Curvature Bounds and Casorati Pinching for Submanifolds in Kähler Product Manifolds

Abstract

In this paper, we establish sharp pinching inequalities that relate the generalized $\delta$-Casorati curvatures to the normalized scalar curvature of submanifolds immersed in Kähler product manifolds endowed with a quarter-symmetric metric connection. The results are obtained for a broad range of geometric configurations, encompassing several important classes of submanifolds. Moreover, we prove that the derived inequalities are optimal by completely characterizing the submanifolds for which equality holds, showing that these cases correspond precisely to invariantly quasi-umbilical submanifolds with trivial normal connection.

1. Introduction

The geometry of submanifolds in Kähler manifolds constitutes a central theme in differential geometry, owing to its strong connections with complex geometry, global analysis, and curvature theory. In particular, the study of submanifolds of Kähler product manifolds [1,2,3] has revealed remarkable structural properties that do not appear in the non-product setting. Early foundational work in this direction was carried out by Yano and Kon [4], who investigated submanifolds that are invariant or anti-invariant with respect to the almost product structure F. Their results showed that any F-invariant submanifold of a Kähler manifold admits a product decomposition, thereby establishing a strong rigidity phenomenon tied to the underlying product geometry.

Subsequent research further clarified the behavior of slant and semi-invariant submanifolds [5,6,7] in Kähler product manifolds. In particular, Şahin proved that an F-invariant slant submanifold of a Kähler product manifold $\overline{\mathcal{M}}={\overline{\mathcal{M}}}^m\times {\overline{\mathcal{M}}}^n$ can be locally expressed as a Riemannian product $\mathcal{M}1\times \mathcal{M}2$, where $\mathcal{M}1$ and $\mathcal{M}2$ are slant submanifolds of ${\overline{\mathcal{M}}}^m$ and ${\overline{\mathcal{M}}}^n$, respectively [8]. This result demonstrates that slant geometry is compatible with the ambient product structure and that the slant angle is preserved componentwise. In a related direction, Şahin [9] analyzed semi-invariant submanifolds of Riemannian product manifolds, while Shahid [10] studied CR-submanifolds in Kähler product manifolds [1,11], both contributing to a deeper understanding of how invariant distributions interact with product geometries.

Parallel to these structural investigations, a fundamental objective in submanifold theory, motivated by Nash’s celebrated embedding theorem [12], has been to understand how intrinsic geometry restricts the extrinsic realization of a Riemannian manifold within a given ambient space. Although Nash’s theorem ensures the existence of isometric embeddings into Euclidean spaces, significant limitations arise when the ambient manifold is required to satisfy geometric constraints such as constant curvature, special holonomy, or non-standard connections. These limitations underscore the necessity of identifying precise relationships between intrinsic invariants of a submanifold and its extrinsic curvature quantities [13,14,15].

To address this challenge, Chen introduced a systematic program aimed at establishing optimal inequalities connecting intrinsic invariants, such as scalar curvature, to extrinsic invariants derived from the second fundamental form. As part of this framework, Chen [7,13,14] proposed several new curvature quantities that are more sensitive to the geometry of submanifolds than classical invariants. Among these, the Casorati curvature [16,17] plays a prominent role. Defined as the normalized squared norm of the second fundamental form, the Casorati curvature serves as a refined extrinsic invariant that captures how a submanifold bends in the ambient space, providing an effective alternative to sectional curvature or squared mean curvature.

The notion of Casorati curvature, originally introduced by Casorati for submanifolds of Riemannian manifolds, can be interpreted as an extension of the concept of principal directions for hypersurfaces [16]. Over the past decade, this invariant has been extensively employed to derive sharp curvature inequalities in a variety of geometric contexts [18,19,20,21,22,23,24]. Notably, optimal Casorati-type inequalities have been established for submanifolds of generalized space forms equipped with semi-symmetric metric connections (SSMC) [25], as well as for quaternionic space forms endowed with a quarter-symmetric metric connection (QSMC) [26]. These studies illustrate that Casorati curvature remains a powerful tool even in non-Levi-Civita settings, where additional torsion or symmetry properties influence curvature behavior.

More recently, attention has shifted toward generalized normalized $\delta$-Casorati curvatures (GNDCC) [27,28,29], which refine classical Casorati invariants by incorporating distributional information and allowing sharper curvature pinching estimates. Such generalized invariants are particularly effective when studying submanifolds in ambient manifolds with product structures or non-standard connections, where different tangent components exhibit distinct geometric behavior.

Motivated by these developments, the present paper aims to establish upper bounds for the GNDCC of submanifolds immersed in Kähler product manifolds endowed with a QSMC. By combining techniques from Kähler geometry, product manifolds, and curvature pinching theory under QSMC, we extend existing Casorati-type inequalities to a broader geometric framework and contribute new insights into the interaction between product structures and extrinsic curvature.

2. Preliminaries

Let $(\overline{\mathcal{M}},g)$ be an $(m+n)$-dimensional Riemannian manifold endowed with a Riemannian metric g. A tensor field J of type $(1,1)$ on $\overline{\mathcal{M}}$ is said to define an almost complex structure if it satisfies $J^2=-I$ at each point $p\in \overline{\mathcal{M}}$, where I denotes the identity transformation on $T_p\overline{\mathcal{M}}$. In this case, $\overline{\mathcal{M}}$ is called an almost complex manifold.

The Nijenhuis torsion tensor $N_J$ associated with the almost complex structure J is defined by [8]

$$\begin{array}{c}\hfill N_J(U,V)=[JU,JV]-[U,V]-J[U,JV]-J[JU,V],\end{array}$$

for all $U,V\in \Gamma \left(T\overline{\mathcal{M}}\right)$, where $\Gamma \left(T\overline{\mathcal{M}}\right)$ denotes the space of smooth sections of the tangent bundle $T\overline{\mathcal{M}}$. If the tensor $N_J$ vanishes identically on $\overline{\mathcal{M}}$, then J defines an integrable complex structure, and $\overline{\mathcal{M}}$ becomes a complex manifold.

A Riemannian metric g on $\overline{\mathcal{M}}$ is called a Hermitian metric if it satisfies

$$\begin{array}{c}\hfill g(U,V)=g(JU,JV),\forall U,V\in T\overline{\mathcal{M}}.\end{array}$$

(1)

An almost complex manifold equipped with a Hermitian metric is referred to as an almost Hermitian manifold and is denoted by $(\overline{\mathcal{M}},g,J)$. Let $\overline{\nabla }$ be the Levi–Civita connection on $\overline{\mathcal{M}}$ corresponding to g. The manifold $\overline{\mathcal{M}}$ is said to be a Kähler manifold if the almost complex structure J is parallel with respect to $\overline{\nabla }$; that is,

$$\begin{array}{c}\hfill \left({\overline{\nabla }}_{U}J\right)=0,\forall U\in T\overline{\mathcal{M}}.\end{array}$$

(2)

Let $\widetilde{\overline{\mathcal{M}}}$ be a Kähler product manifold. Then, there exists an almost product structure F on $\widetilde{\overline{\mathcal{M}}}$ satisfying $F^2=I$ and

$$g(FU,V)=g(FV,U),$$

for all vector fields $U,V$ on $\widetilde{\overline{\mathcal{M}}}$.

