Some Optimal Bounds for δ-Casorati Curvatures with Slant Factor in Trans-Sasakian Manifolds

Abstract

In this article, we derive some optimal inequalities for slant submanifolds on trans-Sasakian manifolds coupled with quarter-symmetric non-metric connection $\left(qsnmc\right)$, utilizing generalized normalized $\delta$-Casorati curvatures. Additionally, we describe submanifolds for which the equality cases are true.

1. Introduction

Hayden [1] deserves recognition for popularizing the understanding of a metric connection with non-vanishing torsion in a Riemannian manifold. Numerous researchers have examined the characteristics of Riemannian manifolds $\mathcal{B}$ with semi-symmetric (symmetric) and non-metric connections ([2,3,4,5,6]). Golab [7] addressed the fundamental conception of quarter-symmetric linear connections ($qsmc)$.

Definition 1. 

The torsion tensor $\overline{\mathbb{T}}$ of a linear connection is said to be $qsm$ if it takes the following form

$$\begin{array}{c}\hfill \overline{\mathbb{T}}(\omega ,{\rm Y})=\gamma \left(\omega \right)⌀\omega -\gamma \left(\omega \right)⌀{\rm Y},\end{array}$$

(1)

for all vector fields $\omega ,{\rm Y}\in \chi \left(\mathcal{B}\right)$, where ⌀ denotes a $(1,1)$-type tensor and γ is a $1-$ form. $⌀=I$ converts the $qsm$ to a semi-symmetric connection.

As a result, $qsm$ can be described as an extrapolation of semi-symmetric connections. $\overline{\nabla }$ is a metric connection if g, a Riemannian metric, exists in $\mathcal{B}$ such that $\overline{\nabla }g=0$; otherwise, it is non-metric.

The authors established a semi-symmetric metric connection in the nearly contact manifold [8] by defining

$$\begin{array}{c}\hfill \overline{\mathbb{T}}(\omega ,{\rm Y})=\gamma ({\rm Y})\omega -\gamma \left(\omega \right){\rm Y}.\end{array}$$

(2)

However, Chinae and Gonzales’ classification of almost-contact metric structures [9] naturally led to trans-Sasakian manifolds $\left(TSM\right)$. Oubina [10] introduced the trans-Sasakian manifold in $1985,$ a novel kind of almost-contact Riemannian manifold.

If a product manifold $\mathcal{B}\times R$ is a member of the class $W_4$ [11], then an almost-contact metric structure on a manifold M is recognized as a trans-Sasakian structure [12]. The class of trans-Sasakian structures of kind $(\alpha ,\beta )$ overlaps with the class $C_6\oplus C_5$.

In contrast, the principal curvatures of surface and extrinsic invariants are defined by the second fundamental form alongside the first fundamental form. In the submanifolds theory, one of the most important tasks is to connect intrinsic and extrinsic invariants; those are extremely beneficial resources for learning submanifolds in the Riemannian manifold $\mathcal{B}$.

Chen [13] introduced a new tool called Chen’s invariants, also termed $\delta$-invariants, and added a new chapter in the study of the association between extrinsic and intrinsic invariants with the help of a strong inequality (see [14,15,16]). Numerous researchers ([17,18,19,20,21,22,23,24,25,26]) who examined optimal inequalities in various ambient spaces used this concept.

Furthermore, optimal inequalities in terms of Casorati curvature were made possible by Felice Casorati [27], who modified the conventional Gauss curvature with the Casorati curvature. Felice Casorati [27] adopted this curvature over the Gauss curvature since it best fits the common concept of curvature by allowing both principal curvatures of a surface on $E^3$ to be zero if and only if the Casorati curvature vanishes. Many researchers have found that the concept of Casorati curvature offers a novel techniques for studying optimal inequalities for various submanifolds [28,29,30,31,32].

Objective and Motivation

The topic is relevant and somewhat original. Casorati-type inequalities are popular subjects in submanifold geometry, and applying Casorati curvatures to slant submanifolds in the context of trans-Sasakian manifolds with $qsnmc$ fills an unusual gap. This study fills the gap by extending generalized Casorati curvature results to this geometric setting and determining equality instances under $qsnmc$, which has not been extensively studied in the literature [14,15,16,33].

The manuscript extends known optimal inequalities (previously obtained in various ambient spaces and connections) to the trans-Sasakian endowed with $qsnmc$ context, gives versions for Casorati curvatures and provides conditions for equality for the invariantly quasi-umbilical submanifolds with trivial normal connection. These findings increase the variety of ambient areas where sharp Casorati limits are known.

2. Preliminaries

Consider $\mathcal{B}$ as an almost-contact metric manifold of dimension $n(=2m+1)$ with an almost-contact metric structure $(⌀,\zeta ,\gamma ,\mathcal{G})$, where ⌀ is a $(1,1)$ tensor field, $\zeta$ is a vector field, $\gamma$ is a 1-form, and $\mathcal{G}$ is a Riemannian metric, such that

$${⌀}^2\omega =-\omega +\gamma \left(\omega \right)\zeta ,\gamma \left(\zeta \right)=1,⌀\left(\zeta \right)=0,\gamma o⌀=0,$$

(3)

$$\mathcal{G}(⌀\omega ,⌀{\rm Y})=\mathcal{G}(\omega ,{\rm Y})-\gamma \left(\omega \right)\gamma ({\rm Y}),$$

(4)

$$\mathcal{G}(\omega ,⌀{\rm Y})=-\mathcal{G}(⌀\omega ,{\rm Y}),\mathcal{G}(\omega ,\zeta )=\gamma \left(\omega \right),$$

(5)

for all $\omega ,{\rm Y}\in \chi \left(\mathcal{B}\right)$.

Definition 2. 

An almost-contact metric structure $(⌀,\zeta ,\gamma ,\mathcal{G})$ on $\mathcal{B}$ is referred to as a $TSM$ if

$$({\nabla }_{\omega }⌀)({\rm Y})=\alpha (\mathcal{G}(\omega ,{\rm Y})\zeta -\gamma ({\rm Y})\omega )+\beta (\mathcal{G}(⌀\omega ,{\rm Y})\zeta -\gamma ({\rm Y})⌀\omega ),$$

(6)

for some smooth functions α and β on $\mathcal{B}$. Also, we state that the $TSM$ is of type $(\alpha ,\beta ).$

We recognize that trans-Sasakian structures of type $(0,0)$ are cosymplectic, those of type $(\alpha ,0)$ are $\alpha$-Sasakian, and those of type $(0,\beta )$ are $\beta$-Kenmotsu [11].

The study of $TSM$ [12] yielded the following findings:

$${\nabla }_{\omega }\zeta =-\alpha ⌀\omega +\beta (\omega -\gamma \left(\omega \right)\zeta )$$

(7)

$$\left({\nabla }_{\omega }\gamma \right)({\rm Y})=-\alpha \mathcal{G}(\Phi \omega ,{\rm Y})+\beta \mathcal{G}(⌀\omega ,⌀{\rm Y})$$

(8)

$$\begin{array}{ccc}\hfill \mathfrak{R}(\omega ,{\rm Y})\zeta & =& ({\alpha }^2-{\beta }^2)[\gamma ({\rm Y})\omega -\gamma \left(\omega \right){\rm Y}]-\left(\omega \alpha \right)⌀{\rm Y}-\left(\omega \beta \right){⌀}^2{\rm Y}\hfill \\ & & +2\alpha \beta [\gamma ({\rm Y})⌀\omega -\gamma \left(\omega \right)⌀{\rm Y}]+({\rm Y}\alpha )⌀\omega +({\rm Y}\beta ){⌀}^2\omega \hfill \end{array}$$

(9)

$${\mathfrak{S}}_{ric}(\omega ,\zeta )=[(n-1)({\alpha }^2-{\beta }^2)-\xi \beta ]\gamma \left(\omega \right)-(⌀\omega )\alpha -(n-2)\left(\omega \beta \right)$$

(10)

when $⌀\left(grad\alpha \right)=(n-2)grad\beta$, in view of (10), the following holds

$${\mathfrak{S}}_{ric}(\omega ,\zeta )=(n-1)({\alpha }^2-{\beta }^2)\gamma \left(\omega \right),$$

(11)

$$Scal=n(n-1)({\alpha }^2-{\beta }^2)$$

(12)

for all vector fields $\omega ,{\rm Y}\in \chi \left(\mathcal{B}\right).$

3. Characteristics of Curvatures Conceding a $\mathit{qsnmc}$

This section will reveal the link between $\widehat{\mathfrak{R}}$ and $\mathfrak{R}$. For the $qsnmc$ $\widehat{\nabla }$ and the Levi–Civita connection ∇ on a $TSM$ $\mathcal{B}$, let $\widehat{\mathfrak{R}}$ and $\mathfrak{R}$ be the curvature tensors, respectively.

