DDVV Inequality on Submanifolds Coupled with a Slant Factor in Quaternionic Kaehler Manifolds

Abstract

This work aims to provide generalized Wintgen inequalities for slant submanifolds embedded in quaternionic space forms, taking into consideration both semi-symmetric metric and semi-symmetric non-metric connections. Moreover, we discuss the same inequality for totally real, anti-invariant, and invariant submanifolds on quaternionic space forms, endowed with both semi-symmetric metric and semi-symmetric non-metric connections. We also characterized the equality case through specific forms of shape operators of Wintgen inequalities for these classes of submanifolds in quaternionic space forms, admitting a semi-symmetric metric connection and a semi-symmetric non-metric connection.

1. Introduction

The second fundamental form of a surface establishes its fundamental curvatures and extrinsic invariants. A smooth submanifold immersed in a Riemannian manifold has an assigned quadratic form.

A fundamental aspect of submanifold theory is connecting intrinsic and extrinsic invariants, essential for investigating submanifold geometry inside Riemannian manifolds. P. Wintgen introduced a novel method in 1979 by examining this relationship using a notable inequality, often referred to as the Wintgen inequality.

Wintgen inequality is a geometric inequality combining extrinsic and intrinsic invariants for surfaces in four-dimensional Euclidean space. In [1], for ${\mathfrak{D}}^2$ surface in $E^4$, authors proved that

$${\left|\right|\Pi \left|\right|}^2\ge \mathcal{S}+\left|{\mathcal{S}}^{\perp }\right|,$$

in which Gauss, normal, and mean curvatures are expressed, respectively, by $\mathcal{S}$, ${\mathcal{S}}^{\perp }$, and $\Pi$. If and only if ${\mathfrak{D}}^2$ in $E^4$ has a circle-shaped curvature ellipse, and the equality holds if the ellipse of the curvature of a surface in $E^4$ is a circle. Simply put, it says that the intrinsic curvature cannot be greater than the extrinsic curvature measurements. Subsequently [2], generalized this result to arbitrary codimension n in ${\overline{\mathfrak{D}}}^{n+2}\left(c\right)$ by

$${\left|\right|\Pi \left|\right|}^2+c\ge \mathcal{S}+\left|{\mathcal{S}}^{\perp }\right|,$$

where c indicates the constant sectional curvature.

The DDVV conjecture (a conjecture for the Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen, and L. Vrancken), demonstrated by the article [3], was a conjecture of the paper [4]. In recent studies, the same inequality has been established for numerous submanifolds within various ambient manifolds [5,6,7,8,9,10].

On the other edge, there is an important class of manifolds known as quaternionic Kaehler manifolds that have attracted the attention of many geometers due to their rich geometrical importance. The notion of the quaternionic manifold was given by S. Ishihara [11]. In the last three decades, the geometry of submanifolds in quaternionic manifolds has been researched intensively by many geometers. In [12], authors explored totally real submanifolds within a quaternionic projective space. Additionally, quaternionic CR submanifolds were examined in [13]. The field of slant submanifolds in quaternionic Kaehler manifolds was further advanced by the noteworthy work of B. Sahin [14].

Hayden [15] initially proposed the semi-symmetric metric connection (or SSMC) on Riemannian manifolds in 1932 while studying subspaces in torsion spaces. The study [16] showed that for a manifold to be conformally flat, it is necessary and sufficient to possess the SSMC with a vanishing curvature tensor. In 1976, Z. Nako [17] explored submanifolds of Riemannian manifolds with these types of connections and derived equations similar to Gauss and Codazzi–Mainardi. Moreover, inequalities for these types of submanifolds were obtained in [18,19,20,21,22]. On the other side, the concept of this type of connection emerged from the works [23,24] concerning the semi-symmetric non-metric connection (in short, SSNMC). The applicability of Chen’s inequalities with this connection type was demonstrated in a 2012 work by Ozgur [25].

This study proposes generalized Wintgen inequalities for quaternionic space forms (in short, QSF) equipped with either SSMC or SSNMC with a slant factor. Additionally, we examine the particular conditions of these inequalities for various types of submanifolds in the same ambient space.

2. Quaternionic Space Forms

Quaternionic space forms are a class of Riemannian manifolds, and Kaehler extends the concept of manifolds to the quaternionic environment. In differential geometry, any complex manifold having a Hermitian metric compatible with its complex structure is referred to as the Kaehler manifold. This compatibility leads to a rich interaction between the complex structure of the manifold and its metric properties. Quaternionic manifolds are defined as manifolds that have a quaternionic structure. The quaternionic structure generalizes complex and symplectic structures and provides a framework for the study of geometric objects with richer algebraic properties.

Slant submanifolds are defined in such a way that at every point of the main manifold, there is a submanifold of its orthogonal complement. But they also have a subclass of curvature vectors, namely those in which they satisfy a certain relation regarding the Ricci curvature. This relationship means that the curvature vector of the manifold lies at a certain angle concerning the normal vectors of the main manifold. Slant manifolds are significant in several studies for their geometric and algebraic properties.

Assume that $\overline{\mathfrak{D}}$ is a differentiable manifold and $End\left(T\overline{\mathfrak{D}}\right)$ has a rank 3 subbundle $\sigma$ provided that for a local basis $\{{\mathbb{J}}_1,{\mathbb{J}}_2,{\mathbb{J}}_3\}$ on sections of b, we have

$$\begin{array}{c}\hfill {\mathbb{J}}_1^2={\mathbb{J}}_2^2={\mathbb{J}}_3^2=-I,\\ \hfill {\mathbb{J}}_1={\mathbb{J}}_2{\mathbb{J}}_3=-{\mathbb{J}}_3{\mathbb{J}}_2,\\ \hfill {\mathbb{J}}_2={\mathbb{J}}_3{\mathbb{J}}_1=-{\mathbb{J}}_1{\mathbb{J}}_3,\\ \hfill {\mathbb{J}}_3={\mathbb{J}}_1{\mathbb{J}}_2=-{\mathbb{J}}_2{\mathbb{J}}_1,\end{array}$$

Here, I is the type $(1,1)$ identity tensor field on $\overline{\mathfrak{D}}$. Then, $(\overline{\mathfrak{D}},b)$ is referred to as an almost quaternionic manifold with almost quaternionic structure b. Furthermore, it can be observed that $(\overline{\mathfrak{D}},b)$ has a dimension of $4m,m\ge 1$.