Suppose that $\mathcal{M}$ is an invariant submanifold of the Kähler product manifold $\tilde{\mathcal{M}}={\tilde{\mathcal{M}}}^m\times {\tilde{\mathcal{M}}}^n$. Then, $\mathcal{M}$ admits a product structure of the form $\mathcal{M}={\mathcal{M}}_1\times {\mathcal{M}}_2$, where ${\mathcal{M}}_1$ is a submanifold of ${\tilde{\mathcal{M}}}^m$ and ${\mathcal{M}}_2$ is a submanifold of ${\tilde{\mathcal{M}}}^n$. According to [8], the following decompositions hold:

$$JU=J_{m}U=F_{1}U+{\omega }_{1}U,J{N}_1=J_m{N}_1=B_1{N}_1+C_1{N}_1,$$

where $F_{1}U,B_1{N}_1\in \Gamma \left(T{\mathcal{M}}_1\right)$ and ${\omega }_{1}U,C_1{N}_1\in \Gamma \left(T{\mathcal{M}}_1^{\perp }\right)$. Similarly, for any $V\in \Gamma \left(T{\mathcal{M}}_2\right)$ and $N_2\in \Gamma \left(T{\mathcal{M}}_2^{\perp }\right)$, we have

$$JV=J_{n}V=F_{2}V+{\omega }_{2}V,J{N}_2=J_n{N}_2=B_2{N}_2+C_2{N}_2.$$

Since $\mathcal{M}$ is a Riemannian product manifold, the tangent bundles of its components satisfy

$$\begin{array}{cc}\hfill T{\mathcal{M}}_1& =\{U\in \Gamma (T\mathcal{M})\mid FU=U\},\hfill \\ \hfill T{\mathcal{M}}_2& =\{U\in \Gamma (T\mathcal{M})\mid FU=-U\}.\hfill \end{array}$$

We now recall the notions of slant and bi-slant submanifolds.

Definition 1. 

Let $\mathcal{M}$ be a submanifold of an almost Hermitian manifold $\overline{\mathcal{M}}$. The submanifold $\mathcal{M}$ is said to be a slant submanifold if, for every point $p\in \mathcal{M}$ and every nonzero tangent vector $U\in T_p\mathcal{M}$, the angle formed between the vector $JU$ and the tangent space $T_p\mathcal{M}$ remains constant. This constant angle, denoted by $\theta \in \left[0,\frac{\pi }{2}\right]$, is referred to as the slant angle of $\mathcal{M}$ [4].

Definition 2. 

A slant submanifold $\mathcal{M}$ of an almost Hermitian manifold $\overline{\mathcal{M}}$ is called an invariant submanifold if its slant angle $\theta =0$. This is equivalent to $J\left(T_p\mathcal{M}\right)\subset T_p\mathcal{M}$ for every point $p\in \mathcal{M}$, meaning the almost complex structure J leaves the tangent space of $\mathcal{M}$ invariant.

Definition 3. 

A slant submanifold $\mathcal{M}$ of an almost Hermitian manifold $\overline{\mathcal{M}}$ is called an invariant submanifold if its slant angle $\theta =\frac{\pi }{2}$. This is equivalent to $J\left(T_p\mathcal{M}\right)\subset T_p^{\perp }\mathcal{M}$ for every point $p\in \mathcal{M}$, meaning the almost complex structure J sends tangent vectors into the normal space.

Definition 4. 

A submanifold $\mathcal{M}$ of an almost Hermitian manifold $\overline{\mathcal{M}}$ is called a bi-slant submanifold if its tangent bundle admits an orthogonal decomposition $T\mathcal{M}=D_1\oplus D_2$, where each distribution $D_i$ is a slant distribution with an associated slant angle ${\theta }_i$, for $i=1,2$ [6].

It is known that the notion of a bi-slant submanifold provides a unified framework that naturally encompasses several well-studied classes of submanifolds in almost Hermitian geometry. By imposing suitable conditions on the distributions $D_1$ and $D_2$ and their associated slant angles, one recovers, as particular cases, semi-slant submanifolds, hemi-slant submanifolds, CR-submanifolds, and ordinary slant submanifolds. The relationships among these subclasses are determined by whether the distributions are invariant, anti-invariant, or proper slant, as well as by the values of the corresponding slant angles, and are summarized in

Table 1. Definition.

S.N.	$\overline{\mathcal{M}}$	$\mathcal{M}$	${\mathit{D}}_1$	${\mathit{D}}_2$	${\mathit{\theta }}_1$	${\mathit{\theta }}_2$
(1)	$\overline{\mathcal{M}}$	bi-slant	slant	slant	slant angle	slant angle
(2)	$\overline{\mathcal{M}}$	semi-slant	invariant	slant	0	slant angle
(3)	$\overline{\mathcal{M}}$	hemi-slant	slant	anti-invariant	slant angle	$\frac{\pi }{2}$
(4)	$\overline{\mathcal{M}}$	CR	invariant	anti-invariant	0	$\frac{\pi }{2}$
(5)	$\overline{\mathcal{M}}$	slant	either $D_1=0$ or $D_2=0$		either ${\theta }_1={\theta }_2=\theta$ or ${\theta }_1={\theta }_2\ne \theta$

(see [28]).

When the slant angle satisfies $0<\theta <\frac{\pi }{2}$, the submanifold $\mathcal{M}$ is said to be a proper slant submanifold. In an analogous manner, a bi-slant submanifold is called proper if both of its associated slant angles ${\theta }_1$ and ${\theta }_2$ belong to the interval $\left(0,\frac{\pi }{2}\right)$.

Let $\tilde{\mathcal{M}}$ be an $(m+n)$-dimensional Riemannian manifold equipped with a Riemannian metric g, and denote by $\tilde{\overline{\nabla }}$ the Levi–Civita connection associated with g. According to [30], one may introduce a linear connection $\overline{\nabla }$ on $\overline{\mathcal{M}}$ defined by

$$\begin{array}{c}\hfill {\overline{\nabla }}_{U}V={\tilde{\overline{\nabla }}}_{U}V+{\Lambda }_1\pi \left(V\right)U-{\Lambda }_{2}g(U,V)\mathcal{P},\end{array}$$

(3)

for any vector fields $U,V$ on $\overline{\mathcal{M}}$, where ${\Lambda }_1,{\Lambda }_2\in \mathbb{R}$ are fixed real constants, $\mathcal{P}$ denotes a prescribed vector field, and $\pi$ is the 1-form given by $\pi \left(U\right)=g(U,\mathcal{P})$. The connection $\overline{\nabla }$ is referred to as a QSMC whenever $\overline{\nabla }g=0$ and as a QSNMC when this condition is not satisfied.

The following special cases of (3) are noteworthy:

• If ${\Lambda }_1={\Lambda }_2=1$, then $\overline{\nabla }$ reduces to a SSMC.
• If ${\Lambda }_1=1$ and ${\Lambda }_2=0$, then $\overline{\nabla }$ reduces to a SSNMC.

The curvature tensor associated with $\overline{\nabla }$ is defined by

$$\begin{array}{c}\hfill \overline{\mathcal{R}}(U,V)Z_3={\overline{\nabla }}_U{\overline{\nabla }}_V{Z}_3-{\overline{\nabla }}_V{\overline{\nabla }}_U{Z}_3-{\overline{\nabla }}_{[U,V]}{Z}_3.\end{array}$$

(4)

Using (3), the curvature tensor can be expressed as [30]

$$\begin{array}{ccc}\hfill \overline{\mathcal{R}}(U,V,Z_3,Z_4)& =& \widetilde{\overline{\mathcal{R}}}(U,V,Z_3,Z_4)+{\Lambda }_1\alpha (U,Z_3)g(V,Z_4)-{\Lambda }_1\alpha (V,Z_3)g(U,Z_4)\hfill \\ & & +{\Lambda }_2\alpha (V,Z_4)g(U,Z_3)-{\Lambda }_2\alpha (U,Z_4)g(V,Z_3)\hfill \\ & & +{\Lambda }_2({\Lambda }_1-{\Lambda }_2)\left[g(U,Z_3)\beta (V,Z_4)-g(V,Z_3)\beta (U,Z_4)\right],\hfill \end{array}$$

(5)

where

$$\begin{array}{ccc}\hfill \alpha (U,V)& =& ({\tilde{\overline{\nabla }}}_U\pi )\left(V\right)-{\Lambda }_1\pi \left(U\right)\pi \left(V\right)+{\displaystyle \frac{{\Lambda }_2}{2}}g(U,V)\pi \left(\mathcal{P}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \beta (U,V)& =& {\displaystyle \frac{\pi \left(\mathcal{P}\right)}{2}}g(U,V)+\pi \left(U\right)\pi \left(V\right),\hfill \end{array}$$

for any vector fields $U,V,Z_3,Z_4$ on $\overline{\mathcal{M}}$.