Additionally, we shall ascertain the correlation between $\widehat{{\mathfrak{S}}_{ric}}$ and ${\mathfrak{S}}_{ric}$, where ${\mathfrak{S}}_{ric}$ represents the Ricci tensor about the Levi–Civita connection ∇ on $\mathcal{B}$, and $\widehat{{\mathfrak{S}}_{ric}}$ represents the Ricci tensor concerning the $qsnmc$ $\widehat{\nabla }$ on $\mathcal{B}$. Tripathi outlined the relationship as follows in [34]:

$${\widehat{\nabla }}_{\omega }{\rm Y}={\nabla }_{\omega }{\rm Y}-\gamma \left(\omega \right)⌀{\rm Y}-\gamma \left(\omega \right){\rm Y}-\gamma ({\rm Y})\omega +\mathcal{G}(\omega ,{\rm Y})\zeta ,$$

(13)

where $\widehat{\nabla }$ is $qsnmc$. By applying Definition 2 and Formula (7), we arrive at

$$({\widehat{\nabla }}_{\omega }⌀)({\rm Y})=\alpha [\mathcal{G}(\omega ,{\rm Y})\zeta -\gamma ({\rm Y})\omega ]+(\beta -1)[\mathcal{G}(⌀\omega ,{\rm Y})\zeta -\gamma ({\rm Y})⌀\omega ],$$

(14)

$${\widehat{\nabla }}_{\omega }\zeta =-\alpha ⌀\omega +(\beta -1)\omega -\beta \gamma \left(\omega \right)\zeta .$$

(15)

With regard to the $qsnmc$ $\widehat{\nabla }$, we will now extract the scalar curvature, the curvature tensor, and the Ricci tensor as follows:

For a $TSM$ $\mathcal{B}$ with the $qsnmc$ $\widehat{\nabla }$, the following equality is provided, for any vector field $\omega ,{\rm Y},\varrho \in \chi \left(\mathcal{B}\right)$

$$\begin{array}{ccc}\widehat{\mathfrak{R}}(\omega ,{\rm Y})\varrho & =& \mathfrak{R}(\omega ,{\rm Y})\varrho +\alpha \left[\right(\mathcal{G}({\rm Y},\varrho )\gamma \left(\omega \right)-\mathcal{G}(\omega ,\varrho )\gamma ({\rm Y}))\zeta +(\gamma ({\rm Y})\gamma \left(\varrho \right)-\mathcal{G}(⌀{\rm Y},\varrho ))\omega \hfill \\ & & +(-\eta (\omega \left)\gamma \right(\varrho )+\mathcal{G}(⌀\omega ,\varrho \left)\right){\rm Y}-2\mathcal{G}(\omega ,⌀{\rm Y})\varrho -\mathcal{G}({\rm Y},\varrho )⌀\omega +\mathcal{G}(\omega ,\varrho )⌀{\rm Y}\hfill \\ & & -2\mathcal{G}(\omega ,⌀{\rm Y})⌀]+(\beta -1\left)\right[\left(\mathcal{G}\right(⌀{\rm Y},\varrho \left)\gamma \right(\omega )-\mathcal{G}({\rm Y},\varrho \left)\gamma \right(\omega )-\mathcal{G}(⌀\omega ,\varrho \left)\gamma \right({\rm Y})\hfill \\ & & +\mathcal{G}(\omega ,\varrho )\mathcal{G}({\rm Y}))\zeta -(\gamma ({\rm Y})\gamma \left(\varrho \right)-\mathcal{G}({\rm Y},\varrho ))\omega +(\gamma \left(\omega \right)\gamma \left(\varrho \right)-\mathcal{G}(\omega ,\varrho )){\rm Y}\hfill \\ & & +\gamma ({\rm Y})\gamma \left(\varrho \right)⌀\omega -\gamma \left(\omega \right)\gamma \left(\varrho \right)\phi {\rm Y}]+\beta [\mathcal{G}({\rm Y},\varrho )\omega -\mathcal{G}(\omega ,\varrho ){\rm Y}]\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\widehat{{\mathfrak{S}}_{ric}}(\omega ,{\rm Y})& =& {\mathfrak{S}}_{ric}(\omega ,{\rm Y})+\alpha [-\mathcal{G}(\omega ,{\rm Y})+\gamma \left(\omega \right)\gamma ({\rm Y})-\mathcal{G}(⌀\omega ,{\rm Y})]\hfill \\ & +& (\beta -1)\left[\mathcal{G}\right(⌀\omega ,{\rm Y})+(n-2\left)\mathcal{G}\right(⌀\omega ,⌀{\rm Y}\left)\right]+\beta (n-1)\mathcal{G}(\omega ,{\rm Y}),\hfill \end{array}$$

(17)

$$\widehat{Scal}=2\beta {(n-1)}^2-(n-1)(n-2)+Scal,$$

(18)

where ${\mathfrak{S}}_{ric}$ and $Scal$ are Ricci tensor and scalar curvature with respect to Levi–Civita connection ∇ on $\mathcal{B}$, respectively.

Let $\Sigma$ be m-dimensional submanifold of $n=(2n+1)$-dimensional $TSM$ manifolds $\mathcal{B}$. Let ∇ and ${\nabla }^{\perp }$ denotes the induced connections on the tangent bundle $\mathcal{T}\Sigma$ and $\mathcal{T}{\Sigma }^{\perp }$ of $\Sigma$ in $\mathcal{B}$, respectively, and indicates by ð the second fundamental form of $\Sigma$ for all $\omega ,{\rm Y}\in \Gamma (\mathcal{T}\Sigma )$ and $\aleph \in \Gamma \left(\mathcal{T}{\Sigma }^{\perp }\right)$ in $\mathcal{B}.$

Recall the Weingarten and Gauss formulas by

$${\widehat{\nabla }}_{\omega }\aleph =-{\Lambda }_{\aleph }\omega +{\nabla }_{\omega }^{\perp }\aleph ,$$

(19)

$${\widehat{\nabla }}_{\omega }{\rm Y}={\nabla }_{\omega }{\rm Y}+ð(\omega ,{\rm Y}),$$

(20)

where the shape operator of $\mathcal{B}$ with respect to ℵ is notated using ${\Lambda }_{\aleph }$. The following formula is widely recognized:

$$\mathcal{G}({\Lambda }_{\aleph }\omega ,{\rm Y})=\mathcal{G}(ð(\omega ,{\rm Y}),\aleph ),forall\omega ,{\rm Y}\in \Gamma (\mathcal{T}\Sigma ),\aleph \in \Gamma ({\mathcal{T}}^{\perp }\Sigma ).$$

Let $\mathfrak{R}$ represent the Riemannian curvature tensor of $TSM$ $\mathcal{B}$. We then review the Gauss formula for immersion, which is provided by

$$\overline{\mathfrak{R}}(\omega ,{\rm Y},\varrho ,\nu )=\widehat{\mathfrak{R}}(\omega ,{\rm Y},\varrho ,\nu )-\mathcal{G}(ð(\omega ,\nu ),ð({\rm Y},\varrho ))+\mathcal{G}(ð(\omega ,\varrho ),ð({\rm Y},\nu ))$$

(21)

for all $\omega ,{\rm Y},\varrho ,\nu \in \Gamma (\mathcal{T}\Sigma ).$

4. Slant Submanifolds of $\mathit{TSM}$ with $\mathit{qsnmc}$

Definition 3 

([35]). Let Σ be a submanifold of a $TSM$ $(\mathcal{B},⌀,\gamma ,\zeta ,\mathcal{G})$. For each nonzero vector $\mathcal{U}$ tangent to Σ at any point $p\in \Sigma$. It is claimed that Σ is slant submanifold if the slant angle between $\mathcal{T}\Sigma$ and $⌀\mathcal{U}$ is irrespective of the p selection.

