Curvature Pinching Conditions in Quaternionic Manifolds Under Quarter-Symmetric Metric Connections

Abstract

This work presents a pair of sharp geometric inequalities that connect the normalized scalar curvature with the generalized normalized $\delta$-Casorati curvature for $\theta$-slant submanifolds immersed in quaternionic space forms endowed with a quarter-symmetric metric connection (QSMC). Alongside establishing these estimates, we rigorously describe the geometric conditions under which equality is achieved. The results not only generalize prior findings related to Casorati curvature but also offer new insights into the extrinsic geometry of submanifolds under non-standard connections. To conclude, we propose several open problems that invite further exploration in this direction.

1. Introduction

The study of geometric inequalities involving curvature invariants of submanifolds has long been an active and evolving area in differential geometry. A significant breakthrough in this domain came with the introduction of the $\delta$-invariants by B.-Y. Chen, which provided a powerful framework for investigating the relationship between intrinsic and extrinsic curvature properties of submanifolds embedded in various ambient spaces [1,2,3]. These invariants, together with the resulting optimal inequalities known as Chen inequalities, offered deep insights into the geometry of submanifolds by connecting scalar curvature with the norm of the second fundamental form.

Over time, the theory has been extended and adapted to a wide array of ambient geometries, including real space forms [4]; complex space forms [5,6]; contact space forms [7,8]; and, more recently, quaternionic space forms [9,10,11]. Each of these ambient manifolds brings with it additional geometric structure, allowing for the refinement and generalization of Chen-type inequalities. Among these refinements, the Casorati curvature, a measure based on the norm of the second fundamental form, has been particularly valuable [12,13,14]. It quantifies the deviation of a submanifold from being totally geodesic and complements other extrinsic invariants like mean curvature. The introduction of normalized and generalized $\delta$-Casorati curvatures has made it possible to derive more precise and sharper inequalities [15,16].

In parallel, quaternionic geometry has gained prominence due to its applications in both mathematics and theoretical physics. Quaternionic Kähler manifolds and quaternionic space forms serve as natural generalizations of complex and real space forms, with the quaternionic structure providing a rich algebraic and geometric framework [17,18,19,20,21]. Within this context, slant submanifolds, those whose tangent spaces form a constant angle with the quaternionic structure, offer a generalization of both totally real and invariant submanifolds, and thus provide a flexible setting for studying curvature inequalities [22,23].

Adding to the complexity and interest of the geometric setting is the presence of non-Levi-Civita connections, such as quarter-symmetric metric connections. These connections preserve the metric but introduce torsion in a specific, structured way [4,24,25]. Unlike the Levi-Civita connection, quarter-symmetric metric connections alter the behavior of curvature tensors and influence both the intrinsic and extrinsic geometries of submanifolds. The presence of such a connection leads to modified versions of the Gauss, Codazzi, and Ricci equations, which in turn affect the formulation and proof of curvature inequalities.

In this paper, we focus on slant submanifolds in quaternionic space forms that are equipped with a QSMC. Our objective is to derive two optimal inequalities involving the generalized normalized $\delta$-Casorati curvature and the normalized scalar curvature of such submanifolds. These results generalize several existing inequalities known for Riemannian and almost Hermitian settings and provide new characterizations for submanifolds achieving equality. Specifically, we identify the geometric structure of those submanifolds for which equality holds and show that they are invariantly quasi-umbilical with a trivial normal connection under suitable orthonormal frames [17,26,27].

By integrating the quaternionic structure with the effects of a QSMC, this study not only expands the scope of curvature inequalities but also deepens our understanding of how torsion and ambient geometry collectively shape the behavior of embedded submanifolds. The techniques employed herein may also be of independent interest in the study of generalized connections and their applications to submanifold theory.

2. Preliminaries

This section outlines the foundational concepts and notations that support the framework of our investigation. We consider a Riemannian submanifold $M^q$ of an q-dimensional ambient Riemannian manifold $({\overline{M}}^m,g)$, where the induced metric on M is also denoted by g. For a plane section $\pi \subset T_{p}M$ at a point $p\in M$, the sectional curvature is denoted by $K\left(\pi \right)$. Given an orthonormal basis $\{u_1,\dots ,u_q\}$ of $T_{p}M$, the scalar curvature $\tau$ at the point p is expressed as

$$\tau \left(p\right)=\sum _{1\le i<j\le q}K(u_i\wedge u_j),$$

and the normalized scalar curvature $\rho$ of M is defined by

$$\rho ={\displaystyle \frac{2\tau }{q(q-1)}}.$$

Let $\overline{\nabla }$ be the Levi-Civita connection on $\overline{M}$, and ∇ the induced connection on M. Then the classical Gauss and Weingarten formulae are given as

$${\overline{\nabla }}_{Z_1}{Z}_2={\nabla }_{Z_1}{Z}_2+h(Z_1,Z_2),\hspace{1em}{\overline{\nabla }}_{Z_1}N=-A_N{Z}_1+{\nabla }_{Z_1}^{\perp }N,$$

for any $Z_1,Z_2\in \Gamma \left(TM\right)$ and $N\in \Gamma \left(T^{\perp }M\right)$, where h denotes the second fundamental form, ${\nabla }^{\perp }$ is the normal connection, and $A_N$ is the shape operator associated with the normal vector field N. The Gauss equation relating the curvatures of M and $\overline{M}$ is then

$$\begin{array}{cc}\hfill R(Z_1,Z_2,Z_3,Z_4)& =\overline{R}(Z_1,Z_2,Z_3,Z_4)\hfill \\ \hfill & +g(h(Z_1,Z_4),h(Z_2,Z_3))-g(h(Z_1,Z_3),h(Z_2,Z_4)).\hfill \end{array}$$

