Sharp Curvature Inequalities for Submanifolds in Conformal Sasakian Space Forms Equipped with Quarter-Symmetric Metric Connection

Abstract

This study focuses on submanifolds embedded in a conformal Sasakian space form (CSSF) equipped with a quarter-symmetric metric connection (QSMC). Utilizing the framework of generalized normalized $\delta$-Casorati curvature (GNDCC) alongside scalar curvature, we derive sharp optimal inequalities that characterize the intrinsic and extrinsic geometry of the submanifolds. Additionally, we examine the geometric behavior of these submanifolds under conformal deformations of the ambient manifold. To substantiate the theoretical developments, we construct an explicit example of a conformal Sasakian manifold that is not Sasakian, thereby confirming the validity and applicability of the derived results.

1. Introduction

The study of the intrinsic and extrinsic geometry of submanifolds in Riemannian and semi-Riemannian manifolds continues to be one of the central themes in differential geometry. In particular, the relationships between intrinsic invariants such as scalar and Ricci curvatures and extrinsic invariants like mean curvature and the second fundamental form have captured the interest of many geometers. A noteworthy contribution in this direction was made in [1], where the authors examined submanifolds tangent to the structure vector field $\tilde{\xi }$ within the context of conformal Sasakian manifolds. Their findings revealed compelling relationships between the squared mean curvature, a primary extrinsic invariant, and intrinsic quantities such as scalar and Ricci curvatures. These results form the primary motivation for the present work, where we aim to derive optimal inequalities involving the GNDCC of submanifolds in CSSF.

The idea of Casorati curvature originates in classical differential geometry as a way of measuring how a submanifold bends inside its ambient space. It is calculated as the normalized square length of the submanifold’s second fundamental form, making it an important indicator of extrinsic geometry. First formally introduced by Casorati in 1999 [2], this curvature concept has since been revisited and generalized by many geometers. In particular, Deszcz et al. [3] highlighted that the Casorati curvature can be viewed as an extension of the classical notion of principal directions, originally used for hypersurfaces, to the more general setting of submanifolds of higher codimension. Since then, a growing body of work has explored optimal inequalities involving the Casorati curvature and its generalizations in various geometric settings. For instance, studies by Aquib et al. [4], Aquib [5], Lone [6], Ghasoui [7], and Vîlcu [8,9] have extended these investigations to broader contexts including contact, almost contact, and statistical manifolds.

Simultaneously, the framework of conformal geometry has offered a rich backdrop for the analysis of geometric inequalities. A significant class within this framework is the conformal Sasakian manifolds, which arise as a generalization of Sasakian manifolds wherein the metric undergoes a smooth conformal deformation while preserving essential contact structures. The concept of conformal modifications of almost contact metric structures was first introduced in the seminal work of Vaisman [10] in 1980. Building upon this foundation, Libermann [11] and Banaru [12] provided a detailed classification of sixteen types of nearly Hermitian manifolds based on specific tensorial conditions, within which the locally conformal Kähler and conformal Sasakian structures have attracted substantial attention.

Recent developments have significantly expanded the study of conformal and related almost contact structures. For instance, investigations into quasi-Sasakian manifolds with specialized curvature conditions, such as those endowed with vanishing pseudo-quasi-conformal curvature tensors [13], and their generalizations to Sasakian space forms [14], have provided deeper insights into curvature behavior under conformal deformations. Parallel studies on new classes of almost contact metric manifolds of Kenmotsu type [15] and analyses of the conharmonic curvature tensor in locally conformal cosymplectic manifolds [16] have further enriched the theoretical landscape. These works collectively highlight the versatility and importance of conformal-type structures in modern differential geometry, both from a purely theoretical perspective and in view of their potential applications.

From another perspective, the evolution of Chen invariants has significantly enriched the geometric analysis of submanifolds. Initiated by Chen in the early 1990s [17], this theory introduced a new set of inequalities that relate the scalar curvature and sectional curvature of a submanifold to its mean curvature and shape operator. These inequalities are not only elegant in form but also provide sharp characterizations for certain types of submanifolds. In particular, Chen’s $\delta$-invariants have inspired numerous studies aimed at exploring the optimal bounds for various curvature quantities. The normalized $\delta$-Casorati curvature (NDCC), being one of the most prominent generalizations, offers a finer tool to measure the geometric behavior of submanifolds under specific curvature constraints.

In a recent work [18], the first author investigated sharp Casorati-type inequalities for bi-slant submanifolds in S-space forms equipped with a semi-symmetric metric connection. That study focused on a different ambient geometry and connection type, as well as a more specialized class of submanifolds, leading to inequalities involving the generalized normalized $\delta$-Casorati curvature in that setting. The current work is fundamentally distinct in three key aspects: (i) we work in the broader framework of conformal Sasakian space forms instead of S-space forms, (ii) we employ a quarter-symmetric metric connection rather than a semi-symmetric one, and (iii) we consider a general class of submanifolds rather than restricting our attention to bi-slant ones. These changes introduce new structural features, modify curvature identities, and necessitate different analytical techniques, which in turn yield genuinely new inequalities and equality characterizations not obtainable from the earlier setting.

In the current study, we investigate submanifolds lying in CSSF that are equipped with a QSMC. This type of connection extends the familiar Levi-Civita and semi-symmetric connections, allowing us to analyze manifolds with richer geometric structures. Within this framework, we establish new relationships between the GNDCC and the scalar curvature. We also study the conditions under which these relationships become equalities, leading to a classification of certain special submanifolds, specifically, those that are invariantly quasi-umbilical and have trivial normal connections.

Furthermore, to support the theoretical results and demonstrate their practical relevance, we present illustrative examples of submanifolds that satisfy the equality conditions. Notably, we consider examples where the ambient space is a conformal Sasakian manifold that does not reduce to a Sasakian manifold, thereby showcasing the broader applicability of our results.

The organization of the paper is as follows: In Section 2, we provide the necessary preliminaries on conformal Sasakian manifolds and QSMC. Section 3 is devoted to the derivation of curvature inequalities involving the GNDCC. In Section 4, we explore the equality cases and geometric consequences. Section 5 presents illustrative examples to substantiate our theoretical findings. Finally, we conclude the paper with a discussion on the significance of the results and possible future directions for research.

2. Preliminaries

Consider a Riemannian manifold $\tilde{\mathcal{M}}$ equipped with a Riemannian metric g. A QSMC is defined as a linear connection $\overline{\nabla }$ on $\tilde{\mathcal{M}}$ whose torsion tensor $\mathcal{T}$, given by

$$\begin{array}{c}\hfill \mathcal{T}(S,T)={\overline{\nabla }}_{S}T-{\overline{\nabla }}_{T}S-[S,T]\end{array}$$

and satisfies the condition

$$\begin{array}{c}\hfill \mathcal{T}(S,T)=\zeta \left(T\right)\phi S-\zeta \left(S\right)\phi T,\end{array}$$

where ${\overline{\nabla }}_{S}T$ is called the covariant derivative of T along S with respect to $\overline{\nabla }$ and $[S,T]$ is the Lie bracket, for all vector fields S and T on $\tilde{\mathcal{M}}$. Here, $\zeta$ is a 1-form defined by $\zeta \left(S\right)=g(S,\mathcal{V})$, where $\mathcal{V}$ is a fixed vector field on $\tilde{\mathcal{M}}$, and $\phi$ is a $(1,1)$-type tensor field.

If the connection $\overline{\nabla }$ preserves the metric, i.e., $\overline{\nabla }g=0$, it is called a QSMC. If $\overline{\nabla }g\ne 0$, then $\overline{\nabla }$ is referred to as a quarter-symmetric non-metric connection.