Examine that if the slant angle is $\ominus =0$ and $\ominus ={\displaystyle \frac{\pi }{2}}$, respectively, then the submanifold ∑ becomes ⌀-invariant and ⌀-anti-invariant. Proper slant (or ⊖-slant proper) submanifold is a slant submanifold that is neither anti-invariant nor invariant.

The following are characteristics of slant submanifolds in $TSM$ $(\mathcal{B},⌀,\gamma ,\zeta ,\mathcal{G})$ of $(\alpha ,\beta )$ type, according to the findings of [35,36].

Theorem 1 

([36]). Let Σ be a submanifolds of an almost-contact metric manifold $(\mathcal{B},⌀,\gamma ,\zeta ,\mathcal{G})$ such that $\zeta \in \mathcal{T}\Sigma$. Then,

1.

Σ is slant if and only if there exists a constant $\lambda \in [0,1]$ such that

$${\mathcal{P}}^2=-\lambda (I-\gamma \otimes \zeta ).$$

Furthermore, if the ⊖ is the slant angle of ∑, then $\lambda =Co{s}^2\ominus$.

2.

$\mathcal{G}(\mathcal{P}\omega ,\mathcal{P}{\rm Y})=Co{s}^2\ominus [\mathcal{G}(\omega ,{\rm Y})-\gamma \left(\omega \right)\gamma ({\rm Y})]$,       for any $\omega ,{\rm Y}\in \Gamma (\mathcal{T}\sum )$.

Corollary 1. 

If Σ is an m-dimensional ⊖-slant submanifold of a $TSM$ $(\mathcal{B},⌀,\gamma ,\zeta ,\mathcal{G})$ of $(\alpha ,\beta )$ type with a $qsnmc$, then, with a $qsnmc$, the scalar curvature is

$$\begin{array}{c}\hfill \widehat{Scal}=2m({\alpha }^2-{\beta }^2)-Co{s}^2\ominus (m-2)(m-1)\end{array}$$

(22)

$$+\beta (m-1)[(m-2)Co{s}^2\ominus +m].$$

Proof. 

Following contraction over $\omega$ and ${\rm Y}$, we arrive to (22) in light of (17) and Theorem (1). □

We now examine the Casorati curvature of a m-dimensional ⊖-slant submanifold $\Sigma$ of a $(2m+1)$-dimensional $TSM$ $(\mathcal{B},⌀,\gamma ,\zeta ,\mathcal{G})$ with a $qsnmc$. We obtain two sharp inequalities concerning the generalized normalized $\delta$-Casorati curvatures (in brief GN-$\delta$-CC) for the submanifold $\Sigma$ in $\mathcal{B}$.

By the local existence theorem for orthonormal frames on a Riemannian manifold, $LOTF$ and $LONF$ can always be chosen in some neighborhood of any point $p\in \sum .$ The local orthonormal tangent frame ($LOTF$) of the tangent bundle $\mathcal{T}\Sigma$ of $\Sigma$ is represented by $\{{\Im }_1,\dots ,{\Im }_m\}$, and the local orthonormal normal frame ($LONF$) of the normal bundle ${\mathcal{T}}^{\perp }\Sigma$ of $\Sigma$ in some neighborhood of any point $p\in \sum$ in $\mathcal{B}$ is represented by $\{{\Im }_{m+1},\dots ,{\Im }_{2m+1}\}$.

Taking the second fundamental form of ∑ as represented by ð, we may obtain the normalized scalar curvature ${\mathfrak{R}}_{Nor}$ as follows:

$$\begin{array}{c}\hfill {\mathfrak{R}}_{Nor}={\displaystyle \frac{2\widehat{Scal}}{m(m-1)}}\end{array}$$

and the mean curvature by the formula

$$\begin{array}{c}\hfill \mho =\sum _{i=1}^m{\displaystyle \frac{1}{m}}ð({\Im }_i,{\Im }_i).\end{array}$$

Let ${ð}_{ij}^a=g(ð({\Im }_i,{\Im }_j),{\Im }_a)$ for $i,j=\{1,\dots ,m\}$ and $a=\{m+1,\dots ,2m+1\}$.

Next, a squared norm of the mean curvature vector is provided by

$$\begin{array}{c}\hfill {\left|\right|\mho \left|\right|}^2={\displaystyle \frac{1}{m^2}}\sum _{a=m+1}^{2m+1}{\left(\sum _{i=1}^m{ð}_{ii}^a\right)}^2.\end{array}$$

Also, the expression for the second fundamental form ð is

$$\begin{array}{c}\hfill {\left|\right|ð\left|\right|}^2=\sum _{a=m+1}^{2m+1}\sum _{i,j=1}^m{\left({ð}_{ij}^a\right)}^2.\end{array}$$

(23)

Definition 4. 

The Casorati curvature $\mathfrak{C}$ of a submanifold ∑ can be represented as follows using the second fundamental form ð.

$$\begin{array}{c}\hfill \mathfrak{C}={\displaystyle \frac{1}{m}}{\left|\right|ð\left|\right|}^2.\end{array}$$

(24)

Let $\{{\Im }_1,\dots ,{\Im }_q\}$ indicate an orthonormal basis of q-dimensional subspace $\mathfrak{L}$ of $T\sum ,q\ge 2$.

Next, we express the q-plane section’s scalar curvature.

$$\begin{array}{c}\hfill \widehat{Scal}\left(\mathcal{L}\right)=\sum _{1\le i<j\le q}\mathfrak{R}({\Im }_i,{\Im }_j,{\Im }_j,{\Im }_i).\end{array}$$

Therefore, the Casorati curvature of $\mathfrak{L}$ is as follows:

$$\begin{array}{c}\hfill \mathfrak{C}\left(\mathfrak{L}\right)={\displaystyle \frac{1}{q}}\sum _{a=m+1}^{2m+1}\sum _{i,j=1}^q{\left({ð}_{ij}^a\right)}^2.\end{array}$$

The normalized δ-Casorati curvatures ${\delta }_c(m-1)$ and ${\widehat{\delta }}_c(m-1)$ are defined as

$$\begin{array}{c}\hfill {\left[{\delta }_c(m-1)\right]}_p={\displaystyle \frac{1}{2}}{\mathfrak{C}}_p+{\displaystyle \frac{m+1}{2m}}\mathrm{Inf}\left\{\mathfrak{C}\left(\mathfrak{L}\right)\right|\mathfrak{L}:\mathrm{a}\mathrm{hyperplane}\mathrm{of}{\mathcal{T}}_p\Sigma \}\end{array}$$

and

$$\begin{array}{c}\hfill {\left[{\widehat{\delta }}_c(m-1)\right]}_p=2{\mathfrak{C}}_p-{\displaystyle \frac{2m-1}{2m}}\mathrm{Sup}\left\{\mathfrak{C}\left(\mathfrak{L}\right)\right|\mathfrak{L}:\mathrm{a}\mathrm{hyperplane}\mathrm{of}{\mathcal{T}}_p\Sigma \}.\end{array}$$

