Casorati-Type Inequalities for Submanifolds in S-Space Forms with Semi-Symmetric Connection

Abstract

The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized $\delta$-Casorati curvatures ${\delta }_c(t;q_1+q_2-1)$ and ${\widehat{\delta }}_c(t;q_1+q_2-1)$ for bi-slant submanifolds. The cases in which equality holds are thoroughly examined, offering a deeper understanding of the geometric structure underlying such submanifolds. In addition, we present several immediate applications that highlight the relevance of our findings, and we support the article with illustrative examples.

1. Introduction

Submanifold theory has long been a fundamental area of investigation in differential geometry due to its ability to capture the intricate relationships between intrinsic and extrinsic properties of geometric structures [1]. At the core of this theory lies the study of curvature invariants, which encode essential information about the shape, bending, and embedding of submanifolds within ambient Riemannian or semi-Riemannian manifolds. The interplay between intrinsic quantities, such as scalar curvature, and extrinsic measures, such as the mean curvature and the second fundamental form, has given rise to a variety of important geometric inequalities [2,3,4].

One of the most influential contributions in this context is due to B.-Y. Chen, who introduced a set of inequalities, now known as Chen inequalities, that provide bounds on scalar curvature in terms of extrinsic invariants. These inequalities not only serve as sharp analytical tools but also provide characterizations of special types of submanifolds, such as totally geodesic, minimal, or pseudo-umbilical submanifolds [2,3,5]. Over the years, many generalizations and refinements of Chen’s original inequalities have been proposed, which are motivated by applications in geometric analysis [6,7], mathematical physics [8,9], and global Riemannian geometry [10,11,12].

Among these generalizations, the concept of $\delta$-Casorati curvature has emerged as a refined measure of extrinsic geometry that takes into account the squared norms of the second fundamental form components [13,14,15]. Unlike mean curvature, which involves only the trace of the second fundamental form, the Casorati curvature reflects the total distribution of curvature across all normal directions. The introduction of normalized $\delta$-Casorati curvature (NDCC) and its generalized forms has significantly expanded the scope of curvature inequalities [16,17,18], enabling sharper estimates and a deeper geometric understanding of submanifolds under various curvature constraints [19,20,21,22].

On another front, S-manifolds provide a rich class of almost-contact metric manifolds, which include as special cases Sasakian manifolds when the number of structure vector fields is one and Kähler manifolds when this number is zero [23,24,25]. These manifolds are equipped with a $(1,1)$-tensor field $\mathsf{\Phi }$, a Riemannian metric g, and a family of vector fields and associated 1-forms that interact according to a well-defined algebraic structure. The presence of multiple-structure vector fields allows S-manifolds to model more complex geometric behaviors than their Sasakian counterparts, making them particularly suitable for generalizing contact geometry and its applications [26,27].

An important modification in this setting arises when the Levi–Civita connection is replaced by an SSMC. This type of connection incorporates torsion in a structured way while preserving metric compatibility and has been studied in various contexts due to its applicability in both geometry and physics [19,21,28]. The interaction between submanifolds and SSMCs has been shown to yield new geometric phenomena not captured under standard Riemannian assumptions.

In this work, we examine a special class of submanifolds, namely, bi-slant submanifolds, within the setting of S-space forms endowed with SSMCs. A bi-slant submanifold is defined by the property that its tangent bundle admits an orthogonal decomposition into two slant distributions, with each associated with a distinct slant angle [24,29,30]. This generalizes several well-known types of submanifolds, such as invariant, anti-invariant, slant, semi-slant, and CR-submanifolds. Bi-slant geometry has gained attention due to its ability to unify different geometric behaviors within a single framework.

Our primary goal in this article is to derive two sharp geometric inequalities involving generalized normalized $\delta$-Casorati curvatures (GNDCCs) for bi-slant submanifolds in S-space forms with SSMCs. These inequalities not only generalize and sharpen existing results in the literature but also provide new characterizations of equality cases. In particular, we identify the conditions under which equality holds and show that they correspond to totally geodesic submanifolds, forming an important geometric configuration.

To further support our theoretical findings, we explore the special cases where bi-slant submanifolds reduce to semi-slant, hemi-slant, CR-, invariant, and anti-invariant submanifolds, and we demonstrate how our results apply in each setting. Several explicit examples are also constructed to illustrate the importance of this study.

The remainder of this paper is organized as follows: In Section 2, we review fundamental definitions and formulas related to S-manifolds and their submanifolds, and we also provide a concrete example of an S-manifold. Section 3 is devoted to the foundational aspects of Casorati geometry for submanifolds within S-space forms. In Section 4, we establish new optimal inequalities for bi-slant submanifolds in this setting. Section 5 extends the discussion to related results for other classes of submanifolds. Finally, in Section 6, we illustrate several examples to support our article.

2. Preliminaries

The notion of an S-manifold, introduced by D. E. Blair, extends several well-known geometric structures in differential geometry, including Sasakian and Kähler manifolds [23]. An S-manifold is a Riemannian manifold equipped with a specific tensor field and a set of structure vector fields that together define its rich geometric framework.

Let $(\overline{M},g)$ be a $(2r+q_2)$-dimensional Riemannian manifold. It is said to possess an S-structure if there exists a $(1,1)$-type tensor field $\mathsf{\Phi }$ of rank $2r$, along with $q_2$ globally defined vector fields ${\xi }_1,\dots ,{\xi }_{q_2}$, called structure vector fields, and their associated 1-forms ${\mu }^1,\dots ,{\mu }^{q_2}$, such that the following conditions hold:

$$\begin{array}{cc}\hfill \mathsf{\Phi }{\xi }_{\alpha }& =0,{\mu }^{\alpha }\circ \mathsf{\Phi }=0,\hfill \\ \hfill {\mathsf{\Phi }}^2& =-I+\sum _{\alpha =1}^{q_2}{\mu }^{\alpha }\otimes {\xi }_{\alpha },\hfill \\ \hfill g(U,V)& =g(\mathsf{\Phi }U,\mathsf{\Phi }U)+\sum _{\alpha =1}^{q_2}{\mu }^{\alpha }\left(U\right){\mu }^{\alpha }\left(V\right),\hfill \end{array}$$

for all vector fields $U,V\in \mho \left(T\overline{M}\right)$. Additionally, the structure must satisfy the integrability condition

$$[\mathsf{\Phi },\mathsf{\Phi }]+2\sum _{\alpha =1}^{q_2}d{\mu }^{\alpha }\otimes {\xi }_{\alpha }=0,$$

and the non-degeneracy condition

$${\mu }^1\wedge \cdots \wedge {\mu }^{q_2}\wedge {\left(d{\mu }^{\alpha }\right)}^n\ne 0,d{\mu }^{\alpha }=F,\forall \alpha =1,\dots ,q_2,$$

where F stands for the fundamental 2-form corresponding to the structure tensor $\mathsf{\Phi }$, which satisfies $F(U,V)=g(U,\mathsf{\Phi }V)$, and $[\mathsf{\Phi },\mathsf{\Phi }]$ denotes the Nijenhuis tensor of $\mathsf{\Phi }$.

On an S-manifold, the Levi–Civita connection $\tilde{\nabla }$ satisfies the identities below:

$$\begin{array}{cc}\hfill {\tilde{\nabla }}_U{\xi }_{\alpha }& =-\mathsf{\Phi }U,\hfill \\ \hfill \left({\tilde{\nabla }}_U\mathsf{\Phi }\right)\left(V\right)& =\sum _{\alpha =1}^p\left[g(\mathsf{\Phi }U,\mathsf{\Phi }V){\xi }_{\alpha }+{\mu }^{\alpha }\left(V\right){\mathsf{\Phi }}^{2}U\right],\hfill \end{array}$$

for all $U,V\in \mho \left(T\overline{M}\right)$ and for each $\alpha =1,\dots ,q_2$.

