ${\mathrm{\Theta }}_2$, respectively.

Let $\mathbb{B}$ be a $QBSS$ of a $QKM$$\tilde{\mathbb{Q}}$. Also, let dim $\mathbb{B}=m=(4m_1+4m_2)$ and $\{{\nu }_1,\dots ,{\nu }_m\}$ be an orthonormal basis of $T_p\mathbb{B}$ at point p in $\mathbb{B}$ such as

$$g^2({\mathbb{J}}_{\alpha }{\nu }_{i+1},{\nu }_i)=\left\{\begin{array}{c}Co{s}^2{\mathrm{\Theta }}_1,fori=1,2,\dots ,4m_1-1,\alpha =1,2,3\hfill \\ Co{s}^2{\mathrm{\Theta }}_2,fori=4m_1+1,\dots ,4m_1+4m_2-1,\alpha =1,2,3.\hfill \end{array}\right.$$

Hence, we have

$$\begin{array}{c}\hfill \sum _{\alpha =1}^3\sum _{i,j=1}^m{g}^2\left({\mathbb{J}}_{\alpha }{\nu }_i,{\nu }_j\right)=12(m_{1}Co{s}^2{\mathrm{\Theta }}_1+m_{2}Co{s}^2{\mathrm{\Theta }}_2).\end{array}$$

(16)

Remark 1. 

The quaternion Kaehler and totally real special instances of the slant submanifold are $\mathrm{\Theta }=0$ and $\mathrm{\Theta }={\displaystyle \frac{\pi }{2}}$, respectively. When $0<\mathrm{\Theta }<{\displaystyle \frac{\pi }{2}}$ and ${\mathrm{\Theta }}_i$ fall between 0 and ${\displaystyle \frac{\pi }{2}}$, the slant submanifold is referred to as a proper slant and a proper bi-slant submanifold, respectively [21].

4. Intrinsic and Extrinsic Invariants in $\mathit{QKM}$

For quaternion bi-slant submanifolds of dimension m in a $4n$-dimensional quaternion space form $\tilde{\mathbb{Q}}$.

Let the local orthonormal tangent frame $\{{\nu }_1,\dots ,{\nu }_m\}$ of the tangent bundle $T\mathbb{B}$ of $\mathbb{B}$ and a local orthonormal normal frame $\{{\nu }_{m+1},\dots ,{\nu }_{4n}\}$ of the normal bundle $T^{\perp }\mathbb{B}$ of $\mathbb{B}$ in $\tilde{\mathbb{Q}}$.

At any $p\in \mathbb{B}$ the scalar curvature $\delta$ at that point is given by

$$\begin{array}{c}\hfill \delta =\sum _{1\le i<j\le m}{\mathbb{R}}_{em}({\nu }_i,{\nu }_j,{\nu }_j,{\nu }_i).\end{array}$$

(17)

The mean curvature $\mathrm{\Lambda }$ of submanifold is express as

$$\mathrm{\Lambda }=\sum _{i=1}^{m}h({\nu }_i,{\nu }_i),$$

(18)

and ℏ, the second fundamental form, is given by has the equation

$$\begin{array}{c}\hfill {\left|\right|\hslash \left|\right|}^2=\sum _{a=1}^{4n-m}\sum _{i,j=1}^m{\left({\hslash }_{ij}^a\right)}^2.\end{array}$$

(19)

Moreover, the divergence of any vector field V on $\mathrm{\Gamma }\left(T\mathbb{B}\right)$ is denoted by $div\left(V\right)$ and defined by

$$div\left(V\right)=\sum _{i=1}^{m}g({\nabla }_{{\nu }_i}V,{\nu }_i),$$

(20)

where $\{{\nu }_1,\dots ,{\nu }_m\}$ a local orthonormal tangent frame of the tangent bundle $T\mathbb{B}$ of $\mathbb{B}$.

Example 1. 

Consider the geodesic sphere ${\mathbb{S}}^{4m-1}\left(r\right)$ of radius $r\in (0,{\displaystyle \frac{\pi }{2}})$ in the quaternionic Euclidean space ${\mathbb{R}}^{4m}(={\mathbb{H}}^m)$. We note that ${\mathbb{S}}^{4m-1}\left(r\right)$ is the curvature-adapted hypersurface of the quaternionic projective space ${\mathbb{HP}}^m\left(4\right)$. Furthermore, it can be observed that the principal curvatures of a geodesic sphere of radius r are

$${\lambda }_1=cotr\mathit{with}\mathit{multiplicity}{m}_1=4(m-1)$$

$${\lambda }_2=2cot\left(2r\right)\mathit{with}\mathit{multiplicity}{m}_2=3.$$

With the help of principal curvatures, the mean curvature vector and the squared length of the second fundamental form are

$$\mathrm{\Lambda }={\displaystyle \frac{4(m-1)}{4m-1}}cotr+{\displaystyle \frac{3}{4m-1}}(cotr-tanr),$$

$${\left|\right|\hslash \left|\right|}^2=4(m-1){cot}^{2}r+3{tan}^{2}r-6$$

are obtained in [15]. So, we conclude that ${\mathbb{S}}^{4m-1}\left(r\right)$ can never be totally geodesic.