(1)

In the case where the ambient manifold $\overline{M}$ is equipped with a quarter-symmetric metric connection (QSMC) $\widehat{\nabla }$, the corresponding curvature tensor $\widehat{R}$ is modified. The associated expression for the curvature becomes [28]

$$\begin{array}{cc}\hfill \overline{\mathcal{R}}(Z_1,Z_2,Z_3,Z_4)& =\widehat{\mathcal{R}}(Z_1,Z_2,Z_3,Z_4)\hfill \\ \hfill & +{\Xi }_1\alpha (Z_1,Z_3)g(Z_2,Z_4)-{\Xi }_1\alpha (Z_2,Z_3)g(Z_1,Z_4)\hfill \\ \hfill & +{\Xi }_2\alpha (Z_2,Z_4)g(Z_1,Z_3)-{\Xi }_2\alpha (Z_1,Z_4)g(Z_2,Z_3)\hfill \\ \hfill & +{\Xi }_2({\Xi }_1-{\Xi }_2)g(Z_1,Z_3)\beta (Z_2,Z_4)\hfill \\ \hfill & -{\Xi }_2({\Xi }_1-{\Xi }_2)g(Z_2,Z_3)\beta (Z_1,Z_4),\hfill \end{array}$$

(2)

where

$$\kappa (Z_1,Z_2)=\left({\widehat{\nabla }}_{Z_1}\pi \right)\left(Z_2\right)-{\Xi }_1\pi \left(Z_1\right)\pi \left(Z_2\right)+{\displaystyle \frac{{\Xi }_2}{2}}g(Z_1,Z_2)\pi (\Pi ),$$

$$\mu (Z_1,Z_2)={\displaystyle \frac{1}{2}}\pi \left(P\right)g(Z_1,Z_2)+\pi \left(Z_1\right)\pi \left(Z_2\right),$$

for vector fields $Z_1,Z_2,Z_3,Z_4\in \Gamma \left(T\overline{M}\right)$, and $\pi$ is the 1-form associated with the vector field P.

The parameters $({\Xi }_1,{\Xi }_2)$ determine the nature of the connection. For reference, the principal cases are summarized in

Table 1. Special cases of the quarter–symmetric metric connection

Connection Type	${\mathbf{\Xi }}_1,{\mathbf{\Xi }}_2$	Remarks
Quarter–symmetric metric connection	${\Xi }_1,{\Xi }_2\in \mathbb{R}$	General case with nonzero torsion; metric compatibility preserved.
Semi–symmetric metric connection	$(1,1)$	Torsion linear in $\pi$; metric–compatible.
Semi–symmetric non–metric connection	$(1,0)$	Torsion present; metric non–compatibility.
Levi–Civita connection	$(0,0)$	Torsion free and metric compatible.

.

Let H be the mean curvature vector defined by

$$H\left(p\right)={\displaystyle \frac{1}{q}}\sum _{i=1}^{q}h(u_i,u_i),$$

and denote the component functions of the second fundamental form by

$$h_{ij}^{\omega }=g(h(u_i,u_j),u_{\alpha }).$$

Then the squared norm of the mean curvature vector is

$${\parallel H\parallel }^2={\displaystyle \frac{1}{q^2}}\sum _{\omega =q+1}^{4m}{\left(\sum _{i=1}^q{h}_{ii}^{\omega }\right)}^2,$$

and the Casorati curvature is given by

$$C={\displaystyle \frac{1}{q}}\sum _{\omega =q+1}^{4m}\sum _{i,j=1}^q{\left(h_{ij}^{\omega }\right)}^2.$$

A submanifold M is said to be invariantly quasi-umbilical if there exists an orthonormal basis of normal vectors $\{{\xi }_{q+1},\dots ,{\xi }_m\}$ such that each corresponding shape operator has an eigenvalue of multiplicity $q-1$, with the same distinguished eigendirection across all ${\xi }_{\omega }$.

Given an s-dimensional subspace $L\subset T_{p}M$, with orthonormal basis $\{u_1,\dots ,u_s\}$, its scalar curvature and Casorati curvature are respectively

$$\tau \left(L\right)=\sum _{1\le i<j\le s}K(u_i\wedge u_j),\hspace{1em}C\left(L\right)={\displaystyle \frac{1}{s}}\sum _{\omega =q+1}^m\sum _{i,j=1}^s{\left(h_{ij}^{\omega }\right)}^2.$$