Example 1 

(S-Manifolds). Let ${\mathbb{R}}^{2r+s}$ be the Euclidean space with global coordinates

$$(x_1,y_1,\dots ,x_n,y_n,z_1,\dots ,z_s).$$

The structure vector fields ${\xi }_{\alpha }={\displaystyle \frac{\partial }{\partial z_{\alpha }}}$ for $\alpha =1,\dots ,s$, 1-forms ${\mu }^{\alpha }=d{z}_{\alpha }$ for $\alpha =1,\dots ,s$, and tensor field Φ of type $(1,1)$ are defined by

$$\mathsf{\Phi }\left({\displaystyle \frac{\partial }{\partial x_i}}\right)={\displaystyle \frac{\partial }{\partial y_i}},\mathsf{\Phi }\left({\displaystyle \frac{\partial }{\partial y_i}}\right)=-{\displaystyle \frac{\partial }{\partial x_i}},\mathsf{\Phi }\left({\displaystyle \frac{\partial }{\partial z_{\alpha }}}\right)=0,$$

for $i=1,\dots ,n$ and $\alpha =1,\dots ,s$, and the Riemannian metric is

$$g=\sum _{i=1}^n(d{x}_i^2+d{y}_i^2)+\sum _{\alpha =1}^{s}d{z}_{\alpha }^2.$$

It is easy to verify that these satisfy the defining properties of an S-manifold as

$${\mathsf{\Phi }}^2=-I+\sum _{\alpha =1}^s{\mu }^{\alpha }\otimes {\xi }_{\alpha },\mathsf{\Phi }\left({\xi }_{\alpha }\right)=0,{\mu }^{\alpha }\left({\xi }_{\beta }\right)={\delta }_{\beta }^{\alpha },{\mu }^{\alpha }\circ \mathsf{\Phi }=0,$$

$$g(\mathsf{\Phi }U,\mathsf{\Phi }V)=g(U,V)-\sum _{\alpha =1}^s{\mu }^{\alpha }\left(U\right){\mu }^{\alpha }\left(V\right),$$

$$\left({\overline{\nabla }}_U\mathsf{\Phi }\right)\left(V\right)=\sum _{\alpha =1}^s\left\{g(\mathsf{\Phi }U,V){\xi }_{\alpha }-{\mu }^{\alpha }\left(V\right)\mathsf{\Phi }U\right\}.$$

Hence, $({\mathbb{R}}^{2r+s},\mathsf{\Phi },\left\{{\xi }_{\alpha }\right\},\left\{{\mu }^{\alpha }\right\},g)$ is an S-manifold.

The Gauss equation for a submanifold M of the S-manifolds expresses the curvature of M in terms of the ambient curvature and the second fundamental form $\pi$. It is given by

$$\overline{R}(U,V,K,L)=R(U,V,K,L)+g(\pi (U,K),\pi (V,L))-g(\pi (U,L),\pi (V,K)),$$

(1)

where $U,V,K,L$ are tangent vector fields on M, and g denotes the Riemannian metric.

When the ambient manifold $\overline{M}$ is equipped with an SSMC $\widehat{\nabla }$, the associated curvature tensor $\widehat{R}$ is modified accordingly. Its expression is

$$\begin{array}{cc}\hfill \widehat{R}(U,V,K,L)& =\overline{R}(U,V,K,L)-\varpi (V,K)g(U,L)+\varpi (U,K)g(V,L)\hfill \\ \hfill & -\varpi (U,L)g(V,K)+\varpi (V,L)g(U,K),\hfill \end{array}$$

(2)

for any vector fields $U,V,K,L\in \chi \left(M\right)$, where $\varpi$ is a symmetric $(0,2)$-tensor defined by

$$\varpi (U,V)=\left({\overline{\nabla }}_U\mathsf{\Phi }\right)V-\mathsf{\Phi }\left(U\right)\mathsf{\Phi }\left(V\right)+{\displaystyle \frac{1}{2}}\mathsf{\Phi }\left(P\right)g(U,V),$$

(3)

and P is a vector field satisfying $g(P,U)=\mathsf{\Phi }\left(U\right)$ for all $U\in \chi \left(\mathcal{M}\right)$.

A manifold with an S-structure and constant $\mathsf{\Phi }$-sectional curvature c is known as an S-space form, which is denoted by $\overline{M}\left(c\right)$. When M is a submanifold of $\overline{M}\left(c\right)$, its ambient curvature tensor R is expressed as [21,26]

$$\begin{array}{cc}\hfill R(U,& V,K,L)=\sum _{\alpha ,\beta =1}^{q_2}\{g(\mathsf{\Phi }U,\mathsf{\Phi }L){\mu }^{\alpha }\left(V\right){\mu }^{\beta }\left(K\right)-g(\mathsf{\Phi }U,\mathsf{\Phi }K){\mu }^{\alpha }\left(V\right){\mu }^{\beta }\left(L\right)\hfill \\ \hfill & +g(\mathsf{\Phi }V,\mathsf{\Phi }K){\mu }^{\alpha }\left(U\right){\mu }^{\beta }\left(L\right)-g(\mathsf{\Phi }V,\mathsf{\Phi }L){\mu }^{\alpha }\left(U\right){\mu }^{\beta }\left(K\right)\}\hfill \\ \hfill & +{\displaystyle \frac{c+3q_2}{4}}\left\{g(\mathsf{\Phi }U,\mathsf{\Phi }L)g(\mathsf{\Phi }V,\mathsf{\Phi }K)-g(\mathsf{\Phi }U,\mathsf{\Phi }K)g(\mathsf{\Phi }V,\mathsf{\Phi }L)\right\}\hfill \\ \hfill & +{\displaystyle \frac{c-q_2}{4}}\left\{g(U,\mathsf{\Phi }L)g(V,\mathsf{\Phi }K)-g(U,\mathsf{\Phi }K)g(V,\mathsf{\Phi }L)-2g(U,\mathsf{\Phi }V)g(K,\mathsf{\Phi }L)\right\}\hfill \\ \hfill & +g\left(\pi \right(U,L),\pi (V,K\left)\right)-g\left(\pi \right(U,K),\pi (V,L\left)\right)-\varpi (V,K)g(U,L)\hfill \\ \hfill & +\varpi (U,K)g(V,L)-\varpi (U,L)g(V,K)+\varpi (V,L)g(U,K),\hfill \end{array}$$

(4)

where $\pi$ is the second fundamental form of M, and $U,V,K,L\in \mho \left(TN\right)$.

For any $U\in \mho \left(TN\right)$, the action of $\mathsf{\Phi }$ on U can be decomposed into tangential and normal components as

$$\mathsf{\Phi }U=PU+FU,$$

where $PU=\tan \left(\mathsf{\Phi }U\right)$, and $FU=\mathrm{nor}\left(\mathsf{\Phi }U\right)$.

Slant submanifolds were introduced in the literature by B.-Y. Chen in the setting of complex manifolds as a natural extension of both holomorphic and totally real submanifolds [29]. This idea was later adapted to the context of almost-contact metric manifolds by Lotta [30] and further extended to S-manifolds by Carriazo and collaborators [25].

Definition 1 

(Slant submanifold [25]).  Let $\overline{M}$ be an S-manifold and M be a submanifold such that all structure vector fields ${\xi }_1,\dots ,{\xi }_{q_2}$ are tangent to M. The submanifold M is called a slant submanifold if for every point $x\in M$ and every vector $U\in T_{x}M$ orthogonal to ${\xi }_1,\dots ,{\xi }_{q_2}$, the angle θ between $\mathsf{\Phi }U$ and the tangent space $T_{x}M$ is constant. The constant $\theta \in [0,{\displaystyle \frac{\pi }{2}}]$ is known as the slant angle of M.

The concept of bi-slant submanifolds was first introduced by Carriazo et al. [24] in almost-contact metric geometry and can be naturally extended to the framework of S-manifolds that follows.

Definition 2 

(Bi-slant submanifold). Let $\overline{M}$ be an S-manifold and M be a submanifold of $\overline{M}$ such that all structure vector fields ${\xi }_1,\dots ,{\xi }_{q_2}$ lie in $TM$. Denote by $⟨\xi ⟩$ the distribution generated by these structure vector fields. The submanifold M is said to be a bi-slant submanifold if there exist two orthogonal distributions ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ on M such that

(i) 

The tangent bundle splits orthogonally as $TM={\mathcal{D}}_1\oplus {\mathcal{D}}_2\oplus ⟨\xi ⟩$;

(ii) 

Each distribution ${\mathcal{D}}_i$ is a slant distribution with constant slant angle ${\theta }_i$, for $i=1,2$.

It is worth noting that several well-known classes of submanifolds, such as semi-slant, hemi-slant, CR-submanifolds, and slant submanifolds, can be viewed as special cases of bi-slant submanifolds. These particular cases are illustrated in the following Table 1 [20].