Theorem 1. 

Let $\mathbb{B}$ be an m-dimensional $QBSS$ in a $QKM$$\tilde{\mathbb{Q}}$ of dimension $4n$, then the scalar curvature is

$$\begin{array}{c}\hfill \tilde{\delta }={\displaystyle \frac{c}{4}}\left[1+{\displaystyle \frac{36}{m(m-1)}}(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right].\end{array}$$

(21)

Proof. 

From Equation (12), we have

$$\begin{array}{ccc}\hfill {\tilde{\mathbb{R}}}_{em}(G,K,H,L)& =& {\displaystyle \frac{c}{4}}[g(K,H)g(E,L)-g(G,H)g(K,L)\hfill \\ & +& g({\mathbb{J}}_{1}K,H)g({\mathbb{J}}_{1}G,L)-g({\mathbb{J}}_{1}G,H)g({\mathbb{J}}_{1}K,L)+2g(G,{\mathbb{J}}_{1}K)g({\mathbb{J}}_{1}H,L)\hfill \\ & +& g({\mathbb{J}}_{2}K,H)g({\mathbb{J}}_{2}G,L)-g({\mathbb{J}}_{2}G,H)g({\mathbb{J}}_{2}K,L)+2g(G,{\mathbb{J}}_{2}K)g({\mathbb{J}}_{2}H,L)\hfill \\ & +& g({\mathbb{J}}_{3}K,H)g({\mathbb{J}}_{3}G,L)-g({\mathbb{J}}_{3}G,H)g({\mathbb{J}}_{3}K,L)+2g(G,{\mathbb{J}}_{3}K)g({\mathbb{J}}_{3}H,L)],\hfill \end{array}$$

or

$$\begin{array}{ccc}\hfill {\tilde{\mathbb{R}}}_{em}(G,K,H,L)& =& {\displaystyle \frac{c}{4}}[g(K,H)g(G,L)-g(G,H)g(K,L)+\sum _{\alpha =1}^{3}g({\mathbb{J}}_{\alpha }K,H)g({\mathbb{J}}_{\alpha }G,L)\hfill \\ & -& g({\mathbb{J}}_{\alpha }G,H)g({\mathbb{J}}_{\alpha }K,L)+2g(G,{\mathbb{J}}_{\alpha }K)g({\mathbb{J}}_{\alpha }H,L)].\hfill \end{array}$$

(22)

Let $\{{\nu }_1,\dots ,{\nu }_m\}$ and $\{{\nu }_{m+1},\dots ,{\nu }_{4n}\}$ be orthonormal tangent frames and orthonormal normal frames on $\mathcal{M}$, respectively. Putting $G=L={\nu }_i$, $K=G={\nu }_j$, $i\ne j$ in Equation (22) and, using Equation (14), we obtain

$$\begin{array}{ccc}\hfill \widetilde{{\mathbb{R}}_{em}}({\nu }_i,{\nu }_j,{\nu }_j,{\nu }_i)& =& {\displaystyle \frac{c}{4}}[g({\nu }_j,{\nu }_j)g({\nu }_i,{\nu }_i)-g({\nu }_i,{\nu }_j)g({\nu }_j,{\nu }_i)]\hfill \\ & +& \sum _{\alpha =1}^3[g({\mathbb{J}}_{\alpha }{\nu }_j,{\nu }_j){\mathbb{J}}_{\alpha }{\nu }_i-g({\mathbb{J}}_{\alpha }{\nu }_i,{\nu }_j){\mathbb{J}}_{\alpha }{\nu }_j\hfill \\ & +& 2g({\nu }_i,{\mathbb{J}}_{\alpha }{\nu }_j){\mathbb{J}}_{\alpha }{\nu }_j].\hfill \end{array}$$

(23)

After applying contraction and taking summation $1\le i<j\le m$ of Equation (23) and using Equation (14), we have

$$\begin{array}{cc}\hfill \sum _{1\le i<j\le m}\widetilde{{\mathbb{R}}_{em}}({\nu }_i,{\nu }_j,{\nu }_j,{\nu }_i)& ={\displaystyle \frac{c}{4}}\left\{m(m-1)+3\sum _{\alpha =1}^3\left|\right|{\mathcal{T}}_{\alpha }{\left|\right|}^2\right\}.\hfill \end{array}$$