Finally, let $\mathcal{M}$ be a $(q_1+q_2)$-dimensional submanifold of an $(m+n)$-dimensional Kähler product manifold $\tilde{\mathcal{M}}(k_1,k_2)$. We denote by ∇ and $\tilde{\nabla }$ the induced QSMC and induced Levi–Civita connection on $\mathcal{M}$, respectively. Decompose the vector field $\mathcal{P}$ into its tangential and normal components as $\mathcal{P}={\mathcal{P}}^T+{\mathcal{P}}^{\perp }$. Then, the Gauss formulas with respect to ∇ and $\tilde{\nabla }$ are given by

$$\begin{array}{c}\hfill {\overline{\nabla }}_{U}V={\nabla }_{U}V+h(U,V),U,V\in \Gamma \left(T\mathcal{M}\right),\end{array}$$

(6)

$$\begin{array}{c}\hfill {\tilde{\overline{\nabla }}}_{U}V={\tilde{\nabla }}_{U}V+h(U,V),U,V\in \Gamma \left(T\mathcal{M}\right),\end{array}$$

(7)

where $\tilde{h}$ denotes the second fundamental form of $\mathcal{M}$ in $\tilde{\mathcal{M}}$ and

$$h(U,V)=\tilde{h}(U,V)-{\Lambda }_{2}g(U,V){\mathcal{P}}^{\perp }.$$

In ${\tilde{\mathcal{M}}}^{m+n}(k_1,k_2)$, we can choose an orthonormal basis $\{u_1,\dots ,u_k,u_1^{\prime },\dots ,u_l^{\prime }\}$ for $T_p\mathcal{M}$ such that $\{u_1,\dots ,u_k\}$ is tangent to $M_1$ and $\{u_1^{\prime },\dots ,u_l^{\prime }\}$ is tangent to $M_2$.

Let $\tilde{\mathcal{M}}(k_1,k_2)$ be an $(m+n)$-dimensional Kähler product manifold endowed with a QSMC satisfying (3). With respect to the Levi–Civita connection $\tilde{\overline{\nabla }}$, the curvature tensor $\tilde{\overline{R}}$ of the Kähler product manifold $\tilde{\mathcal{M}}(k_1,k_2)$ is given by [8].

$$\begin{array}{ccc}\hfill g(\widetilde{\overline{\mathcal{R}}}(U,V)Z_3,Z_4)& =& {\displaystyle \frac{1}{16}}(k_1+k_2)[g(V,Z_3)g(U,Z_4)-g(U,Z_3)g(V,Z_4)\hfill \\ & & +g(JV,Z_3)g(JU,Z_4)-g(JU,Z_3)g(JV,Z_4+2g(U,JV)g(J{Z}_3,Z_4)\hfill \\ & & +2g(FV,Z_3)g(FU,Z_4)-g(FU,Z_3)g(FV,Z_4)\hfill \\ & & +g(JV,F{Z}_3)g(JU,F{Z}_4)-g(JU,F{Z}_3)g(JV,F{Z}_4)\hfill \\ & & +2g(FU,JV)g(J{Z}_3,F{Z}_4)]+\frac{1}{16}(k_1-k_2)[g(FV,Z_3)g(U,Z_4)\hfill \\ & & -g(FU,Z_3)g(V,Z_4)+g(V,Z_3)g(FU,Z_4)-g(U,Z_3)g(FV,Z_4)\hfill \\ & & +g(JV,F{Z}_3)g(JU,Z_4)-g(JU,F{Z}_3)g(JV,Z_4)\hfill \\ & & +g(JV,Z_3)g(JU,F{Z}_4)-g(JU,Z_3)g(JV,F{Z}_4)\hfill \\ & & +2g(FU,JV)g(J{Z}_3,Z_4)-2g(U,JV)g(F{Z}_3,J{Z}_4)].\hfill \end{array}$$

(8)

By (3) and (8), we have

$$\begin{array}{ccc}\hfill g(\overline{\mathcal{R}}(U,V)Z_3,Z_4)& =& \frac{1}{16}(k_1+k_2)[g(V,Z_3)g(U,Z_4)-g(U,Z_3)g(V,Z_4)\hfill \\ & & +g(JV,Z_3)g(JU,Z_4)-g(JU,Z_3)g(JV,Z_4+2g(U,JV)g(J{Z}_3,Z_4)\hfill \\ & & +2g(FV,Z_3)g(FU,Z_4)-g(FU,Z_3)g(FV,Z_4)\hfill \\ & & +g(JV,F{Z}_3)g(JU,F{Z}_4)-g(JU,F{Z}_3)g(JV,F{Z}_4)\hfill \\ & & +2g(FU,JV)g(J{Z}_3,F{Z}_4)]+\frac{1}{16}(k_1-k_2)[g(FV,Z_3)g(U,Z_4)\hfill \\ & & -g(FU,Z_3)g(V,Z_4)+g(V,Z_3)g(FU,Z_4)-g(U,Z_3)g(FV,Z_4)\hfill \\ & & +g(JV,F{Z}_3)g(JU,Z_4)-g(JU,F{Z}_3)g(JV,Z_4)\hfill \\ & & +g(JV,Z_3)g(JU,F{Z}_4)-g(JU,Z_3)g(JV,F{Z}_4)\hfill \\ & & +2g(FU,JV)g(J{Z}_3,Z_4)-2g(U,JV)g(F{Z}_3,J{Z}_4)]\hfill \\ & & +{\Lambda }_1\alpha (U,Z_3)g(V,Z_4)-{\Lambda }_1\alpha (V,Z_3)g(U,Z_4)\hfill \\ & & +{\Lambda }_{2}g(U,Z_3)\alpha (V,Z_4)-{\Lambda }_{2}g(V,Z_3)\alpha (U,Z_4)\hfill \\ & & +{\Lambda }_2({\Lambda }_1-{\Lambda }_2)g(U,Z_3)\beta (V,Z_4)\hfill \\ & & -{\Lambda }_2({\Lambda }_1-{\Lambda }_2)g(V,Z_3)\beta (U,Z_4).\hfill \end{array}$$

(9)

Similarly to [30], we have the Gauss equation

$$\begin{array}{ccc}\hfill g(\overline{\mathcal{R}}(U,V)Z_3,Z_4)& =& g(\mathcal{R}(U,V)Z_3,Z_4)-g(h(U,Z_4),h(V,Z_3))+g(h(V,Z_4),h(U,Z_3))\hfill \\ & & +({\Lambda }_1-{\Lambda }_2)g(h(V,Z_3),{\mathcal{P}}^{\perp })g(U,Z_4)\hfill \\ & & +({\Lambda }_2-{\Lambda }_1)g(h(U,Z_3),{\mathcal{P}}^{\perp })g(V,Z_4).\hfill \end{array}$$

(10)

3. Casorati Curvatures

In this section, we recall the curvature quantities required for the study of submanifolds of Kähler product manifolds. Let $\mathcal{M}$ be a $(q_1+q_2)$-dimensional submanifold isometrically immersed in an $(m+n)$-dimensional Kähler product manifold $\overline{\mathcal{M}}(k_1,k_2)$. We denote by $\{u_1,\dots ,u_{q_1+q_2}\}$ a local orthonormal frame of the tangent bundle $T\mathcal{M}$ and by $\{u_{q_1+q_2+1},\dots ,u_{m+n}\}$ a local orthonormal frame of the normal bundle $T^{\perp }\mathcal{M}$.

For a point $p\in \mathcal{M}$, the scalar curvature $\tau$ of $\mathcal{M}$ is expressed as

$$\begin{array}{c}\hfill \tau =\sum _{1\le i<j\le q_1+q_2}R(u_i,u_j,u_j,u_i),\end{array}$$

(11)

where R denotes the Riemannian curvature tensor of $\mathcal{M}$. The corresponding normalized scalar curvature $\rho$ is defined by

$$\begin{array}{c}\hfill \rho ={\displaystyle \frac{2\tau }{(q_1+q_2)(q_1+q_2-1)}}.\end{array}$$

(12)

The mean curvature vector field $\mathcal{H}$ of $\mathcal{M}$ is given by

$$\begin{array}{c}\hfill \mathcal{H}={\displaystyle \frac{1}{q_1+q_2}}\sum _{i=1}^{q_1+q_2}h(u_i,u_i),\end{array}$$

where h denotes the second fundamental form of the immersion.