Table 1. Definitions of cases.

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

Let $\overline{M}$ be a $(2r+q_2)$-dimensional S-manifold and M be a $(q_1+q_2)$-dimensional bi-slant submanifold in $\overline{M}$. Then, for the orthonormal basis $\{u_1,\dots ,u_{q_1+q_2}\}$ of the tangent space $T_{x}M$, the squared norm of P at $x\in M$ is given by [31,32]

$$\begin{array}{c}\hfill {\parallel P\parallel }^2=\sum _{𝚤,j}^{q_1+q_2}{\mathit{g}}^2(P{u}_{𝚤},u_{𝚥})=2({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2+\parallel P_1{\parallel }^2+\parallel P_2{\parallel }^2),\end{array}$$

(5)

where $2d_1$ and $2d_2$ are dimensions of $D_1$ and $D_2$, respectively, and

$$\parallel P_1{\parallel }^2=\sum _{i=1}^{2d_1}\parallel P_{2}T{e}_i{\parallel }^2,\parallel P_2{\parallel }^2=\sum _{j=1}^{2d_2}{\parallel P_{1}T{e}_j\parallel }^2.$$

3. Casorati Geometry of Submanifolds Within $\mathit{S}$-Space Forms

Let $\overline{M}$ be a $(2r+q_2)$-dimensional S-manifold, and let M be an $(q_1+q_2)$-dimensional Riemannian submanifold of $\overline{M}$. Consider a point $x\in M$, and let $\{u_1,\dots ,u_{q_1+q_2}\}$ be an orthonormal basis of the tangent space $T_{x}M$, while $\{u_{q_1+q_2+1},\dots ,u_{2r+q_2}\}$ forms an orthonormal basis of the normal space $T_x^{\perp }M$.

The scalar curvature $\mathsf{\Lambda }\left(x\right)$ of M at the point x is defined as the sum of sectional curvatures over all independent pairs of tangent vectors as follows:

$$\mathsf{\Lambda }\left(x\right)=\sum _{1\le i<j\le q_1+q_2}K(u_i\wedge u_j),$$

where $K(u_i\wedge u_j)=R(u_i,u_j,u_i,u_j)$ denotes the sectional curvature of the plane spanned by $u_i$ and $u_j$.

The corresponding normalized scalar curvature $\mathsf{\Omega }$ is given by

$$\mathsf{\Omega }={\displaystyle \frac{2\mathsf{\Lambda }}{(q_1+q_2)(q_1+q_2-1)}}.$$

(6)

The vector H, representing the mean curvature of M, is given by

$$H={\displaystyle \frac{1}{q_1+q_2}}\sum _{i=1}^{q_1+q_2}\pi (u_i,u_i),$$

where $\pi$ is the second fundamental form of M. The squared norm of H is then

$${\parallel H\parallel }^2={\displaystyle \frac{1}{{(q_1+q_2)}^2}}\sum _{a=q_1+q_2+1}^{2r+q_2}{\left(\sum _{i=1}^{q_1+q_2}{\pi }_{ii}^a\right)}^2,$$

(7)

with ${\pi }_{ij}^a=g(\pi (u_i,u_j),u_a)$ for $1\le i,j\le q_1+q_2$ and $q_1+q_2+1\le a\le 2r+q_2$.

The Casorati curvature C of the submanifold M at x is defined by [33,34,35]

$$C={\displaystyle \frac{1}{q_1+q_2}}\sum _{a=q_1+q_2+1}^{2r+q_2}\sum _{i,j=1}^{q_1+q_2}{\left({\pi }_{ij}^a\right)}^2.$$

(8)

Now, let ℧ be an r-dimensional subspace of $T_{x}M$ for $r\ge 2$, with orthonormal basis $\{u_1,\dots ,u_r\}$. Then, the scalar curvature and Casorati curvature of ℧ are defined respectively as

$$\mathsf{\Lambda }(\mho )=\sum _{1\le i<j\le r}K(u_i\wedge u_j),C(\mho )={\displaystyle \frac{1}{r}}\sum _{a=q_1+q_2+1}^{2r+q_2}\sum _{i,j=1}^r{\left({\pi }_{ij}^a\right)}^2.$$

The following curvature invariants are known as the NDCCs:

$${\left[{\delta }_c(q_1+q_2-1)\right]}_x={\displaystyle \frac{1}{2}}{C}_x+{\displaystyle \frac{q_1+q_2+1}{2(q_1+q_2)}}inf\{C(\mho )\mid \mho \mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\},$$

$${\left[{\widehat{\delta }}_c(q_1+q_2-1)\right]}_x=2C_x+{\displaystyle \frac{2(q_1+q_2)-1}{2(q_1+q_2)}}sup\{C(\mho )\mid \mho \mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\}.$$

To further generalize these quantities, define a function $f\left(t\right)$ for a positive real number $t\ne (q_1+q_2)(q_1+q_2-1)$ as

$$f\left(t\right)={\displaystyle \frac{(q_1+q_2-1)(q_1+q_2+t)[{(q_1+q_2)}^2-(q_1+q_2)-t]}{(q_1+q_2)t}}.$$

Then, the GNDCCs at x are given by

$$\begin{array}{cc}\hfill & {\left[{\delta }_c(t;q_1+q_2-1)\right]}_x=t{C}_x+f\left(t\right)inf\{C(\mho )\mid \mho \mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\},\hfill \\ \hfill & 0<t<{(q_1+q_2)}^2-(q_1+q_2),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\left[{\widehat{\delta }}_c(t;q_1+q_2-1)\right]}_x=t{C}_x+f\left(t\right)sup\{C(\mho )\mid \mho \mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{x}M\},\hfill \\ \hfill & t>{(q_1+q_2)}^2-(q_1+q_2).\hfill \end{array}$$

These generalized invariants play a central role in curvature inequalities, particularly those relating to the intrinsic and extrinsic geometry of submanifolds in S-space forms.

4. Optimal Inequalities for Bi-Slant Submanifolds

In this section, we derive sharp inequalities that relate the normalized scalar curvature of bi-slant submanifolds in S-space forms to GNDCCs and NDCCs. These bounds capture the interaction between intrinsic and extrinsic geometric quantities and serve as natural extensions of classical Chen-type inequalities. The inequalities established here not only generalize earlier results for special submanifold classes but also provide conditions for equality, offering a deeper understanding of the geometric structure under an SSMC.

We state the following inequalities.

Theorem 1. 

Let M be a bi-slant submanifold of dimension $(q_1+q_2)$ immersed in a $(2r+q_2)$-dimensional S-space form $\overline{M}\left(c\right)$ endowed with an SSMC. Then, the following results hold:

(i)

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

$$\begin{array}{ccc}\hfill \mathsf{\Omega }& \le & {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right)\hfill \\ & +& {\displaystyle \frac{q_1(q_1-1)(c+3q_2)+8q_1{q}_2+3(c-q_2)({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2+\parallel P_1{\parallel }^2+\parallel P_2{\parallel }^2)}{4(q_1+q_2)(q_1+q_2-1)}},\hfill \end{array}$$

(9)

where t is any real parameter lying strictly between 0 and $(q_1+q_2)(q_1+q_2-1)$.

(ii)

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

$$\begin{array}{ccc}\hfill \mathsf{\Omega }& \le & {\displaystyle \frac{\widehat{{\delta }_c}(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right)\hfill \\ & +& {\displaystyle \frac{q_1(q_1-1)(c+3q_2)+8q_1{q}_2+3(c-q_2)({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2+\parallel P_1{\parallel }^2+\parallel P_2{\parallel }^2)}{4(q_1+q_2)(q_1+q_2-1)}},\hfill \end{array}$$

(10)

where t denotes any real number exceeding $(q_1+q_2)(q_1+q_2-1)$.

Equality in the inequalities (9) and (10) characterizes the case where the submanifold M is totally geodesic.

Proof. 