When the metric g on $\overline{\mathfrak{D}}$ satisfies

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

$\forall K^{♯},G^{♯}\in \left(\overline{\mathfrak{D}}\right)$, then g is adapted to b, and $(\overline{\mathfrak{D}},\sigma ,g)$ is referred as an almost quaternionic Hermitian manifold. In addition, $(\overline{\mathfrak{D}},b,g)$ is known as the quaternionic Kaehler manifold $(q.k.m)$, provided b is parallel for $\overline{\nabla }$ of g. Comparatively, we have ${\gamma }_1,{\gamma }_2,{\gamma }_3$ (1-forms) holding

$${\overline{\nabla }}_{K^{♯}}{\mathbb{J}}_{\alpha }={\gamma }_{\alpha +2}\left(K^{♯}\right){\mathbb{J}}_{\alpha +1}-{\gamma }_{\alpha +1}\left(K^{♯}\right){\mathbb{J}}_{\alpha +2},\forall \alpha \in \{1,2,3\},$$

$\forall K^{♯}\in \overline{\mathfrak{D}}$.

When $K^{♯}$ is a vector field that is not null on $\overline{\mathfrak{D}}$, the subspace $\mathcal{Q}\left(K^{♯}\right)$ (dimension = 4) that is spanned by $\{B^{♯},{\mathbb{J}}_1{B}^{♯},{\mathbb{J}}_2{B}^{♯},{\mathbb{J}}_3{B}^{♯}\}$ is known as a quaternionic four-plane.

A quaternionic plane is any two-dimensional subspace contained in $\mathcal{Q}\left(\ddot{E}\right)$. Quaternionic sectional curvature is the sectional curvature of a quaternionic plane.

Let ${\overline{\mathfrak{D}}}^{4m}\left(c\right)$ be a QSF with a quaternionic sectional curvature c. Following that, the curvature of $\overline{\mathfrak{D}}\left(c\right)$ is provided by

$$\begin{array}{ccc}\hfill {\overline{\Delta }}^{\prime }(K^{♯},G^{♯})L^{♯}& =& {\displaystyle \frac{c}{4}}\{g(L^{♯},G^{♯})K^{♯}-g(K^{♯},L^{♯})G^{♯}+\sum _{\alpha =1}^3[g(L^{♯},{\mathbb{J}}_{\alpha }{G}^{♯}){\mathbb{J}}_{\alpha }{K}^{♯}\hfill \\ & & -g(L^{♯},{\mathbb{J}}_{\alpha }{K}^{♯}){\mathbb{J}}_{\alpha }{G}^{♯}+2g(K^{♯},{\mathbb{J}}_{\alpha }{G}^{♯}){\mathbb{J}}_{\alpha }{L}^{♯}\left]\right\}\hfill \end{array}$$

(1)

$\forall K^{♯},G^{♯},L^{♯}\in \Gamma \left(T\overline{\mathfrak{D}}\right).$

Now, we discuss a few definitions.

Definition 1

([13]). Consider a Riemannian manifold $\mathfrak{D}$ immersed isometrically within quaternionic manifold $\overline{\mathfrak{D}}$. The distribution

$$\mathbb{D}:p\to {\mathbb{D}}_p\subseteq T_p\mathfrak{D},$$

is referred to as quaternion distribution when

$${\mathbb{J}}_{\alpha }\left(\mathbb{D}\right)\subseteq \mathbb{D},q\alpha =1,2,3.$$

Definition 2

([26]). The submanifold $\mathfrak{D}$ of a $q.k.m$$(\overline{\mathfrak{D}},b,g)$ is referred to as slant if the angle $\Theta \left(K^{♯}\right)$ between ${\mathbb{J}}_{\alpha }\left(K^{♯}\right),\alpha \in \{1,2,3\}$ and $T_p\mathfrak{D}$ is constant, meaning that $\Theta \left(K^{♯}\right)$ remains unchanged regardless of the selection of $p\in \mathfrak{D}$ and $K^{♯}\in T_p\mathfrak{D}$. This $\Theta \in [0,{\displaystyle \frac{\pi }{2}}]$ is referred as slant angle of $\mathfrak{D}$ in $\overline{\mathfrak{D}}$.

A submanifold $\mathfrak{D}$ with $\Theta ={\displaystyle \frac{\pi }{2}}$ and $\Theta =0$ characterizes quaternionic submanifolds and totally real submanifolds, respectively. If $\mathfrak{D}$ is neither totally real nor quaternionic, then it is proper or $\theta$-slant proper. Sahin [14] demonstrated in 2007 that a $q.k.m$$\overline{\mathfrak{D}}$ has slant submanifolds $\mathfrak{D}$ iff for $\Lambda \in [-1,0]$, we have

$$\begin{array}{c}\hfill P_{\alpha }{P}_{\beta }{K}^{♯}=-Co{s}^2\Theta K^{♯},\forall K^{♯}\in \Gamma \left(T\mathfrak{D}\right),\alpha ,\beta \in \{1,2,3\},\end{array}$$

(2)

here $P_{\alpha }{K}^{♯}$ is the tangential component of ${\mathbb{J}}_{\alpha }{K}^{♯}$. In view of (2), we have

$$\begin{array}{c}\hfill g(P_{\beta }{K}^{♯},P_{\alpha }{G}^{♯})=Co{s}^2\Theta g(K^{♯},G^{♯}),\forall K^{♯},G^{♯}\in \Gamma \left(T\mathfrak{D}\right),\alpha ,\beta \in \{1,2,3\}.\end{array}$$

(3)

3. Riemannian Manifolds Endowed with SSMC

An SSMC connection is the combination with a metric tensor and has a special symmetry. Suppose ${\overline{\mathfrak{D}}}^m$ is a Riemannian manifold. For a linear connection $\overline{\nabla }\in \overline{\mathfrak{D}}$, we present the torsion tensor $\overline{\tau }$ by

$$\begin{array}{c}\hfill \overline{\tau }(\overline{K^{♯}},\overline{G^{♯}})={\overline{\nabla }}_{\overline{K^{♯}}}\overline{G^{♯}}-{\overline{\nabla }}_{\overline{G^{♯}}}\overline{K^{♯}}-[\overline{K^{♯}},\overline{G^{♯}}],\forall \overline{K^{♯}},\overline{G^{♯}}\in \Gamma \left(T\overline{\mathfrak{D}}\right).\end{array}$$

(4)

For a 1-form $\psi$ on $\overline{\mathfrak{D}}$, $\overline{\nabla }$ is known as follows:

• SSC if $\overline{\tau }(\overline{K^{♯}},\overline{G^{♯}})=\psi \left(\overline{K^{♯}}\right)\overline{G^{♯}}-\psi \left(\overline{K^{♯}}\right)\overline{G^{♯}};$
• SSMC if $\overline{\nabla }g=0$

. Further, suppose that ${\overline{\nabla }}^{\prime }$ refers to the L.V. connection compatible to g. The SSMC $\overline{\nabla }$ on $\overline{\mathfrak{D}}$ follows [16]:

$${\overline{\nabla }}_{\overline{K^{♯}}}\overline{G^{♯}}={\overline{\nabla }}_{\overline{K^{♯}}}^{\prime }\overline{G^{♯}}+\psi \left(\overline{G^{♯}}\right)\overline{K^{♯}}-g(\overline{K^{♯}},\overline{G^{♯}})V,\forall \overline{K^{♯}},\overline{G^{♯}}\in \Gamma \left(T\overline{\mathfrak{D}}\right),$$

where vector field V holds

$$\overline{g}(K^{♯},\overline{K^{♯}})=\psi \left(\overline{K^{♯}}\right).$$

Consider a submanifold ${\mathfrak{D}}^n$ immersed in Riemannian manifold ${\overline{\mathfrak{D}}}^m$ equipped with the SSMC $\overline{\nabla }$. On $\mathfrak{D}$, let ∇ and ${\nabla }^{\prime }$ represent an SSMC and the Levi–Civita connection, respectively. Furthermore, consider a $(0,2)$ tensor $\pi$ on $\mathfrak{D}$ and let ${\pi }^{\prime }$ represent the second fundamental form of $\mathfrak{D}$ in $\overline{\mathfrak{D}}$.

Next, we recall the following relations:

$$\begin{array}{c}\hfill {\overline{\nabla }}_{K^{♯}}{G}^{♯}={\nabla }_{K^{♯}}{G}^{♯}+\pi (K^{♯},G^{♯}),{\overline{\nabla }}_{K^{♯}}^{\prime }{G}^{♯}={\nabla }_{K^{♯}}^{\prime }{G}^{♯}+{\pi }^{\prime }(K^{♯},G^{♯}),\end{array}$$

$\forall K^{♯},G^{♯}\in \Gamma \left(T\mathfrak{D}\right).$

We also observe that $\pi$ is symmetric [17]. Suppose that $\overline{\Delta }$ and ${\overline{\Delta }}^{\prime }$ represent curvature tensors of $\overline{\mathfrak{D}}$ based on $\overline{\nabla }$ and ${\overline{\nabla }}^{\prime }$, respectively; likewise, $\Delta$ and ${\Delta }^{\prime }$ signify curvature tensors of $\mathfrak{D}$ based on ∇ and ${\nabla }^{\prime }$, respectively. The $(0,2)$ tensor field $\omega$ is provided as

$$\begin{array}{c}\hfill \omega (K^{♯},G^{♯})=\left({\overline{\nabla }}_{K^{♯}}^{\prime }\psi \right)G^{♯}-\psi \left(K^{♯}\right)\psi \left(G^{♯}\right)+{\displaystyle \frac{1}{2}}\psi \left(V\right)g(K^{♯},G^{♯}),\forall K^{♯},G^{♯}\in \Gamma \left(T\mathfrak{D}\right).\end{array}$$

The curvature tensor $\overline{\Delta }$ on $\overline{\mathfrak{D}}$ is then written by [27] with regard to SSMC $\overline{\nabla }$:

$$\begin{array}{ccc}\hfill \overline{\Delta }(K^{♯},G^{♯},L^{♯},O^{♯})& =& {\overline{\Delta }}^{\prime }(K^{♯},G^{♯},L^{♯},O^{♯})-\omega (G^{♯},L^{♯})g(K^{♯},O^{♯})+\omega (K^{♯},L^{♯})g(G^{♯},O^{♯})\hfill \\ & & -\omega (K^{♯},O^{♯})g(G^{♯},L^{♯})+\omega (G^{♯},O^{♯})g(K^{♯},L^{♯}),\hfill \end{array}$$

(5)

$\forall K^{♯},G^{♯},L^{♯},O^{♯}\in \Gamma \left(T\mathfrak{D}\right).$ For $K^{♯}\in \Gamma \left(T\mathfrak{D}\right)$, one writes

$$\mathbb{J}{K}^{♯}=P{K}^{♯}+Q{K}^{♯},$$

Here, $Q,$P denote the normal and tangential components of $\mathbb{J}{K}^{♯},$ respectively.

Now, assume that the normal bundle $T^{\perp }\mathfrak{D}$ of $\mathfrak{D}$ in $\overline{\mathfrak{D}}$ has a local orthonormal normal frame $\{B_{n+1},\cdots ,B_{4m}\}$ and that the local orthonormal tangent frame of the tangent bundle $T\mathfrak{D}$ of $\mathfrak{D}$ is $\{B_1,\cdots ,B_n\}$.

Next, the following gives the mean curvature vector of $\mathfrak{D}$:

$$\begin{array}{c}\hfill \Pi =\sum _{i=1}^n{\displaystyle \frac{1}{n}}\pi (B_i,B_i)\end{array}$$

and

$$\begin{array}{c}\hfill {\left|\right|\pi \left|\right|}^2=\sum _{i,j=1}^{n}g\left(\pi (B_i,B_j),\pi (B_i,B_j)\right).\end{array}$$

Additionally, we establish

$$\begin{array}{c}\hfill {\left|\right|P\left|\right|}^2=\sum _{i,j=1}^n{g}^2({\mathbb{J}}_{\alpha }{B}_i,B_j)\end{array}$$

(6)

and define the scalar curvature, which is represented as

$$\begin{array}{c}\hfill \mu =\sum _{1\le i<j\le n}\Delta (B_i,B_j,B_j,B_i).\end{array}$$

(7)

We define the normalized scalar curvature ${\mathcal{N}}_{scal}$ of $\mathfrak{D}$ as follows:

$$\begin{array}{ccc}\hfill {\mathcal{N}}_{scal}& =& {\displaystyle \frac{2\mu }{n(n-1)}}={\displaystyle \frac{2}{n(n-1)}}\sum _{1\le i<j\le n}\mathcal{S}(B_i\wedge B_j)\hfill \end{array}$$

(8)

wherein $\mathcal{S}$ represents sectional curvature function on $\mathfrak{D}$. Scalar normal curvature ${\mathcal{S}}_N$ and normalized scalar normal curvature ${\mathcal{N}}_{scal}$, respectively, are expressed as [7].