The normalized $\delta$-Casorati curvatures are defined by [29]

$${\left[{\delta }_C(q-1)\right]}_p={\displaystyle \frac{1}{2}}{C}_p+{\displaystyle \frac{q+1}{2q}}inf\{C\left(L\right):L \mathrm{a} \mathrm{hyperplane} \mathrm{in} T_{p}M\},$$

$${\left[{\widehat{\delta }}_C(q-1)\right]}_p=2C_p+{\displaystyle \frac{2q-1}{2q}}sup\{C\left(L\right):L \mathrm{a} \mathrm{hyperplane} \mathrm{in} T_{p}M\}.$$

Moreover, for any real number $d\ne q(q-1)$, the generalized versions are given by [29]

$${\left[{\delta }_C(d;q-1)\right]}_p=d{C}_p+{\displaystyle \frac{(q-1)(q+d)(q^2-q-d)}{qd}}inf\left\{C\left(L\right)\right\}, \mathrm{for} 0<d<q^2-q,$$

$${\left[{\widehat{\delta }}_C(d;q-1)\right]}_p=d{C}_p+{\displaystyle \frac{(q-1)(q+d)(d-q^2+q)}{qd}}sup\left\{C\left(L\right)\right\}, \mathrm{for} d>q^2-q.$$

Now consider a Riemannian manifold M of dimension $4m$ endowed with a rank-3 subbundle $\sigma \subset \mathrm{End}\left(TM\right)$, admitting a local basis $\{J_1,J_2,J_3\}$ satisfying:

$$J_{\alpha }^2=-\mathrm{Id}, J_{\alpha }{J}_{\alpha +1}=-J_{\alpha +1}{J}_{\alpha }=J_{\alpha +2}, g(J_{\alpha }{Z}_1,J_{\alpha }{Z}_2)=g(Z_1,Z_2),$$

with indices taken modulo 3. Then $(M,\sigma ,g)$ is an almost quaternionic Hermitian manifold. If $\sigma$ is parallel with respect to the Levi-Civita connection, the structure becomes quaternionic Kähler.

A quaternionic Kähler manifold with constant quaternionic sectional curvature c is called a quaternionic space form, denoted $M\left(c\right)$. Its curvature tensor satisfies

$$\begin{array}{cc}\hfill \overline{R}& (Z_1,Z_2)Z_3={\displaystyle \frac{c}{4}}[g(Z_2,Z_3)Z_1-g(Z_1,Z_3)Z_2\hfill \\ \hfill & +\sum _{\alpha =1}^3\left(g(Z_3,J_{\alpha }{Z}_2)J_{\alpha }{Z}_1-g(Z_3,J_{\alpha }{Z}_1)J_{\alpha }{Z}_2+2g(Z_1,J_{\alpha }{Z}_2)J_{\alpha }{Z}_3\right)].\hfill \end{array}$$

(3)

A submanifold M of such a space is called a slant submanifold if the angle between $J_{\alpha }X$ and the tangent space $T_{p}M$ is constant for all nonzero $X\in T_{p}M$ and for each $\alpha \in \{1,2,3\}$. If this angle is zero, M is quaternionic; if it is ${\displaystyle \frac{\pi }{2}}$, then M is totally real. Any slant submanifold that is neither of these is said to be proper. Such submanifolds necessarily have even dimension $q=2s>2$ and admit a canonical adapted orthonormal slant frame of the form:

$$\{u_1,u_2=sec\theta P_{\alpha }{u}_1,\dots ,u_{2s-1},u_{2s}=sec\theta P_{\alpha }{u}_{2s-1}\},$$

where $P_{\alpha }{u}_{2k-1}$ denotes the tangential projection of $J_{\alpha }{u}_{2k-1}$ onto $TM$ [23].

Notation and Conventions

For the reader’s convenience, we summarize the main notation and conventions used throughout the manuscript in the

Table 2. Notation and conventions used throughout the paper.

Symbol/Convention	Meaning
q	Dimension of the submanifold M ($q=dimM$).
$4m$	Dimension of the ambient quaternionic space form $M^{4m}\left(c\right)$.
s	Half-dimension of a proper slant submanifold: $s={\displaystyle \frac{q}{2}}$.
$i,j,k,\dots$	Tangential indices ranging over $\{1,\dots ,q\}$.
$\omega$	Normal index ranging over $\{q+1,\dots ,4m\}$.
${\Xi }_1,{\Xi }_2$	Parameters of the quarter-symmetric metric connection (QSMC). Special cases: $(0,0)$: Levi-Civita connection, $(1,1)$: semi-symmetric metric connection, $(1,0)$: semi-symmetric non-metric connection.
$\pi$	The 1-form associated with the vector field P defining the QSMC torsion structure.
P	The vector field dual to $\pi$.
$\kappa ,\mu$	Auxiliary $(0,2)$-tensors associated with the curvature expression under QSMC (see Equation (2)).
$h_{ij}^{\omega }$	Components of the second fundamental form: $h_{ij}^{\omega }=g(h(u_i,u_j),{\xi }_{\omega })$.
$\tau$	Scalar curvature: $\tau \left(p\right)={\sum }_{1\le i<j\le q}K(u_i\wedge u_j)$.
$\rho$	Normalized scalar curvature: $\rho ={\displaystyle \frac{2\tau }{q(q-1)}}$.
C	Casorati curvature: $C={\displaystyle \frac{1}{q}}{\sum }_{\omega =q+1}^{4m}{\sum }_{i,j=1}^q{\left(h_{ij}^{\omega }\right)}^2$.
$C\left(L\right)$	Casorati curvature of a hyperplane L.
H	Mean curvature vector.
A	Shape operator.
${\delta }_C(q-1)$	Normalized $\delta$-Casorati curvature.
${\widehat{\delta }}_C(q-1)$	Dual normalized $\delta$-Casorati curvature.
${\delta }_C(d;q-1)$, ${\widehat{\delta }}_C(d;q-1)$	Generalized $\delta$-Casorati curvatures depending on parameter d.
Curvature sign convention	We use $$R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z.$$ This convention ensures that quaternionic space forms of constant curvature c satisfy Formula (3).