Let $\{u_1,\dots ,u_m,u_{m+1}={\xi }_1,\dots ,u_{q_1+q_2}={\xi }_p\}$ and $\{u_{q_1+q_2+1},\dots ,u_{2r+q_2}\}$ be orthonormal bases of $T_{x}M$ and $T_x^{\perp }M$, respectively, at any point $x\in M$. Then, from (5), (7), (8), and (4) we have

$$\begin{array}{ccc}\hfill 2\mathsf{\Lambda }& =& {(q_1+q_2)}^2{\parallel H\parallel }^2-(q_1+q_2)C+q_1(q_1-1){\displaystyle \frac{c+3q_2}{4}}+2q_1{q}_2\hfill \\ & +& {\displaystyle \frac{3(c-q_2)}{4}}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2+\parallel P_1{\parallel }^2+\parallel P_2{\parallel }^2)-2(q_1+q_2-1)tr\left(\varpi \right).\hfill \end{array}$$

(11)

We introduce a quadratic expression $\mathsf{\Pi }$, formulated using the components of the second fundamental form, as follows:

$$\begin{array}{ccc}\hfill \mathsf{\Pi }& =& tC+f\left(t\right)C(\mho )-2\mathsf{\Lambda }+q_1(q_1-1){\displaystyle \frac{c+3q_2}{4}}+2q_1{q}_2\hfill \\ & +& {\displaystyle \frac{3(c-q_2)}{4}}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2+\parallel P_1{\parallel }^2+\parallel P_2{\parallel }^2)-2(q_1+q_2-1)tr\left(\varpi \right),\hfill \end{array}$$

(12)

where ℧ is a hyperplane of $T_{x}M$.

Assume that the subspace ℧ is defined by the span of $\{u_1,\dots ,u_{q_1+q_2-1}\}$; then, Equation (12) gives

$$\begin{array}{ccc}\hfill \mathsf{\Pi }& =& {\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}\sum _{a=q_1+q_2+1}^{2r+q_2}\sum _{𝚤,𝚥=1}^{q_1+q_2}{\left({\pi }_{𝚤𝚥}^a\right)}^2+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\sum _{a=q_1+q_2+1}^{2r+q_2}\sum _{𝚤,𝚥=1}^{q_1+q_2-1}{\left({\pi }_{𝚤𝚥}^a\right)}^2\hfill \\ & -& \sum _{a=q_1+q_2+1}^{2r+q_2}{\left(\sum _{𝚤=1}^{q_1+q_2}{\pi }_{𝚤𝚤}^a\right)}^2.\hfill \end{array}$$

The above equation is equivalently expressed as

$$\begin{array}{ccc}\hfill \mathsf{\Pi }& =& \sum _{a=q_1+q_2+1}^{2r+q_2}\sum _{𝚤=1}^{q_1+q_2-1}[\left({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\right){\left({\pi }_{𝚤𝚤}^a\right)}^2\hfill \\ & +& {\displaystyle \frac{2(q_1+q_2+t)}{q_1+q_2}}{\left({\pi }_{𝚤(q_1+q_2)}^a\right)}^2]\hfill \\ & +& \sum _{q_1+q_2+1}^{2r+q_2}[2\left({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\right)\sum _{(𝚤<𝚥)=1}^{q_1+q_2}{\left({\pi }_{𝚤𝚥}^a\right)}^2\hfill \\ & -& 2\sum _{(𝚤<𝚥)=1}^{q_1+q_2}{\pi }_{𝚤𝚤}^a{\pi }_{𝚥𝚥}^a+{\displaystyle \frac{t}{q_1+q_2}}{\left({\pi }_{(q_1+q_2)(q_1+q_2)}^a\right)}^2].\hfill \end{array}$$

(13)

It is observed that the solution to the following system of homogeneous equations is

$$\left\{\begin{array}{c}\frac{\partial \mathsf{\Pi }}{\partial {\pi }_{𝚤𝚤}^a}=2(\frac{q_1+q_2+t}{q_1+q_2}+\frac{f\left(t\right)}{q_1+q_2-1})\left({\pi }_{𝚤𝚤}^a\right)-2{\sum }_{k=1}^{q_1+q_2}{\pi }_{kk}^a=0\hfill \\ \frac{\partial \mathsf{\Pi }}{\partial {\pi }_{(q_1+q_2)(q_1+q_2)}^a}=\frac{2t}{q_1+q_2}{\pi }_{(q_1+q_2)(q_1+q_2)}^a-2{\sum }_{k=1}^{q_1+q_2-1}{\pi }_{kk}^a=0\hfill \\ \frac{\partial \mathsf{\Pi }}{\partial {\pi }_{𝚤𝚥}^a}=4\left(\frac{q_1+q_2+t}{q_1+q_2}+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\right){\pi }_{𝚤𝚥}^a=0\hfill \\ \frac{\partial \mathsf{\Pi }}{\partial {\pi }_{𝚤(q_1+q_2)}^a}=4\left({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}\right)\left({\pi }_{𝚤(q_1+q_2)}^a\right)=0,\hfill \end{array}\right.$$

(14)

$𝚤,𝚥=\{1,2,\dots ,q_1+q_2-1\},𝚤\ne 𝚥$ and $a\in \{q_1+q_2+1,\dots ,2r+q_2\}$ are the critical points

$${\pi }^c=({\pi }_{11}^{q_1+q_2+1},{\pi }_{12}^{q_1+q_2+1},\dots ,{\pi }_{(q_1+q_2)(q_1+q_2)}^{q_1+q_2+1},\dots ,{\pi }_{11}^{2r+q_2},\dots ,{\pi }_{(q_1+q_2)(q_1+q_2)}^{2r+q_2})$$

of $\mathsf{\Pi }$ from (13).

As a result, every solution ${\pi }^c$ satisfies ${\pi }_{𝚤𝚥}^a=0$ whenever $𝚤\ne 𝚥$. Additionally, the determinant associated with the first two equations in (14) vanishes. The Hessian matrix of the quadratic form $\mathsf{\Pi }$ then takes the following block-diagonal structure:

$$\begin{array}{c}\hfill Hess\left(\mathsf{\Pi }\right)=\left(\begin{array}{ccc}{\mathsf{\Xi }}_1& \mathbf{N}& \mathbf{N}\\ \mathbf{N}& {\mathsf{\Xi }}_2& \mathbf{N}\\ \mathbf{N}& \mathbf{N}& {\mathsf{\Xi }}_3\end{array}\right),\end{array}$$

where $\mathbf{N}$ denotes a null matrix of appropriate dimension, and the matrices ${\mathsf{\Xi }}_1$, ${\mathsf{\Xi }}_2$, and ${\mathsf{\Xi }}_3$ are defined in the following.

$$\begin{array}{c}\hfill {\mathsf{\Xi }}_1=\left(\begin{array}{cccc}2\left({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\right)-2& \cdots & -2& -2\\ -2& \cdots & -2& -2\\ ⋮& \ddots & ⋮& ⋮\\ -2& \cdots & 2\left({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\right)-2& -2\\ -2& \dots & -2& {\displaystyle \frac{2t}{q_1+q_2}}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\mathsf{\Xi }}_2=diag\left(4\left({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\right),\dots ,4\left({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\right)\right),\end{array}$$

and

$${\mathsf{\Xi }}_3=diag\left({\displaystyle \frac{4(q_1+q_2+t)}{q_1+q_2}},\dots ,{\displaystyle \frac{4(q_1+q_2+t)}{q_1+q_2}}\right).$$

Also, $Hess\left(\mathsf{\Pi }\right)$ has the following eigenvalues (see [12,21]):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mu }_{11}=0, {\mu }_{22}=2\left({\displaystyle \frac{2t}{q_1+q_2}}+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\right),\hfill \\ {\mu }_{33}=\cdots ={\mu }_{q_1+q_1{q}_2+q_2}=2\left({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\right),\hfill \\ {\mu }_{𝚤𝚥}=4\left({\displaystyle \frac{q_1+q_2+t}{q_1+q_2}}+{\displaystyle \frac{f\left(t\right)}{q_1+q_2-1}}\right),\hfill \\ {\mu }_{𝚤q_1}={\displaystyle \frac{4(q_1+q_2+t)}{q_1+q_2}}, \forall 𝚤,𝚥\in \{1,2,\dots ,q_1+q_2-1\}, 𝚤\ne 𝚥.\hfill \end{array}\right.\end{array}$$

(15)