In this framework, a unique QSMC can be explicitly constructed using the formula [19]:

$$\begin{array}{c}\hfill {\overline{\nabla }}_{S}T={\tilde{\overline{\nabla }}}_{S}T+{\mu }_1\zeta \left(T\right)S-{\mu }_{2}g(S,T)\mathcal{V},\end{array}$$

(1)

where $\tilde{\overline{\nabla }}$ denotes the Levi-Civita connection on $\tilde{\mathcal{M}}$, and ${\mu }_1$ and ${\mu }_2$ are real constants.

The curvature tensor associated with the quarter-symmetric connection $\overline{\nabla }$ is defined by

$$\overline{\mathcal{R}}(S,T)U={\overline{\nabla }}_S{\overline{\nabla }}_{T}U-{\overline{\nabla }}_T{\overline{\nabla }}_{S}U-{\overline{\nabla }}_{[S,T]}U,$$

(2)

for all vector fields $S,T,U$ on the Riemannian manifold $\tilde{\mathcal{M}}$.

The curvature tensor $\widetilde{\overline{\mathcal{R}}}$ associated with the Levi-Civita connection $\tilde{\overline{\nabla }}$ is defined analogously. Now, consider the auxiliary $(0,2)$-type tensors $\alpha$ and $\beta$ given by

$$\begin{array}{cc}\hfill \alpha (S,T)& =\left({\tilde{\overline{\nabla }}}_S\zeta \right)\left(T\right)-{\mu }_1\zeta \left(S\right)\zeta \left(T\right)+{\displaystyle \frac{{\mu }_2}{2}}g(S,T)\zeta \left(\mathcal{V}\right),\hfill \\ \hfill \beta (S,T)& ={\displaystyle \frac{1}{2}}\zeta \left(\mathcal{V}\right)g(S,T)+\zeta \left(S\right)\zeta \left(T\right),\hfill \end{array}$$

where $\zeta$ is a 1-form defined by $\zeta \left(X\right)=g(X,\mathcal{V})$, and $\mathcal{V}$ is a fixed vector field on $\tilde{\mathcal{M}}$.

Let $a=\mathrm{tr}\left(\alpha \right)$ and $b=\mathrm{tr}\left(\beta \right)$ be the traces of the tensors $\alpha$ and $\beta$, respectively. Then, following [20], the curvature tensor $\overline{\mathcal{R}}$ with respect to the quarter-symmetric metric connection $\overline{\nabla }$ is given by

$$\begin{array}{cc}\hfill \overline{\mathcal{R}}(S,T,U,V)& =\widetilde{\overline{\mathcal{R}}}(S,T,U,V)+{\mu }_1\alpha (S,U)g(T,V)-{\mu }_1\alpha (T,U)g(S,V)\hfill \\ \hfill & +{\mu }_2\alpha (T,V)g(S,U)-{\mu }_2\alpha (S,V)g(T,U)\hfill \\ \hfill & +{\mu }_2({\mu }_1-{\mu }_2)g(S,U)\beta (T,V)-{\mu }_2({\mu }_1-{\mu }_2)g(T,U)\beta (S,V),\hfill \end{array}$$

(3)

for all vector fields $S,T,U,V$ on $\tilde{\mathcal{M}}$.

Suppose that $\mathcal{M}$ is a $\mathrm{m}$-dimensional submanifold of a Riemannian manifold $\tilde{\mathcal{M}}$. Then, let $\tilde{\overline{\nabla }}$, $\tilde{\nabla }$ represent the Levi-Civita connection and the induced QSMC, respectively. The Gauss formula is therefore provided by:

$$\begin{array}{c}\hfill {\overline{\nabla }}_{S}T={\nabla }_{S}T+h(S,T),\end{array}$$

(4)

$$\begin{array}{c}\hfill {\tilde{\overline{\nabla }}}_{S}T={\tilde{\nabla }}_{S}T+\tilde{h}(S,T).\end{array}$$

(5)

In this case, $\tilde{h}$ represents the second fundamental form of $\mathcal{M}$ in $\tilde{\mathcal{M}}$ with respect to $\tilde{\overline{\nabla }}$ and h is the second fundamental form of $\mathcal{M}$ in $\tilde{\mathcal{M}}$ with respect to $\tilde{\nabla }$, as provided by $h(S,T)=\tilde{h}(S,T)-{\mu }_{2}g(S,T){\mathcal{V}}^{\perp }$.

In addition, Wang [20] provides the Gauss equations.

$$\begin{array}{ccc}\hfill \overline{\mathcal{R}}(S,T,U,V)& =& \mathcal{R}(S,T,U,V)-g\left(h\right(S,V),h(T,U\left)\right)+g\left(h\right(T,V),h(S,U\left)\right)\hfill \\ & & +({\mu }_1-{\mu }_2)g(h(T,U),{\mathcal{V}}^{\perp })g(S,V)\hfill \\ & & +({\mu }_2-{\mu }_1)g(h(S,U),{\mathcal{V}}^{\perp })g(T,V),\hfill \end{array}$$

(6)

for all $S,T,U,V\in \Gamma \left(\mathcal{TM}\right)$.

Let $\tilde{\mathcal{M}}$ be a Riemannian manifold of odd dimension $(2n+1)$. Suppose that there exist a $(1,1)$-type tensor field $\phi$, a vector field $\xi$, and a 1-form $\zeta$ satisfying the following conditions:

$${\phi }^2\left(S\right)=-S+\zeta \left(S\right)\xi ,g(S,\xi )=\zeta \left(S\right),$$

$$g(\phi S,\phi T)=g(S,T)-\zeta \left(S\right)\zeta \left(T\right),$$

(7)

for all vector fields $S,T$ on $\tilde{\mathcal{M}}$. Then, $\tilde{\mathcal{M}}$ is said to possess an almost contact metric structure denoted by $(\phi ,\xi ,\zeta ,g)$ [21,22].

The associated 2-form $\Phi$, defined by $\Phi (S,T)=g(S,\phi T)$, is referred to as the fundamental 2-form. If this form coincides with the exterior derivative of $\zeta$, i.e., $\Phi =d\zeta$, then the manifold is known as a contact metric manifold.

A contact metric manifold becomes a Sasakian manifold when the following condition holds for all vector fields $S,T$ on $\tilde{\mathcal{M}}$:

$$\left({\nabla }_S\phi \right)\left(T\right)=g(S,T)\xi -\zeta \left(T\right)S.$$

This characterizes Sasakian geometry through a specific interaction between the Levi-Civita connection ∇ and the tensor field $\phi$.

Let $\tilde{\mathcal{M}}$ be a $(2n+1)$-dimensional Riemannian manifold endowed with an almost contact metric structure $(\phi ,\xi ,\zeta ,g)$, which is said to be a conformal Sasakian manifold if for a smooth function $f:\tilde{\mathcal{M}}\to \mathbb{R}$, there exist (see [1]):

$$\begin{array}{c}\hfill \tilde{g}=e^{f}g,\tilde{\phi }=\phi ,\tilde{\xi }=e^{-{\displaystyle \frac{f}{2}}}\xi ,\tilde{\zeta }=e^{{\displaystyle \frac{f}{2}}}\zeta ,\end{array}$$

(8)

such that $(\tilde{\mathcal{M}},\tilde{g},\tilde{\phi },\tilde{\zeta },\tilde{\xi })$ is a Sasakian manifold.

We obtain the following relationship between the connections $\tilde{\nabla }$ and ∇ using Koszul formula.

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{S}T={\nabla }_{S}T+{\displaystyle \frac{1}{2}}\{\omega \left(S\right)T+\omega \left(T\right)S-g(S,T){\omega }^{\#}\},\end{array}$$

(9)

for all vector fields $S,T$ on $\tilde{\mathcal{M}}$, such that $\omega \left(S\right)=S\left(f\right)$ and ${\omega }^{\#}$ are vector fields that are metrically equivalent to 1-form $\omega$, specifically $g({\omega }^{\#},S)=\omega \left(S\right)$. The Lee vector field of conformal Sasakian manifold $\tilde{\mathcal{M}}$ is a vector field ${\omega }^{\#}=gradf$.