(24)

Using Equations (15) and (17), we obtain

$$\begin{array}{cc}\hfill \tilde{\delta }& ={\displaystyle \frac{c}{4}}\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\}.\hfill \end{array}$$

(25)

□

5. Ricci Solitons Inequality on Quaternion Bi-Slant Submanifolds

To obtain a relationship between the intrinsic and extrinsic invariants, we give the scalar curvature of submanifold $\mathbb{B}$ of Ricci soliton $(\mathbb{Q},g,\mathcal{F},\lambda )$ in this section. Then, to describe such a submanifold $\mathbb{B}$, we establish an inequality for the Ricci soliton $(\mathbb{Q},g,\mathcal{F},\lambda )$ and gradient Ricci soliton.

Let $(\mathbb{Q},g,\mathcal{F},\mathrm{\Lambda })$ be a $4n$-dimensional $QKM$ and $\phi :{\mathbb{B}}^m\to {\mathbb{Q}}^{4n}$ be an isometric immersion from an m-dimensional quaternion manifold $({\mathbb{B}}^m,g)$ into $({\mathbb{Q}}^{4n},g)$ manifold.

Then, the Ricci tensor $S_{ric}$ can be written as

$${\tilde{S}}_{ric}(G,K)={\tilde{S}}_{ric}{{|}_{T_p\mathbb{B}}(G,K)+{\tilde{S}}_{ric}|}_{T_p^{\perp }\mathbb{B}}(G,K)$$

(26)

for any $G,K\in T_p\mathbb{B}.$

Since ($\tilde{\mathbb{Q}},g,\mathcal{F},\lambda )$ is a $RS$. Then in view of Equation (26), we get

$${\displaystyle \frac{1}{2}}\sum _{i=1}^m\{g({\nabla }_{{\nu }_i}\mathcal{F},{\nu }_i)+g({\nu }_i,{\nabla }_{{\nu }_i}\mathcal{F})\}+\sum _{i=1}^m{\tilde{S}}_{ric}({\nu }_i,{\nu }_i)+\sum _{i=1}^m\lambda g({\nu }_i,{\nu }_i)=0,$$

(27)

where $\{{\nu }_1,\dots ,{\nu }_m\}$ a local orthonormal tangent frame of the tangent bundle $T\mathbb{B}$ of $\mathbb{B}$.

Then, again applying contraction and using Equations (19), (26) and (20) in Equation (27) we turn up

$$\widetilde{div}\left(\mathcal{F}\right)+{\tilde{S}}_{ric}({\nu }_i,{\nu }_i){|}_{T_p\mathbb{B}}({\nu }_i,{\nu }_i)+\lambda \sum _{i=1}^{m}g({\nu }_i,{\nu }_i)=0.$$

(28)

Adopting Equations (14) and (21) in Equation (28) we get

$$\widetilde{div}\left(\mathcal{F}\right)+2\tilde{\delta }-\sum _{i,j=1}^m\{g(\hslash ({\nu }_i,{\nu }_i),\hslash ({\nu }_i,{\nu }_i))-g(\hslash ({\nu }_j,{\nu }_i),\hslash ({\nu }_i,{\nu }_i))\}$$

(29)

$$+\lambda \sum _{j=1}^{m}g({\nu }_j,{\nu }_j)=0.$$

Now, in view of Equations (18), (19), and (21), we get

$$\begin{array}{cc}\hfill \widetilde{\mathrm{div}\left(\mathcal{F}\right)}& ={\displaystyle \frac{c}{4}}\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}+m^2{\parallel \mathrm{\Lambda }\parallel }^2-{\parallel \hslash \parallel }^2-m\lambda .\hfill \end{array}$$

(30)

Thus, we can articulate the following:

Theorem 2. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F},\lambda )$ is an $RS$ and $\mathbb{B}$ is an m-dimensional quaternion bi-slant submanifold of a $4n$-dimensional $QKM$$(\tilde{\mathbb{Q}},g)$, then we have

$$\begin{array}{ccc}\hfill \widetilde{\mathrm{div}\left(\mathcal{F}\right)}& =& {\displaystyle \frac{c}{4}}\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}+m^2{\parallel \mathrm{\Lambda }\parallel }^2-{\parallel \hslash \parallel }^2-m\lambda .\hfill \end{array}$$

We state the following as a result of Equation (30).