.

3. Main Results

In this section, we present optimal inequalities involving the generalized normalized $\delta$-Casorati curvatures for $\theta$-slant proper submanifolds immersed in quaternionic space forms equipped with a quarter-symmetric metric connection (QSMC). These results provide sharp upper bounds for the normalized scalar curvature in terms of the extrinsic Casorati curvature invariants while also characterizing the cases in which the equality conditions are attained. Our findings generalize and extend existing curvature inequalities to the setting of quaternionic geometry under non-Riemannian connections.

Theorem 1.

Let $M^q$ be a proper θ-slant submanifold of a quaternionic space form $M^{4m}\left(c\right)$ equipped with a quarter-symmetric metric connection. Then the following statements hold:

(i) 

For any real number d satisfying $0<d<q(q-1)$, the normalized scalar curvature ρ satisfies the inequality:

$$\begin{array}{cc}\hfill \rho & \le {\displaystyle \frac{{\delta }_C(d;q-1)}{q(q-1)}}+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)\hfill \\ \hfill & -{\displaystyle \frac{{\Xi }_1+{\Xi }_2}{q}}{t}_1-{\displaystyle \frac{{\Xi }_2({\Xi }_1-{\Xi }_2)}{q}}{t}_2+({\Xi }_2-{\Xi }_1)\pi \left(H\right),\hfill \end{array}$$

(4)

where $t_1=trace\left(\kappa \right)$, $t_2=trace\left(\mu \right)$, and $\pi \left(H\right)=g(P^{\perp },H)$ denotes the projection of the mean curvature vector onto the direction of the vector field Π.

(ii) 

For any real number $d>q(q-1)$, the inequality involving the dual δ-Casorati curvature holds:

$$\begin{array}{cc}\hfill \rho & \le {\displaystyle \frac{{\widehat{\delta }}_C(d;q-1)}{q(q-1)}}+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)\hfill \\ \hfill & -{\displaystyle \frac{{\Xi }_1+{\Xi }_2}{q}}{t}_1-{\displaystyle \frac{{\Xi }_2({\Xi }_1-{\Xi }_2)}{q}}{t}_2+({\Xi }_2-{\Xi }_1)\pi \left(H\right).\hfill \end{array}$$

(5)

Moreover, the equalities in both cases are achieved if and only if the submanifold $M^q$ is invariantly quasi-umbilical with a flat normal connection, and the corresponding shape operators $A_d=A_{{\xi }_r}$, for $d\in \{q+1,\dots ,4m\}$ take the form:

$$A_{q+1}=\left(\begin{array}{cccc}a& 0& \dots & 0\\ 0& a& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\displaystyle \frac{q(q-1)}{d}}a\end{array}\right),A_{q+2}=\dots =A_{4m}=0,$$

(6)

with respect to a suitable orthonormal tangent frame $\{u_1,\dots ,u_q\}$ and a normal orthonormal frame $\{{\xi }_{q+1},\dots ,{\xi }_{4m}\}$.

Proof. 

Since M is assumed to be a proper $\theta$-slant submanifold, it follows from the structure of slant geometry (cf. [19]) that the composition of the tangential parts $P_{\alpha }$ and $P_{\beta }$ of the almost quaternionic structures satisfies

$$P_{\beta }{P}_{\alpha }{Z}_1=-{cos}^2\theta Z_1,\forall Z_1\in \Gamma \left(TM\right),\alpha ,\beta \in \{1,2,3\}.$$

(7)

From this identity, one can immediately deduce the relation

$$g(P_{\alpha }{Z}_1,P_{\beta }{Z}_2)={cos}^2\theta g(Z_1,Z_2),\forall Z_1,Z_2\in \Gamma \left(TM\right),\alpha ,\beta \in \{1,2,3\}.$$

(8)

Now, let us consider the ambient space $M^{4m}\left(c\right)$ to be a quaternionic space form endowed with a quarter-symmetric metric connection. Then, making use of the Gauss equation along with the expressions for the curvature tensors under this connection structure (see Equations (1)–(3)), we obtain:

$$\begin{array}{cc}\hfill q^2{\parallel H\parallel }^2& =2\tau \left(p\right)+{\parallel h\parallel }^2-{\displaystyle \frac{q(q-1)c}{4}}-{\displaystyle \frac{3c}{4}}\sum _{\beta =1}^3\sum _{i,j=1}^q{g}^2(P_{\beta }{u}_i,u_j)\hfill \\ \hfill & +({\Xi }_1+{\Xi }_2)(q-1)t_1+{\Xi }_2({\Xi }_1-{\Xi }_2)(q-1)t_2\hfill \\ \hfill & -({\Xi }_2-{\Xi }_1)q(q-1)\pi \left(H\right).\hfill \end{array}$$