The curvature tensor of the almost contact metric manifold $(\tilde{\mathcal{M}},\phi ,\xi ,\zeta ,g)$ satisfies [1]:

$$\begin{array}{ccc}\hfill g(\widetilde{\overline{\mathcal{R}}}(S,T)U,V)& =& e^f\{\left({\displaystyle \frac{k+3}{4}}\right)\left(g(T,U)g(S,V)-g(S,U)g(T,V)\right)\hfill \\ & & +\left({\displaystyle \frac{k-1}{4}}\right)(\zeta \left(S\right)\zeta \left(U\right)g(T,V)-\zeta \left(T\right)\zeta \left(U\right)g(S,V)\hfill \\ & & +g(S,U)g(\xi ,V)\zeta \left(T\right)-g(T,U)g(\xi ,V)\zeta \left(S\right)\hfill \\ & & -g(\varphi T,Z)g(\varphi S,V)-g(\varphi S,U)g(\varphi T,V)\hfill \\ & & -2g(\varphi S,T)g(\varphi U,V))\}-{\displaystyle \frac{1}{2}}(B(S,U)g(T,V)\hfill \\ & & -B(T,U)g(S,V)+B(T,V)g(T,U)-B(S,V)g(T,U))\hfill \\ & & -{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2\left(g(S,U)g(T,V)-g(S,V)g(T,U)\right),\hfill \end{array}$$

(10)

where $B=\nabla \omega -{\displaystyle \frac{1}{2}}\omega \otimes \omega$, $\left|\right|{\omega }^{\#}\left|\right|$ is the norm of ${\omega }^{\#}$ with respect to the Riemannian metric g, and this relation holds for all vector fields $S,T,U,$ and V on $\tilde{\mathcal{M}}$. In this context, $\tilde{\mathcal{M}}$ is referred to as a CSSF.

From (3) and (10), we get [23]:

$$\begin{array}{ccc}\hfill g(\overline{\mathcal{R}}(S,T)U,V)& =& e^f\{\left({\displaystyle \frac{k+3}{4}}\right)\left(g(T,U)g(S,V)-g(S,U)g(T,V)\right)\hfill \\ & & +\left({\displaystyle \frac{k-1}{4}}\right)(\zeta \left(S\right)\zeta \left(U\right)g(T,V)-\zeta \left(T\right)\zeta \left(U\right)g(S,V)\hfill \\ & & +g(S,U)g(\xi ,V)\zeta \left(T\right)-g(T,U)g(\xi ,V)\zeta \left(S\right)\hfill \\ & & -g(\varphi T,U)g(\varphi S,V)-g(\varphi S,U)g(\varphi T,V)\hfill \\ & & -2g(\varphi S,T)g(\varphi U,V))\}-{\displaystyle \frac{1}{2}}(B(S,U)g(T,V)\hfill \\ & & -B(T,U)g(S,V)+B(T,V)g(T,U)-B(S,V)g(T,U))\hfill \\ & & -{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2\left(g(S,U)g(T,V)-g(S,V)g(T,U)\right)\hfill \\ & & +{\mu }_1\alpha (S,U)g(T,V)-{\mu }_1\alpha (T,U)g(S,V)\hfill \\ & & +{\mu }_{2}g(S,U)\alpha (T,V)-{\mu }_{2}g(T,U)\alpha (S,V)\hfill \\ & & +{\mu }_2({\mu }_1-{\mu }_2)g(S,U)\beta (T,V)-{\mu }_2({\mu }_1-{\mu }_2)g(T,U)\beta (S,V),\hfill \end{array}$$

(11)

for any vector fields $S,T,U,V$ on $\tilde{\mathcal{M}}$.

3. Casorati Curvatures

The Casorati curvature of a submanifold $\mathcal{M}$ of dimension $\mathrm{m}$ in a conformal Sasakian manifold $\tilde{\mathcal{M}}$ that is (2n+1)-dimensional is examined in this section. Think about a local orthonormal normal frame $\{X_{\mathrm{m}+l},...,X_{2n+1}\}$ of the normal bundle ${\mathcal{T}}^{\perp }\mathcal{M}$ of $\mathcal{M}$ in $\tilde{\mathcal{M}}$ and a local orthonormal tangent frame $\{X_1,...,X_{\mathrm{m}}\}$ of the tangent bundle $\mathcal{TM}$ of $\mathcal{M}$. Given the scalar curvature $\tau$ at any $p\in \mathcal{M}$,

$$\begin{array}{ccc}\hfill \tau & =& \sum _{1\le i<j\le \mathrm{m}}\mathcal{R}(X_i,X_j,X_j,X_i).\hfill \end{array}$$

(12)

Furthermore, the normalized scalar curvature $\rho$ of $\mathcal{M}$ is described as

$$\begin{array}{ccc}\hfill \rho & =& {\displaystyle \frac{2\tau }{\mathrm{m}(\mathrm{m}-1)}}.\hfill \end{array}$$

(13)

The mean curvature $\mathcal{H}$ of $\mathcal{M}$ can be found using

$$\begin{array}{ccc}\hfill \mathcal{H}& =& {\displaystyle \frac{1}{\mathrm{m}}}\sum _{i=1}^{\mathrm{m}}h(X_i,X_i).\hfill \end{array}$$

Easily, let us place

$$\begin{array}{ccc}\hfill h_{ij}^r& =& g(h(X_i,X_i),X_r)\hfill \end{array}$$

for every $i,j=\{1,...,\mathrm{m}\}$ and $r=\{\mathrm{m}+1,...,2n+1\}$. Next, the squared norm of the mean curvature is as follows:

$$\begin{array}{c}\hfill {\left|\right|\mathcal{H}\left|\right|}^2={\displaystyle \frac{1}{{\mathrm{m}}^2}}\sum _{r=\mathrm{m}+1}^{2n+1}{\left(\sum _{i,j=1}^{\mathrm{m}}{h}_{ii}^r\right)}^2\end{array}$$

(14)

then $\mathcal{C}$, which is defined in [24], represents the squared norm of the second fundamental form h.

$$\begin{array}{c}\hfill \mathcal{C}={\displaystyle \frac{1}{\mathrm{m}}}{\left|\right|h\left|\right|}^2\end{array}$$

(15)

where

$$\begin{array}{ccc}\hfill {\left|\right|h\left|\right|}^2& =& {\displaystyle \frac{1}{\mathrm{m}}}\sum _{\mathrm{m}+1}^{2n+1}\sum _{i,j=1}^{\mathrm{m}}{\left(h_{ij}^r\right)}^2,\hfill \end{array}$$

referred to as Casorati curvature $\mathcal{C}$ of $\mathcal{M}$.

Consider the following: $\{X_1,...,X_s\}$ is an orthonormal basis of ℧ and ℧ is an s-dimensional subspace of $\mathcal{TM}$, $s\ge 2$. The scalar curvature of the s-plane section ℧ is given as

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

and the subspace ℧ has the following Casorati curvature $\mathcal{C}$:

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

A point $p\in \mathcal{M}$ on a submanifold is called an invariantly quasi-umbilical if there are exactly $2n+1-\mathrm{m}$ mutually orthogonal unit normal vectors ${\xi }_{\mathrm{m}+1},\dots ,{\xi }_{2n+1}$ such that the corresponding shape operators ${\mathcal{S}}_{{\xi }_{\alpha }}$, for each $\alpha =\mathrm{m}+1,\dots ,2n+1$, share a common eigenvector and have one distinct eigenvalue with multiplicity $\mathrm{m}-1$. If this condition holds at every point on the submanifold, then the submanifold itself is referred to as an invariantly quasi-umbilical submanifold.