To simplify notation, the components of the second fundamental form are defined as

$$\begin{array}{c}\hfill h_{ij}^r=g\left(h(u_i,u_j),u_r\right),\end{array}$$

with $1\le i,j\le q_1+q_2$ and $q_1+q_2+1\le r\le m+n$. Using this notation, the squared norm of the mean curvature vector can be written as

$$\begin{array}{c}\hfill {\parallel \mathcal{H}\parallel }^2={\displaystyle \frac{1}{{(q_1+q_2)}^2}}\sum _{r=q_1+q_2+1}^{m+n}{\left(\sum _{i=1}^{q_1+q_2}{h}_{ii}^r\right)}^2.\end{array}$$

(13)

The squared norm of the second fundamental form h is given by

$$\begin{array}{c}\hfill {\parallel h\parallel }^2={\displaystyle \frac{1}{q_1+q_2}}\sum _{r=q_1+q_2+1}^{m+n}\sum _{i,j=1}^{q_1+q_2}{\left(h_{ij}^r\right)}^2.\end{array}$$

The Casorati curvature of the submanifold $\mathcal{M}$ is then defined by

$$\begin{array}{c}\hfill \mathcal{C}={\displaystyle \frac{1}{q_1+q_2}}{\parallel h\parallel }^2.\end{array}$$

(14)

Let ⋁ be an s-dimensional linear subspace of $T_p\mathcal{M}$ with $s\ge 2$, and let $\{u_1,\dots ,u_s\}$ be an orthonormal basis of ⋁. The scalar curvature associated with the s-plane section ⋁ is given by

$$\begin{array}{c}\hfill \tau =\sum _{1\le i<j\le s}\kappa (u_i\wedge u_j),\end{array}$$

where $\kappa$ denotes the sectional curvature. The Casorati curvature of ⋁ is defined as

$$\begin{array}{c}\hfill \mathcal{C}(\bigvee )={\displaystyle \frac{1}{s}}\sum _{r=q_1+q_2+1}^{m+n}\sum _{i,j=1}^s{\left(h_{ij}^r\right)}^2.\end{array}$$

A point $p\in \mathcal{M}$ is called an invariantly quasi-umbilical point if there exist $m+n-q_1-q_2-1$ mutually orthogonal unit normal vectors ${\xi }_{q_1+q_2+1},\dots ,{\xi }_{m+n}$ such that, for each normal direction ${\xi }_{\alpha }$, the corresponding shape operator admits an eigenvalue of multiplicity $(q_1+q_2-1)$ and the associated eigenvector is common for all ${\xi }_{\alpha }$. If this condition holds at every point of $\mathcal{M}$, then $\mathcal{M}$ is said to be invariantly quasi-umbilical.

The normalized $\delta$-Casorati curvatures (NDCC) ${\delta }_c(q_1+q_2-1)$ and ${\tilde{\delta }}_c(q_1+q_2-1)$ are defined, respectively, by

$$\begin{array}{c}\hfill {\left[{\delta }_c(q_1+q_2-1)\right]}_p={\displaystyle \frac{1}{2}}{\mathcal{C}}_p+{\displaystyle \frac{q_1+q_2+1}{2(q_1+q_2)}}inf\{\mathcal{C}(\bigvee )\mid \bigvee \mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_p\mathcal{M}\},\end{array}$$

(15)

and

$$\begin{array}{c}\hfill {\left[{\tilde{\delta }}_c(q_1+q_2-1)\right]}_p={\displaystyle \frac{1}{2}}{\mathcal{C}}_p+{\displaystyle \frac{2(q_1+q_2)-1}{2(q_1+q_2)}}sup\{\mathcal{C}(\bigvee )\mid \bigvee \mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_p\mathcal{M}\}.\end{array}$$

(16)

For a real parameter $t>0$ with $t\ne (q_1+q_2)(q_1+q_2-1)$, we define

$$\begin{array}{c}\hfill b\left(t\right)={\displaystyle \frac{1}{(q_1+q_2)t}}(q_1+q_2-1)(q_1+q_2+t)\left({(q_1+q_2)}^2-(q_1+q_2)-t\right).\end{array}$$

(17)

The GNDCC ${\delta }_c(t;q_1+q_2-1)$ and ${\tilde{\delta }}_c(t;q_1+q_2-1)$ are defined, respectively, by

$$\begin{array}{c}\hfill {\left[{\delta }_c(t;q_1+q_2-1)\right]}_p=t{\mathcal{C}}_p+b\left(t\right)inf\{\mathcal{C}(\bigvee )\mid \bigvee \mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_p\mathcal{M}\},\end{array}$$

for $0<t<{(q_1+q_2)}^2-(q_1+q_2)$, and

$$\begin{array}{c}\hfill {\left[{\tilde{\delta }}_c(t;q_1+q_2-1)\right]}_p=t{\mathcal{C}}_p+b\left(t\right)sup\{\mathcal{C}(\bigvee )\mid \bigvee \mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_p\mathcal{M}\},\end{array}$$

for $t>{(q_1+q_2)}^2-(q_1+q_2)$.

4. Main Results

In this section, we derive sharp inequalities relating the generalized Normalized $\delta$-Casorati Curvature (GNDCC) to the normalized scalar curvature of submanifolds in Kähler product manifolds endowed with a QSMC. The results are established for bi-slant submanifolds in full generality and yield, as particular cases, corresponding inequalities for several important classes of submanifolds. We also characterize the equality cases.

First of all, we prove the following:

Theorem 1. 

Let $\mathcal{M}$ be a $(q_1+q_2)$-dimensional bi-slant submanifold in Kähler product manifold $\tilde{\mathcal{M}}(k_1,k_2)={\tilde{\mathcal{M}}}^m\left(k_1\right)\times {\tilde{\mathcal{M}}}^n\left(k_2\right)$ endowed with a QSMC. Then, we have the following:

(i) 

The GNDCC ${\delta }_c(t;q_1+q_2-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]\hfill \\ & & +{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\}\hfill \\ & & -{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right),\hfill \end{array}$$

(18)

for any real number t such that $0<t<(q_1+q_2)(q_1+q_2-1)$, and the normalized scalar curvature ρ.

(ii) 

The GNDCC $\widetilde{{\delta }_c}(t;q_1+q_2-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\tilde{\delta }}_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]\hfill \\ & & +{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\}\hfill \\ & & -{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right),\hfill \end{array}$$

(19)

for any real number $t>(q_1+q_2)(q_1+q_2-1)$.

Additionally, the equalities are satisfied in (18) and (19) only when $\mathcal{M}$ exhibits invariable quasi-umbilical characteristics with a trivial normal connection in $\tilde{\mathcal{M}}$. This holds true under the condition that, in relation to a suitable orthonormal frame comprised of tangential vectors $\{u_1,\dots ,u_{q_1+q_2}\}$ and a normal orthonormal frame $\{u_{q_1+q_2+1},\dots ,u_{m+n}\}$, the shape operator $\Phi r\equiv \Phi u_r$, $r\in q_1+q_2+1,\dots ,m+n$ takes on the following structure:

$$\begin{array}{c}\hfill {\Phi }_{q_1+q_2+1}=\left(\begin{array}{cccccccc}b& 0& 0& .& .& .& 0& 0\\ 0& b& 0& .& .& .& 0& 0\\ 0& 0& b& .& .& .& 0& 0\\ .& .& .& .& & & .& .\\ .& .& .& & .& & .& .\\ .& .& .& & & .& .& .\\ 0& 0& 0& .& .& .& b& .\\ 0& 0& 0& .& .& .& 0& \frac{(q_1+q_2)(q_1+q_2-1)}{t}b\end{array}\right),{\Phi }_{q_1+q_2+2}=\dots ={\Phi }_{m+n}=0.\end{array}$$

(20)

Proof. 