(9)

The term $\pi \left(H\right)$ represents the projection of the mean curvature vector H along the vector field $P^{\perp }$ and can be expressed as

$$\pi \left(H\right)={\displaystyle \frac{1}{q}}\sum _{j=1}^q\pi \left(h(u_j,u_j)\right)=g(P^{\perp },H).$$

Now, consider an adapted orthonormal basis

$$\{u_1,e_2=sec\theta P_{\alpha }{u}_1,\dots ,u_{2s-1},u_{2s}=sec\theta P_{\alpha }{u}_{2s-1}\}$$

for the tangent bundle $TM$, valid locally at a point $p\in M$ such that $q=2s$. Using this basis and applying the previously established identities, we find:

$$\left\{\begin{array}{cc}\hfill & g^2(P_{\beta }{u}_i,u_{i+1})={cos}^2\theta ,i=1,2,\dots ,2s-1\hfill \\ \hfill & g(P_{\beta }{u}_i,u_j)=0\mathrm{for} \mathrm{all} (i,j)\notin \{(2l-1,2l),(2l,2l-1)\},l\in \{1,2,\dots ,s\}.\hfill \end{array}\right.$$

(10)

Substituting this information into the previous expression for $q^2{\parallel H\parallel }^2$ and isolating $2\tau \left(p\right)$, we obtain the relation:

$$\begin{array}{cc}\hfill 2\tau \left(p\right)& =q^2{\parallel H\parallel }^2-q{\parallel h\parallel }^2+qc+{\displaystyle \frac{c}{4}}[q(q-1)+9q{cos}^2\theta ]\hfill \\ \hfill & -({\Xi }_1+{\Xi }_2)(q-1)t_1-{\Xi }_2({\Xi }_1-{\Xi }_2)(q-1)t_2\hfill \\ \hfill & +({\Xi }_2-{\Xi }_1)q(q-1)\pi \left(H\right).\hfill \end{array}$$

(11)

This expression serves as the foundation for deriving upper bounds for normalized scalar curvature in terms of $\delta$-Casorati curvatures.

We now construct a quadratic polynomial $\Pi$ involving the components of the second fundamental form, which will be instrumental in establishing an upper bound for the normalized scalar curvature $\rho$. Specifically, we define:

$$\begin{array}{cc}\hfill \Pi & =dC+\left({\displaystyle \frac{q+1}{qd}}\right)(q+d)(q^2-q-d)C\left(L\right)-2d\tau \left(p\right)\hfill \\ \hfill & +{\displaystyle \frac{c}{4}}\left[q(q-1)+9q{cos}^2\theta \right]({\Xi }_1+{\Xi }_2)(q-1)t_1\hfill \\ \hfill & -{\Xi }_2({\Xi }_1-{\Xi }_2)(q-1)t_2+({\Xi }_2-{\Xi }_1)q(q-1)\pi \left(H\right),\hfill \end{array}$$

(12)

where L is an arbitrary hyperplane in the tangent space $T_{p}M$ and $C\left(L\right)$ denotes the Casorati curvature restricted to L. Without a loss of generality, we may take L to be spanned by the vectors $\{u_1,\dots ,u_{q-1}\}$.

Using the standard form of C and $C\left(L\right)$, we express $\Pi$ in terms of the components $h_{ij}^{\omega }$ of the second fundamental form:

$$\begin{array}{cc}\hfill \Pi & ={\displaystyle \frac{d}{q}}\sum _{\omega =q+1}^{4m}\sum _{i,j=1}^q{\left(h_{ij}^{\omega }\right)}^2+{\displaystyle \frac{(q+d)(q^2-q-d)}{qd}}\sum _{\omega =q+1}^{4m}\sum _{i,j=1}^{q-1}{\left(h_{ij}^{\omega }\right)}^2\hfill \\ \hfill & -2d\tau \left(p\right)+{\displaystyle \frac{c}{4}}\left[q(q-1)+9q{cos}^2\theta \right]-({\Xi }_1+{\Xi }_2)(q-1)t_1\hfill \\ \hfill & -{\Xi }_2({\Xi }_1-{\Xi }_2)(q-1)t_2+({\Xi }_2-{\Xi }_1)q(q-1)\pi \left(H\right).\hfill \end{array}$$

(13)