The GN-$\delta$-CC ${\delta }_C(\Xi ;m-1)$ and ${\widehat{\delta }}_C(\Xi ;m-1)$ of the submanifold $\Sigma$ are expressed for any positive real number $\Xi \ne m(m-1)$ as

$$\begin{array}{ccc}\hfill {\left[{\delta }_c(\Xi ;m-1)\right]}_p& =& \Xi {\mathfrak{C}}_p\hfill \\ & +& {\displaystyle \frac{(m^2-m-\Xi )(m-1)(m+\Xi )}{\Xi m}}\mathrm{Inf}\left\{\mathfrak{C}\left(\mathfrak{L}\right)\right|\mathfrak{L}:\mathrm{a}\mathrm{hyperplane}\mathrm{of}{\mathcal{T}}_p\Sigma \}\hfill \end{array}$$

when $0<\Xi <m(m-1),$ and

$$\begin{array}{ccc}\hfill {\left[{\widehat{\delta }}_c(\Xi ;m-1)\right]}_p& =& \Xi {\mathfrak{C}}_p\hfill \\ & +& {\displaystyle \frac{(m^2-m-\Xi )(m-1)(m+\Xi )}{\Xi m}}\mathrm{Sup}\left\{\mathfrak{C}\left(\mathfrak{L}\right)\right|\mathfrak{L}:\mathrm{a}\mathrm{hyperplane}\mathrm{of}{\mathcal{T}}_p\Sigma \},\hfill \end{array}$$

when $\Xi >m(m-1).$ The parameter $\Xi$ in the context of Casorati curvature usually refers to specific values that arise in Casorati inequalities, which are optimum constraints linking a submanifold’s intrinsic curvature to its extrinsic Casorati curvature. These constants, along with the Casorati curvature, have a geometric meaning that relates to how a surface or submanifold bends and deviates from being flat in its ambient space. Moreover, the constants $\Xi$ in related theorems and inequalities, generally indicated as $\left({\delta }_c\right)$, are precise numerical values used to construct optimal bounds or criteria (invariants) for geometric features.

The constants help define the specific conditions that characterize these ideal geometric shapes (submanifolds that achieve the equality in these optimal inequalities are called ideal submanifolds). For example, in a specific inequality for a hypersurface in Euclidean space, equality holds if and only if the point is an umbilical point (where all principal curvatures are equal).

Remark 1. 

According to [37], any m-dimensional submanifold Σ of a n-dimensional Riemannian manifold is invariantly quasi-umbilical ($IQU$), if there exist $n-m$ orthogonal unit normal vectors $\{{\Im }_{m+1},\dots ,{\Im }_{2m+1}\}$, such that the shape operators with respect to all directions ${\Im }_a$ have an eigenvalue of multiplicity $m-1$, and that for each ${\Im }_a$ the distinguished eigenvector direction is the same.

5. Based Example

Example 1. 

Let a three-dimensional manifold $\mathcal{B}=[(x,y,z)\in R^3\mid z\ne 0],$ where $(x,y,z)$ are the Cartesian coordinates in $R^3$. Consider the vector fields

$${\Im }_1=-z\left({\displaystyle \frac{\partial }{\partial x}}+y{\displaystyle \frac{\partial }{\partial z}}\right),{\Im }_2=-z{\displaystyle \frac{\partial }{\partial y}},{\Im }_3={\displaystyle \frac{\partial }{\partial z}},$$

which are linearly independent at each point of $\mathcal{B}.$ Let g be the Riemannian metric defined by

$$\mathcal{G}({\Im }_1,{\Im }_3)=\mathcal{G}({\Im }_2,{\Im }_3)=\mathcal{G}({\Im }_2,{\Im }_2)=0,\mathcal{G}({\Im }_1,{\Im }_1)=\mathcal{G}({\Im }_2,{\Im }_2)=g({\Im }_3,{\Im }_3)=1,$$

and γ be the 1-form such that $\gamma \left(Z\right)=\mathcal{G}(\omega ,{\Im }_3)$ for any vector field ω on tangent space $T\mathcal{B}$. Consider ⌀ be the $(1,1)$ tensor field defined by

$$⌀\left({\Im }_1\right)=-{\Im }_2,⌀\left({\Im }_2\right)=-{\Im }_1,⌀\left({\Im }_3\right)=0.$$

Using the linearity of ⌀ and $\mathcal{G}$, we have

$${⌀}^2\omega =-\omega +\gamma \left(\omega \right){\Im }_3,\gamma \left({\Im }_3\right)=1\mathrm{and}\mathcal{G}(⌀\omega ,⌀{\rm Y})=\mathcal{G}(\omega ,{\rm Y})-\gamma \left(\omega \right)\gamma ({\rm Y})$$

for any vector fields $\omega ,{\rm Y}$ on $\mathcal{B}$. Thus, $(⌀,\zeta ,\gamma ,\mathcal{G})$ shows an almost-contact metric structure on $\mathcal{B}$.

Let ∇ be the Levi–Civita connection with respect to the metric g. Then, we have

$$[{\Im }_1,{\Im }_2]=-y{\Im }_2-z^2{\Im }_3,[{\Im }_1,{\Im }_3]=-{\displaystyle \frac{1}{z}}{\Im }_1,[{\Im }_2,{\Im }_3]=-{\displaystyle \frac{1}{z}}{\Im }_2.$$

The Riemannian connection ∇ in relation to the metric $\mathcal{G}$ is given by

$$2\mathcal{G}({\nabla }_{U}V,W)=U\mathcal{G}(V,W)+V\mathcal{G}(W,U)-W\mathcal{G}(U,V)+\mathcal{G}([U,V],W)-\mathcal{G}([V,W],U)$$

$$+\mathcal{G}\left(\right[W,U],V).$$

The above equation, often known as Koszul’s formula, yields

$${\nabla }_{{\Im }_1}{\Im }_3=-{\displaystyle \frac{1}{z}}{\Im }_1+{\displaystyle \frac{1}{2}}{z}^2{\Im }_2,{\nabla }_{{\Im }_2}{\Im }_3=-{\displaystyle \frac{1}{2}}{z}^2{\Im }_1-{\displaystyle \frac{1}{z}}{\Im }_2,{\nabla }_{{\Im }_3}{\Im }_3=0,$$

(25)

$${\nabla }_{{\Im }_1}{\Im }_2=-{\displaystyle \frac{1}{2}}{z}^2{\Im }_3,{\nabla }_{{\Im }_2}{\Im }_2=-y{\Im }_1+{\displaystyle \frac{1}{z}}{\Im }_3,{\nabla }_{{\Im }_3}{\Im }_2=-{\displaystyle \frac{1}{2}}{z}^2{\Im }_1,$$

$${\nabla }_{{\Im }_1}{\Im }_1={\displaystyle \frac{1}{z}}{\Im }_3,{\nabla }_{{\Im }_2}{\Im }_1=y{\Im }_2+{\displaystyle \frac{1}{2}}{z}^2{\Im }_3,{\nabla }_{{\Im }_3}{\Im }_1={\displaystyle \frac{1}{2}}{z}^2{\Im }_2.$$

Using the above relations, for any vector field p on Θ, we have

$${\nabla }_{\omega }\zeta =-\alpha ⌀\omega +\left\{\beta (\omega -\gamma (\omega )\zeta \right\},$$

for $\zeta \in {\Im }_3$, here, $\alpha =-{\displaystyle \frac{1}{2}}{z}^2$ and $\beta =-{\displaystyle \frac{1}{z}}$. Hence, the manifold $\mathcal{B}$ under consideration is a trans-Sasakian manifold of type $(-{\displaystyle \frac{1}{2}}{z}^2,-{\displaystyle \frac{1}{z}})$ manifold of dimension 3.