Assume $\mathcal{M}$ is a submanifold that remains invariant under the action of F within a Kähler product manifold denoted as $\tilde{\mathcal{M}}(k_1,k_2)={\tilde{\mathcal{M}}}^m\left(k_1\right)\times {\tilde{\mathcal{M}}}^n\left(k_2\right)$. By carefully selecting an orthonormal basis, denoted as $\{u_1,\dots ,u_k,u_1^{\prime },\dots ,u_l^{\prime }\}$, from the tangent space $T_p\mathcal{M}$, we can ensure that $\{u_1,\dots ,u_k\}$ represents tangent vectors associated with ${\mathcal{M}}_1$, while $\{u_1^{\prime },\dots ,u_l^{\prime }\}$ corresponds to tangent vectors related to ${\mathcal{M}}_2$. By substituting $U=Z_4=u_i$ and $V=Z_3=u_j$, with the condition $i\ne j$, into Equation (10) and utilizing Equation (9), we obtain

$$\begin{array}{ccc}\hfill \mathcal{R}(u_i,u_j,u_j,u_i)& =& {\displaystyle \frac{1}{16}}(k_1+k_2)[g(u_j,u_j)g(u_i,u_i)-g(u_i,u_j)g(u_j,u_i)\hfill \\ & & +g(J{u}_j,u_j)g(J{u}_i,u_i)-g(J{u}_i,u_j)g(J{u}_j,u_i)\hfill \\ & & +2g(u_i,J{u}_j)g(J{u}_j,u_i)+2g(F{u}_j,u_j)g(F{u}_i,u_i)\hfill \\ & & -g(F{u}_i,u_j)g(F{u}_j,u_i)+g(J{u}_j,F{u}_j)g(J{u}_i,F{u}_i)\hfill \\ & & -g(J{u}_i,F{u}_j)g(J{u}_j,F{u}_i)+2g(F{u}_i,J{u}_j)g(J{u}_j,F{u}_i)]\hfill \\ & & +{\displaystyle \frac{1}{16}}(k_1-k_2)[g(F{u}_j,u_j)g(u_i,u_i)-g(F{u}_i,u_j)g(u_j,u_i)\hfill \\ & & +g(u_j,u_j)g(F{u}_i,u_i)-g(u_i,u_j)g(F{u}_j,u_i)\hfill \\ & & +g(J{u}_j,F{u}_j)g(J{u}_i,u_i)-g(J{u}_i,F{u}_j)g(J{u}_j,u_i)\hfill \\ & & +g(J{u}_j,u_j)g(J{u}_i,F{u}_i)-g(J{u}_i,u_j)g(J{u}_j,F{u}_i)\hfill \\ & & +2g(F{u}_i,J{u}_j)g(J{u}_j,u_i)-2g(u_i,J{u}_j)g(F{u}_j,J{u}_i)]\hfill \\ & & +{\Lambda }_1\alpha (u_i,u_j)g(u_j,u_i)-{\Lambda }_1\alpha (u_j,u_j)g(u_i,u_i)\hfill \\ & & +{\Lambda }_{2}g(u_i,u_j)\alpha (u_j,u_i)-{\Lambda }_{2}g(u_j,u_j)\alpha (u_i,u_i)\hfill \\ & & +{\Lambda }_2({\Lambda }_1-{\Lambda }_2)g(u_i,u_j)\beta (u_j,u_i)\hfill \\ & & -{\Lambda }_2({\Lambda }_1-{\Lambda }_2)g(u_j,u_i)\beta (u_i,u_i)\hfill \\ & & -({\Lambda }_1-{\Lambda }_2)g(h(u_j,u_j),{\mathcal{P}}^{\perp })g(u_i,u_i)\hfill \\ & & -({\Lambda }_2-{\Lambda }_1)g(h(u_i,u_j),{\mathcal{P}}^{\perp })g(u_j,u_i)\hfill \\ & & +g(h(u_i,u_i),h(u_j,u_j))-g(h(u_i,u_j),h(u_i,u_j)).\hfill \end{array}$$

(21)

By summing over the indices i and j in the range $1\le i,j\le q_1+q_2$, we find

$$\begin{array}{ccc}\hfill 2\tau & =& {\displaystyle \frac{k_1}{8}}\left(2q_1^2-2q_1+q_2^2+6\left|\right|P_1{\left|\right|}^2\right)+{\displaystyle \frac{k_1}{8}}\left(2q_2^2-2q_2+q_1^2+6\left|\right|P_2{\left|\right|}^2\right)\hfill \\ & & +{(q_1+q_2)}^2{\left|\right|\mathcal{H}\left|\right|}^2-(q_1+q_2){\left|\right|h\left|\right|}^2-({\Lambda }_1-{\Lambda }_2)(q_1+q_2-1)a\hfill \\ & & -{\Lambda }_2({\Lambda }_1-{\Lambda }_2)(q_1+q_2-1)b+({\Lambda }_2-{\Lambda }_1)(q_1+q_2)(q_1+q_2-1)\pi \left(\mathcal{H}\right),\hfill \end{array}$$

(22)

where

$$\begin{array}{c}\hfill \left|\right|P_1{\left|\right|}^2=\sum _{i,j=1}^k{g}^2(u_i,F_1{u}_j)=q_{1}co{s}^2{\theta }_1,\left|\right|P_2{\left|\right|}^2=\sum _{\alpha ,\beta =1}^l{g}^2(u_{\alpha },F_1{u}_{\beta })=q_{2}co{s}^2{\theta }_2,\end{array}$$

(23)

$\dim D_1=2d_1=q_1$, $\dim D_2=2d_2=q_2$, $tr\left(\alpha \right)=a$ and $tr\left(\beta \right)=b$ and

$$\begin{array}{c}\hfill \pi \left(\mathcal{H}\right)={\displaystyle \frac{1}{q_1+q_2}}\sum _{j=1}^{q_1+q_2}\pi \left(h(u_j,u_j)\right)=g({\mathcal{P}}^{\perp },\mathcal{H}).\end{array}$$

Let us introduce a function denoted as $\Xi$, which is defined as a quadratic polynomial involving the components of the second fundamental form

$$\begin{array}{ccc}\hfill \Xi & =& t\mathcal{C}+b\left(t\right)(\bigvee )-2\tau +{\displaystyle \frac{k_1}{8}}\left(2q_1^2-2q_1+q_2^2+6\left|\right|P_1{\left|\right|}^2\right)+{\displaystyle \frac{k_1}{8}}(2q_2^2-2q_2\hfill \\ & & +q_1^2+6\left|\right|P_2{\left|\right|}^2)-({\Lambda }_1+{\Lambda }_2)(q_1+q_2-1)a-{\Lambda }_2({\Lambda }_1-{\Lambda }_2)(q_1+q_2-1)b\hfill \\ & & +({\Lambda }_2-{\Lambda }_1)(q_1+q_2)(q_1+q_2-1)\bigvee \left(\mathcal{H}\right).\hfill \end{array}$$

(24)

Without loss of generality, we suppose that ⋁ is spanned by $\{u_1,\dots ,u_{q_1+q_2-1}\}$. Then, (24) gives

$$\begin{array}{ccc}\hfill \Xi & =& \sum _{r=q_1+q_2+1}^{m+n}\sum _{i,j=1}^{q_1+q_2-1}\left[\left({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{b\left(t\right)}{q_1+q_2-1}}-1\right){\left(h_{ii}^r\right)}^2+2{\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}{\left(h_{i{q}_1+q_2}^r\right)}^2\right]\hfill \\ & & +\sum _{r=q_1+q_2+1}^{m+n}[2({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{b\left(t\right)}{q_1+q_2-1}})\sum _{i<j}^{q_1+q_2-1}{\left(h_{ij}^r\right)}^2-2\sum _{i<j}^{q_1+q_2}{h}_{ii}^r{h}_{jj}^r\hfill \\ & & +{\displaystyle \frac{t}{q_1+q_2}}{\left(h_{q_1+q_2{q}_1+q_2}^r\right)}^2].\hfill \end{array}$$