Recalling the earlier expression for $\tau \left(p\right)$ derived in the proof, we now substitute it to eliminate $\tau \left(p\right)$ from the polynomial. This yields:

$$\begin{array}{cc}\hfill \Pi & ={\displaystyle \frac{q+d}{q}}\sum _{\omega =q+1}^{4m}\sum _{j=1}^q{\left(h_{jj}^{\omega }\right)}^2\hfill \\ \hfill & +{\displaystyle \frac{(q+d)(q^2-q-d)}{qd}}\sum _{\omega =q+1}^{4m}\sum _{i,j=1}^{q-1}{\left(h_{ij}^{\omega }\right)}^2-2\sum _{\omega =q+1}^{4m}{\left(\sum _{i=1}^q{h}_{ii}^{\omega }\right)}^2,\hfill \end{array}$$

which simplifies further into the following explicit quadratic form:

$$\begin{array}{cc}\hfill \Pi & =\sum _{\omega =q+1}^{4m}\sum _{i=1}^{q-1}\left[{\displaystyle \frac{q^2+q(d-1)}{qd}}{\left(h_{ii}^{\omega }\right)}^2+{\displaystyle \frac{2(q+d)}{q}}{h}_{ii}^{\omega }{h}_{qq}^{\omega }\right]\hfill \\ \hfill & +\sum _{\omega =q+1}^{4m}\left[{\displaystyle \frac{2(q+d)(q-1)}{qd}}\sum _{1\le i<j\le q-1}{\left(h_{ij}^{\omega }\right)}^2-2\sum _{1\le i<j\le q}{h}_{ii}^{\omega }{h}_{jj}^{\omega }+{\displaystyle \frac{d}{q}}{\left(h_{qq}^{\omega }\right)}^2\right].\hfill \end{array}$$

(14)

To find the minimum of $\Pi$, we consider the critical points of this polynomial with respect to the variables $h_{ij}^{\omega }$. Let us denote these components collectively as:

$$h^c=\left(h_{11}^{q+1},h_{12}^{q+1},\dots ,h_{qq}^{q+1},\dots ,h_{11}^{4m},h_{12}^{4m},\dots ,h_{qq}^{4m}\right).$$

The necessary conditions for critical points are obtained by taking the partial derivatives of $\Pi$ with respect to each $h_{ij}^{\omega }$ and setting them equal to zero. This yields the following system of equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\partial \Pi }{\partial h_{ii}^{\omega }}=\frac{2(q+d)(q-1)}{r}{h}_{ii}^{\omega }-2\sum _{k=1}^q{h}_{kk}^{\omega }=0,}\hfill & i=1,\dots ,q-1,\hfill \\ {\displaystyle \frac{\partial \Pi }{\partial h_{qq}^{\omega }}=\frac{2d}{q}{h}_{qq}^{\omega }-2\sum _{k=1}^{q-1}{h}_{kk}^{\omega }=0,}\hfill \\ {\displaystyle \frac{\partial \Pi }{\partial h_{ij}^{\omega }}=\frac{4(q+d)(q-1)}{qd}{h}_{ij}^{\omega }=0,}\hfill & i\ne j,i,j\le q-1,\hfill \\ {\displaystyle \frac{\partial \Pi }{\partial h_{iq}^{\omega }}=\frac{4(q+d)}{qd}{h}_{iq}^{\omega }=0,}\hfill & i=1,\dots ,q-1,\hfill \end{array}\right.\end{array}$$

(15)

where $\omega \in \{q+1,\dots ,4m\}$.

This system reveals that off-diagonal components of the second fundamental form must vanish for a minimum, and diagonal components are constrained by linear combinations. We will now use this structure to evaluate $\Pi$ at its minimum and proceed to finalize the proof.

From the critical point system in (15), we observe that all off-diagonal components of the second fundamental form vanish, i.e., $h_{ij}^{\omega }=0$ for all $i\ne j$. Furthermore, the homogeneous linear system involving the diagonal entries admits non-trivial solutions, indicating that the submanifold is not necessarily totally geodesic.

To analyze the nature of the extremum, we examine the Hessian matrix $\mathcal{H}(\Pi )$ of the polynomial $\Pi$. This matrix decomposes into a block diagonal form:

$$\mathcal{H}(\Pi )=\left(\begin{array}{ccc}■& ⊚& ⊚\\ ⊚& ♦& ⊚\\ ⊚& ⊚& ★\end{array}\right),$$

where $■,♦$ and ★ correspond to partial second derivatives with respect to a subset of the components $h_{ij}^{\omega }$ and ⊚ denotes a null matrix of appropriate dimensions.

The matrix ■ represents the second derivatives with respect to diagonal entries of $h_{ij}^{\omega }$, and it is symmetric with structure:

$$■=2\left(\begin{array}{ccccc}{\displaystyle \frac{(q+d)(q-1)}{d}}-1& -1& \dots & -1& -1\\ -1& {\displaystyle \frac{(q+d)(q-1)}{d}}-1& \dots & -1& -1\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -1& -1& \dots & {\displaystyle \frac{(q+d)(q-1)}{d}}-1& {\displaystyle \frac{d}{q}}\end{array}\right)$$

Meanwhile, ♦ and ★ are diagonal matrices with constant positive entries:

$$\begin{array}{cc}\hfill & ♦=\mathrm{diag}\left({\displaystyle \frac{4(q+d)(q-1)}{d}},\dots ,{\displaystyle \frac{4(q+d)(q-1)}{d}}\right),\hfill \\ \hfill & ★=\mathrm{diag}\left({\displaystyle \frac{4(q+d)}{q}},\dots ,{\displaystyle \frac{4(q+d)}{q}}\right).\hfill \end{array}$$

Also, $\mathcal{H}(\Pi )$ has the following eigenvalues (see [7,8]):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varkappa }_{11}=0,{\varkappa }_{22}={\displaystyle \frac{2(q^3-q^2+d^2)}{qd}},\hfill \\ {\varkappa }_{33}=\dots ={\varkappa }_{qq}={\displaystyle \frac{2(q+d)(q-1)}{d}},\hfill \\ {\varkappa }_{ıj}={\displaystyle \frac{4(q+d)(q-1)}{d}},\hfill \\ {\varkappa }_{ım}={\displaystyle \frac{4(q+d)}{q}},\forall ı,j\in \{1,2,\dots ,q-1\},ı\ne j.\hfill \end{array}\right.\end{array}$$

(16)

Therefore, $\Pi$ achieves a global minimum at the critical points $h^c$ defined by the solution of system (15). Substituting these conditions into the expression for $\Pi$ yields $\Pi \left(h^c\right)=0$. This shows that $\Pi \ge 0$ for all possible values, with equality only at the critical configuration.