It follows from Equation (15) that the function $\mathsf{\Pi }$ is parabolic in nature and attains its minimum value at $\mathsf{\Pi }\left({\pi }^c\right)=0$, where ${\pi }^c$ denotes a solution of the system (14). Consequently, the inequality $\mathsf{\Pi }\ge 0$ holds, leading to

$$\begin{array}{ccc}\hfill 2\mathsf{\Lambda }& \le & tC+f\left(t\right)C(\mho )+q_1(q_1-1){\displaystyle \frac{c+3q_2}{4}}+2q_1{q}_2\hfill \\ & & +{\displaystyle \frac{3(c-q_2)}{4}}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2+\parallel P_1{\parallel }^2+\parallel P_2{\parallel }^2)-2(q_1+q_2-1)tr\left(\varpi \right).\hfill \end{array}$$

(16)

Therefore, using (16), it follows that for every tangent hyperplane ℧ of M, we have

$$\begin{array}{ccc}\hfill \mathsf{\Omega }& \le & {\displaystyle \frac{t}{(q_1+q_2)(q_1+q_2-1)}}C+{\displaystyle \frac{f\left(t\right)}{(q_1+q_2)(q_1+q_2-1)}}C(\mho )\hfill \\ & +& {\displaystyle \frac{q_1(q_1-1)(c+3q_2)}{4(q_1+q_2)(q_1+q_2-1)}}+{\displaystyle \frac{2q_1{q}_2}{(q_1+q_2)(q_1+q_2-1)}}\hfill \\ & +& {\displaystyle \frac{3(c-q_2)}{4(q_1+q_2)(q_1+q_2-1)}}({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2+\parallel P_1{\parallel }^2+\parallel P_2{\parallel }^2)-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right).\hfill \end{array}$$

Evaluating the infimum over all tangent hyperplanes ℧ leads directly to the result. The condition for equality is satisfied if and only if

$$\begin{array}{c}\hfill {\pi }_{𝚤𝚥}^a=0, \forall 𝚤,𝚥\in \{1,\dots ,q_1+q_2\}, 𝚤\ne 𝚥; a\in \{q_1+q_2+1,\dots ,2r+q_2\}\end{array}$$

(17)

and

$$\begin{array}{c}\hfill {\pi }_{(q_1+q_2)(q_1+q_2)}^a={\displaystyle \frac{(q_1+q_2)(q_1+q_2-1)}{t}}{\pi }_{11}^a=\dots =\end{array}$$

(18)

$$={\displaystyle \frac{(q_1+q_2)(q_1+q_2-1)}{t}}{\pi }_{(q_1+q_2-1)(q_1+q_2-1)}^a,$$

for all $a\in \{q_1+q_2+1,\dots ,2r+q_2\}$.

On the other hand, since $u_{q_1+q_2}={\xi }_{q_2}$, one has $\pi (u_{q_1+q_2},u_{q_1+q_2})=0$. Then, (18) implies that M is totally geodesic.

Using a comparable method, one can establish the geometric inequality presented in (ii).

□

Corollary 1. 

Let M be a bi-slant submanifold of dimension $(q_1+q_2)$ immersed in a $(2r+q_2)$-dimensional S-space form $\overline{M}\left(c\right)$ endowed with an SSMC. Then, the following results hold:

(i)

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

$$\begin{array}{ccc}\hfill \mathsf{\Omega }& \le & {\delta }_c(q_1+q_2-1)-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right)\hfill \\ & +& {\displaystyle \frac{q_1(q_1-1)(c+3q_2)+8q_1{q}_2+3(c-q_2)({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2+\parallel P_1{\parallel }^2+\parallel P_2{\parallel }^2)}{4(q_1+q_2)(q_1+q_2-1)}}.\hfill \end{array}$$

(19)

Equality in the inequality characterizes the case where the submanifold M is totally geodesic.

(ii)

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

$$\begin{array}{ccc}\hfill \mathsf{\Omega }& \le & {\widehat{\delta }}_c(q_1+q_2-1)-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right)\hfill \\ & +& {\displaystyle \frac{q_1(q_1-1)(c+3q_2)+8q_1{q}_2+3(c-q_2)({\mathit{d}}_{1}co{s}^2{\theta }_1+{\mathit{d}}_{2}co{s}^2{\theta }_2+\parallel P_1{\parallel }^2+\parallel P_2{\parallel }^2)}{4(q_1+q_2)(q_1+q_2-1)}}.\hfill \end{array}$$

(20)

Equality in the inequality characterizes the case where the submanifold M is totally geodesic.

Proof. 

It is straightforward to verify that

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

(21)

$$(q_1+q_2)(q_1+q_2-1){\left[{\delta }_c(q_1+q_2-1)\right]}_x,$$

for all $x\in M$. Inserting $t={\displaystyle \frac{(q_1+q_2)(q_1+q_2-1)}{2}}$ into (9) and taking into account (21), we have our assertion. Using a comparable method, one can establish the geometric inequality presented in (ii). Further, equality holds in the inequality (19) if and only if

$$\begin{array}{c}\hfill {\pi }_{𝚤𝚥}^a=0, \forall 𝚤,𝚥\in \{1,\dots ,q_1+q_2\}, 𝚤\ne 𝚥; a\in \{q_1+q_2+1,\dots ,2r+q_2\}\end{array}$$

(22)

and

$$\begin{array}{c}\hfill 2{\pi }_{(q_1+q_2)(q_1+q_2)}^a={\pi }_{11}^a=\dots ={\pi }_{(q_1+q_2-1)(q_1+q_2-1)}^a,\end{array}$$

(23)

for all $a\in \{q_1+q_2+1,\dots ,2r+q_2\}$.

It follows that equality in (20) holds exactly when (22) is satisfied with

$$\begin{array}{c}\hfill {\pi }_{(q_1+q_2)(q_1+q_2)}^a=2{\pi }_{11}^a=\dots =2{\pi }_{(q_1+q_2-1)(q_1+q_2-1)}^a,\end{array}$$

(24)

for all $a\in \{q_1+q_2+1,\dots ,2r+q_2\}$.

Because $\pi (u_{q_1+q_2},u_{q_1+q_2})=0$, consequently, the validity of equalities (23) and (24) each leads to the conclusion that M must be a totally geodesic submanifold.

□

5. Results in Other Classes of Submanifolds

In this section, we explore how the established inequalities specialize to notable classes of submanifolds such as semi-slant, hemi-slant, CR-, invariant, and anti-invariant submanifolds. Furthermore, we examine the conditions necessary for equality to hold. We further demonstrate that the equality case characterizes a specific geometric configuration, namely, totally geodesic submanifolds.

Theorem 2. 

Let M be a semi-slant submanifold of dimension $(q_1+q_2)$ immersed in a $(2r+q_2)$-dimensional S-space form $\overline{M}\left(c\right)$ endowed with an SSMC. Then, the following results hold:

(i)

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

$$\begin{array}{cc}\hfill \mathsf{\Omega }& \le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right)\hfill \\ \hfill & +{\displaystyle \frac{q_1(q_1-1)(c+3q_2)+8q_1{q}_2+3(c-q_2)({\mathit{d}}_1+{\mathit{d}}_{2}co{s}^2{\theta }_2)}{4(q_1+q_2)(q_1+q_2-1)}},\hfill \end{array}$$

(25)

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

(ii)

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

$$\begin{array}{cc}\hfill \mathsf{\Omega }& \le {\displaystyle \frac{\widehat{{\delta }_c}(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right)\hfill \\ \hfill & +{\displaystyle \frac{q_1(q_1-1)(c+3q_2)+8q_1{q}_2+3(c-q_2)({\mathit{d}}_1+{\mathit{d}}_{2}co{s}^2{\theta }_2)}{4(q_1+q_2)(q_1+q_2-1)}},\hfill \end{array}$$

(26)

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

Equality in the inequalities (25) and (26) characterizes the case where the submanifold M is totally geodesic.

Theorem 3. 

Let M be a pseudo-slant submanifold of dimension $(q_1+q_2)$ immersed in a $(2r+q_2)$-dimensional S-space form $\overline{M}\left(c\right)$ endowed with an SSMC. Then, the following results hold:

(i)

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

$$\begin{array}{cc}\hfill \mathsf{\Omega }& \le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right)\hfill \\ \hfill & +{\displaystyle \frac{q_1(q_1-1)(c+3q_2)+8q_1{q}_2+3(c-q_2)({\mathit{d}}_{1}co{s}^2{\theta }_1+\parallel P_1{\parallel }^2)}{4(q_1+q_2)(q_1+q_2-1)}},\hfill \end{array}$$

(27)

where t is any real parameter lying strictly between 0 and $(q_1+q_2)(q_1+q_2-1)$.