Theorem 3. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F},\lambda )$ is an $RS$ and $\mathbb{B}$ is an m-dimensional quaternion bi-slant submanifold of a $4n$-dimensional $QKM$$(\tilde{\mathbb{Q}},g)$ admits an $RS$ with a potential vector field $\mathcal{F}\in T\mathbb{B}$ of Ricci soliton. Then the Ricci soliton on $QBSS$$\mathbb{B}$ is expanding, steady, and shrinking according as

1.

${\displaystyle \frac{c}{4m}}\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}+m{\parallel \mathrm{\Lambda }\parallel }^2>{\displaystyle \frac{\widetilde{\mathrm{div}\left(\mathcal{F}\right)}}{m}}+{\displaystyle \frac{{\parallel \hslash \parallel }^2}{m}},$

2.

${\displaystyle \frac{c}{4m}}\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}+m{\parallel \mathrm{\Lambda }\parallel }^2={\displaystyle \frac{\widetilde{\mathrm{div}\left(\mathcal{F}\right)}}{m}}+{\displaystyle \frac{{\parallel \hslash \parallel }^2}{m}},$

3.

${\displaystyle \frac{c}{4m}}\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}+m{\parallel \mathrm{\Lambda }\parallel }^2<{\displaystyle \frac{\widetilde{\mathrm{div}\left(\mathcal{F}\right)}}{m}}+{\displaystyle \frac{{\parallel \hslash \parallel }^2}{m}},$ respectively.

Remark 2. 

Theorem 3 shows that the condition for expanding/steady/shrinking solitons and these conditions will present as three separate inequalities. Therefore, for the sake of simplicity and to maintain the lower bound inequality, from Equation (30), we are using the expanding Ricci soliton for $\lambda >0.$ Thus, throughout the study we consider the expanding Ricci soliton.

At this point, we recall the following lemma from [33].

Lemma 1. 

If $r_1,r_2\dots r_m$ for $m>1$, are real numbers, then

$${\displaystyle \frac{1}{m}}{\left\{\sum _{i=1}^n{r}_i\right\}}^2\le \sum _{i=1}^m{r}_i^2$$

with equality holding if and only if $r_1=r_2=\dots r_m.$

Considering Equations (30), (19), and (20), we arrive at

$$\begin{array}{ccc}\hfill {\displaystyle \frac{c}{4}}\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}& =& -\widetilde{\mathrm{div}\left(\mathcal{F}\right)}+m^2{\parallel \mathrm{\Lambda }\parallel }^2-{\parallel \hslash \parallel }^2-m\lambda .\hfill \end{array}$$

$$\begin{array}{l}={\displaystyle \frac{c}{4}}\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}\\ =-\widetilde{\mathrm{div}\left(\mathcal{F}\right)}+m^2{\parallel \mathrm{\Lambda }\parallel }^2-m\lambda -{\displaystyle \sum _{a=m+1}^{4n}}{\displaystyle \sum _{i=1}^m}{\left({\hslash }_{ii}^a\right)}^2-{\displaystyle \sum _{a=m+1}^{4n}}{\displaystyle \sum _{i,j\ne 1}^m}{\left({\hslash }_{ij}^a\right)}^2\\ \hfill {\displaystyle \frac{c}{4}}\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}\hfill \\ \le -\widetilde{\mathrm{div}\left(\mathcal{F}\right)}+m^2{\parallel \mathrm{\Lambda }\parallel }^2-m\lambda -{\displaystyle \frac{m^2{\parallel \mathrm{\Lambda }\parallel }^2}{m}}-{\displaystyle \sum _{a=m+1}^{4n}}{\displaystyle \sum _{i,j\ne 1}^m}{\left({\hslash }_{ij}^a\right)}^2\\ \le -\widetilde{\mathrm{div}\left(\mathcal{F}\right)}-m\lambda -m(m-1){\parallel \mathrm{\Lambda }\parallel }^2-{\displaystyle \sum _{a=m+1}^{4n}}{\displaystyle \sum _{i,j\ne 1}^m}{\left({\hslash }_{ij}^a\right)}^2\end{array}$$

is found then

$$\begin{array}{c}{\displaystyle \frac{c}{4}}\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}\\ \le -\widetilde{\mathrm{div}\left(\mathcal{F}\right)}-m\lambda -m(m-1){\parallel \mathrm{\Lambda }\parallel }^2\end{array}$$

(31)

is acquired, providing us with Equation (31). $\mathbb{B}$ is totally umbilical if the equality of Equation (31) is met.

Now, we can state the following outcome

Theorem 4. 