Thus, we infer the following upper bound:

$$\begin{array}{c}2\tau \left(p\right)\le dC+{\displaystyle \frac{(q-1)(q+d)(q^2-q-d)}{qd}}C\left(L\right)+{\displaystyle \frac{c}{4}}\left[q(q-1)+9q{cos}^2\theta \right]\hfill \\ -({\Xi }_1+{\Xi }_2)(q-1)t_1-{\Xi }_2({\Xi }_1-{\Xi }_2)(q-1)t_2+({\Xi }_2-{\Xi }_1)q(q-1)\pi \left(H\right).\hfill \end{array}$$

(17)

Dividing both sides by $q(q-1)$ and recalling that $\rho ={\displaystyle \frac{2\tau }{q(q-1)}}$, we obtain the desired inequality:

$$\begin{array}{cc}\hfill \rho & \le {\displaystyle \frac{d}{q(q-1)}}C+{\displaystyle \frac{(q+d)(q^2-q-d)}{d{q}^2}}C\left(L\right)+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)\hfill \\ \hfill & -{\displaystyle \frac{({\Xi }_1+{\Xi }_2)}{q}}{t}_1-{\displaystyle \frac{{\Xi }_2({\Xi }_1-{\Xi }_2)}{q}}{t}_2+({\Xi }_2-{\Xi }_1)\pi \left(H\right),\hfill \end{array}$$

(18)

which holds for every hyperplane L in $T_{p}M$.

Regarding the equality case, it follows from the structure of the critical points that equality in the above inequality occurs if and only if:

$$h_{ij}^{\omega }=0\mathrm{for} \mathrm{all} i\ne j,$$

(19)

and

$$h_{qq}^{\omega }={\displaystyle \frac{q(q-1)}{d}}{h}_{ii}^{\omega },\mathrm{for} i=1,\dots ,q-1.$$

(20)

These conditions characterize a submanifold with a flat normal connection, whose shape operators take a block-diagonal form with precisely prescribed eigenvalue multiplicities. Hence, the submanifold $M^q$ must be invariantly quasi-umbilical in $M^{4m}\left(c\right)$ with a trivial normal connection, and the shape operators must be of the form described earlier in the statement of the theorem. □

Corollary 1.

Let $M^q$ be a proper θ-slant submanifold of a quaternionic space form $M^{4m}\left(c\right)$ equipped with a quarter-symmetric metric connection. Then the following bounds for normalized δ-Casorati curvature invariants hold:

(i) 

The normalized δ-Casorati curvature ${\delta }_c(q-1)$ satisfies the inequality:

$$\begin{array}{cc}\hfill \rho & \le {\delta }_C(q-1)+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)\hfill \\ \hfill & -{\displaystyle \frac{{\Xi }_1+{\Xi }_2}{q}}{t}_1-{\displaystyle \frac{{\Xi }_2({\Xi }_1-{\Xi }_2)}{q}}{t}_2+({\Xi }_2-{\Xi }_1)\pi \left(H\right).\hfill \end{array}$$

(21)

Equality is attained if and only if the submanifold $M^q$ is invariantly quasi-umbilical with a flat normal connection. In this case, the shape operator $A_{{\xi }_d}$ corresponding to the normal vector fields ${\xi }_d$, $d\in \{q+1,\dots ,4m\}$ satisfies:

$$A_{q+1}=\left(\begin{array}{cccc}a& 0& \dots & 0\\ 0& a& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & a\\ 0& 0& \dots & 2a\end{array}\right),A_{q+2}=\dots =A_{4m}=0.$$

(22)

(ii) 

The normalized dual δ-Casorati curvature ${\widehat{\delta }}_c(q-1)$ satisfies:

$$\begin{array}{cc}\hfill \rho & \le {\widehat{\delta }}_C(q-1)+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)\hfill \\ \hfill & -{\displaystyle \frac{{\Xi }_1+{\Xi }_2}{q}}{t}_1-{\displaystyle \frac{{\Xi }_2({\Xi }_1-{\Xi }_2)}{q}}{t}_2+({\Xi }_2-{\Xi }_1)\pi \left(H\right).\hfill \end{array}$$

(23)

The equality case occurs precisely when $M^q$ is invariantly quasi-umbilical with a flat normal connection, and the shape operators $A_{{\xi }_d}$, $d\in \{q+1,\dots ,4m\}$ have the structure:

$$A_{q+1}=\left(\begin{array}{cccc}2a& 0& \dots & 0\\ 0& 2a& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 2a\\ 0& 0& \dots & a\end{array}\right),A_{q+2}=\dots =A_{4m}=0.$$

(24)

Proof. 