(25)

From (25), we can see the critical points

$$\begin{array}{c}\hfill h^c=\left\{h_{11}^{q_1+q_2+1},h_{12}^{q_1+q_2+1},\dots ,h_{q_1+q_2{q}_1+q_2}^{q_1+q_2+1},\dots ,h_{11}^{m+n},\dots ,h_{q_1+q_2{q}_1+q_2}^{m+n}\right\}\end{array}$$

of $\Xi$ are the solutions of the followings system of homogeneous equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\frac{\partial \Xi }{\partial h_{ii}}=2(\frac{q_1+q_2+t}{q_1+q_2}+\frac{b\left(t\right)}{q_1+q_2-1}-1)h_{ii}^r-2\sum _{q=1}^{q_1+q_2}{h}_{qq}^r=0\hfill \\ \\ \frac{\partial \Xi }{\partial h_{q_1+q_2{q}_1+q_2}}=\frac{2t}{q_1+q_2}{h}_{q_1+q_2{q}_1+q_2}^r-2\sum _{q=1}^{q_1+q_2-1}{h}_{qq}^r=0\hfill \\ \\ \frac{\partial \Xi }{\partial h_{ij}}=4(\frac{q_1+q_2+t}{q_1+q_2}+\frac{b\left(t\right)}{q_1+q_2-1})h_{ij}^r\hfill \\ \\ \frac{\partial \Xi }{\partial h_{i{q}_1+q_2}}=4(\frac{q_1+q_2+t}{q_1+q_2})h_{i{q}_1+q_2}^r,\hfill \end{array}\right.\end{array}$$

(26)

where $i,j=\{1,2,\dots ,q_1+q_2-1\}$, $i\ne j$ and $r\in \{q_1+q_2+1,\dots ,m+n\}$.

As a result, for every solution $h^c$ and $h_{ij}^r=0$ when $i\ne j$, the determinant associated with the first two equations of the aforementioned system becomes zero. Furthermore, the Hessian matrix of function $\Xi$ exhibits the following structure:

$$\begin{array}{c}\hfill \mathbb{H}(\Xi )=\left(\begin{array}{ccc}{\Pi }_1& O& O\\ O& {\Pi }_2& O\\ O& O& {\Pi }_3\end{array}\right),\end{array}$$

where

$$\begin{array}{c}\hfill {\Pi }_1=\left(\begin{array}{ccccccc}\Omega -2& -2& .& .& .& -2& -2\\ -2& \Omega -2& .& .& .& -2& -2\\ .& .& .& & & .& .\\ .& .& & .& & .& .\\ .& .& & & .& .& .\\ -2& -2& .& .& .& \Omega -2& -2\\ -2& -2& .& .& .& -2& \frac{2t}{q_1+q_2}\end{array}\right),\end{array}$$

(27)

$\Omega =2(\frac{q_1+q_2+t}{q_1+q_2}+\frac{b\left(t\right)}{q_1+q_2-1}-1)$, O is the null matrix of the respective dimensions and ${\Pi }_2$ and ${\Pi }_3$ are the next diagonal matrices

$$\begin{array}{ccc}\hfill {\Pi }_2& =& diag[4({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{b\left(t\right)}{q_1+q_2-1}}-1),4({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{b\left(t\right)}{q_1+q_2-1}}-1),\dots \hfill \\ & & ,4({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{b\left(t\right)}{q_1+q_2-1}}-1)].\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\Pi }_3& =& diag\left[4{\displaystyle \frac{(q_1+q_2+t)}{q_1+q_2}},4{\displaystyle \frac{(q_1+q_2+t)}{q_1+q_2}},\dots ,4{\displaystyle \frac{(q_1+q_2+t)}{q_1+q_2}}\right].\hfill \end{array}$$

Consequently, it follows that the eigenvalues of $\mathbb{H}(\Xi )$ are as follows:

${\lambda }_{11}=0$, ${\lambda }_{22}=2(\frac{2t}{q_1+q_2}+\frac{b\left(t\right)}{q_1+q_2-1})$, ${\lambda }_{33}=,\dots ={\lambda }_{q_1+q_2{q}_1+q_2}=2(\frac{2t}{q_1+q_2}+\frac{b\left(t\right)}{q_1+q_2-1})$,

${\lambda }_{ij}=4(\frac{q_1+q_2+t}{q_1+q_2}+\frac{b\left(t\right)}{q_1+q_2-1})$, ${\lambda }_{i{q}_1+q_2}=4(\frac{q_1+q_2+t}{q_1+q_2})$, $\forall i,j\in \{1,2,\dots ,q_1+q_2-1\}$, $i\ne j$.

Therefore, we can conclude that $\Xi$ is a parabolic function and attains its minimum value at $\Xi \left(h^c\right)=0$ for a certain solution $h^c$ of the system (26). As a result, it follows that $\Xi \ge 0$, thereby satisfying the inequality

$$\begin{array}{ccc}\hfill 2\tau & \le & t\mathcal{C}+b\left(t\right)\mathcal{C}(\bigvee )+{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2+6q_{1}co{s}^2{\theta }_1\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2+6q_{2}co{s}^2{\theta }_2\right]\hfill \\ & & -({\Lambda }_1+{\Lambda }_2)(q_1+q_2-1)a-{\Lambda }_2({\Lambda }_1-{\Lambda }_2)(q_1+q_2-1)b\hfill \\ & & +({\Lambda }_2-{\Lambda }_1)(q_1+q_2)(q_1+q_2-1)\pi \left(\mathcal{H}\right),\hfill \end{array}$$

whereby we obtain

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\mathcal{C}+{\displaystyle \frac{b\left(t\right)}{(q_1+q_2)(q_1+q_2-1)}}\mathcal{C}(\bigvee )+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\hfill \\ & & \left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2+6q_{1}co{s}^2{\theta }_1\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2+6q_{2}co{s}^2{\theta }_2\right]\right\}\hfill \\ & & -{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right),\hfill \end{array}$$

for any hyperplane L within $\mathcal{M}$, the statement holds true. When we consider the infimum over all tangent hyperplanes L, the outcome follows immediately and without difficulty, i.e.,

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\delta }_c(t,q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2+6q_{1}co{s}^2{\theta }_1\right]\hfill \\ & & +{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2+6q_{2}co{s}^2{\theta }_2\right]\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}\hfill \\ & & -({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)\hfill \\ & =& {\displaystyle \frac{{\delta }_c(t,q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]\hfill \\ & & +{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}\hfill \\ & & -{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right).\hfill \end{array}$$

Furthermore, the equality holds if and only if

$$\begin{array}{c}\hfill h_{ij}^r=0,\forall i,j\in \{1,\dots ,q_1+q_2\},i\ne jandr\in \{q_1+q_2+1,\dots ,m+n\},\end{array}$$

(28)

and

$$\begin{array}{c}\hfill h_{q_1+q_2{q}_1+q_2}^r={\displaystyle \frac{(q_1+q_2)(q_1+q_2-1)}{t}}{h}_{11}^r=\dots ={\displaystyle \frac{(q_1+q_2)(q_1+q_2-1)}{t}}{h}_{q_1+q_2-1q_1+q_2-1}^r,\end{array}$$

(29)

$\forall r\in \{q_1+q_2+1,\dots ,m+n\}$.

By examining (28) and (29), we can deduce that the equality is valid exclusively when the submanifold possesses invariantly quasi-umbilical properties alongside a trivial normal connection within $\overline{\mathcal{M}}$. This condition necessitates that the shape operator, in relation to orthonormal frames for both tangential and normal vectors, adopts the specific structure outlined in (20).

Similarly, one can establish the proof for (ii). □

Next, we give sharp inequalities involving the NDCC for bi-slant submanifold in Kähler product manifold $\tilde{\mathcal{M}}(k_1,k_2)$ with QSMC.

Corollary 1. 