Let $(\tilde{\mathbb{Q}},g,\mathcal{F},\lambda )$ be a $RS$ and $\mathbb{B}$ be a m-dimensional $QBSS$ of a $4n$-dimensional $QKM.$ Then we have

$$\begin{array}{c}\widetilde{\mathrm{div}\left(\mathcal{F}\right)}\le m(m-1){\parallel \mathrm{\Lambda }\parallel }^2-m\lambda \hfill \\ -{\displaystyle \frac{c}{4}}\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\},\hfill \end{array}$$

(32)

for $\mathcal{F}\in \mathrm{\Gamma }\left(T\mathbb{B}\right)$. If the equality of Equation (32) holds, then $\mathbb{B}$ is totally umbilical.

Let $\mathcal{F}$ be a gradient-type soliton vector field, meaning that $\mathcal{F}=\nabla f$, where f is a smooth function on $\mathbb{B}$. Thus, we state the following in light of Equations (3) and (30).

Theorem 5. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a soliton vector field $\mathcal{F}\in T\mathbb{B}$ of gradient type and $\mathbb{B}$ an m-dimensional $QBSS$ of a $4n$-dimensional $QKM$, then we have

$$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2.$$

Corollary 1. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a soliton vector field $\mathcal{F}\in T\mathbb{B}$ of gradient type and $\mathbb{B}$ an m-dimensional totally umbilical bi-slant submanifold of a $4n$-dimensional $QKM$, then we have

$$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+m{\lambda }^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2.$$

(33)

6. Some Applications of Theorem 5

In this section, we turn up the lower bound inequities for semi-slant submanifolds, hemi-slant submanifolds, quaternion $CR$ submanifolds, slant submanifolds, invariant and totally real submanifolds of $QKM$.

In the light of Theorem 5 and

Table 1. Classification of various submanifolds in $QKM$ with respect to the angles [32].

S.N	${\mathbb{B}}^{\mathit{m}}$	${\mathbb{D}}_{{\mathbf{\Theta }}_1}$	${\mathbb{D}}_{{\mathbf{\Theta }}_2}$	${\mathbf{\Theta }}_1$	${\mathbf{\Theta }}_2$
(1)	bi-slant	slant distribution	slant distribution	slant angle	slant angle
(2)	semi-slant	invariant distribution	slant distribution	0	slant angle
(3)	hemi-slant	slant distribution	totally real distribution	slant angle	${\displaystyle \frac{\pi }{2}}$
(4)	quaternion $CR$	invariant distribution $\mathbb{D}$ under ${\mathbb{J}}_{\alpha }$, that is, ${\mathbb{J}}_{\alpha }\mathbb{D}\subseteq \mathbb{D}$	totally real distribution ${\mathbb{D}}^{\perp }$, that is, ${\mathbb{J}}_{\alpha }{\mathbb{D}}^{\perp }\subseteq T^{\perp }\mathbb{B}$	0	${\displaystyle \frac{\pi }{2}}$
(5)	slant	either ${\mathbb{D}}_{{\mathrm{\Theta }}_1}=0$ or ${\mathbb{D}}_{{\mathrm{\Theta }}_2}=0$		either ${\mathrm{\Theta }}_1={\mathrm{\Theta }}_2=\mathrm{\Theta }$ or ${\mathrm{\Theta }}_1={\mathrm{\Theta }}_2\ne \mathrm{\Theta }$

, we obtain the lower bound of various quaternion submanifolds of $4n$-dimensional $QKM$ in terms of an expanding $GRs$ with a soliton vector field ($SVF$) $\mathcal{F}\in T\mathbb{B}$.

Corollary 2. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a $SVF$$\mathcal{F}\in T\mathbb{B}$ of gradient type and ($\mathbb{B},\mathrm{\Theta }={\mathrm{\Theta }}_1=0)$ an m-dimensional quaternion semi-slant submanifold of a $4n$-dimensional $QKM$, then we have

$$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36(m_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2.$$

Corollary 3. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a $SVF$$\mathcal{F}\in T\mathbb{B}$ of gradient type and ($\mathbb{B},{\mathrm{\Theta }}_1=\mathrm{\Theta },{\mathrm{\Theta }}_2={\displaystyle \frac{\pi }{2}})$ an m-dimensional quaternion hemi-slant submanifold of a $4n$-dimensional $QKM$, then we have

$$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36\left(m_{1}co{s}^2{\mathrm{\Theta }}_1\right)\right\}}^2.$$