When we evaluate this structure for $qsnmc$-connection from (25) and (13), we get

$${\widehat{\nabla }}_{{\Im }_1}{\Im }_1=0,{\widehat{\nabla }}_{{\Im }_1}{\Im }_2=0,{\widehat{\nabla }}_{{\Im }_{\Im }}{\Im }_3=0,$$

(26)

$${\widehat{\nabla }}_{{\Im }_2}\Im =0,{\widehat{\nabla }}_{{\Im }_2}{\Im }_2=-y{\Im }_1,{\widehat{\nabla }}_{{\Im }_3}{\Im }_2=-{\displaystyle \frac{1}{2}}{z}^2{\Im }_1,$$

$${\widehat{\nabla }}_{{\Im }_3}\Im =0,{\widehat{\nabla }}_{{\Im }_2}{\Im }_1=y{\Im }_2,{\widehat{\nabla }}_{{\Im }_3}{\Im }_1={\displaystyle \frac{1}{2}}{z}^2{\Im }_2.$$

The non-vanishing Riemannian, Ricci curvature, and scalar curvature tensor components with regard to the $qsnmc$ are as follows:

$$\widehat{\mathfrak{R}}({\Im }_1,{\Im }_2){\Im }_2=-\left(y^2+{\displaystyle \frac{1}{2}}{z}^4\right){\Im }_1,,\widehat{\mathfrak{R}}({\Im }_1,{\Im }_2){\Im }_1=\left(y^2+{\displaystyle \frac{1}{2}}{z}^4\right){\Im }_2,$$

$$\widehat{\mathfrak{R}}({\Im }_1,{\Im }_3){\Im }_1=-y{z}^2{\Im }_2,\widehat{\mathfrak{R}}({\Im }_1,{\Im }_3){\Im }_2=y{z}^2{\Im }_1,$$

$$\overline{\Re }({\Im }_2,{\Im }_3){\Im }_1={\displaystyle \frac{y}{z}}{e}_2,\overline{\Re }({\Im }_2,{\Im }_3){\Im }_2=-{\displaystyle \frac{y}{z}}{\Im }_1.$$

$$\begin{array}{c}\hfill \widehat{{\mathfrak{S}}_{ric}}{({\Im }_i,{\Im }_i)}_{i=1,2,3}=\left[\begin{array}{ccc}-\left({\displaystyle \frac{1}{2}}{z}^4+y^2\right)& 0& 0\\ 0& -\left({\displaystyle \frac{1}{2}}{z}^4+y^2\right)& 0\\ 0& 0& 0\end{array}\right],\end{array}$$

$$\widehat{Scal}==-(2y^2+z^4).$$

The normalized scalar curvature σ with respect to the $SVK$-connection is computed as

$$\widehat{\sigma }=-{\displaystyle \frac{(2y^2+z^4)}{3}}.$$

Let Σ be a two-dimensional $IQU$ submanifold of a three-dimensional almost-contact manifold $\mathcal{B}$, which is spanned by $\{{\Im }_1,{\Im }_2\}$. Since $\widehat{\nabla }$ is a $qsnmc$. Thus, from (26) we get

$${\widehat{\nabla }}_{{\Im }_1}{\Im }_1=0,{\widehat{\nabla }}_{{\Im }_2}{\Im }_3=0,{\widehat{\nabla }}_{{\Im }_2}{\Im }_2=-y{\Im }_2.$$

(27)

Additionally, the second fundamental forms in relation to the $SVK$-connection are provided by

$$\begin{array}{c}\hfill \widehat{ð}{({\Im }_i,{\Im }_i)}_{i=1,2}=\left[\begin{array}{cc}{\displaystyle \frac{1}{2}}z{\Im }_3& 0\\ 0& {\displaystyle \frac{1}{2}}z{\Im }_3\end{array}\right].\end{array}$$

(28)

From this, we can easily compute the mean curvature with respect to the $qsnmc$ as

$$\widehat{\mho }={\displaystyle \frac{1}{2}}\sum _{1=1}^2\widehat{ð}({\Im }_i,{\Im }_i)={\displaystyle \frac{1}{2}}z{\Im }_3.$$

(29)

According to (24), the Casorati Curvature $\mathfrak{C}$ with the $qsnmc$ is provided by

$$\widehat{\mathfrak{C}}={\displaystyle \frac{1}{m}}{\left|\right|ð\left|\right|}^2={\displaystyle \frac{z^2}{4}}.$$

(30)

Therefore, the normalized δ-CC ${\delta }_c(m-1)$ and ${\widehat{\delta }}_c(m-1)$ are expressed as

$$\begin{array}{c}\hfill {\left[{\delta }_c\left(1\right)\right]}_p={{\displaystyle \frac{z^2}{8}}}_p+{\displaystyle \frac{3}{4}}\mathit{inf}\{{\displaystyle \frac{z^2}{4}}\left(\mathfrak{L}\right)|\mathfrak{L}:a\mathit{hyperplane}\mathit{of}{T}_p\Sigma \}\end{array}$$

and similarly for ${\widehat{\delta }}_c{\left(1\right)}_p$.

Real-world Applications: The theoretical results of inequalities using Casorati curvature have practical applications in a variety of domains, owing to its easier to understand and effective measure of curvature than typical Gauss curvature. The key application areas include the following [38]:

1.

In computer graphics and mechanics, the Casorati curvature is recognized as the “bending energy” of surfaces, which is critical for modeling deformable objects and understanding their physical behavior.

2.

The theoretical results of inequalities involving Casorati curvature is utilized as a natural measure of normal deviations from a straight line to help create algorithms for 3D shape analysis and recognition.

3.

The study of some optimumal inequalities involving Casorati curvatures for immersion, Riemannian maps, and submersions is useful in mechanics, relativity, spacetime, and theories that incorporate these mathematical structures, such as supergravity and string theory [39].

Theorem 2. 

Let Σ be an m-dimensional ⊖-slant submanifold of a $(2m+1)$-dimensional $TSM$ $(\mathcal{B},⌀,\gamma ,\zeta ,\mathcal{G})$ of $(\alpha ,\beta )$ type with a $QSNMC$ $\widehat{\nabla }$:

(i) 

the GN-δ-CC ${\delta }_c(\Xi ;m-1)$ satisfies

$$\begin{array}{ccc}\hfill {\mathfrak{R}}_{Nor}& \le & {\displaystyle \frac{{\delta }_c(\Xi ;m-1)}{m(m-1)}}+{\displaystyle \frac{2}{(m-1)}}({\alpha }^2-{\beta }^2)\hfill \\ & & +{\displaystyle \frac{\beta }{m}}[(m-2)Co{s}^2\ominus +m]-{\displaystyle \frac{(m-2)}{m}}Co{s}^2\ominus ,\hfill \end{array}$$

(31)

for any real number Ξ such that $0<\Xi <m(m-1)$;

(ii) 

the GN-δ-CC ${\widehat{\delta }}_c(\Xi ;m-1)$ satisfies

$$\begin{array}{ccc}\hfill {\mathfrak{R}}_{Nor}& \le & {\displaystyle \frac{{\widehat{\delta }}_c(\Xi ;m-1)}{m(m-1)}}+{\displaystyle \frac{2}{(m-1)}}({\alpha }^2-{\beta }^2)\hfill \\ & & +{\displaystyle \frac{\beta }{m}}[(m-2)Co{s}^2\ominus +m]-{\displaystyle \frac{(m-2)}{m}}Co{s}^2\ominus ,\hfill \end{array}$$

(32)

for any real number $\Xi >m(m-1).$

Remark 2. 

Furthermore, equality is maintained by (31) and (32) if and only if the submanifold Σ is $IQU$ with trivial normal connection in $\mathcal{B}$, such that the shape operators ${\Lambda }_a$, $a\in \{m+1,\dots ,2m+1\}$ with respect to some $LOTF$ $\{{\Im }_1,\dots ,{\Im }_n\}$ and $LONF$ $\{{\Im }_{m+1},\dots ,{\Im }_{2m+1}\}$ take the following forms:

$$\begin{array}{c}\hfill {\Lambda }_{m+1}=\left[\begin{array}{cccccc}\sigma & 0& 0& \dots & 0& 0\\ 0& \sigma & 0& \dots & 0& 0\\ 0& 0& \sigma & \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & \sigma & 0\\ 0& 0& 0& \dots & 0& {\displaystyle \frac{m(m-1)}{\Xi }}b\end{array}\right],{\Lambda }_{m+2}=\dots ={\Lambda }_{2m+1}=0.\end{array}$$

(33)

Proof. 