$$\begin{array}{c}\hfill {\mathcal{S}}_N=\sum _{n+1\le \alpha <\beta \le 4m}\sum _{1\le i<j\le n}{(\sum _{k=1}^n{\pi }_{jk}^{\alpha }{\pi }_{ik}^{\beta }-{\pi }_{jk}^{\alpha }{\pi }_{ik}^{\beta })}^2\end{array}$$

(9)

and

$$\begin{array}{c}\hfill {\mathcal{N}}_{scal}={\displaystyle \frac{2}{n(n-1)}}\sqrt{{\mathcal{S}}_N}.\end{array}$$

(10)

4. Wintgen Inequality for Slant Submanifolds of QSF Equipped with SSMC

Moving forward, $\overline{\mathfrak{D}}$ will be fixed for quaternionic space forms equipped with SSMC. Given (5) and (1), curvature tensor $\overline{\Delta }$ of $\overline{\mathfrak{D}}$ is obtained as follows:

$$\begin{array}{ccc}\hfill \overline{\Delta }(K^{♯},G^{♯},L^{♯},O^{♯})& =& {\displaystyle \frac{c}{4}}\{g(L^{♯},G^{♯})g(K^{♯},O^{♯})-g(K^{♯},L^{♯})g(G^{♯},O^{♯})\hfill \\ & & +\sum _{\alpha =1}^3[g(L^{♯},{\mathbb{J}}_{\alpha }{G}^{♯})g({\mathbb{J}}_{\alpha }{K}^{♯},O^{♯})\hfill \\ & & -g(L^{♯},{\mathbb{J}}_{\alpha }{K}^{♯})g({\mathbb{J}}_{\alpha }{G}^{♯},O^{♯})+2g(K^{♯},{\mathbb{J}}_{\alpha }{G}^{♯})g({\mathbb{J}}_{\alpha }{L}^{♯},O^{♯})\left]\right\}\hfill \\ & & -\omega (G^{♯},L^{♯})g(K^{♯},O^{♯})+\omega (K^{♯},L^{♯})g(G^{♯},O^{♯})\hfill \\ & & -\omega (K^{♯},O^{♯})g(G^{♯},L^{♯})+\omega (G^{♯},O^{♯})g(K^{♯},L^{♯}).\hfill \end{array}$$

The established outcomes are as follows:

Theorem 1.

Consider a Θ-slant proper submanifold ${\mathfrak{D}}^n$ immersed in ${\overline{\mathfrak{D}}}^{4m}\left(c\right)$. Then,

$$\begin{array}{ccc}\hfill {\mathcal{N}}_{scal}& \le & {\left|\right|\Pi \left|\right|}^2-2({\displaystyle \frac{n\mu }{n-1}}-{\displaystyle \frac{c}{4}})+{\displaystyle \frac{9c}{2(n-1)}}Co{s}^2\Theta -{\displaystyle \frac{4}{n}}tr\left(\omega \right).\hfill \end{array}$$

(11)

The equality sign is valid in (11) if and only if the shape operators A, as determined by any orthonormal tangent frame $\{B_1,\dots ,B_n\}$ and orthonormal normal frame $\{B_{n+1},\dots ,B_{4m}\}$, are as follows:

$$\begin{array}{c}\hfill A_{n+1}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_1& \Psi & 0& \cdots & 0& 0\\ \Psi & {\mathsf{{\rm Y}}}_1& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_1& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_1\end{array}\right),\end{array}$$

(12)

$$\begin{array}{c}\hfill A_{n+2}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_2+\Psi & 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_2-\Psi & 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_2& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_2& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_2\end{array}\right),\end{array}$$

(13)

$$\begin{array}{c}\hfill A_{n+3}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_3& 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_3& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_3& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_3& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_3\end{array}\right),A_{n+4}=\cdots =A_{4m}=0,\end{array}$$

(14)

where ${\mathsf{{\rm Y}}}_1,{\mathsf{{\rm Y}}}_2,{\mathsf{{\rm Y}}}_3$, and Ψ are real functions on $\mathfrak{D}$.

Proof. 

We can select a local orthonormal normal frame $\{B_{n+1},\cdots ,B_{4m}\}$ and local orthonormal tangent frame $\{B_1,\cdots ,B_n\}$ on $\mathfrak{D}$, respectively, for the quaternionic space form ${\overline{\mathfrak{D}}}^{4m}\left(c\right)$. Next, we obtain the following by using the Gauss equation and (1):

$$\begin{array}{ccc}\hfill \Delta (K^{♯},G^{♯},L^{♯},O^{♯})& =& {\displaystyle \frac{c}{4}}\{g(L^{♯},G^{♯})g(K^{♯},O^{♯})-g(K^{♯},L^{♯},g(G^{♯},O^{♯})\hfill \\ & & +\sum _{\alpha =1}^3[g(L^{♯},{\mathbb{J}}_{\alpha }{G}^{♯})g({\mathbb{J}}_{\alpha }{K}^{♯},O^{♯})\hfill \\ & & -g(L^{♯},{\mathbb{J}}_{\alpha }{K}^{♯})g({\mathbb{J}}_{\alpha }{G}^{♯},O^{♯})+2g(K^{♯},{\mathbb{J}}_{\alpha }{G}^{♯})g({\mathbb{J}}_{\alpha }{L}^{♯},O^{♯})\left]\right\}\hfill \\ & & -\omega (G^{♯},L^{♯})g(K^{♯},O^{♯})+\omega (K^{♯},L^{♯})g(G^{♯},O^{♯})\hfill \\ & & -\omega (K^{♯},O^{♯})g(G^{♯},L^{♯})+\omega (G^{♯},O^{♯})g(K^{♯},L^{♯})\hfill \\ & & +g(\pi (K^{♯},O^{♯}),\pi (G^{♯},L^{♯}))-g(\pi (K^{♯},L^{♯}),\pi (G^{♯},O^{♯}))\hfill \end{array}$$

for all vector fields $K^{♯},L^{♯},G^{♯},O^{♯}$ tangent to $\mathfrak{D}$.