The NDCC ${\delta }_k(\mathrm{m}-1)$ and $\widehat{{\delta }_k}(\mathrm{m}-1)$ are defined as [25]:

$$\begin{array}{c}\hfill {\left[{\delta }_k(\mathrm{m}-1)\right]}_p={\displaystyle \frac{1}{2}}{\mathcal{C}}_p+{\displaystyle \frac{\mathrm{m}+1}{2\mathrm{m}}}inf\left\{\mathcal{C}\left(\mathcal{L}\right)\right|\mho :ahyperplaneof{\mathcal{T}}_p\mathcal{M}\}\end{array}$$

(16)

and

$$\begin{array}{c}\hfill {\left[\widehat{{\delta }_k}(\mathrm{m}-1)\right]}_p={\displaystyle \frac{1}{2}}{\mathcal{C}}_p+{\displaystyle \frac{2\mathrm{m}-1}{2\mathrm{m}}}sup\left\{\mathcal{C}\left(\mathcal{L}\right)\right|\mho :ahyperplaneof{\mathcal{T}}_p\mathcal{M}\}.\end{array}$$

(17)

$t\ne \mathrm{m}(\mathrm{m}-1)$ is a positive real number; we have

$$\begin{array}{ccc}\hfill b\left(t\right)& =& {\displaystyle \frac{1}{\mathrm{m}t}}(\mathrm{m}-1)(\mathrm{m}+t)({\mathrm{m}}^2-\mathrm{m}-t);\hfill \end{array}$$

(18)

the GNDCC ${\widehat{\delta }}_k(t;\mathrm{m}-1)$ and ${\delta }_k(t;\mathrm{m}-1)$ are subsequently provided as [26]:

$$\begin{array}{c}\hfill {\left[{\delta }_k(t;\mathrm{m}-1)\right]}_p=t{\mathcal{C}}_p+b\left(t\right)inf\left\{\mathcal{C}(\mho )\right|\mho :ahyperplaneof{\mathcal{T}}_p\mathcal{M}\},\end{array}$$

if $0<t<\mathrm{m}(\mathrm{m}-1)$

and

$$\begin{array}{c}\hfill {\left[{\widehat{\delta }}_k(t;\mathrm{m}-1)\right]}_p=t{\mathcal{C}}_p+b\left(t\right)sup\left\{\mathcal{C}(\mho )\right|\mho :ahyperplaneof{\mathcal{T}}_p\mathcal{M}\},\end{array}$$

if $t>\mathrm{m}(\mathrm{m}-1).$

4. Main Results

In this section, we present the principal findings of our investigation concerning submanifolds in a CSSF endowed with a QSMC. Building upon the foundational definitions and curvature identities discussed in the preliminary section, we derive a set of optimal inequalities involving the GNDCC and the scalar curvature. These results extend and refine existing inequalities in the literature and reveal deeper geometric relationships influenced by the conformal and quarter-symmetric structures of the ambient manifold. The theorems stated below are established under specific geometric conditions, and the equality cases are also characterized precisely.

Theorem 1.

Let $\mathcal{M}$ be an $\mathrm{m}$-dimensional submanifold of CSSF $\tilde{\mathcal{M}}\left(k\right)$ of dimension $(2n+1)$, endowed with QSMC. Then, the following results hold:

(i) 

The GNDCC ${\delta }_k(t;\mathrm{m}-1)$ satisfies the inequality:

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\delta }_k(t;\mathrm{m}-1)}{\mathrm{m}(\mathrm{m}-1)}}+e^f\left\{{\displaystyle \frac{k+3}{4}}-{\displaystyle \frac{k-1}{2\mathrm{m}}}+{\displaystyle \frac{3(k-1)}{4\mathrm{m}(\mathrm{m}-1)}}{\left|\right|P\left|\right|}^2\right\}+{\displaystyle \frac{trB}{\mathrm{m}}}\hfill \\ & & +{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2-\left\{({\mu }_1+{\mu }_2){\displaystyle \frac{a}{\mathrm{m}}}+{\mu }_2({\mu }_1-{\mu }_2){\displaystyle \frac{b}{\mathrm{m}}}-({\mu }_2-{\mu }_1)\pi \left(\mathcal{H}\right)\right\},\hfill \end{array}$$

(19)

for any real number t satisfying $0<t<\mathrm{m}(\mathrm{m}-1)$.

(ii) 

The GNDCC ${\widehat{\delta }}_k(t;\mathrm{m}-1)$ satisfies the inequality:

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\widehat{\delta }}_k(t;\mathrm{m}-1)}{\mathrm{m}(\mathrm{m}-1)}}+e^f\left\{{\displaystyle \frac{k+3}{4}}-{\displaystyle \frac{k-1}{2\mathrm{m}}}+{\displaystyle \frac{3(k-1)}{4\mathrm{m}(\mathrm{m}-1)}}{\left|\right|P\left|\right|}^2\right\}+{\displaystyle \frac{trB}{\mathrm{m}}}\hfill \\ & & +{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2-\left\{({\mu }_1+{\mu }_2){\displaystyle \frac{a}{\mathrm{m}}}+{\mu }_2({\mu }_1-{\mu }_2){\displaystyle \frac{b}{\mathrm{m}}}-({\mu }_2-{\mu }_1)\pi \left(\mathcal{H}\right)\right\},\hfill \end{array}$$

(20)

for all real numbers $t>\mathrm{m}(\mathrm{m}-1)$.

Moreover, equality in either of the inequalities (19) or (20) holds if and only if the submanifold $\mathcal{M}$ is invariantly quasi-umbilical with trivial normal connection in $\tilde{\mathcal{M}}$. In such a case, there exist orthonormal tangent and normal frames $\{X_1,\dots ,X_{\mathrm{m}}\}$ and $\{X_{\mathrm{m}+1},\dots ,X_{2n+1}\}$, respectively, such that the shape operators ${\mathcal{S}}_r\equiv \mathcal{S}{X}_r$ for $r\in \mathrm{m}+1,\dots ,2n+1$ take the form:

$$\begin{array}{c}\hfill {\mathcal{S}}_{\mathrm{m}+1}=diag{\left(b,b,\dots ,b,{\displaystyle \frac{\mathrm{m}(\mathrm{m}-1)}{t}}b\right)}_{\mathrm{m}\times \mathrm{m}},{\mathcal{S}}_{\mathrm{m}+2}=\dots ={\mathcal{S}}_{2n+1}=0.\end{array}$$

(21)

Proof. 