(i) Consider that $\mathcal{B}$ is a $(2m+1)$-dimensional $TSM$ $(\mathcal{B},\Phi ,\eta ,\zeta ,g)$ of $(\alpha ,\beta )$ type with a $QSNMC$ $\widehat{\nabla }$. Further, let $\{{\Im }_1,\dots ,{\Im }_m\}$ be a $LOTF$ of the tangent bundle $\mathcal{T}\Sigma$ of $\Sigma$ and let $\{{\Im }_{m+1},\dots ,{\Im }_{2m+1}\}$ be a $LONF$ of the normal bundle $\mathcal{T}{\Sigma }^{\perp }$ of $\Sigma$ in $\mathcal{B}$. Then, in view of (22) and the Gauss equation, we get

$$\begin{array}{ccc}\hfill 2\widehat{Scal}& =& 2m({\alpha }^2-{\beta }^2)+\beta (m-1)[(m-2)Co{s}^2\ominus +m]\hfill \\ & & -Co{s}^2\ominus (m-2)(m-1)+m^2{\left|\right|\mho \left|\right|}^2-m\mathfrak{C},\hfill \end{array}$$

(34)

where we have utilized (24).

Suppose $\mathfrak{L}$ represents a hyperplane of ${\mathcal{T}}_p\Sigma$ and use $\mathfrak{Q}$ for a quadratic polynomial in the elements of the second fundamental form:

$$\begin{array}{ccc}\hfill \mathfrak{Q}& =& \Xi \mathfrak{C}+{\displaystyle \frac{(m-1)(m+\Xi )(m^2-m-\Xi )}{\Xi m}}\mathfrak{C}\left(\mathfrak{L}\right)-2\widehat{Scal}\left(p\right)+2m({\alpha }^2-{\beta }^2)\hfill \\ & & +\beta (m-1)[(m-2)Co{s}^2\ominus +m]-Co{s}^2\ominus (m-2)(m-1).\hfill \end{array}$$

(35)

Suppose that $\mathfrak{L}$ is spanned by $\{{\Im }_1,\dots ,{\Im }_{m-1}\}$.

$$\begin{array}{ccc}\hfill \mathfrak{Q}& =& {\displaystyle \frac{\Xi }{m}}\sum _{a=m+1}^{2m+1}\sum _{i,j=1}^m{\left({ð}_{ij}^a\right)}^2+{\displaystyle \frac{(m+\Xi )(m^2-m-\Xi )}{\Xi m}}\sum _{a=m+1}^{2m+1}\sum _{i,j=1}^{m-1}{\left({ð}_{ij}^a\right)}^2\hfill \\ & & -2\widehat{Scal}\left(p\right)+2m({\alpha }^2-{\beta }^2)+\beta (m-1)[(m-2)Co{s}^2\ominus +m]\hfill \\ & & -Co{s}^2\ominus (m-2)(m-1).\hfill \end{array}$$

(36)

Substitute definition (36) into (35) and expand $\mathfrak{Q}$:

$$\begin{array}{c}\hfill \mathfrak{Q}=\Xi {\mathfrak{C}}_{ij}{ð}_{ij}^a{ð}_{ij}^a+{\displaystyle \frac{(m+\Xi )(m^2-m-\Xi )}{\Xi m}}{\mathfrak{C}}_{ij}{\mathfrak{L}}_{ij}.\end{array}$$

Gather the coefficients of the second-order derivatives:

$$\begin{array}{c}\hfill ={ð}_{ij}^a{ð}_{kl}^a[\Xi {\mathfrak{C}}_{ij}+{\displaystyle \frac{(m+\Xi )(m^2-m-\Xi )}{\Xi m}}{\mathfrak{C}}_{ij}{\mathfrak{L}}_{ij}]\end{array}$$

$$\begin{array}{c}\hfill ={ð}_{ij}^a{ð}_{kl}^a[{\displaystyle \frac{{\Xi }^{2}m+(m+\Xi )(m^2-m-\Xi )}{\Xi m}}{\mathfrak{C}}_{ij}{\delta }_{ik}{\delta }_{jl}],\end{array}$$

which collapses to the coefficient given in (36). Using (34) and (36), we get

$$\begin{array}{ccc}\hfill \mathfrak{Q}& =& {\displaystyle \frac{m+\Xi }{m}}\sum _{a=m+1}^{2m+1}\sum _{i,j=1}^m{\left({ð}_{ij}^a\right)}^2+{\displaystyle \frac{(m+\Xi )(m^2-m-\Xi )}{\Xi m}}\sum _{a=m+1}^{2m+1}\sum _{i,j=1}^{m-1}{\left({ð}_{ij}^a\right)}^2\hfill \\ & & -\sum _{a=m+1}^{2m+1}{(\sum _{i=1}^m{ð}_{ii}^a)}^2.\hfill \end{array}$$

Easy computations transform the equation above into

$$\begin{array}{ccc}\hfill \mathfrak{Q}& =& \sum _{a=m+1}^{2m+1}\sum _{i=1}^{m-1}\left[{\displaystyle \frac{m^2+m(\Xi -1)-2\Xi }{\Xi }}{\left({ð}_{ii}^a\right)}^2+{\displaystyle \frac{2(m+\Xi )}{m}}{\left({ð}_{im}^a\right)}^2\right]\hfill \\ & & +\sum _{a=m+1}^{2m+1}[{\displaystyle \frac{2(m+\Xi )(m-1)}{\Xi }}\sum _{i<j=1}^{m-1}{\left({ð}_{ij}^a\right)}^2-2\sum _{i<j=1}^m{ð}_{ii}^a{ð}_{jj}^a\hfill \\ & & +{\displaystyle \frac{\Xi }{m}}{\left({ð}_{mm}^a\right)}^2].\hfill \end{array}$$

(37)

Utilizing (37), the crucial points

$$\begin{array}{c}\hfill {ð}^c=({ð}_{11}^{m+1},{ð}_{12}^{m+1},\dots ,{ð}_{mm}^{m+1},\dots ,{ð}_{11}^{2m+1},\dots ,{ð}_{mm}^{2m+1})\end{array}$$

of $\mathfrak{Q}$ are the solutions of homogeneous linear system

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial \mathfrak{Q}}{\partial {ð}_{ii}^a}}& =& {\displaystyle \frac{2(m+\Xi )(m-1)}{\Xi }}{ð}_{ii}^a-2\sum _{l=1}^m{ð}_{ll}^a=0,\hfill \\ \hfill {\displaystyle \frac{\partial \mathfrak{Q}}{\partial {ð}_{mm}^a}}& =& {\displaystyle \frac{2\Xi }{m}}{ð}_{mm}^a-2\sum _{l=1}^{m-1}{ð}_{ll}^a=0,\hfill \\ \hfill {\displaystyle \frac{\partial \mathfrak{Q}}{\partial {ð}_{ij}^a}}& =& {\displaystyle \frac{4(m+\Xi )(m-1)}{\Xi }}{ð}_{ij}^a=0,\hfill \\ \hfill {\displaystyle \frac{\partial \mathfrak{Q}}{\partial {ð}_{im}^a}}& =& {\displaystyle \frac{4(m+\Xi )}{m}}{ð}_{im}^a=0,\hfill \end{array}$$

(38)

$\forall i,j=\{1,2,\dots ,m-1\},i\ne j,a\in \{m+1,\dots ,2m+1\}$.