Putting $K^{♯}=O^{♯}=B_i$, $G^{♯}=L^{♯}=B_j$, $i\ne j$ and computing the summation for $1\le i<\le j\le n$, we obtain

$$\begin{array}{ccc}\hfill \sum _{1\le i<j\le n}\Delta (B_i,B_j,B_j,B_i)& =& {\displaystyle \frac{n(n-1)c}{4}}+{\displaystyle \frac{3c}{4}}\sum _{\beta =1}^3\sum _{i,j=1}^n{g}^2(P_{\beta }{B}_i,B_j)\hfill \\ & -& 2(n-1)tr\left(\omega \right)+\sum _{\alpha =n+1}^{4m}\sum _{1\le i<j\le n}[{\pi }_{ii}^{\alpha }{\pi }_{jj}^{\alpha }-{\left({\pi }_{ij}^{\alpha }\right)}^2],\hfill \end{array}$$

(15)

where (6) has been considered.

A canonical orthonormal local frame of $T_p\mathfrak{D}$, referred to as an adapted slant frame, can be selected for $\mathfrak{D}$ in the following manner: $\{B_1,B_2={\displaystyle \frac{1}{Cos\Theta }}{P}_{\beta }{B}_1,\cdots ,B_{2r-1},B_{2r}={\displaystyle \frac{1}{Cos\Theta }}{P}_{\beta }{B}_{2r-1}\}$, where $2r=n$. Then, in the light of (2) and (3), we have

$$\begin{array}{c}\hfill g^2(P_{\alpha }{B}_i,B_{i+1})=g^2(P_{\alpha }{B}_{i+1},B_i)=Co{s}^2\Theta ,\mathrm{for}i\in \{1,3,...,2r-1\}\end{array}$$

(16)

and

$$\begin{array}{c}\hfill g(P_{\alpha }{B}_i,B_j)=0,\mathrm{for}(i,j)\notin \{(2k-1,2k),(2k,2k-1)|k\in \{1,2,\cdots ,r\}\}.\end{array}$$

(17)

Taking (16) and (17) in (15), we obtain

$$\begin{array}{ccc}\hfill \mu & =& {\displaystyle \frac{c}{4}}[n(n-1)+9nCo{s}^2\Theta ]-2(n-1)tr\left(\omega \right)\hfill \\ & +& \sum _{\alpha =n+1}^{4m}\sum _{1\le i<j\le n}[{\pi }_{ii}^{\alpha }{\pi }_{jj}^{\alpha }-{\left({\pi }_{ij}^{\alpha }\right)}^2],\hfill \end{array}$$

(18)

where (7) has been considered.

Additionally, note that

$$\begin{array}{ccc}\hfill n^2{\left|\right|\Pi \left|\right|}^2& =& \sum _{\alpha =n+1}^{2m-n}(\sum _{i=1}^n{\pi }_{ii}^{\alpha }{)}^2\hfill \\ & =& {\displaystyle \frac{1}{n-1}}\sum _{\alpha =n+1}^{2m-n}\sum _{1\le i<j\le n}{({\pi }_{ii}^{\alpha }-{\pi }_{jj}^{\alpha })}^2+{\displaystyle \frac{2n}{n-1}}\sum _{\alpha =n+1}^{4m}\sum _{1\le i<j\le n}{\pi }_{ii}^{\alpha }{\pi }_{jj}^{\alpha }.\hfill \end{array}$$

(19)

We have from [28]

$$\begin{array}{c}\hfill \sum _{\alpha =n+1}^{2m-n}\sum _{1\le i<j\le n}{({\pi }_{ii}^{\alpha }-{\pi }_{jj}^{\alpha })}^2+2n\sum _{\alpha =n+1}^{4m}\sum _{1\le i<j\le n}{\left({\pi }_{ij}^{\alpha }\right)}^2\ge \\ \hfill 2n[\sum _{n+1\le \alpha <\beta \le 2m-n}\sum _{1\le i<j\le n}{\left(\sum _{k=1}^n({\pi }_{jk}^{\alpha }{\pi }_{ik}^{\beta }-{\pi }_{ik}^{\alpha }{\pi }_{jk}^{\beta })\right)}^2{]}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

(20)

Combining (19) and (20), we obtain

$$\begin{array}{c}\hfill n^2{\left|\right|\Pi \left|\right|}^2-n^2{\mathcal{N}}_{scal}\ge {\displaystyle \frac{2n}{n-1}}\sum _{\alpha =n+1}^{4m}\sum _{1\le i<j\le n}[{\pi }_{ii}^{\alpha }{\pi }_{jj}^{\alpha }-{\left({\pi }_{ij}^{\alpha }\right)}^2],\end{array}$$

(21)

where (9) have been taken into view.

Finally, considering (10), (18) and using (21), we find

$$\begin{array}{ccc}\hfill {\mathcal{N}}_{scal}-{\left|\right|\Pi \left|\right|}^2& \le & {\displaystyle \frac{c}{2}}-{\displaystyle \frac{2n}{n-1}}\mu +{\displaystyle \frac{9c}{2(n-1)}}Co{s}^2\Theta -{\displaystyle \frac{4}{n}}tr\left(\omega \right).\hfill \end{array}$$

□

Using Theorem 1, we express the Wintgen inequalities for various types of submanifolds in ${\overline{\mathfrak{D}}}^{4m}\left(c\right)$. The results are as follows:

Theorem 2.

Consider a totally real submanifold ${\mathfrak{D}}^n$ immersed in ${\overline{\mathfrak{D}}}^{4m}\left(c\right)$. Then,

$$\begin{array}{ccc}\hfill {\mathcal{N}}_{scal}& \le & {\left|\right|\Pi \left|\right|}^2-2({\displaystyle \frac{n\mu }{n-1}}-{\displaystyle \frac{c}{4}})-{\displaystyle \frac{4}{n}}tr\left(\omega \right).\hfill \end{array}$$

(22)

The equality in (22) holds if and only if for $\{B_1,\dots ,B_n\}$ and $\{B_{n+1},\dots ,B_{4m}\}$, A take the following forms:

$$\begin{array}{c}\hfill A_{n+1}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_1& \Psi & 0& \cdots & 0& 0\\ \Psi & {\mathsf{{\rm Y}}}_1& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_1& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_1\end{array}\right),\end{array}$$

(23)

$$\begin{array}{c}\hfill A_{n+2}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_2+\Psi & 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_2-\Psi & 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_2& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_2& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_2\end{array}\right),\end{array}$$

(24)

$$\begin{array}{c}\hfill A_{n+3}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_3& 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_3& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_3& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_3& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_3\end{array}\right),A_{n+4}=\cdots =A_{4m}=0,\end{array}$$

(25)

${\mathsf{{\rm Y}}}_1,{\mathsf{{\rm Y}}}_2,{\mathsf{{\rm Y}}}_3$, and Ψ represent the real functions on $\mathfrak{D}$.