(ii)

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

$$\begin{array}{cc}\hfill \mathsf{\Omega }& \le {\displaystyle \frac{\widehat{{\delta }_c}(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right)\hfill \\ \hfill & +{\displaystyle \frac{q_1(q_1-1)(c+3q_2)+8q_1{q}_2+3(c-q_2)({\mathit{d}}_{1}co{s}^2{\theta }_1+\parallel P_1{\parallel }^2)}{4(q_1+q_2)(q_1+q_2-1)}},\hfill \end{array}$$

(28)

where t denotes any real number exceeding $(q_1+q_2)(q_1+q_2-1)$.

Equality in the inequalities (27) and (28) characterizes the case where the submanifold M is totally geodesic.

Theorem 4. 

Let M be a CR-submanifold of dimension $(q_1+q_2)$ immersed in a $(2r+q_2)$-dimensional S-space form $\overline{M}\left(c\right)$ endowed with an SSMC. Then, the following results hold:

(i)

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

$$\begin{array}{cc}\hfill \mathsf{\Omega }& \le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right)\hfill \\ \hfill & +{\displaystyle \frac{q_1(q_1-1)(c+3q_2)+8q_1{q}_2+3(c-q_2)\left({\mathit{d}}_1\right)}{4(q_1+q_2)(q_1+q_2-1)}},\hfill \end{array}$$

(29)

where t is any real parameter lying strictly between 0 and $(q_1+q_2)(q_1+q_2-1)$.

(ii)

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

$$\begin{array}{cc}\hfill \mathsf{\Omega }& \le {\displaystyle \frac{\widehat{{\delta }_c}(t;q_1+q_2-1)}{(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right)\hfill \\ \hfill & +{\displaystyle \frac{q_1(q_1-1)(c+3q_2)+8q_1{q}_2+3(c-q_2)\left({\mathit{d}}_1\right)}{4(q_1+q_2)(q_1+q_2-1)}},\hfill \end{array}$$

(30)

where t denotes any real number exceeding $(q_1+q_2)(q_1+q_2-1)$.

Equality in the inequalities (29) and (30) characterizes the case where the submanifold M is totally geodesic.

Theorem 5. 

Let M be a slant submanifold of dimension $(q_1+q_2)$ immersed in a $(2r+q_2)$-dimensional S-space form $\overline{M}\left(c\right)$ endowed with an SSMC. Then, the following results hold:

(i)

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

$$\begin{array}{cc}\hfill \mathsf{\Omega }& \le {\displaystyle \frac{4{\delta }_c(t;q_1+q_2-1)+8q_1{q}_2+3(c-q_2)(q_1+q_2)co{s}^2\theta }{4(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right),\hfill \end{array}$$

(31)

where t is any real parameter lying strictly between 0 and $(q_1+q_2)(q_1+q_2-1)$.

(ii)

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

$$\begin{array}{cc}\hfill \mathsf{\Omega }& \le {\displaystyle \frac{4\widehat{{\delta }_c}(t;q_1+q_2-1)+8q_1{q}_2+3(c-q_2)(q_1+q_2)co{s}^2\theta }{4(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right),\hfill \end{array}$$

(32)

where t denotes any real number exceeding $(q_1+q_2)(q_1+q_2-1)$.

Equality in the inequalities (31) and (32) characterizes the case where the submanifold M is totally geodesic.

Theorem 6. 

Let M be an invariant submanifold of dimension $(q_1+q_2)$ immersed in a $(2r+q_2)$-dimensional S-space form $\overline{M}\left(c\right)$ endowed with an SSMC. Then, the following results hold:

(i)

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

$$\begin{array}{cc}\hfill \mathsf{\Omega }& \le {\displaystyle \frac{4{\delta }_c(t;q_1+q_2-1)+8q_1{q}_2+3(c-q_2)(q_1+q_2)}{4(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right),\hfill \end{array}$$

(33)

where t is any real parameter lying strictly between 0 and $(q_1+q_2)(q_1+q_2-1)$.

(ii)

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

$$\begin{array}{cc}\hfill \mathsf{\Omega }& \le {\displaystyle \frac{4\widehat{{\delta }_c}(t;q_1+q_2-1)+8q_1{q}_2+3(c-q_2)(q_1+q_2)}{4(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right),\hfill \end{array}$$

(34)

where t denotes any real number exceeding $(q_1+q_2)(q_1+q_2-1)$.

Equality in the inequalities (33) and (34) characterizes the case where the submanifold M is totally geodesic.

Theorem 7. 

Let M be an anti-invariant submanifold of dimension $(q_1+q_2)$ immersed in a $(2r+q_2)$-dimensional S-space form $\overline{M}\left(c\right)$ endowed with an SSMC. Then, the following results hold:

(i)

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

$$\begin{array}{c}\hfill \mathsf{\Omega }\le {\displaystyle \frac{{\delta }_c(t;q_1+q_2-1)+2q_1{q}_2}{(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right),\end{array}$$

(35)

where t is any real parameter lying strictly between 0 and $(q_1+q_2)(q_1+q_2-1)$.

(ii)

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

$$\begin{array}{c}\hfill \mathsf{\Omega }\le {\displaystyle \frac{\widehat{{\delta }_c}(t;q_1+q_2-1)+2q_1{q}_2}{(q_1+q_2)(q_1+q_2-1)}}-{\displaystyle \frac{2}{q_1+q_2}}tr\left(\varpi \right),\end{array}$$

(36)

where t denotes any real number exceeding $(q_1+q_2)(q_1+q_2-1)$.

Equality in the inequalities (35) and (36) characterizes the case where the submanifold M is totally geodesic.

Remark 1. 

The first four theorems follow directly from Table 1 in conjunction with Theorem 1. The remaining two results correspond to the special cases of slant submanifolds when the slant angle is set to $\theta =0$ and $\theta ={\displaystyle \frac{\pi }{2}}$, which yield the cases of invariant and anti-invariant submanifolds, respectively.

Remark 2. 

We can obtain results similar to Theorem 2, Theorem 3, Theorem 4, Theorem 5, Theorem 6, and Theorem 7 corresponding to the Corollary 1.

Remark 3. 

The results presented in this article naturally extend to Sasakian and Kähler manifolds as special cases of the S-manifold framework. In particular, by taking $q_2=1$, the structure reduces to that of a Sasakian manifold, whereas setting $q_2=0$ yields the classical Kähler case.

6. Illustrative Examples

In this section, we present explicit constructions of bi-slant submanifolds within an S-manifold equipped with the standard S-structure. These examples are motivated by the framework developed by Carriazo and collaborators [25] and serve to demonstrate the geometric features discussed in the preceding sections.

Example 2. 

Let ${\overline{M}}^{8+s}\left(c\right)$ be an S-space form of dimension $8+s$ with $(\varphi ,{\xi }_{\alpha },{\eta }^{\alpha },g)$-structure and constant ϕ-sectional curvature $c\in \mathbb{R}$. The structure satisfies

$$\begin{array}{cc}\hfill {\varphi }^2& =-I+\sum _{\alpha =1}^s{\eta }^{\alpha }\otimes {\xi }_{\alpha },{\eta }^{\alpha }\left({\xi }_{\beta }\right)={\delta }_{\beta }^{\alpha },g(\varphi X,\varphi Y)=g(X,Y)-\sum _{\alpha =1}^s{\eta }^{\alpha }\left(X\right){\eta }^{\alpha }\left(Y\right).\hfill \end{array}$$

We consider the ambient coordinates

$$(x^1,x^2,x^3,x^4,x^5,x^6,z^1,\dots ,z^s)$$

and define the structure tensors

$$\varphi \left({\displaystyle \frac{\partial }{\partial x^i}}\right)={\displaystyle \frac{\partial }{\partial x^{i+3}}},\varphi \left({\displaystyle \frac{\partial }{\partial x^{i+3}}}\right)=-{\displaystyle \frac{\partial }{\partial x^i}},i=1,2,3,\varphi \left({\displaystyle \frac{\partial }{\partial z^{\alpha }}}\right)=0.$$

Let ${\xi }_{\alpha }=2{\displaystyle \frac{\partial }{\partial z^{\alpha }}}$ and ${\eta }^{\alpha }={\displaystyle \frac{1}{2}}\left(d{z}^{\alpha }+{\sum }_{i=1}^6{x}^{i+3}d{x}^i\right)$.