Corollary 4. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a $SVF$$\mathcal{F}\in T\mathbb{B}$ of gradient type and ($\mathbb{B},{\mathrm{\Theta }}_1=0,{\mathrm{\Theta }}_2={\displaystyle \frac{\pi }{2}})$ an m-dimensional quaternion $CR$ submanifold of a $4n$-dimensional $QKM$, then we have

$$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36m_1\right\}}^2.$$

A bi-slant submanifold reduces to a general slant submanifold if the two distributions have the same non-zero, non-$\pi /2$ constant slant angle (${\mathrm{\Theta }}_1={\mathrm{\Theta }}_2=\mathrm{\Theta }$). This generalizes both invariant and anti-invariant cases (totally real submanifolds). Invariant submanifolds occur when the bi-slant angles are both zero (${\mathrm{\Theta }}_1={\mathrm{\Theta }}_2=0$). The connection here is that the tangent space is entirely invariant under the action of the quaternionic structure. Anti-invariant submanifolds are defined by both bi-slant angles being $\pi /2$ (${\mathrm{\Theta }}_1={\mathrm{\Theta }}_2=\pi /2$). In this case, the quaternionic structure maps the entire tangent space into the normal space, meaning there is no tangential component [20].

Therefore, in the light of the Theorem 5, we can state the following Corollaries:

Corollary 5. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a $SVF$$\mathcal{F}\in T\mathbb{B}$ of gradient type and ($\mathbb{B},{\mathrm{\Theta }}_1={\mathrm{\Theta }}_2=\mathrm{\Theta })$ an m-dimensional quaternion slant submanifold of a $4n$-dimensional $QKM$, then we have

$$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2-{\displaystyle \frac{c^2}{16}}{\left\{(m-1)+36co{s}^2\mathrm{\Theta }\right\}}^2.$$

Corollary 6. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a $SVF$$\mathcal{F}\in T\mathbb{B}$ of gradient type and ($\mathbb{B},\mathrm{\Theta }=0)$ an m-dimensional invariant submanifold of a $4n$-dimensional $QKM$, then we have

$$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2-{\displaystyle \frac{1}{16}}\left\{c^2{(m-1)}^2+81\right\}.$$

Corollary 7. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a $SVF$$\mathcal{F}\in T\mathbb{B}$ of gradient type and ($\mathbb{B},\mathrm{\Theta }={\displaystyle \frac{\pi }{2}})$ an m-dimensional totally real submanifold of a $4n$-dimensional $QKM$, then we have

$$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2-{\displaystyle \frac{c^2{(m-1)}^2}{256}}.$$

Moreover, in view of Table 1 and Corollary 1, we gain the following inequalities classification in the case of different totally umbilical bi-slant submanifolds of a $4n$-dimensional $QKM$. Then one can summarize the results in the form of the following table:

Corollary 8. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a $SVF$$\mathcal{F}\in T\mathbb{B}$ of gradient type and $\mathbb{B}$ an m-dimensional totally umbilical submanifold of a $4n$-dimensional $QKM$, then we have

Table 2. Classification solitonic inequality for totally umbilical submanifolds with ${\mathrm{\Theta }}_1$ and ${\mathrm{\Theta }}_2$.

S.N	Submanifold $\mathbb{B}$	Inequality
(1)	semi-slant	$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36(m_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2+m{\lambda }^2$
(2)	hemi-slant	$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36m_{1}co{s}^2{\mathrm{\Theta }}_1\right\}}^2+m{\lambda }^2$
(3)	quaternion $CR$	$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36m_1\right\}}^2+m{\lambda }^2$
(4)	slant	$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2-{\displaystyle \frac{c^2}{16}}{\left\{(m-1)+36co{s}^2\mathrm{\Theta }\right\}}^2+m{\lambda }^2$
(5)	invariant	$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2-{\displaystyle \frac{c^2}{16}}{\left\{(m-1)+36\right\}}^2+m{\lambda }^2$
(6)	totally real	$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2-{\displaystyle \frac{c^2{(m-1)}^2}{16}}+m{\lambda }^2$

.

7. Double Inequality for Expanding Gradient Ricci Solitons on Submanifolds in Quaternion Manifolds

In this section, we compute a double inequality for the scalar curvature of semi-slant submanifolds, hemi-slant submanifolds, quaternion CR submanifolds, slant submanifolds, and invariant and totally real submanifolds of quaternion Kahler manifolds in terms of expanding gradient Ricci solitons. Moreover, we also deduced the same double inequality with a scalar concircular field $\left(SCF\right)$.

Because of the rich, rigid geometric properties of quaternion Kähler manifolds and the significant interaction between concircular fields and Ricci flow phenomena, studying scalar concircular fields in quaternion Kähler manifolds is a natural and promising extension of research into gradient Ricci solitons. The gradient Ricci solitons $S_{ric}+\mathcal{H}ess\left(f\right)=\lambda g,$ or a potential function f analyzing how a scalar concircular field, which specifies certain conformal symmetries, interacts with this soliton equation on a quaternion Kähler manifold, which has an Einstein structure. Quaternion Kähler manifolds with positive scalar curvature are strongly confined and thought to be symmetric spaces. The presence of a scalar concircular field adds another level of symmetry and constraint.