To demonstrate the presence of solutions for non-totally geodesic submanifolds, using (38), any solution ${ð}^c$ for $i\ne j$ reveals ${ð}_{ij}^a=0$, and the related determinant to the first two sets of equations of (38) is zero. In addition, $\mathfrak{Q}$’s Hessian matrix ${\mho }^{\prime }\left(\mathfrak{Q}\right)$ is provided by

$$\begin{array}{c}\hfill {\mho }^{\prime }\left(\mathfrak{Q}\right)=\left[\begin{array}{ccc}{\mho }_1& 0& 0\\ 0& {\mho }_2& 0\\ 0& 0& {\mho }_3\end{array}\right],\end{array}$$

where

$$\begin{array}{c}\hfill {\mho }_1=\left[\begin{array}{ccccc}{\displaystyle \frac{2(n+\Xi )(m-1)}{\Xi }}-2& -2& \dots & -2& -2\\ -2& {\displaystyle \frac{2(m+\Xi )(m-1)}{\Xi }}-2& \dots & -2& -2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -2& -2& \dots & {\displaystyle \frac{2(m+\Xi )(m-1)}{\Xi }}-2& -2\\ -2& -2& \dots & -2& {\displaystyle \frac{2\Xi }{m}}\end{array}\right],\end{array}$$

0 denotes the null matrices of corresponding sizes and the diagonal matrices ${\mho }_2$, ${\mho }_3$ are given as follows

$$\begin{array}{c}\hfill {\mho }_2=\mathrm{diag}({\displaystyle \frac{2(m+\Xi )(m-1)}{\Xi }},{\displaystyle \frac{2(m+\Xi )(m-1)}{\Xi }},\dots ,{\displaystyle \frac{2(m+\Xi )(m-1)}{\Xi }}),\end{array}$$

$$\begin{array}{c}\hfill {\mho }_3=\mathrm{diag}({\displaystyle \frac{2(m+\Xi )}{m}},{\displaystyle \frac{2(m+\Xi )}{m}},\dots ,{\displaystyle \frac{2(m+\Xi )}{m}}).\end{array}$$

As a result, $\mathfrak{Q}$ is parabolic and reaches a minimum $\mathfrak{Q}\left({ð}^c\right)$ for any solution ${ð}^c$ of (38). Using (37) and (38), we determine that $\mathfrak{Q}\left({ð}^c\right)=0$, demonstrating that $\mathfrak{Q}\ge 0$ and thus

$$\begin{array}{c}\hfill \begin{array}{ccc}2\widehat{Scal}\left(p\right)& \le & \Xi \mathfrak{C}+{\displaystyle \frac{(m-1)(m+\Xi )(m^2-m-\Xi )}{\Xi m}}\mathfrak{C}\left(\mathfrak{L}\right)+2m({\alpha }^2-{\beta }^2)\\ & & +\beta (m-1)[(m-2)Co{s}^2\ominus +m]-Co{s}^2\ominus (m-2)(m-1),\end{array}\end{array}$$

whereby we obtain

$$\begin{array}{c}\hfill \begin{array}{ccc}{\mathfrak{R}}_{Nor}& \le & {\displaystyle \frac{\Xi }{m(m-1)}}\mathfrak{C}+{\displaystyle \frac{(m+\Xi )(m^2-m-\Xi )}{\Xi m^2}}C\left(\mathfrak{L}\right)+{\displaystyle \frac{2}{(m-1)}}({\alpha }^2-{\beta }^2)\\ & & +{\displaystyle \frac{\beta }{m}}[(m-2)Co{s}^2\ominus +m]-{\displaystyle \frac{(m-2)}{m}}Co{s}^2\ominus ,\end{array}\end{array}$$

for any $\mathfrak{L}$ tangent hyperplane of ${\mathcal{T}}_p\Sigma$, so that (31) clearly holds in the context of the equation above. Furthermore, the equality’s sign is maintained in (31) if and only if

$$\begin{array}{c}\hfill {ð}_{ij}^a=0,\forall i,j\in \{1,\dots ,m\},i\ne j\end{array}$$

(39)

and

$$\begin{array}{c}\hfill {ð}_{mm}^a={\displaystyle \frac{m(m-1)}{\Xi }}{ð}_{11}^a={\displaystyle \frac{m(m-1)}{\Xi }}{B}_{22}^a\dots ={\displaystyle \frac{m(m-1)}{\Xi }}{ð}_{m-1m-1}^a,\\ \hfill \forall a\in \{m+1,\dots ,2m+1\}.\end{array}$$

(40)

As a result, (39) and (40) allow one to obtain the equality in (31) if and only if the submanifold $\Sigma$ is $IQU$ with trivial normal connection in $TSM$ $\mathcal{B}$, meaning that the shape operators satisfy (33) with respect to appropriate local orthonormal tangent and orthonormal normal frames.

(ii) In an analogous manner (32) can be proven. □

Remark 3. 

The shape of the operator (also known as the Weingarten map) is a matrix that represents how a surface curves inside the ambient space. Its eigenvalues represent the primary curvatures. The Hessian of an function defined on a manifold or connected to the second fundamental form can be linked to this operator. The condition that the Hessian be positive definite is common criteria in optimization for establishing a local minimum. The relationship is frequently found in the fact that the second fundamental form is the Hessian of the embedding in a given coordinate system.

The Hessian matrix (32) is both necessary and sufficient for a critical point to be a local minimum. When coupled to the shape-operator matrix (which is proportional to the Hessian of the second fundamental form in appropriate coordinates), this criterion implies that all primary curvatures must be positive. This assures that the quadratic form $\mathfrak{Q}$ is locally reduced, providing precise and concise mathematical verification of the exact criteria given in the Equation (32).

Then, for submanifolds $\Sigma$ of a $(2m+1)$ $TSM$ $\mathcal{B}$ with the $QSNMC$ $\widehat{\nabla }$, we gain sharp inequalities involving the normalized $\delta$-Casorati curvatures.

Corollary 2. 

For a m-dimensional θ-slant submanifold Σ in a $(2m+1)$-dimensional $TSM$ $\mathcal{B}$ with a $QSNMC$ $\widehat{\nabla }$. Then

(i) 

the normalized δ-Casorati curvature ${\delta }_c(m-1)$ holds

$$\begin{array}{ccc}\hfill {\mathfrak{R}}_{Nor}& \le & {\delta }_c(m-1)+{\displaystyle \frac{2}{(m-1)}}({\alpha }^2-{\beta }^2)\hfill \\ & & +{\displaystyle \frac{\beta }{m}}[(m-2)Co{s}^2\ominus +m]-{\displaystyle \frac{(m-2)}{m}}Co{s}^2\ominus .\hfill \end{array}$$

(41)

(ii) 

the normalized δ-Casorati curvature ${\widehat{\delta }}_c(m-1)$ holds

$$\begin{array}{ccc}\hfill {\mathfrak{R}}_{Nor}& \le & {\widehat{\delta }}_c(m-1)+{\displaystyle \frac{2}{(m-1)}}({\alpha }^2-{\beta }^2)\hfill \\ & & +{\displaystyle \frac{\beta }{m}}[(m-2)Co{s}^2\ominus +m]-{\displaystyle \frac{(m-2)}{m}}Co{s}^2\ominus .\hfill \end{array}$$

(42)

Additionally, the equalities in the relations (41) and (42) obey if and only if Σ is an $IQU$ submanifold with trivial normal connection in $\mathcal{B}$ such that the shape operators ${\Lambda }_a$ satisfy the following with some $LOTF$ $\{{\Im }_1,\dots ,{\Im }_n\}$ of the tangent bundle $\mathcal{T}\Sigma$ of Σ, and $LONF$ $\{{\Im }_{m+1},\dots ,{\Im }_{2m+1}\}$ of the normal bundle ${\mathcal{T}}^{\perp }\sigma$ of Σ in $\mathcal{B}$.

$$\begin{array}{c}\hfill {\Lambda }_{m+1}=\left[\begin{array}{cccccc}\sigma & 0& 0& \dots & 0& 0\\ 0& \sigma & 0& \dots & 0& 0\\ 0& 0& \sigma & \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & \sigma & 0\\ 0& 0& 0& \dots & 0& 2\sigma \end{array}\right],{\Lambda }_{m+2}=\dots ={\Lambda }_{2m+1}=0\end{array}$$

(43)

and

$$\begin{array}{c}\hfill {\Lambda }_{m+1}=\left[\begin{array}{cccccc}2\sigma & 0& 0& \dots & 0& 0\\ 0& 2\sigma & 0& \dots & 0& 0\\ 0& 0& 2\sigma & \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & 2\sigma & 0\\ 0& 0& 0& \dots & 0& \sigma \end{array}\right],{\Lambda }_{m+2}=\dots ={\Lambda }_{2m+1}=0.\end{array}$$

(44)

As an implementation of Theorem 2, we produce the subsequent outcomes.