Theorem 3.

Consider a quaternionic submanifold ${\mathfrak{D}}^n$ immersed in ${\overline{\mathfrak{D}}}^{4m}\left(c\right)$. Then,

$$\begin{array}{ccc}\hfill {\mathcal{N}}_{scal}& \le & {\left|\right|\Pi \left|\right|}^2-2({\displaystyle \frac{n\mu }{n-1}}-{\displaystyle \frac{c}{4}})+{\displaystyle \frac{9c}{2(n-1)}}-{\displaystyle \frac{4}{n}}tr\left(\omega \right).\hfill \end{array}$$

(26)

The equality sign is also valid in (26) iff according to $\{B_1,\cdots ,B_n\}$ and $\{B_{n+1},\cdots ,B_{4m}\}$, operators A have the following forms:

$$\begin{array}{c}\hfill A_{n+1}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_1& \Psi & 0& \cdots & 0& 0\\ \Psi & {\mathsf{{\rm Y}}}_1& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_1& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_1\end{array}\right),\end{array}$$

(27)

$$\begin{array}{c}\hfill A_{n+2}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_2+\Psi & 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_2-\Psi & 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_2& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_2& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_2\end{array}\right),\end{array}$$

(28)

$$\begin{array}{c}\hfill A_{n+3}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_3& 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_3& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_3& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_3& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_3\end{array}\right),A_{n+4}=\cdots =A_{4m}=0,\end{array}$$

(29)

where ${\mathsf{{\rm Y}}}_1,{\mathsf{{\rm Y}}}_2,{\mathsf{{\rm Y}}}_3$, and Ψ represent the real functions on $\mathfrak{D}$.

Remark 1.

Theorem 2 and Theorem 3 have similar proof as Theorem 1. In fact, we obtain inequalities (22) and (26) by putting $\Theta ={\displaystyle \frac{\pi }{2}}$ and $\Theta =0$ in inequality (11), respectively.

5. Riemannian Manifolds Concerning SSNMC

The SSNMC is not metric tensor compatible, meaning it is not associated with or compatible with the metric tensor. However, it is semi-symmetric, meaning that the curvature tensor of the connection has a certain type of symmetry. Such connections have various applications in differential geometry and general relativity. In particular, non-metric connections can be used to study different geometric structures or physical phenomena under certain theoretical frameworks.

Let $\overline{\mathfrak{D}}$ be the quaternionic space form. For linear connection $\overline{\nabla }$ on $\overline{\mathfrak{D}}$, we introduced the torsion tensor $\overline{\tau }$ in Equation (4). Furthermore, we observe that $\overline{\nabla }$ is SSC if it fulfills

$$\overline{\tau }(\overline{K^{♯}},\overline{G^{♯}})=\psi \left(\overline{G^{♯}}\right)\overline{K^{♯}}-\psi \left(\overline{K^{♯}}\right)\overline{G^{♯}}$$

and SSNMC if

$$\overline{\nabla }g\ne 0,$$

where $\psi$ is a 1-form on $\overline{\mathfrak{D}}$. We write the v.f. E by

$$\overline{g}(E,\overline{K^{♯}})=\psi \left(\overline{K^{♯}}\right),$$

$\forall \overline{K^{♯}}\in \Gamma \left(T\overline{\mathfrak{D}}\right)$. Furthermore [23], characterizes the SSNMC $\overline{\nabla }$ on $\overline{\mathfrak{D}}$ if ${\overline{\nabla }}^{\prime }$ satisfies

$${\overline{\nabla }}_{\overline{K^{♯}}}\overline{G^{♯}}={\overline{\nabla }}_{\overline{K^{♯}}}^{\prime }\overline{G♯}+\psi \left(\overline{G^{♯}}\right)\overline{K^{♯}},\forall \overline{K^{♯}},\overline{K^{♯}}\in \Gamma \left(T\overline{\mathfrak{D}}\right).$$

Consider a submanifold ${\mathfrak{D}}^n$ immersed in a Riemannian manifold ${\overline{\mathfrak{D}}}^m$ equipped with SSNMC $\overline{\nabla }$. Let us represent the induced L.V. connection on $\mathfrak{D}$ and the induced SSNMC as ∇ and ${\nabla }^{\prime }$, respectively. The following is an expression for the Gauss formulas:

$$\begin{array}{c}\hfill {\overline{\nabla }}_{K^{♯}}{G}^{♯}={\nabla }_{K^{♯}}{G}^{♯}+\pi (K^{♯},G^{♯}),{\overline{\nabla }}_{K^{♯}}^{\prime }{G}^{♯}={\nabla }_{K^{♯}}^{\prime }{G}^{♯}+{\pi }^{\prime }(K^{♯},G^{♯}),\end{array}$$

Here, ${\pi }^{\prime }$ and $\pi$ represent second fundamental form of $\mathfrak{D}$ in $\overline{\mathfrak{D}}$ and $(0,2)$ tensor on $\mathfrak{D}$, respectively. Additionally, we see that $\pi ={\pi }^{\prime }$ [24].

Let $\overline{\Delta }$ and ${\overline{\Delta }}^{\prime }$ represent curvature of $\overline{\mathfrak{D}}$ with regard to $\overline{\nabla }$ and ${\overline{\nabla }}^{\prime }$, respectively. Similarly, let $\Delta$ and ${\Delta }^{\prime }$ represent the curvature of $\mathfrak{D}$ corresponding to ∇ and ${\nabla }^{\prime }$, respectively. Consequently, the Gauss equation for $\mathfrak{D}$ immersed in ${\overline{\mathfrak{D}}}^m$ can be expressed using the SSNMC as shown below [24]:

$$\begin{array}{ccc}\hfill {\overline{\Delta }}^{\prime }(K^{♯},G^{♯},L^{♯},O^{♯})& =& {\Delta }^{\prime }(K^{♯},G^{♯},L^{♯},O^{♯})-g({\pi }^{\prime }(K^{♯},O^{♯}),{\pi }^{\prime }(G^{♯},L^{♯}))\hfill \\ & & +g({\pi }^{\prime }(K^{♯},L^{♯}),{\pi }^{\prime }(G^{♯},O^{♯}))+g(E,{\pi }^{\prime }(G^{♯},L^{♯}))g(K^{♯},O^{♯})\hfill \\ & & -g(E,{\pi }^{\prime }(K^{♯},L^{♯}))g(G^{♯},O^{♯}),\forall K^{♯},G^{♯},L^{♯},O^{♯}\in \Gamma \left(T\mathfrak{D}\right).\hfill \end{array}$$