Let $\mathcal{M}$ be a $(q_1+q_2)$-dimensional bi-slant submanifold in Kähler product manifold $\tilde{\mathcal{M}}(k_1,k_2)={\tilde{\mathcal{M}}}^m\left(k_1\right)\times {\tilde{\mathcal{M}}}^n\left(k_2\right)$ with QSMC. Then, we have

(i) 

The NDCC ${\delta }_c(q_1+q_2-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\delta }_c(q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]\hfill \\ & & +{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\}\hfill \\ & & -{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right).\hfill \end{array}$$

(30)

(ii) 

The NDCC ${\tilde{\delta }}_c(q_1+q_2-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\overline{\delta }}_c(q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]\hfill \\ & & +{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\}\hfill \\ & & -{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right).\hfill \end{array}$$

(31)

Additionally, the equalities are satisfied in (30) and (31) only when $\mathcal{M}$ exhibits invariable quasi-umbilical characteristics with a trivial normal connection in $\tilde{\mathcal{M}}$. This holds true under the condition that, in relation to a suitable orthonormal frame comprised of tangential vectors $\{u_1,\dots ,u_{q_1+q_2}\}$ and a normal orthonormal frame $\{u_{q_1+q_2+1},\dots ,u_{m+n}\}$, the shape operator takes a specific form.

For the equality case of (i), the shape operators are given by:

$$\begin{array}{c}\hfill {\Phi }_{q_1+q_2+1}=\left(\begin{array}{cccccccc}b& 0& 0& .& .& .& 0& 0\\ 0& b& 0& .& .& .& 0& 0\\ 0& 0& b& .& .& .& 0& 0\\ .& .& .& .& & & .& .\\ .& .& .& & .& & .& .\\ .& .& .& & & .& .& .\\ 0& 0& 0& .& .& .& b& 0\\ 0& 0& 0& .& .& .& 0& 2b\end{array}\right){\Phi }_{q_1+q_2+2}=\dots ={\Phi }_{m+n}=0.\end{array}$$

(32)

For the equality case of (ii), the shape operators are given by:

$$\begin{array}{c}\hfill {\Phi }_{q_1+q_2+1}=\left(\begin{array}{cccccccc}2b& 0& 0& .& .& .& 0& 0\\ 0& 2b& 0& .& .& .& 0& 0\\ 0& 0& 2b& .& .& .& 0& 0\\ .& .& .& .& & & .& .\\ .& .& .& & .& & .& .\\ .& .& .& & & .& .& .\\ 0& 0& 0& .& .& .& 2b& 0\\ 0& 0& 0& .& .& .& 0& b\end{array}\right){\Phi }_{q_1+q_2+2}=\dots ={\Phi }_{m+n}=0.\end{array}$$

(33)

Proof. (i) A routine verification shows that

$$\begin{array}{c}\hfill {\left[{\delta }_c({\displaystyle \frac{(q_1+q_2)(q_1+q_2-1)}{2}}:q_1+q_2-1)\right]}_p=(q_1+q_2)(q_1+q_2-1)\left[{\delta }_c(q_1+q_2-1)\right],\end{array}$$

(34)

at any point $p\in \mathcal{M}$. Therefore, substituting $t=\frac{(q_1+q_2)(q_1+q_2-1)}{2}$ in (18), the proof is completed by invoking (34).

Using an argument analogous to that above, we obtain (ii). □

Moreover, we have the following generalized normalized inequalities, which are the particular cases of bi-slant submanifolds from the above theorem, as follows:

Corollary 2. 

Let $\mathcal{M}$ be a $(q_1+q_2)$-dimensional bi-slant submanifold of a Kähler product manifold $\tilde{\mathcal{M}}(k_1,k_2)={\tilde{\mathcal{M}}}^m\left(k_1\right)\times {\tilde{\mathcal{M}}}^n\left(k_2\right)$ with a QSMC. Then, we have the following table for GNDCC (

Table 2. GNDCC.

S.N.	$\overline{\mathcal{M}}$	$\mathcal{M}$	Inequality
(1)	$\overline{\mathcal{M}}$	semi-slant	$\rho \le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\displaystyle \frac{{\overline{\delta }}_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$
(2)	$\overline{\mathcal{M}}$	hemi-slant	$\rho \le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\displaystyle \frac{{\overline{\delta }}_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$
(3)	$\overline{\mathcal{M}}$	CR	$\rho \le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_1\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\displaystyle \frac{{\overline{\delta }}_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_1\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$
(4)	$\overline{\mathcal{M}}$	slant	$\rho \le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{4}}\left(k_1{q}_1+k_2{q}_2\right)co{s}^2\theta \right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\displaystyle \frac{{\overline{\delta }}_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{4}}\left(k_1{q}_1+k_2{q}_2\right)co{s}^2\theta \right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$
(5)	$\overline{\mathcal{M}}$	Invariant	$\rho \le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}[2q_1^2-2q_1+q_2^2]+{\displaystyle \frac{k_2}{8}}[2q_2^2-2q_2+q_1^2]+{\displaystyle \frac{3}{4}}(k_1{q}_1+k_2{q}_2)-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\displaystyle \frac{{\overline{\delta }}_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}[2q_1^2-2q_1+q_2^2]+{\displaystyle \frac{k_2}{8}}[2q_2^2-2q_2+q_1^2]+{\displaystyle \frac{3}{4}}(k_1{q}_1+k_2{q}_2)-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$
(6)	$\overline{\mathcal{M}}$	Anti-Invariant	$\rho \le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}[2q_2^2-2q_2+q_1^2]\right\}{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\displaystyle \frac{{\overline{\delta }}_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}[2q_1^2-2q_2+q_1^2]\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$

).

Additionally, the equalities are satisfied in the given inequalities only when $\mathcal{M}$ exhibits invariable quasi-umbilical characteristics with a trivial normal connection in $\tilde{\mathcal{M}}$. This holds true under the condition that, in relation to a suitable orthonormal frame comprised of tangential vectors $\{u_1,\dots ,u_{q_1+q_2}\}$ and a normal orthonormal frame $\{u_{q_1+q_2+1},\dots ,u_{m+n}\}$, the shape operator $\Phi r\equiv \Phi u_r$, $r\in \{q_1+q_2+1,\dots ,m+n\}$ takes the form in (20).

Proof. 

The initial four outcomes presented in Corollary 2 can be readily derived by utilizing the information provided in Table 1, in conjunction with the findings of Theorem 1. Furthermore, the subsequent two results of Corollary 2 can be observed by substituting $\theta =0$ and $\theta =\frac{\pi }{2}$ for cases involving invariant and anti-invariant submanifolds, respectively. □

Similarly, we can obtain the NDCC for the particular cases of bi-slant submanifolds.

Corollary 3. 

Let $\mathcal{M}$ be a $(q_1+q_2)$-dimensional bi-slant submanifold of a Kähler product manifold $\tilde{\mathcal{M}}(k_1,k_2)={\tilde{\mathcal{M}}}^m\left(k_1\right)\times {\tilde{\mathcal{M}}}^n\left(k_2\right)$ with a QSMC. Then, we have the following table for NDCC (

Table 3. NDCC.