Next, we have the following conclusions for $TSM$ with the $QSNMC$ whose invariant submanifolds are. Remember that when $\ominus =0$, a slant submanifold turns into an invariant submanifold.

Corollary 3. 

Let Σ be an m-dimensional invariant submanifold of a $(2m+1)$-dimensional $TSM$ $\mathcal{B}$ with a $QSNMC$ $\widehat{\nabla }$. Then

(i) 

the GN δ-CC ${\delta }_c(\Xi ;m-1)$ holds

$$\begin{array}{ccc}\hfill {\mathfrak{R}}_{Nor}& \le & {\displaystyle \frac{{\delta }_c(\Xi ;m-1)}{m(m-1)}}+{\displaystyle \frac{2}{(m-1)}}({\alpha }^2-{\beta }^2)\hfill \\ & & +{\displaystyle \frac{2\beta }{m}}(m-1)-{\displaystyle \frac{(m-2)}{m}},\hfill \end{array}$$

(45)

for any real number Ξ such that $0<\Xi <m(m-1)$;

(ii) 

the generalized normalized δ-Casorati curvature ${\widehat{\delta }}_c(\lambda ;m-1)$ satisfies

$$\begin{array}{ccc}\hfill {\mathfrak{R}}_{Nor}& \le & {\displaystyle \frac{{\widehat{\delta }}_c(\Xi ;m-1)}{m(m-1)}}+{\displaystyle \frac{2}{(m-1)}}({\alpha }^2-{\beta }^2)\hfill \\ & & +{\displaystyle \frac{2\beta }{m}}(m-1)-{\displaystyle \frac{(m-2)}{m}},\hfill \end{array}$$

(46)

for any real number $\Xi >m(m-1).$

Moreover, (45) and (46) maintain the equality if and only if the submanifold Σ is $IQU$ with trivial normal connection in $\mathcal{B}$, such that the shape operators ${\Lambda }_a$, $a\in \{m+1,\dots ,2m+1\}$ with respect to some $LOTF$ $\{{\Im }_1,\dots ,{\Im }_n\}$ and $LONF$ $\{{\Im }_{m+1},\dots ,{\Im }_{2m+1}\}$ give the following forms:

$$\begin{array}{c}\hfill {\Lambda }_{m+1}=\left[\begin{array}{cccccc}\sigma & 0& 0& \dots & 0& 0\\ 0& \sigma & 0& \dots & 0& 0\\ 0& 0& \sigma & \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & \sigma & 0\\ 0& 0& 0& \dots & 0& {\displaystyle \frac{m(m-1)}{\Xi }}b\end{array}\right],{\Lambda }_{m+2}=\dots ={\Lambda }_{2m+1}=0.\end{array}$$

(47)

Next, we write the following results with the $QSNMC$ for anti-invariant submanifolds of $TSM$. When $\ominus ={\displaystyle \frac{\pi }{2}}$, we know that a slant submanifold reduces to an anti-invariant submanifold.

Corollary 4. 

Let Σ be an m-dimensional anti-invariant submanifold of a $(2m+1)$-dimensional $TSM$ $\mathcal{B}$ with a $QSNM$ $\widehat{\nabla }$. Then

(i) 

the GN δ-CC ${\delta }_c(\Xi ;m-1)$ holds

$$\begin{array}{ccc}\hfill {\mathfrak{R}}_{Nor}& \le & {\displaystyle \frac{{\delta }_c(\Xi ;m-1)}{m(m-1)}}+{\displaystyle \frac{2}{(m-1)}}({\alpha }^2-{\beta }^2)+\beta ,\hfill \end{array}$$

(48)

for any real number Ξ such that $0<\Xi <m(m-1)$;

(ii) 

the GN δ-CC ${\widehat{\delta }}_c(\Xi ;m-1)$ holds

$$\begin{array}{ccc}\hfill {\mathfrak{R}}_{Nor}& \le & {\displaystyle \frac{{\widehat{\delta }}_c(\Xi ;m-1)}{m(m-1)}}+{\displaystyle \frac{2}{(m-1)}}({\alpha }^2-{\beta }^2)+\beta ,\hfill \end{array}$$

(49)

for any real number $\Xi >m(m-1).$

Moreover, (48) and (49) maintain the equality if and only if the submanifold $M^{\prime }$ is $IQU$ with trivial normal connection in $\mathcal{B}$, such that the shape operators ${\Lambda }_a$, $a\in \{m+1,\dots ,2m+1\}$ with respect to some $LOTF$ $\{{\Im }_1,\dots ,{\Im }_n\}$ and $LONF$ $\{{\Im }_{m+1},\dots ,{\Im }_{2m+1}\}$ give the following forms:

$$\begin{array}{c}\hfill {\Lambda }_{m+1}=\left[\begin{array}{cccccc}\sigma & 0& 0& \dots & 0& 0\\ 0& \sigma & 0& \dots & 0& 0\\ 0& 0& \sigma & \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & \sigma & 0\\ 0& 0& 0& \dots & 0& {\displaystyle \frac{m(m-1)}{\Xi }}b\end{array}\right],{\Lambda }_{m+2}=\dots ={\Lambda }_{2m+1}=0.\end{array}$$

(50)

Remark 4. Open problem: 

It is worth mentioning that any compact submanifold ∑ in almost-contact manifold $(\mathcal{B},\mathcal{G})$ can be seen as minimal submanifolds if and only mean curvature or normal curvature vanish. In differential geometry, inequalities for minimum submanifolds are crucial, with a focus on isoperimetric Sobolev inequalities, which relate area/volume (${\int }_{\partial \sum }\sum >0)$ to boundary size. These inequalities, such as the Michael–Simon inequality, establish sharp constraints for minimum surfaces and hypersurfaces in Euclidean space.

6. Applications of Theorem 2

In the light of (31) and (32) one can classifies the GN-$\delta$-CC for slant submanifold ∑ in the various ambient spaces $\mathcal{B}$ with $qsnmc$ in the following table.

$\sum \mathrm{of}\mathcal{B}$	$\mathit{\alpha }, \mathit{\beta }$	$\mathbf{Inequality}\mathbf{for} {\mathit{\delta }}_{\mathit{c}}(\mathbf{\Xi }\mathbf{;}\mathit{m}\mathbf{-}\mathbf{1})$
Sasakian manifold	$\alpha =1, \beta =0$	${\mathfrak{R}}_{Nor}\le {\displaystyle \frac{{\delta }_c(\Xi ;m-1)}{m(m-1)}}+{\displaystyle \frac{2}{(m-1)}}-{\displaystyle \frac{(m-2)}{m}}Co{s}^2\ominus$
Kenmotsu manifolds	$\alpha =0, \beta =1$	${\mathfrak{R}}_{Nor}\le {\displaystyle \frac{{\delta }_c(\Xi ;m-1)}{m(m-1)}}-{\displaystyle \frac{2}{(m-1)}}+{\displaystyle \frac{1}{m}}[(m-2)Co{s}^2\ominus +m]-{\displaystyle \frac{(m-2)}{m}}Co{s}^2\ominus$
Cosymplectic manifold	$\alpha =\beta =0$	${\mathfrak{R}}_{Nor}\le {\displaystyle \frac{{\delta }_c(\Xi ;m-1)}{m(m-1)}}-{\displaystyle \frac{(m-2)}{m}}Co{s}^2\ominus$

Similarly, we can construct the table of GN-$\delta$-CC ${\widehat{\delta }}_c(\Xi ;m-1)$ for slant submanifold ∑ in the various ambient spaces $\mathcal{B}$ with $qsnmc$.