Equations (6) and (10) are obtained by inserting $S=V=X_i$, $T=U=X_j$, $i\ne j$ into the statement

$$\begin{array}{cc}\hfill g\left(\mathcal{R}\right(X_i,X_j,& X_j,X_i)=e^f\{\left({\displaystyle \frac{k+3}{4}}\right)\left(g(X_j,X_j)g(X_i,X_i)-g(X_i,X_j)g(X_j,X_i)\right)\hfill \\ \hfill & +\left({\displaystyle \frac{k-1}{4}}\right)(\zeta \left(X_i\right)\zeta \left(X_j\right)g(X_j,X_i)-\zeta \left(X_j\right)\zeta \left(X_j\right)g(X_i,X_i)\hfill \\ \hfill & +g(X_i,X_j)g(\xi ,X_i)\zeta \left(X_j\right)-g(X_j,X_j)g(\xi ,X_i)\zeta \left(X_i\right)\hfill \\ \hfill & -g(\varphi X_j,X_j)g(\varphi X_i,X_i)-g(\varphi X_i,X_j)g(\varphi X_j,X_i)\hfill \\ \hfill & -2g(\varphi X_i,X_j)g(\varphi X_j,X_i)\left)\right\}-{\displaystyle \frac{1}{2}}(B(X_i,X_j)g(X_j,X_i)\hfill \\ \hfill & -B(X_j,X_j)g(X_i,X_i)+B(X_j,X_i)g(X_i,X_j)-B(X_i,X_i)g(X_j,X_j))\hfill \\ \hfill & -{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2\left(g(X_i,X_j)g(X_j,X_i)-g(X_j,X_j)g(X_i,X_i)\right)\hfill \\ \hfill & +{\mu }_1\alpha (X_i,X_j)g(X_j,X_i)-{\mu }_1\alpha (X_j,X_j)g(X_i,X_i)+{\mu }_{2}g(X_i,X_j)\alpha (X_j,X_i)\hfill \\ \hfill & -{\mu }_{2}g(X_j,X_j)\alpha (X_i,X_i)+{\mu }_2({\mu }_1-{\mu }_2)g(X_i,X_j)\beta (X_j,X_i)\hfill \\ \hfill & -{\mu }_2({\mu }_1-{\mu }_2)g(X_j,X_j)\beta (X_i,X_i)-({\mu }_1-{\mu }_2)g(h(X_j,X_j),{\mathcal{P}}^{\perp })g(X_i,X_i)\hfill \\ \hfill & -({\mu }_2-{\mu }_1)g(h(X_i,X_j),{\mathcal{P}}^{\perp })g(X_j,X_i)\hfill \\ \hfill & +g(h(X_i,X_i),h(X_j,X_j))-g(h(X_j,X_i),h(X_i,X_j))\hfill \end{array}$$

(22)

Utilizing the Gauss equation with (12) and the summation $1\le i,j\le \mathrm{m}$, we obtain

$$\begin{array}{ccc}\hfill 2\tau & =& e^f\{{\displaystyle \frac{(k+3)}{4}}\mathrm{m}(\mathrm{m}-1)+{\displaystyle \frac{(k-1)}{4}}\left(2-2\mathrm{m}+{3\left|\right|P\left|\right|}^2)\right\}+(\mathrm{m}-1)trB\hfill \\ & & +{\displaystyle \frac{1}{4}}\mathrm{m}(\mathrm{m}-1)\left|\right|{\omega }^{\#}{\left|\right|}^2+({\mu }_1+{\mu }_2)(1-\mathrm{m})a+{\mu }_2({\mu }_1-{\mu }_2)(1-\mathrm{m})b\hfill \\ & & +({\mu }_2-{\mu }_1)\mathrm{m}(\mathrm{m}-1)\pi \left(\mathcal{H}\right)+{\mathrm{m}}^2{\left|\right|\mathcal{H}\left|\right|}^2-\mathrm{m}\mathcal{C},\hfill \end{array}$$

(23)

where

$$\begin{array}{c}\hfill {\left|\right|P\left|\right|}^2=\sum _{i,j=1}^{\mathrm{m}}{g}^2(\varphi X_i,X_j)\mathrm{and}\pi \left(\mathcal{H}\right)={\displaystyle \frac{1}{\mathrm{m}}}\sum _{j=1}^{\mathrm{m}}\pi \left(h(X_j,X_j)\right)=g({\mathcal{V}}^{\perp },\mathcal{H}).\end{array}$$

We introduce the function $\mathcal{Q}$, which expresses a quadratic relationship involving the components of the second fundamental form.

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& t\mathcal{C}+b\left(t\right)\mathcal{C}(\mho )-2\tau +e^f\{{\displaystyle \frac{(k+3)\mathrm{m}(\mathrm{m}-1)}{4}}+{\displaystyle \frac{(k-1)}{4}}\left(2-2\mathrm{m}+{3\left|\right|P\left|\right|}^2)\right\}\hfill \\ & & +(\mathrm{m}-1)trB+{\displaystyle \frac{1}{4}}\mathrm{m}(\mathrm{m}-1)\left|\right|{\omega }^{\#}{\left|\right|}^2+({\mu }_1+{\mu }_2)(1-\mathrm{m})a\hfill \\ & & +{\mu }_2({\mu }_1-{\mu }_2)(1-\mathrm{m})b+({\mu }_2-{\mu }_1)\mathrm{m}(\mathrm{m}-1)\pi \left(\mathcal{H}\right)\hfill \end{array}$$

(24)

where the hyperplane of ${\mathcal{T}}_p\mathcal{M}$ is denoted by ℧. If we assume, without losing generality, that ℧ is spanned by $\{X_1,...,X_{\mathrm{m}-1}\}$, then (24) implies that

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& {\displaystyle \frac{t}{m}}\sum _{r=\mathrm{m}+1}^{2n+1}\sum _{i,j=1}^{\mathrm{m}}{\left(h_{ij}^r\right)}^2+{\displaystyle \frac{b\left(t\right)}{m-1}}\sum _{r=\mathrm{m}+1}^{2n+1}\sum _{i,j=1}^{\mathrm{m}-1}{\left(h_{ij}^r\right)}^2-2\tau \hfill \\ & & +e^f\{{\displaystyle \frac{(k+3)\mathrm{m}(\mathrm{m}-1)}{4}}+{\displaystyle \frac{(k-1)}{4}}\left(2-2\mathrm{m}+{3\left|\right|P\left|\right|}^2)\right\}\hfill \\ & & +(\mathrm{m}-1)trB+{\displaystyle \frac{1}{4}}\mathrm{m}(\mathrm{m}-1)\left|\right|{\omega }^{\#}{\left|\right|}^2+({\mu }_1+{\mu }_2)(1-\mathrm{m})a\hfill \\ & & +{\mu }_2({\mu }_1-{\mu }_2)(1-\mathrm{m})b+({\mu }_2-{\mu }_1)\mathrm{m}(\mathrm{m}-1)\pi \left(\mathcal{H}\right).\hfill \end{array}$$

(25)

Combining (23) and (25), we get

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& {\displaystyle \frac{t}{m}}\sum _{r=\mathrm{m}+1}^{2n+1}\sum _{i,j=1}^{\mathrm{m}}{\left(h_{ij}^r\right)}^2+{\displaystyle \frac{b\left(t\right)}{m-1}}\sum _{r=\mathrm{m}+1}^{2n+1}\sum _{i,j=1}^{\mathrm{m}-1}{\left(h_{ij}^r\right)}^2-{\mathrm{m}}^2{\left|\right|\mathcal{H}\left|\right|}^2+\mathrm{m}\mathcal{C},\hfill \end{array}$$

(26)

which can be further written as

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

(27)

We may observe the critical points from (27).

$$\begin{array}{c}\hfill h^c=\left\{h_{11}^{\mathrm{m}+1},h_{12}^{\mathrm{m}+1},...,h_{\mathrm{m}\mathrm{m}}^{\mathrm{m}+1},...,h_{11}^{2n+1},...,h_{\mathrm{m}\mathrm{m}}^{2n+1}\right\}\end{array}$$

of $\mathcal{Q}$ are the solutions to the subsequent homogeneous equation system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\mathsf{\partial }\mathcal{Q}}{\mathsf{\partial }{h}_{ii}^r}}=2\left({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}}-1\right)h_{ii}^r-2\sum _{q=1}^{\mathrm{m}}{h}_{qq}^r=0\hfill \\ \\ {\displaystyle \frac{\mathsf{\partial }\mathcal{Q}}{\mathsf{\partial }{h}_{\mathrm{m}\mathrm{m}}^r}}={\displaystyle \frac{2t}{\mathrm{m}}}{h}_{\mathrm{m}\mathrm{m}}^r-2\sum _{q=1}^{\mathrm{m}-1}{h}_{qq}^r=0\hfill \\ \\ {\displaystyle \frac{\mathsf{\partial }\mathcal{Q}}{\mathsf{\partial }{h}_{ij}^r}}=4\left({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}}\right)h_{ij}^r=0\hfill \\ \\ {\displaystyle \frac{\mathsf{\partial }\mathcal{Q}}{\mathsf{\partial }{h}_{i\mathrm{m}}^r}}=4\left({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}\right)h_{i\mathrm{m}}^r=0,\hfill \end{array}\right.\end{array}$$