Define a submanifold M of ${\overline{M}}^{8+s}\left(c\right)$ via the immersion

$$\phi (u,v,w,t)=\left(u,v,u+v,u-v,w-t,w+t,c_1,\dots ,c_s\right),$$

where $c_1,\dots ,c_s\in \mathbb{R}$ are fixed constants.

Then, the tangent space of M is spanned by the vector fields

$$\begin{array}{cc}\hfill X_1& ={\displaystyle \frac{\partial }{\partial x^1}}+{\displaystyle \frac{\partial }{\partial x^3}}+{\displaystyle \frac{\partial }{\partial x^4}},X_2={\displaystyle \frac{\partial }{\partial x^2}}+{\displaystyle \frac{\partial }{\partial x^3}}-{\displaystyle \frac{\partial }{\partial x^4}},\hfill \\ \hfill X_3& ={\displaystyle \frac{\partial }{\partial x^5}}-{\displaystyle \frac{\partial }{\partial x^6}},X_4={\displaystyle \frac{\partial }{\partial x^5}}+{\displaystyle \frac{\partial }{\partial x^6}}.\hfill \end{array}$$

Now, define

$${\mathcal{D}}_1=Span\{X_1,X_2\},{\mathcal{D}}_2=Span\{X_3,X_4\}.$$

For all i, $\varphi \left(X_i\right)\in T^{\perp }M$ (i.e., $\varphi \left(X_i\right)$ is normal to M). Therefore, both ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ are anti-invariant distributions. The total tangent bundle $TM={\mathcal{D}}_1\oplus {\mathcal{D}}_2$ is anti-invariant as well.

Thus, M is a bi-slant submanifold of the S-space form ${\overline{M}}^{8+s}\left(c\right)$ corresponding to Case 1 in [32], where both distributions are anti-invariant, and M is also an anti-invariant submanifold.

Example 3. 

Let ${\overline{M}}^{8+s}\left(c\right)$ be an $(8+s)$-dimensional S-space form with structure tensors $(\varphi ,{\xi }_{\alpha },{\eta }^{\alpha },g)$ and constant ϕ-sectional curvature $c\ne 0$. The structure satisfies

$$\begin{array}{cc}\hfill {\varphi }^2& =-I+\sum _{\alpha =1}^s{\eta }^{\alpha }\otimes {\xi }_{\alpha },{\eta }^{\alpha }\left({\xi }_{\beta }\right)={\delta }_{\beta }^{\alpha },\hfill \\ \hfill g(\varphi X,\varphi Y)& =g(X,Y)-\sum _{\alpha =1}^s{\eta }^{\alpha }\left(X\right){\eta }^{\alpha }\left(Y\right).\hfill \end{array}$$

Consider the local coordinates

$$(x^1,x^2,x^3,x^4,y^1,y^2,y^3,y^4,z^1,\dots ,z^s),$$

with the S-structure defined by

$$\begin{array}{cc}\hfill \varphi \left({\displaystyle \frac{\partial }{\partial x^i}}\right)& ={\displaystyle \frac{\partial }{\partial y^i}},\varphi \left({\displaystyle \frac{\partial }{\partial y^i}}\right)=-{\displaystyle \frac{\partial }{\partial x^i}},\varphi \left({\displaystyle \frac{\partial }{\partial z^{\alpha }}}\right)=0,\hfill \\ \hfill {\xi }_{\alpha }& =2{\displaystyle \frac{\partial }{\partial z^{\alpha }}},{\eta }^{\alpha }={\displaystyle \frac{1}{2}}\left(d{z}^{\alpha }+\sum _{i=1}^4{y}^{i}d{x}^i\right).\hfill \end{array}$$

Now, define a submanifold M by the embedding

$$\begin{array}{cc}\hfill \phi (u_1,u_2,v_1,v_2,t_1,\dots ,t_s)=& (u_{1}cos{\theta }_1-u_{2}sin{\theta }_1,u_{1}sin{\theta }_1+u_{2}cos{\theta }_1,0,0,\hfill \\ \hfill & v_{1}cos{\theta }_2-v_{2}sin{\theta }_2,v_{1}sin{\theta }_2+v_{2}cos{\theta }_2,0,0,t_1,\dots ,t_s),\hfill \end{array}$$

for fixed angles ${\theta }_1,{\theta }_2\in (0,\pi /2)$.

Then, the tangent space $TM$ is spanned by the vector fields

$$\begin{array}{cc}\hfill X_1& =cos{\theta }_1{\displaystyle \frac{\partial }{\partial x^1}}+sin{\theta }_1{\displaystyle \frac{\partial }{\partial x^2}},X_2=-sin{\theta }_1{\displaystyle \frac{\partial }{\partial x^1}}+cos{\theta }_1{\displaystyle \frac{\partial }{\partial x^2}},\hfill \\ \hfill X_3& =cos{\theta }_2{\displaystyle \frac{\partial }{\partial y^1}}+sin{\theta }_2{\displaystyle \frac{\partial }{\partial y^2}},X_4=-sin{\theta }_2{\displaystyle \frac{\partial }{\partial y^1}}+cos{\theta }_2{\displaystyle \frac{\partial }{\partial y^2}},\hfill \\ \hfill X_{4+\alpha }& =2{\displaystyle \frac{\partial }{\partial z^{\alpha }}}={\xi }_{\alpha },\alpha =1,\dots ,s.\hfill \end{array}$$

Define distributions

$${\mathcal{D}}_1=Span\{X_1,X_2\},{\mathcal{D}}_2=Span\{X_3,X_4\},\mathcal{L}=Span{\left\{{\xi }_{\alpha }\right\}}_{\alpha =1}^s.$$

Then, ${\mathcal{D}}_1$ is a slant distribution with slant angle ${\theta }_1$. ${\mathcal{D}}_2$ is a slant distribution with slant angle ${\theta }_2\ne {\theta }_1$. $TM={\mathcal{D}}_1\oplus {\mathcal{D}}_2\oplus \mathcal{L}$ is not a slant distribution as a whole, since ${\theta }_1\ne {\theta }_2$. $\mathcal{L}$ is invariant, since $\varphi {\xi }_{\alpha }=0$.

Hence, M is a proper bi-slant submanifold of the S-space form ${\overline{M}}^{8+s}\left(c\right)$ corresponding to Case 2 in [32].

Example 4. 

Let ${\overline{M}}^{8+s}\left(c\right)$ be an S-space form with constant ϕ-sectional curvature c endowed with structure tensors $(\varphi ,{\xi }_{\alpha },{\eta }^{\alpha },g)$ satisfying

$$\begin{array}{cc}\hfill {\varphi }^2& =-I+\sum _{\alpha =1}^s{\eta }^{\alpha }\otimes {\xi }_{\alpha },{\eta }^{\alpha }\left({\xi }_{\beta }\right)={\delta }_{\beta }^{\alpha },\hfill \\ \hfill g(\varphi X,\varphi Y)& =g(X,Y)-\sum _{\alpha =1}^s{\eta }^{\alpha }\left(X\right){\eta }^{\alpha }\left(Y\right).\hfill \end{array}$$

Assume that the local coordinates are

$$(x^1,x^2,x^3,x^4,y^1,y^2,y^3,y^4,z^1,\dots ,z^s),$$

with the S-structure given by

$$\begin{array}{cc}\hfill \varphi \left({\displaystyle \frac{\partial }{\partial x^i}}\right)& ={\displaystyle \frac{\partial }{\partial y^i}},\varphi \left({\displaystyle \frac{\partial }{\partial y^i}}\right)=-{\displaystyle \frac{\partial }{\partial x^i}},\varphi \left({\displaystyle \frac{\partial }{\partial z^{\alpha }}}\right)=0,\hfill \\ \hfill {\xi }_{\alpha }& =2{\displaystyle \frac{\partial }{\partial z^{\alpha }}},{\eta }^{\alpha }={\displaystyle \frac{1}{2}}\left(d{z}^{\alpha }+\sum _{i=1}^4{y}^{i}d{x}^i\right).\hfill \end{array}$$