(30)

Denote the $(0,2)$ tensor field by $\beta$.

$$\begin{array}{c}\hfill \beta (K^{♯},G^{♯})=\left({\overline{\nabla }}_{K^{♯}}^{\prime }\psi \right)G^{♯}-\psi \left(K^{♯}\right)\psi \left(G^{♯}\right),\forall K^{♯},G^{♯}\in \Gamma \left(T\mathfrak{D}\right).\end{array}$$

Subsequently, with regard to $\overline{\nabla }$, the curvature tensor $\overline{\Delta }$ on $\overline{\mathfrak{D}}$ is represented as follows: [23]

$$\begin{array}{c}\hfill \overline{\Delta }(K^{♯},G^{♯},L^{♯},O^{♯})={\overline{\Delta }}^{\prime }(K^{♯},G^{♯},L^{♯},O^{♯})-\beta (G^{♯},L^{♯})g(K^{♯},O^{♯})+\beta (K^{♯},L^{♯})g(G^{♯},O^{♯}),\end{array}$$

(31)

for any $K^{♯},G^{♯},L^{♯},O^{♯}\in \Gamma \left(T\mathfrak{D}\right)$.

6. Wintgen Inequality for Slant Submanifolds of QSF for SSNMC

From now on, we will use $\overline{\mathfrak{D}}$ for quaternionic space forms equipped with SSNMC $\overline{\nabla }$. Referring to (31) and (1), curvature tensor $\overline{\Delta }$ is expressed as follows:

$$\begin{array}{ccc}\hfill \overline{\Delta }(K^{♯},G^{♯},L^{♯},O^{♯})& =& {\displaystyle \frac{c}{4}}\{g(L^{♯},G^{♯})g(K^{♯},O^{♯})-g(K^{♯},L^{♯})g(G^{♯},O^{♯})\hfill \\ & & +\sum _{\alpha =1}^3[g(L^{♯},{\mathbb{J}}_{\alpha }{G}^{♯})g({\mathbb{J}}_{\alpha }{K}^{♯},O^{♯})\hfill \\ & & -g(L^{♯},{\mathbb{J}}_{\alpha }{K}^{♯})g({\mathbb{J}}_{\alpha }{G}^{♯},O^{♯})+2g(K^{♯},{\mathbb{J}}_{\alpha }{G}^{♯})g({\mathbb{J}}_{\alpha }{L}^{♯},O^{♯})\left]\right\}\hfill \\ & & -\beta (G^{♯},L^{♯})g(K^{♯},O^{♯})+\beta (K^{♯},L^{♯})g(G^{♯},O^{♯}).\hfill \end{array}$$

(32)

We demonstrate the subsequent outcome, providing the main theorem.

Theorem 4.

Consider a θ-slant proper submanifold ${\mathfrak{D}}^n$ immersed in ${\overline{\mathfrak{D}}}^{4m}\left(c\right)$. Then,

$$\begin{array}{c}\hfill {\mathcal{N}}_{scal}\le {\displaystyle \frac{c}{2}}[1+{\displaystyle \frac{9}{(n-1)}}Co{s}^2\Theta ]-2[{\displaystyle \frac{n\mu }{n-1}}-{\displaystyle \frac{{\left|\right|\Pi \left|\right|}^2}{2}}]-{\displaystyle \frac{2}{n}}tr\left(\beta \right)-2\psi (\Pi ).\end{array}$$

(33)

The equality in (33) holds iff operator A can have the following forms pertaining to $\{B_1,\dots ,B_n\}$ and $\{B_{n+1},\dots ,B_{4m}\}$:

$$\begin{array}{c}\hfill A_{n+1}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_1& \Psi & 0& \cdots & 0& 0\\ \Psi & {\mathsf{{\rm Y}}}_1& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_1& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_1\end{array}\right),\end{array}$$

(34)

$$\begin{array}{c}\hfill A_{n+2}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_2+\Psi & 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_2-\Psi & 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_2& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_2& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_2\end{array}\right),\end{array}$$

(35)

$$\begin{array}{c}\hfill A_{n+3}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_3& 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_3& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_3& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_3& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_3\end{array}\right),A_{n+4}=\cdots =A_{4m}=0,\end{array}$$

(36)

where ${\mathsf{{\rm Y}}}_1,{\mathsf{{\rm Y}}}_2,{\mathsf{{\rm Y}}}_3$, and Ψ are real functions on $\mathfrak{D}$.

Proof. 

Consider the sets $\{B_1,\cdots ,B_n\}$ and $\{B_{n+1},\cdots ,B_{4m}\}$ as orthonormal bases for $T_p{\mathfrak{D}}^n$ and $T_p^{\perp }{\mathfrak{D}}^n$, respectively, where p belongs to $M^n$.

Setting $K^{♯}=O^{♯}=B_i$, $G^{♯}=L^{♯}=B_j$, $i\ne j,$ from (30) and (32), we obtain

$$\begin{array}{ccc}\hfill \overline{\Delta }(B_i,B_j,B_j,B_i)& =& {\displaystyle \frac{c}{4}}\{g(B_j,B_j)g(B_i,B_i)-g(B_i,B_j)g(B_j,B_i)\hfill \\ & & +\sum _{\alpha =1}^3[g(B_j,{\mathbb{J}}_{\alpha }{B}_j)g({\mathbb{J}}_{\alpha }{B}_i,B_i)-g(B_j,{\mathbb{J}}_{\alpha }{B}_i)g({\mathbb{J}}_{\alpha }{B}_j,B_i)\hfill \\ & & +2g(B_i,{\mathbb{J}}_{\alpha }{B}_j)g({\mathbb{J}}_{\alpha }{B}_j,B_i)\left]\right\}\hfill \\ & & -\beta (B_j,B_j)g(B_i,B_i)+\beta (B_i,B_j)g(B_j,B_i)\hfill \\ & & -g(\pi (B_i,B_j),\pi (B_j,B_i))+g(\pi (B_i,B_i),\pi (B_j,B_j))\hfill \\ & & -g(E,\pi (B_j,B_j))g(B_i,B_i)+g(E,\pi (B_i,B_j)g(B_j,B_i).\hfill \end{array}$$

(37)

Applying summation to $1\le i\le j\le n,$ we turn up

$$\begin{array}{ccc}\hfill \mu & =& {\displaystyle \frac{n(n-1)c}{4}}+{\displaystyle \frac{9nc}{4}}Co{s}^2\Theta -(n-1)tr\left(\beta \right)-n(n-1)\psi (\Pi )\hfill \\ & +& \sum _{\alpha =n+1}^{4m}\sum _{1\le i<j\le n}[{\pi }_{ii}^{\alpha }{\pi }_{jj}^{\alpha }-{\left({\pi }_{ij}^{\alpha }\right)}^2],\hfill \end{array}$$

(38)

where (2) and (3) have been used.