(28)

within $i,j=\{1,2,...,\mathrm{m}-1\}$, $i\ne j$ and $r\in \{\mathrm{m}+1,....,2n+1\}$. This means that the solution to $h^c$ and $h_{ij}^r=0$ for any $i\ne j$ and the related determinant to the system’s first two equations above is zero. It also has a Hessian matrix, which is the following block matrix in the form of $\mathcal{Q}$:

$$\begin{array}{c}\hfill H\left(\mathcal{Q}\right)=\left(\begin{array}{ccc}{\mathcal{H}}_1& O& O\\ O& {\mathcal{H}}_2& O\\ O& O& {\mathcal{H}}_3.\end{array}\right)\end{array}$$

The blocks are in the corresponding dimension’s null matrix, denoted by O.

$$\begin{array}{c}\hfill {\mathcal{H}}_1={\left(\begin{array}{ccccccc}2({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}})-2& -2& .& .& .& -2& -2\\ -2& 2({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}})-2& .& .& .& -2& -2\\ .& .& .& & & .& .\\ .& .& & .& & .& .\\ .& .& & & .& .& .\\ -2& -2& .& .& .& 2({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}})-2& -2\\ -2& -2& .& .& .& -2& {\displaystyle \frac{2t}{\mathrm{m}}}\end{array}\right)}_{\mathrm{m}\times \mathrm{m}},\end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\mathcal{H}}_2& =& diag{\left[4({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}}-1),4({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}}-1),...,4({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}}-1)\right]}_{\mathrm{m}\times \mathrm{m}}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathcal{H}}_3& =& diag{\left[4{\displaystyle \frac{(\mathrm{m}+t)}{\mathrm{m}}},4{\displaystyle \frac{(\mathrm{m}+t)}{\mathrm{m}}},...,4{\displaystyle \frac{(\mathrm{m}+t)}{\mathrm{m}}}\right]}_{\mathrm{m}\times \mathrm{m}}.\hfill \end{array}$$

Thus, we discover that the eigenvalues of $\mathcal{H}\left(\mathcal{Q}\right)$ are as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mu }_{11}=0,\hfill \\ {\mu }_{22}=2({\displaystyle \frac{2t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}}),\hfill \\ {\mu }_{33}=...={\mu }_{\mathrm{m}\mathrm{m}}=2({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}}),\hfill \\ {\mu }_{ij}=4({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}-1}}),\hfill \\ {\mu }_{i\mathrm{m}}=4\left({\displaystyle \frac{\mathrm{m}+t}{\mathrm{m}}}\right),\forall i,j\in \{1,2,...,\mathrm{m}-1\},i\ne j.\hfill \end{array}\right.\end{array}$$

(30)

As a result, for some solution $h^c$ of the system (28), $\mathcal{Q}$ is parabolic and reaches at least $\mathcal{Q}\left(h^c\right)=0$. Thus, $\mathcal{Q}\ge 0$ and consequently

$$\begin{array}{ccc}\hfill 2\tau & \le & t\mathcal{C}+b\left(t\right)\mathcal{C}(\mho )+e^f\left\{{\displaystyle \frac{(k+3)\mathrm{m}(\mathrm{m}-1)}{4}}+{\displaystyle \frac{(k-1)}{4}}\left(2-2\mathrm{m}+{3\left|\right|P\left|\right|}^2\right)\right\}\hfill \\ & & +(\mathrm{m}-1)trB+{\displaystyle \frac{1}{4}}\mathrm{m}(\mathrm{m}-1)\left|\right|{\omega }^{\#}{\left|\right|}^2+({\mu }_1+{\mu }_2)(1-\mathrm{m})a\hfill \\ & & +{\mu }_2({\mu }_1-{\mu }_2)(1-\mathrm{m})b+({\mu }_2-{\mu }_1)\mathrm{m}(\mathrm{m}-1)\pi \left(\mathcal{H}\right),\hfill \end{array}$$

by means of which we acquire

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{1}{\mathrm{m}(\mathrm{m}-1)}}\mathcal{C}+{\displaystyle \frac{b\left(t\right)}{\mathrm{m}(\mathrm{m}-1)}}\mathcal{C}(\mho )+{\displaystyle \frac{e^f}{\mathrm{m}(\mathrm{m}-1)}}\{{\displaystyle \frac{(k+3)\mathrm{m}(\mathrm{m}-1)}{4}}\hfill \\ & & +{\displaystyle \frac{(k-1)}{4}}\left(2-2\mathrm{m}+{3\left|\right|P\left|\right|}^2\right)\}+{\displaystyle \frac{trB}{\mathrm{m}}}+{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2\hfill \\ & & +{\displaystyle \frac{1}{\mathrm{m}(\mathrm{m}-1)}}\left\{({\mu }_1+{\mu }_2)(1-\mathrm{m})a+{\mu }_2({\mu }_1-{\mu }_2)(1-\mathrm{m})b\right\},\hfill \end{array}$$

for each and every hyperplane ℧ of $\mathcal{M}$. The result follows easily if we take the infimum over all tangent hyperplanes ℧; i.e.,

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\delta }_k(t,\mathrm{m}-1)}{\mathrm{m}(\mathrm{m}-1)}}+{\displaystyle \frac{e^f}{\mathrm{m}(\mathrm{m}-1)}}\left\{{\displaystyle \frac{(k+3)\mathrm{m}(\mathrm{m}-1)}{4}}+{\displaystyle \frac{(k-1)}{4}}{(2-2\mathrm{m}+3|\left|P\right||}^2)\right\}\hfill \\ & & +{\displaystyle \frac{trB}{\mathrm{m}}}+{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2-\left\{({\mu }_1+{\mu }_2){\displaystyle \frac{a}{\mathrm{m}}}+{\mu }_2({\mu }_1-{\mu }_2){\displaystyle \frac{b}{\mathrm{m}}}-({\mu }_2-{\mu }_1)\pi \left(\mathcal{H}\right)\right\}.\hfill \end{array}$$

In addition, the ratio remains valid only in the event that

$$\begin{array}{c}\hfill h_{ij}^r=0,\forall i,j\in \{1,...,\mathrm{m}\},i\ne j,\end{array}$$

(31)

and

$$\begin{array}{c}\hfill h_{\mathrm{m}\mathrm{m}}^r={\displaystyle \frac{\mathrm{m}(\mathrm{m}-1)}{t}}{h}_{11}^r=...={\displaystyle \frac{\mathrm{m}(\mathrm{m}-1)}{t}}{h}_{\mathrm{m}-1\mathrm{m}-1}^r,\end{array}$$

(32)

pertaining to every $r\in \{\mathrm{m}+1,...,2n+1\}$.

So, based on the calculations of (31) and (32), we deduce that equality holds if and only if the submanifold in $\tilde{\mathcal{M}}$ is invariantly quasi-umbilical with trivial normal connection, meaning that the shape operators satisfy (21) with respect to appropriate tangent and normal orthonormal frames.

Likewise, we can demonstrate (ii). □

An easy conclusion of the preceding theorem is the following result:

Corollary 1.