Define a submanifold M of ${\overline{M}}^{8+s}\left(c\right)$ by the embedding

$$\phi (u,v,w,t,t_1,\dots ,t_s)=\left({\displaystyle \frac{1}{2}}(u-v),{\displaystyle \frac{1}{2}}(u+v),{\displaystyle \frac{\sqrt{2}}{2}}u,{\displaystyle \frac{\sqrt{2}}{2}}v,w,t,0,0,t_1,\dots ,t_s\right).$$

Then, the tangent space of M is spanned by

$$\begin{array}{cc}\hfill X_1& ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial }{\partial x^1}}+{\displaystyle \frac{\partial }{\partial x^2}}\right)+{\displaystyle \frac{\sqrt{2}}{2}}{\displaystyle \frac{\partial }{\partial x^3}},\hfill \\ \hfill X_2& ={\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{\partial }{\partial x^1}}+{\displaystyle \frac{\partial }{\partial x^2}}\right)+{\displaystyle \frac{\sqrt{2}}{2}}{\displaystyle \frac{\partial }{\partial x^4}},\hfill \\ \hfill X_3& ={\displaystyle \frac{\partial }{\partial y^1}},X_4={\displaystyle \frac{\partial }{\partial y^2}},\hfill \\ \hfill X_{4+\alpha }& =2{\displaystyle \frac{\partial }{\partial z^{\alpha }}}={\xi }_{\alpha },\alpha =1,\dots ,s.\hfill \end{array}$$

Let

$${\mathcal{D}}_1=Span\{X_1,X_2\},{\mathcal{D}}_2=Span\{X_3,X_4\},\mathcal{L}=Span{\left\{{\xi }_{\alpha }\right\}}_{\alpha =1}^s.$$

Then, ${\mathcal{D}}_1$ is a slant distribution (not invariant and not anti-invariant) with slant angle $\theta ={\displaystyle \frac{\pi }{3}}$, ${\mathcal{D}}_2$ is also a slant distribution with the same angle θ, ${\mathcal{D}}_1\oplus {\mathcal{D}}_2$ is again a slant distribution (with same angle θ), and $\mathcal{L}$ is invariant (since $\varphi {\xi }_{\alpha }=0$).

Hence, the submanifold M is a proper bi-slant submanifold of the S-space form ${\overline{M}}^{8+s}\left(c\right)$ corresponding to Case 3 in [32], where ${\mathcal{D}}_1$, ${\mathcal{D}}_2$ are slant distributions, and $TM$ is also a slant distribution with angle $\theta ={\displaystyle \frac{\pi }{3}}$.

Example 5. 

Let $E^{8+s}$ denote the Euclidean space with global coordinates

$$(x^1,x^2,x^3,x^4,y^1,y^2,y^3,y^4,z^1,\dots ,z^s),$$

equipped with a standard S-structure defined by

$$\begin{array}{cc}\hfill {\xi }_{\alpha }& =2{\displaystyle \frac{\partial }{\partial z^{\alpha }}},\hfill \\ \hfill {\eta }^{\alpha }& ={\displaystyle \frac{1}{2}}\left(d{z}^{\alpha }+\sum _{i=1}^4{y}^{i}d{x}^i\right),\hfill \\ \hfill \varphi \left({\displaystyle \frac{\partial }{\partial x^i}}\right)& ={\displaystyle \frac{\partial }{\partial y^i}},\varphi \left({\displaystyle \frac{\partial }{\partial y^i}}\right)=-{\displaystyle \frac{\partial }{\partial x^i}},\varphi \left({\displaystyle \frac{\partial }{\partial z^{\alpha }}}\right)=0.\hfill \end{array}$$

We define a 4-dimensional submanifold M of $E^{8+s}$ by the smooth immersion

$$X(u,v,w,m)=\left(ucos{\theta }_1,usin{\theta }_1,v,0,wcos{\theta }_2,wsin{\theta }_2,m,0,c_1,\dots ,c_s\right),$$

where ${\theta }_1,{\theta }_2\in (0,{\displaystyle \frac{\pi }{2}})$ are fixed slant angles, and $c_1,\dots ,c_s\in \mathbb{R}$ are constants.

The tangent bundle $TM$ of M is spanned by the orthonormal frame

$$\begin{array}{cc}\hfill e_1& =cos{\theta }_1{\displaystyle \frac{\partial }{\partial x^1}}+sin{\theta }_1{\displaystyle \frac{\partial }{\partial x^2}},\hfill \\ \hfill e_2& ={\displaystyle \frac{\partial }{\partial x^3}},\hfill \\ \hfill e_3& =cos{\theta }_2{\displaystyle \frac{\partial }{\partial y^1}}+sin{\theta }_2{\displaystyle \frac{\partial }{\partial y^2}},\hfill \\ \hfill e_4& ={\displaystyle \frac{\partial }{\partial y^3}}.\hfill \end{array}$$

We define two orthogonal distributions as

$${\mathcal{D}}_1=\mathrm{span}\{e_1,e_2\},{\mathcal{D}}_2=\mathrm{span}\{e_3,e_4\}.$$

Now, compute

$$\begin{array}{cc}\hfill \varphi \left(e_1\right)& =cos{\theta }_1{\displaystyle \frac{\partial }{\partial y^1}}+sin{\theta }_1{\displaystyle \frac{\partial }{\partial y^2}},\hfill \\ \hfill \varphi \left(e_2\right)& ={\displaystyle \frac{\partial }{\partial y^3}},\hfill \\ \hfill \varphi \left(e_3\right)& =-cos{\theta }_2{\displaystyle \frac{\partial }{\partial x^1}}-sin{\theta }_2{\displaystyle \frac{\partial }{\partial x^2}},\hfill \\ \hfill \varphi \left(e_4\right)& =-{\displaystyle \frac{\partial }{\partial x^3}}.\hfill \end{array}$$

These vectors $\varphi \left(e_i\right)$ are not tangent to M, but the angle between each $\varphi \left(e_i\right)$ and the tangent space $TM$ is constant (since their projections on $TM$ lie in specific directions depending on fixed ${\theta }_1$ or ${\theta }_2$). Hence, ${\mathcal{D}}_1$ is a slant distribution with slant angle ${\theta }_1$, ${\mathcal{D}}_2$ is a slant distribution with slant angle ${\theta }_2$, and since ${\theta }_1\ne {\theta }_2$, the whole tangent bundle $TM$ is not slant.

Therefore, M is a proper bi-slant submanifold of the S-space form $E^{8+s}$.

The normal bundle $T^{\perp }M$ of M is spanned by the orthonormal vectors

$$\begin{array}{cc}\hfill e_5& =-sin{\theta }_1{\displaystyle \frac{\partial }{\partial x^1}}+cos{\theta }_1{\displaystyle \frac{\partial }{\partial x^2}},\hfill \\ \hfill e_6& ={\displaystyle \frac{\partial }{\partial x^4}},\hfill \\ \hfill e_7& =-sin{\theta }_2{\displaystyle \frac{\partial }{\partial y^1}}+cos{\theta }_2{\displaystyle \frac{\partial }{\partial y^2}},\hfill \\ \hfill e_8& ={\displaystyle \frac{\partial }{\partial y^4}}.\hfill \end{array}$$

This construction satisfies all the required conditions for a bi-slant submanifold in the sense of Case 4 in [32].

7. Conclusions and Future Directions

In this article, we have derived two sharp geometric inequalities involving the generalized normalized $\delta$-Casorati curvatures for bi-slant submanifolds in S-space forms equipped with semi-symmetric metric connections. These inequalities deepen our understanding of the interplay between intrinsic and extrinsic invariants in the context of submanifold geometry and extend several previously known results to a broader class of submanifolds. We have also examined the conditions under which equality is achieved and have provided geometric characterizations for those cases. To support our theoretical findings, illustrative examples were constructed.

Looking ahead, there are several promising directions for future research. One possible extension is to investigate analogous inequalities for other curvature-related invariants, such as scalar or mean curvature functions, in the same geometric setting. Another direction involves exploring similar results in broader ambient spaces, such as generalized S-space forms or statistical manifolds. Additionally, the role of $\delta$-Casorati-type invariants in the context of warped product submanifolds or in almost-contact metric structures with other types of connections (e.g., quarter-symmetric) presents an interesting avenue for further study. These investigations may yield new insights and potential applications in both theoretical differential geometry and mathematical physics.