To prove the first part, recall the definition of the generalized normalized $\delta$-Casorati curvature ${\delta }_C(d;q-1)$, which for $d={\displaystyle \frac{q(q-1)}{2}}$ reduces to the classical normalized $\delta$-Casorati curvature ${\delta }_c(q-1)$. Substituting this value of d into the inequality in Theorem 1 (i), and simplifying, we obtain the inequality stated in part (i) of the corollary.

Similarly, for part (ii), choosing $d=2q(q-1)$ in Theorem 1 (ii) yields the inequality involving the normalized dual $\delta$-Casorati curvature ${\widehat{\delta }}_c(q-1)$. This substitution leads directly to the form of inequality provided in part (ii) of the corollary.

As for the conditions under which equality holds, they directly follow from the equality characterization provided in Theorem 1. That is, the equality case is achieved precisely when the submanifold $M^q$ admits a trivial normal connection and each shape operator takes the specific diagonal form described, making $M^q$ an invariantly quasi-umbilical submanifold. The explicit forms of the shape operators $A_{q+1}$ and vanishing of all others align with the critical configuration identified in the main result. □

4. Results with Other Connection Type

In addition to the findings obtained under the QSMC, it is of interest to examine how similar geometric inequalities behave under different connection types. To this end, we present a comparative overview of related results involving other geometric connections, including semi-symmetric Levi-Civita. The following table summarizes key inequalities and their corresponding geometric settings, offering a broader understanding of curvature behavior across various connection types.

Theorem 2.

Let $M^q$ be a proper θ-slant submanifold of a quaternionic space form $M^{4m}\left(c\right)$. Then the following statements hold:

S.M.	Connection Type	Inequalities
(1)	Semi-Symmetric Metric Connection	(i)  $\rho \le {\displaystyle \frac{{\delta }_C(d;q-1)}{q(q-1)}}+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)-{\displaystyle \frac{2t_1}{q}}$ (ii)  $\rho \le {\displaystyle \frac{{\widehat{\delta }}_C(d;q-1)}{q(q-1)}}+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)-{\displaystyle \frac{2t_1}{q}}$
(2)	Semi-Symmetric Non-Metric Connection	(i)  $\rho \le {\displaystyle \frac{{\delta }_C(d;q-1)}{q(q-1)}}+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)+({\displaystyle \frac{t_1}{q}}-\pi \left(H\right)),$ (ii)  $\rho \le {\displaystyle \frac{{\widehat{\delta }}_C(d;q-1)}{q(q-1)}}+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)-({\displaystyle \frac{t_1}{q}}+\pi \left(H\right))$
(3)	Levi-Civita Connection	(i)  $\rho \le {\displaystyle \frac{{\delta }_C(d;q-1)}{q(q-1)}}+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)$ (ii)  $\rho \le {\displaystyle \frac{{\widehat{\delta }}_C(d;q-1)}{q(q-1)}}+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{9}{q-1}}{cos}^2\theta \right)$

Moreover, the equalities in both cases are achieved if and only if the submanifold $M^q$ is invariantly quasi-umbilical with a flat normal connection, and the corresponding shape operators $A_d=A_{{\xi }_r}$, for $d\in \{q+1,\dots ,4m\}$ take the form:

$$A_{q+1}=\left(\begin{array}{cccc}a& 0& \dots & 0\\ 0& a& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\displaystyle \frac{q(q-1)}{d}}a\end{array}\right),A_{q+2}=\dots =A_{4m}=0,$$

(25)

with respect to a suitable orthonormal tangent frame $\{u_1,\dots ,u_q\}$ and a normal orthonormal frame $\{{\xi }_{q+1},\dots ,{\xi }_{4m}\}$.

Remark 1.

The findings summarized in the table follow as immediate consequences of Theorem 1, in conjunction with the evaluation identities provided in Table 1. Together, they offer a unified framework for deriving optimal inequalities under various geometric conditions dictated by the choice of connections.

Remark 2.

The third result established in Theorem 2, corresponding to the case of the Levi-Civita connection, aligns with the result previously obtained in [29], thereby confirming the consistency of our general framework with known results in the literature.

Remark 3.

A result analogous to Theorem 2 can also be derived in the context of Corollary 1, following a similar approach.

5. Conclusions

In this study, we have derived two sharp inequalities that relate the normalized scalar curvature to the generalized normalized $\delta$-Casorati curvatures for $\theta$-slant submanifolds immersed in quaternionic space forms endowed with a QSMC. These results extend classical curvature pinching inequalities by incorporating the influence of the quarter-symmetric structure, offering refined bounds that generalize previous findings in both Riemannian and quaternionic settings.

Moreover, we have fully classified the geometric structure of the submanifolds that attain equality in these inequalities. It was shown that equality holds precisely when the submanifold is invariantly quasi-umbilical with a flat normal connection and when the shape operators satisfy specific algebraic conditions. This characterization not only deepens our understanding of the submanifold geometry under QSMC but also highlights the role of Casorati-type invariants in describing the interplay between intrinsic and extrinsic properties.