Let $\mathcal{M}$ be an $\mathrm{m}$-dimensional submanifold of a CSSF $\tilde{\mathcal{M}}\left(k\right)$ of dimension $(2n+1)$, endowed with QSMC. Then, the following statements hold:

(i) 

The NDCC ${\delta }_k(t;\mathrm{m}-1)$ satisfies the inequality

$$\begin{array}{ccc}\hfill \rho & \le & {\delta }_k(t;\mathrm{m}-1)+e^f\left\{{\displaystyle \frac{k+3}{4}}-{\displaystyle \frac{k-1}{2\mathrm{m}}}+{\displaystyle \frac{3(k-1)}{4\mathrm{m}(\mathrm{m}-1)}}{\left|\right|P\left|\right|}^2\right\}\hfill \\ & & +{\displaystyle \frac{trB}{\mathrm{m}}}+{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2-\left\{({\mu }_1+{\mu }_2){\displaystyle \frac{a}{\mathrm{m}}}+{\mu }_2({\mu }_1-{\mu }_2){\displaystyle \frac{b}{\mathrm{m}}}-({\mu }_2-{\mu }_1)\pi \left(\mathcal{H}\right)\right\}.\hfill \end{array}$$

(33)

(ii) 

The NDCC ${\widehat{\delta }}_k(t;\mathrm{m}-1)$ satisfies the inequality

$$\begin{array}{ccc}\hfill \rho & \le & {\widehat{\delta }}_k(t;\mathrm{m}-1)+e^f\left\{{\displaystyle \frac{k+3}{4}}-{\displaystyle \frac{k-1}{2\mathrm{m}}}+{\displaystyle \frac{3(k-1)}{4\mathrm{m}(\mathrm{m}-1)}}{\left|\right|P\left|\right|}^2\right\}\hfill \\ & & +{\displaystyle \frac{trB}{\mathrm{m}}}+{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2-\left\{({\mu }_1+{\mu }_2){\displaystyle \frac{a}{\mathrm{m}}}+{\mu }_2({\mu }_1-{\mu }_2){\displaystyle \frac{b}{\mathrm{m}}}-({\mu }_2-{\mu }_1)\pi \left(\mathcal{H}\right)\right\}.\hfill \end{array}$$

(34)

Furthermore, equality in either inequality (33) or (34) holds if and only if the submanifold $\mathcal{M}$ is invariantly quasi-umbilical and has a trivial normal connection in the ambient manifold $\tilde{\mathcal{M}}$. In such a case, there exist orthonormal frames $\{X_1,\dots ,X_{\mathrm{m}}\}$ for the tangent bundle $T\mathcal{M}$ and $\{X_{\mathrm{m}+1},\dots ,X_{2n+1}\}$ for the normal bundle $T^{\perp }\mathcal{M}$, such that the shape operators ${\mathcal{S}}_r$ assume the following diagonal forms:

$$\begin{array}{c}\hfill {\mathcal{S}}_{\mathrm{m}+1}=diag{(b,b,\dots ,b,2b)}_{\mathrm{m}\times \mathrm{m}},{\mathcal{S}}_{\mathrm{m}+2}=\dots ={\mathcal{S}}_{2n+1}=0,\end{array}$$

(35)

or

$$\begin{array}{c}\hfill {\mathcal{S}}_{\mathrm{m}+1}=diag{(2b,2b,\dots ,2b,b)}_{\mathrm{m}\times \mathrm{m}},{\mathcal{S}}_{\mathrm{m}+2}=\dots ={\mathcal{S}}_{2n+1}=0.\end{array}$$

(36)

Proof. 

(i) It is evident that the following relationship exists:

$$\begin{array}{c}\hfill {\left[{\delta }_k({\displaystyle \frac{\mathrm{m}(\mathrm{m}-1)}{2}}:\mathrm{m}-1)\right]}_p=\mathrm{m}(\mathrm{m}-1){\left[{\delta }_k(\mathrm{m}-1)\right]}_p,\end{array}$$

(37)

anywhere $p\in \mathcal{M}$ occurs. Therefore, we have our assumption by using (37) and substituting $t={\displaystyle \frac{\mathrm{m}(\mathrm{m}-1)}{2}}$ in (19).

(ii) It is simple to verify the following relationship:

$$\begin{array}{c}\hfill {\left[{\widehat{\delta }}_k({\displaystyle \frac{\mathrm{m}(\mathrm{m}-1)}{2}}:\mathrm{m}-1)\right]}_p=\mathrm{m}(\mathrm{m}-1){\left[{\widehat{\delta }}_k(\mathrm{m}-1)\right]}_p,\mathrm{for}\mathrm{all}p\in \mathcal{M}.\end{array}$$

(38)

at any point $p\in \mathcal{M}$. Therefore, by replacing $t={\displaystyle \frac{\mathrm{m}(\mathrm{m}-1)}{2}}$ in (20), we can make our claim. □

The straightforward result for the semi-symmetric metric connection ${\mu }_1={\mu }_2=1$ is as follows:

Theorem 2.

Let $\tilde{\mathcal{M}}\left(k\right)$ be a CSSF of dimension $(2n+1)$, and let $\mathcal{M}$ be an $\mathrm{m}$-dimensional submanifold of $\tilde{\mathcal{M}}\left(k\right)$ endowed with a semi-symmetric metric connection. Then, the following inequalities hold:

(i) 

The GNDCC ${\delta }_k(t;\mathrm{m}-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\delta }_k(t;\mathrm{m}-1)}{\mathrm{m}(\mathrm{m}-1)}}+e^f\left\{{\displaystyle \frac{k+3}{4}}-{\displaystyle \frac{k-1}{2\mathrm{m}}}+{\displaystyle \frac{3(k-1)}{4\mathrm{m}(\mathrm{m}-1)}}{\left|\right|P\left|\right|}^2\right\}\hfill \\ & & +{\displaystyle \frac{trB}{\mathrm{m}}}+{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2-{\displaystyle \frac{2a}{\mathrm{m}}},\hfill \end{array}$$

(39)

for all real numbers t such that $0<t<\mathrm{m}(\mathrm{m}-1)$.

(ii) 

The GNDCC ${\widehat{\delta }}_k(t;\mathrm{m}-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\widehat{\delta }}_k(t;\mathrm{m}-1)}{\mathrm{m}(\mathrm{m}-1)}}+e^f\left\{{\displaystyle \frac{k+3}{4}}-{\displaystyle \frac{k-1}{2\mathrm{m}}}+{\displaystyle \frac{3(K-1)}{4\mathrm{m}(\mathrm{m}-1)}}{\left|\right|P\left|\right|}^2\right\}\hfill \\ & & +{\displaystyle \frac{trB}{\mathrm{m}}}+{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2-{\displaystyle \frac{2a}{\mathrm{m}}},\hfill \end{array}$$

(40)

for all real numbers $t>\mathrm{m}(\mathrm{m}-1)$.

Moreover, equality in either inequality (39) or (40) holds if and only if the submanifold $\mathcal{M}$ is invariantly quasi-umbilical and has a trivial normal connection in the ambient manifold $\tilde{\mathcal{M}}$. In this case, there exist orthonormal frames $\{X_1,\dots ,X_{\mathrm{m}}\}$ for the tangent bundle $T\mathcal{M}$ and $\{X_{\mathrm{m}+1},\dots ,X_{2n+1}\}$ for the normal bundle $T^{\perp }\mathcal{M}$, such that the shape operators $\mathcal{S}r$ take the form:

$$\begin{array}{c}\hfill {\mathcal{S}}_{\mathrm{m}+1}=diag{\left(b,b,\dots ,b,{\displaystyle \frac{\mathrm{m}(\mathrm{m}-1)}{t}}b\right)}_{\mathrm{m}\times \mathrm{m}},{\mathcal{S}}_{\mathrm{m}+2}=\dots ={\mathcal{S}}_{2n+1}=0.\end{array}$$

(41)

The semi-symmetric non-metric connection ${\mu }_1=1$ and ${\mu }_2=0$ has the following application as a direct outcome:

Theorem 3.