Both gradient Ricci solitons with a scalar concircular field on quaternion Kähler manifolds have important implications for string theory and supergravity. Investigating their intersection may reveal new insights applicable to various physical theories, such as the development of new metrics or the understanding of moduli spaces.

Now, in the light of Equations (4), (25), and (32), we convert the following double inequality for the submanifolds of quaternion Kahler manifold in terms of gradient Ricci Solitons with scalar curvature $\tilde{\delta }$:

$$\left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2-{\displaystyle \frac{1}{n}}({\widetilde{\mathrm{\Delta }f)}}^2\le \left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\le \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+{\displaystyle \frac{{\tilde{\delta }}^2}{n}}.$$

(34)

Inserting again Equation (25) in Equation (34), we get

$$\begin{array}{cc}\hfill \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2-{\displaystyle \frac{1}{n}}{\left(\widetilde{\mathrm{\Delta }f}\right)}^2& \le \left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\hfill \\ \hfill & \le \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+{\displaystyle \frac{c^2}{16n}}{\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}}^2.\hfill \end{array}$$

(35)

Thus, we state the following results:

Theorem 6. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a $SVF$$\mathcal{F}\in T\mathbb{B}$ and $\mathbb{B}$ a m-dimensional quaternion bi-slant submanifold of a $4n$-dimensional $QKM$, then we have

$$\parallel \widetilde{\mathcal{H}ess\left(f\right)}{\parallel }_g^2-{\displaystyle \frac{1}{n}}{\left(\widetilde{\mathrm{\Delta }f}\right)}^2\le \parallel \widetilde{S_{ric}}{\parallel }_g^2\le {\parallel \widetilde{\mathcal{H}ess\left(f\right)}\parallel }_g^2+{\displaystyle \frac{c^2}{16n}}{\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}}^2.$$

Now, in view of Theorem 5, we obtain the double inequality for the soliton vector field $\mathcal{F}$ of gradient type.

$$\left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2$$

(36)

$$+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2\le \left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\le \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+{\displaystyle \frac{{\tilde{\delta }}^2}{n}}.$$

$$\begin{array}{c}\hfill \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2\end{array}$$

$$\begin{array}{c}\hfill \le \left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\le \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+{\displaystyle \frac{c^2}{16n}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2.\\ \hfill \end{array}$$

(37)

Theorem 7. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a $SVF$$\mathcal{F}\in T\mathbb{B}$ of gradient type and $\mathbb{B}$ a m-dimensional quaternion bi-slant submanifold of a $4n$-dimensional $QKM$, then we have

$$\begin{array}{c}\hfill \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2\le \left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\end{array}$$

$$\begin{array}{c}\hfill \le \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+{\displaystyle \frac{c^2}{8m}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2.\\ \hfill \end{array}$$

Corollary 9. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a $SVF$$\mathcal{F}\in T\mathbb{B}$ of gradient type and $\mathbb{B}$ a m-dimensional totally umbilical bi-slant submanifold of a $4n$-dimensional $QKM$, then we have

$$\begin{array}{c}\parallel \widetilde{\mathcal{H}ess\left(f\right)}{\parallel }_g^2+m{\lambda }^2\le {\parallel \widetilde{S_{ric}}\parallel }_g^2\\ \le \parallel \widetilde{\mathcal{H}ess\left(f\right)}{\parallel }_g^2+{\displaystyle \frac{c^2}{8m}}{\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}}^2.\end{array}$$

(38)

Remark 3. 

In view of Table 1, Theorem 7, and Corollary 9, we can easily obtain the same double inequality for the semi-slant submanifolds, hemi-slant submanifolds, quaternion $CR$-submanifolds, slant submanifolds, and invariant and anti-invariant submanifolds of a $4n$-dimensional $QKM$

Next using Equations (5), (34) and (25) together, we infer

$$\begin{array}{c}\hfill {\mho }^2{\left|\right|G\left|\right|}_g^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2\end{array}$$

$$\begin{array}{c}\hfill \le \left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\le {\mho }^2{\left|\right|G\left|\right|}_g^2+{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36(m_1{cos}^2{\mathrm{\Theta }}_1+m_2{cos}^2{\mathrm{\Theta }}_2)\right\}}^2.\end{array}$$

(39)