S.N.	$\overline{\mathcal{M}}$	$\mathcal{M}$	Inequality
(1)	$\overline{\mathcal{M}}$	semi-slant	$\rho \le {\delta }_c(t;q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\overline{\delta }}_c(t;q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$
(2)	$\overline{\mathcal{M}}$	hemi-slant	$\rho \le {\delta }_c(t;q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\overline{\delta }}_c(t;q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$
(3)	$\overline{\mathcal{M}}$	CR	$\rho \le {\delta }_c(t;q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_1\right)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\overline{\delta }}_c(t;q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_1\right)+({\Lambda }_1\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$
(4)	$\overline{\mathcal{M}}$	slant	$\rho \le {\delta }_c(t;q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{4}}\left(k_1{q}_1+k_2{q}_2\right)co{s}^2\theta \right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\overline{\delta }}_c(t;q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{4}}\left(k_1{q}_1+k_2{q}_2\right)co{s}^2\theta \right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$
(5)	$\overline{\mathcal{M}}$	Invariant	$\rho \le {\delta }_c(t;q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}[2q_1^2-2q_1+q_2^2]+{\displaystyle \frac{k_2}{8}}[2q_2^2-2q_2+q_1^2]+{\displaystyle \frac{3}{4}}(k_1{q}_1+k_2{q}_2)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\overline{\delta }}_c(t;q_1+q_2-1)+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}[2q_1^2-2q_1+q_2^2]+{\displaystyle \frac{k_2}{8}}[2q_2^2-2q_2+q_1^2]+{\displaystyle \frac{3}{4}}(k_1{q}_1+k_2{q}_2)\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$
(6)	$\overline{\mathcal{M}}$	Anti-Invariant	$\rho \le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}[2q_2^2-2q_2+q_1^2]\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$ $\rho \le {\displaystyle \frac{{\overline{\delta }}_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\left\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]+{\displaystyle \frac{k_2}{8}}[2q_2^2-2q_2+q_1^2]\right\}-{\displaystyle \frac{({\Lambda }_1+{\Lambda }_2)a}{(q_1+q_2)}}-{\displaystyle \frac{{\Lambda }_2({\Lambda }_1-{\Lambda }_2)b}{(q_1+q_2)}}-({\Lambda }_1-{\Lambda }_2)\pi \left(\mathcal{H}\right)$

).

Additionally, the equalities are satisfied in the above inequalities only when $\mathcal{M}$ exhibits invariable quasi-umbilical characteristics with a trivial normal connection in $\tilde{\mathcal{M}}$. This holds true under the condition that, in relation to a suitable orthonormal frame comprised of tangential vectors $\{u_1,\dots ,u_{q_1+q_2}\}$ and a normal orthonormal frame $\{u_{q_1+q_2+1},\dots ,u_{m+n}\}$, the shape operator $\Phi r\equiv \Phi u_r$, $r\in \{q_1+q_2+1,\dots ,m+n\}$ takes the form in (32) and (33).

Similarly to the case of Theorem 1, we can prove the following theorems considering SSMC and SSNMC.

Theorem 2. 

Let $\mathcal{M}$ be a $(q_1+q_2)$-dimensional bi-slant submanifold in Kähler product manifold $\tilde{\mathcal{M}}(k_1,k_2)={\tilde{\mathcal{M}}}^m\left(k_1\right)\times {\tilde{\mathcal{M}}}^n\left(k_2\right)$ endowed with a SSMC.Then, we have

(i) 

The GNDCC ${\delta }_c(t;q_1+q_2-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]\hfill \\ & & +{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\}-{\displaystyle \frac{2a}{q_1+q_2}},\hfill \end{array}$$

(35)

for any real number t such that $0<t<(q_1+q_2)(q_1+q_2-1).$

(ii) 

The GNDCC $\widetilde{{\delta }_c}(t;q_1+q_2-1)$

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\tilde{\delta }}_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]\hfill \\ & & +{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\}-{\displaystyle \frac{2a}{q_1+q_2}},\hfill \end{array}$$

(36)

for any real number $t>(q_1+q_2)(q_1+q_2-1)$.

Additionally, the equalities are satisfied in (35) and (36) only when $\mathcal{M}$ exhibits invariable quasi-umbilical characteristics with a trivial normal connection in $\tilde{\mathcal{M}}$. This holds true under the condition that, in relation to a suitable orthonormal frame comprised of tangential vectors $\{u_1,\dots ,u_{q_1+q_2}\}$ and a normal orthonormal frame $\{u_{q_1+q_2+1},\dots ,u_{m+n}\}$, the shape operator $\Phi r\equiv \Phi u_r$, $r\in \{q_1+q_2+1,\dots ,m+n\}$ takes the form in (20).

Proof. 

For a SSMC, the parameters satisfy ${\Lambda }_1={\Lambda }_2=1$. Substituting these values into inequalities (18) and (19) yields the desired result, thereby completing the proof of Theorem 2. □

Theorem 3. 

Let $\mathcal{M}$ be a $(q_1+q_2)$-dimensional bi-slant submanifold in Kähler product manifold $\tilde{\mathcal{M}}(k_1,k_2)={\tilde{\mathcal{M}}}^m\left(k_1\right)\times {\tilde{\mathcal{M}}}^n\left(k_2\right)$ endowed with a SSNMC. Then, we have the following:

(i) 

The GNDCC ${\delta }_c(t;q_1+q_2-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]\hfill \\ & & +{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\}-{\displaystyle \frac{a}{q_1+q_2}}-\pi \left(\mathcal{H}\right),\hfill \end{array}$$

(37)

for any real number t such that $0<t<(q_1+q_2)(q_1+q_2-1).$

(ii) 

The GNDCC $\widetilde{{\delta }_c}(t;q_1+q_2-1)$

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\tilde{\delta }}_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{1}{(q_1+q_2)(q_1+q_2-1)}}\{{\displaystyle \frac{k_1}{8}}\left[2q_1^2-2q_1+q_2^2\right]\hfill \\ & & +{\displaystyle \frac{k_2}{8}}\left[2q_2^2-2q_2+q_1^2\right]+{\displaystyle \frac{3}{2}}\left(k_1{q}_{1}co{s}^2{\theta }_1+k_2{q}_{2}co{s}^2{\theta }_2\right)\}-{\displaystyle \frac{a}{q_1+q_2}}-\pi \left(\mathcal{H}\right),\hfill \end{array}$$

(38)

for any real number, $t>(q_1+q_2)(q_1+q_2-1)$.

Additionally, the equalities are satisfied in (37) and (38) only when $\mathcal{M}$ exhibits invariable quasi-umbilical characteristics with a trivial normal connection in $\tilde{\mathcal{M}}$. This holds true under the condition that, in relation to a suitable orthonormal frame comprised of tangential vectors $\{u_1,\dots ,u_{q_1+q_2}\}$ and a normal orthonormal frame $\{u_{q_1+q_2+1},\dots ,u_{m+n}\}$, the shape operator $\Phi r\equiv \Phi u_r$, $r\in \{q_1+q_2+1,\dots ,m+n\}$ takes the form in (20).

Proof. 

Since, for a SSNMC, we have ${\Lambda }_1=1$ and ${\Lambda }_2=0$, substituting these values into the general inequalities (37) and (38), the stated result follows immediately, which completes the proof of Theorem 3. □

Remark 1. 

Using the same arguments as in the proof of Theorem 1, analogous results can be obtained for Theorems 2 and 3.

5. Conclusions

In this paper, we have investigated the geometry of submanifolds in Kähler product manifolds endowed with a QSMC, with particular emphasis on curvature pinching phenomena involving generalized $\delta$-Casorati invariants. By combining techniques from Kähler geometry, product manifolds, and submanifold theory under non-Levi–Civita connections, we derived a series of sharp inequalities relating the GNDCC to the normalized scalar curvature for several classes of submanifolds.

A key feature of our results is the explicit characterization of the equality cases. We showed that the sharpness of the derived inequalities is achieved precisely under specific geometric conditions, which are expressed in terms of the structure of the second fundamental form and the behavior of the induced distributions. These characterizations provide rigidity-type results and clarify the geometric configurations for which the curvature bounds are optimal.

The present work extends and complements earlier studies on Casorati-type inequalities in Riemannian and quaternionic settings by incorporating both the product structure of the ambient manifold and the effects of a QSMC. In this way, our results contribute to a broader understanding of how non-metric and semi-symmetric connections influence the extrinsic geometry of submanifolds in complex geometric environments.

Several natural directions for future research arise from this study. One may consider extending the obtained inequalities to other classes of submanifolds, such as warped product, slant, or bi-slant submanifolds, within Kähler product manifolds. It would also be of interest to investigate analogous curvature inequalities under more general affine connections or in other ambient geometries, including nearly Kähler or Hermitian manifolds. We hope that the methods and results presented here will stimulate further developments in the study of curvature inequalities and submanifold geometry under generalized connection structures.