But, from the equation (21), we have

$$\begin{array}{c}\hfill n^2{\left|\right|\Pi \left|\right|}^2-n^2{\mathcal{N}}_{scal}\ge {\displaystyle \frac{2n}{n-1}}\sum _{\alpha =n+1}^{4m}\sum _{1\le i<j\le n}[{\pi }_{ii}^{\alpha }{\pi }_{jj}^{\alpha }-{\left({\pi }_{ij}^{\alpha }\right)}^2].\end{array}$$

(39)

Finally, combining (38), (39), and (8), we find

$$\begin{array}{c}\hfill {\mathcal{N}}_{scal}\le {\left|\right|\Pi \left|\right|}^2+{\displaystyle \frac{c}{2}}-{\displaystyle \frac{2}{n(n-1)}}\mu +{\displaystyle \frac{9c}{2(n-1)}}Co{s}^2\Theta -{\displaystyle \frac{2}{n}}tr\left(\beta \right)-2\psi (\Pi ),\end{array}$$

whereby proving the result.

□

The following are outcomes from Theorem 4:

Theorem 5.

Consider a totally real submanifold ${\mathfrak{D}}^n$ immersed in ${\overline{\mathfrak{D}}}^{4m}\left(c\right)$. Then,

$$\begin{array}{c}\hfill {\mathcal{N}}_{scal}\le {\displaystyle \frac{c}{2}}-2[{\displaystyle \frac{n\mu }{n-1}}-{\displaystyle \frac{{\left|\right|\Pi \left|\right|}^2}{2}}]-{\displaystyle \frac{2}{n}}tr\left(\beta \right)-2\psi (\Pi ).\end{array}$$

(40)

The equality in (40) holds iff operators A with respect to $\{B_1,\dots ,B_n\}$ and $\{B_{n+1},\dots ,B_{4m}\}$, reduce to the following forms:

$$\begin{array}{c}\hfill A_{n+1}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_1& \Psi & 0& \cdots & 0& 0\\ \Psi & {\mathsf{{\rm Y}}}_1& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_1& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_1\end{array}\right),\end{array}$$

(41)

$$\begin{array}{c}\hfill A_{n+2}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_2+\Psi & 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_2-\Psi & 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_2& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_2& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_2\end{array}\right),\end{array}$$

(42)

$$\begin{array}{c}\hfill A_{n+3}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_3& 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_3& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_3& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_3& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_3\end{array}\right),A_{n+4}=\cdots =A_{4m}=0,\end{array}$$

(43)

where ${\mathsf{{\rm Y}}}_1,{\mathsf{{\rm Y}}}_2,{\mathsf{{\rm Y}}}_3$, and Ψ are real functions on $\mathfrak{D}$.

Theorem 6.

Assume a quaternionic submanifold ${\mathfrak{D}}^n$ immersed in ${\overline{\mathfrak{D}}}^{4m}\left(c\right)$. Then,

$$\begin{array}{c}\hfill {\mathcal{N}}_{scal}\le {\displaystyle \frac{c}{2}}[1+{\displaystyle \frac{9}{(n-1)}}]-2[{\displaystyle \frac{n\mu }{n-1}}-{\displaystyle \frac{{\left|\right|\Pi \left|\right|}^2}{2}}]-{\displaystyle \frac{2}{n}}tr\left(\beta \right)-2\psi (\Pi ).\end{array}$$

(44)

The sign of equality is valid in (44) iff operators A with respect to $\{B_1,\dots ,B_n\}$ and $\{B_{n+1},\dots ,B_{4m}\}$, reduce to the following forms:

$$\begin{array}{c}\hfill A_{n+1}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_1& \Psi & 0& \cdots & 0& 0\\ \Psi & {\mathsf{{\rm Y}}}_1& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_1& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_1\end{array}\right),\end{array}$$

(45)

$$\begin{array}{c}\hfill A_{n+2}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_2+\Psi & 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_2-\Psi & 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_2& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_2& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_2\end{array}\right),\end{array}$$

(46)

$$\begin{array}{c}\hfill A_{n+3}=\left(\begin{array}{cccccc}{\mathsf{{\rm Y}}}_3& 0& 0& \cdots & 0& 0\\ 0& {\mathsf{{\rm Y}}}_3& 0& \cdots & 0& 0\\ 0& 0& {\mathsf{{\rm Y}}}_3& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & {\mathsf{{\rm Y}}}_3& 0\\ 0& 0& 0& \cdots & 0& {\mathsf{{\rm Y}}}_3\end{array}\right),A_{n+4}=\cdots =A_{4m}=0,\end{array}$$

(47)

where ${\mathsf{{\rm Y}}}_1,{\mathsf{{\rm Y}}}_2,{\mathsf{{\rm Y}}}_3$, and Ψ are real functions on $\mathfrak{D}$.

Remark 2.

The proofs of Theorems 5 and 6 can be expressed in a manner that is comparable with that of Theorem 4. In fact, we can obtain inequalities (40) and (44) by putting $\Theta ={\displaystyle \frac{\pi }{2}}$ and $\Theta =0$ in inequality (33), respectively.

7. Conclusions

Beyond the conventional studies of Riemannian submanifolds on space form, geometric inequality is a relatively new subject of study. Extrinsic and intrinsic curvature relationships with respect to submanifolds have received a lot of attention. In light of the Wintgen inequality or the DDVV conjecture, this work attempts to investigate submanifolds via intrinsic and extrinsic curvatures in quaternionic Kaehler manifolds endowed with a semi-symmetric metric and semi-symmetric non-metric connections.

Even though a generalized Wintgen inequality for submanifolds has been established, interested readers still have a lot of research issues to look into. Slant submanifolds, totally real submanifolds, invariant and anti-invariant sumanifolds of space forms, such as quaternionic Kaehler manifolds coupled with semi-symmetric metric and semi-symmetric non-metric connections, can all be considered in this manner. The equality case is characterized through specific forms of shape operators. It makes a valuable contribution by extending fundamental geometric inequalities to more general settings. The results effectively bridge classical differential geometry with modern developments in submanifold theory.