In addition, for a totally umbilical bi-slant submanifold (Equation (39)) is reduced to

$$\begin{array}{c}\hfill {\mho }^2{\left|\right|G\left|\right|}_g^2-{\displaystyle \frac{c^2}{16m}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2+m{\lambda }^2\end{array}$$

$$\begin{array}{c}\hfill \le \left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\le {\mho }^2{\left|\right|G\left|\right|}_g^2+{\displaystyle \frac{c^2}{16n}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2.\end{array}$$

(40)

Consequently, one can state the following results:

Theorem 8. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a scalar concircular field $\mathcal{F}\in T\mathbb{B}$ and $\mathbb{B}$ an m-dimensional quaternion bi-slant submanifold of a $4n$-dimensional $QKM$, then we have

$$\begin{array}{c}\hfill {\mho }^2{\left|\right|G\left|\right|}_g^2+m^3{\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{m}}+m{\lambda }^2\le \left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\end{array}$$

$$\begin{array}{c}\hfill \le {\mho }^2{\left|\right|G\left|\right|}_g^2+{\displaystyle \frac{c^2}{8m}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2.\\ \hfill \end{array}$$

Corollary 10. 

If $(\tilde{\mathbb{Q}},g,\mathcal{F}=\nabla f,\lambda )$ is an expanding $GRs$ with a scalar concircular field $\mathcal{F}\in T\mathbb{B}$ and $\mathbb{B}$ an m-dimensional totally umbilical bi-slant submanifold of a $4n$-dimensional $QKM$, then we have

$$\begin{array}{c}\hfill {\mho }^2{\left|\right|G\left|\right|}_g^2+m{\lambda }^2\le \left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\le {\mho }^2{\left|\right|G\left|\right|}_g^2+{\displaystyle \frac{c^2}{8m}}{\left\{m(m-1)+36(m_{1}co{s}^2{\mathrm{\Theta }}_1+m_{2}co{s}^2{\mathrm{\Theta }}_2)\right\}}^2.\end{array}$$

Remark 4. 

In view of Table 1, Theorem 8, and Corollary 10, we can easily obtain the same double inequality in terms of expanding $GRs$ with a scalar concircular field for the semi-slant submanifolds, hemi-slant submanifolds, quaternion $CR$-submanifolds, slant submanifolds, invariant and anti-invariant submanifolds of a $4n$-dimensional $QKM$

8. Conclusions

Inequalities in terms of expanding gradient solitons are a significant area of research that provides profound insights into the geometric and analytic features of quaternion Kahler manifolds with positive scalar curvature. Gradient soliton-based inequalities are fundamental mathematical assertions in differential geometry and analysis involving geometric features such as curvatures, which relate extrinsic to intrinsic invariants for submanifolds in quaternion Kahler manifolds with soliton behavior. This paper contributes to the geometric theory of quaternion Kahler manifolds by studying gradient Ricci solitons on quaternion bi-slant submanifolds. We derive lower-bound-type and double inequalities and extend these results to special submanifolds like semi-slant, quaternion $CR$-submanifolds, hemi-slant and slant subamanifolds with a scalar concircular fields.

In the case of slant manifolds for specific slant angles and dimensions, if $c=4$, $m=4$ and ${\mathrm{\Theta }}_1={\mathrm{\Theta }}_2={\displaystyle \frac{\pi }{4}}$, then, in view of Theorem 5, we obtain the following inequality:

$$\left|\right|\widetilde{S_{ric}}{\left|\right|}_g^2\ge \left|\right|\widetilde{\mathcal{H}ess\left(f\right)}{\left|\right|}_g^2+{64\left|\right|\mathrm{\Lambda }\left|\right|}^4+{\displaystyle \frac{{\left|\right|\hslash \left|\right|}^4}{4}}+4{\lambda }^2-39.$$

The following are the limitations of the current study or prospective directions for future extensions:

(a)

Whether the current inequalities are only applicable to compact submanifolds or arbitrary submanifolds (the domain assumption needs to be clearly specified).

(b)

The feasibility of extending the results to pseudo-quaternion Kähler manifolds or quaternion manifolds with torsion.

(c)

The research potential of combining scalar concircular fields with other geometric structures (e.g., almost complex structures, contact structures, and metallic structures). This will provide valuable references for subsequent studies.

(d)

Quaternionic structure and slant angles result in tighter, more restrictive bounds in geometric inequalities than their real or complex counterparts, owing to the more rigid and higher-dimensional limitations imposed by quaternionic geometry.

The constraints in these inequalities are often more stringent since the prerequisites for a submanifold to achieve equality are quite explicit, frequently identifying highly limited submanifolds such as fully geodesic or invariantly quasi-umbilical ones.