Let $\mathcal{M}$ be an $\mathrm{m}$-dimensional submanifold equipped with a semi-symmetric non-metric connection in a CSSF $\tilde{\mathcal{M}}\left(k\right)$ of dimension $(2n+1)$. Then, the following inequalities hold:

(i) 

The GNDCC ${\delta }_k(t;\mathrm{m}-1)$ satisfies:

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\delta }_k(t;\mathrm{m}-1)}{\mathrm{m}(\mathrm{m}-1)}}+e^f\left\{{\displaystyle \frac{k+3}{4}}-{\displaystyle \frac{k-1}{2\mathrm{m}}}+{\displaystyle \frac{3(k-1)}{\mathrm{m}(\mathrm{m}-1)}}{\left|\right|P\left|\right|}^2\right\}\hfill \\ & & +{\displaystyle \frac{trB}{\mathrm{m}}}+{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2-{\displaystyle \frac{a}{\mathrm{m}}}-\pi \left(\mathcal{H}\right),\hfill \end{array}$$

(42)

for all real values of t satisfying $0<t<\mathrm{m}(\mathrm{m}-1)$.

(ii) 

The GNDCC ${\widehat{\delta }}_k(t;\mathrm{m}-1)$ satisfies:

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{{\widehat{\delta }}_k(t;\mathrm{m}-1)}{\mathrm{m}(\mathrm{m}-1)}}+e^f\left\{{\displaystyle \frac{k+3}{4}}-{\displaystyle \frac{k-1}{2\mathrm{m}}}+{\displaystyle \frac{3(k-1)}{\mathrm{m}(\mathrm{m}-1)}}{\left|\right|P\left|\right|}^2\right\}\hfill \\ & & +{\displaystyle \frac{trB}{\mathrm{m}}}+{\displaystyle \frac{1}{4}}\left|\right|{\omega }^{\#}{\left|\right|}^2-{\displaystyle \frac{a}{\mathrm{m}}}-\pi \left(\mathcal{H}\right),\hfill \end{array}$$

(43)

for any real number $t>\mathrm{m}(\mathrm{m}-1)$.

Moreover, equality in either inequality (42) or (43) holds if and only if the submanifold $\mathcal{M}$ is invariantly quasi-umbilical and has a trivial normal connection in the ambient space $\tilde{\mathcal{M}}$. In such a case, there exist orthonormal frames $\{X_1,\dots ,X_{\mathrm{m}}\}$ for the tangent bundle $T\mathcal{M}$ and $\{X_{\mathrm{m}+1},\dots ,X_{2n+1}\}$ for the normal bundle $T^{\perp }\mathcal{M}$, such that the shape operators ${\mathcal{S}}_r$ assume the diagonal form:

$$\begin{array}{c}\hfill {\mathcal{S}}_{\mathrm{m}+1}=diag{\left(b,b,\dots ,b,{\displaystyle \frac{\mathrm{m}(\mathrm{m}-1)}{t}}b\right)}_{\mathrm{m}\times \mathrm{m}},{\mathcal{S}}_{\mathrm{m}+2}=\cdots ={\mathcal{S}}_{2n+1}=0.\end{array}$$

(44)

5. Example

To bolster our findings, this section builds and examines an instance of a conformal Sasakian manifold that is not a Sasakian manifold.

Example 1.

Take ${\tilde{\mathcal{M}}}^3=\{(x,y,z)\in {\mathbb{R}}^3:z>0\}$ whose linearity independent vector fields are defined as

$$X_1=2({\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}+y{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}),X_2=2{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }y}},X_3=2e^{{\displaystyle \frac{z}{2}}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}.$$

Let the matrix for the Riemannian metric g be defined by

$$\begin{array}{c}\hfill \left(g_{ij}\right)=diag{\left[e^{-z},e^{-z},1\right]}_{3\times 3}.\end{array}$$

Then, the 1-form ζ is given by

$$\begin{array}{c}\hfill \zeta \left(X_i\right)=\left\{\begin{array}{cc}0,\hfill & \mathit{for}i=1,2;\hfill \\ 1,\hfill & \mathit{for}i=3\hfill \end{array}\right.\end{array}$$

Then, taking $X_3=\xi ,(\varphi ,\xi ,\zeta ,g)$ defines an almost contact metric structure on $\tilde{\mathcal{M}}$, where ϕ satisfies

$$\begin{array}{c}\hfill \varphi X_1=X_2,\varphi X_2=-X_1\mathit{and}\varphi X_3=0.\end{array}$$

Also, we get

$$\begin{array}{c}\hfill [X_1,X_2]=(-2e^{{\displaystyle \frac{-z}{2}}})X_3,[X_1,X_3]=y{E}_3,[X_2,X_3]=0.\end{array}$$

Next, by the virtue of the Koszul formula, we get

$$\begin{array}{ccc}\hfill {\nabla }_{X_1}{X}_2& =& y_2-e^{{\displaystyle \frac{z}{2}}}{X}_3,{\nabla }_{X_1}{X}_3=-e^{{\displaystyle \frac{z}{2}}}(X_1-X_2),\hfill \\ \hfill {\nabla }_{X_2}{X}_3& =& e^{{\displaystyle \frac{z}{2}}}(X_1+X_2),{\nabla }_{X_2}{X}_2=y{X}_1+e^{{\displaystyle \frac{z}{2}}}{X}_3,\hfill \\ \hfill {\nabla }_{X_3}{X}_3& =& y{e}^z{X}_1,{\nabla }_{X_3}{X}_2=-e^{{\displaystyle \frac{z}{2}}}(X_1+X_2),\hfill \\ \hfill {\nabla }_{X_1}{X}_1& =& y{X}_1+e^{{\displaystyle \frac{z}{2}}}{X}_3,{\nabla }_{X_3}{X}_1=e^{{\displaystyle \frac{z}{2}}}(X_1-X_2)y{X}_3,\hfill \\ \hfill {\nabla }_{X_2}{X}_1& =& -y{X}_2+e^{{\displaystyle \frac{z}{2}}}{X}_3.\hfill \end{array}$$

Further, we have a Sasakian manifold $(\tilde{\mathcal{M}},\tilde{\varphi },\tilde{\xi },\tilde{\zeta },\tilde{g})$ that fulfills [27]

$$\begin{array}{c}\hfill \tilde{g}=e^{x}g,\tilde{\xi }=e^{{\displaystyle \frac{-x}{2}}}\xi ,\tilde{\zeta }=e^{{\displaystyle \frac{x}{2}}}\zeta ,\tilde{\varphi }=\varphi .\end{array}$$

Accordingly, $\tilde{\mathcal{M}}$ is not a Sasakian manifold, but it is a conformal Sasakian manifold.

6. Conclusions and Future Work

In this work, we have examined submanifolds of CSSF equipped with a quarter-symmetric connection and established several optimal inequalities involving the GNDCC and scalar curvature. These results provide new insights into the interaction between intrinsic and extrinsic geometric invariants under conformal deformations in this setting. A significant feature of our study is the validation of the theoretical findings through a concrete example of a conformal Sasakian manifold that is not Sasakian, thereby illustrating the generality and applicability of our approach.

Future research can focus on extending these results to more specialized types of submanifolds, such as slant, semi-slant, or CR-submanifolds within CSSF. Another possible direction is the study of other $\delta$-invariants, such as $\delta$-Ricci or generalized Chen–Ricci inequalities, in the same ambient structure. Furthermore, the exploration of physical interpretations of quarter-symmetric connections, particularly in geometric models arising in theoretical physics, may also present fruitful avenues for interdisciplinary applications. These extensions would not only broaden the scope of submanifold geometry in contact metric manifolds but also strengthen the connection between curvature theory and its applications.