Optimal Inequalities on (α,β)-Type Almost Contact Manifold with the Schouten–Van Kampen Connection

Abstract

In the current research, we develop optimal inequalities for submanifolds in trans-Sasakian manifolds or $(\alpha ,\beta )$-type almost contact manifolds endowed with the Schouten–Van Kampen connection ($SVK$-connection), including generalized normalized $\delta$-Casorati Curvatures ($\delta$-CC). We also discuss submanifolds on which the equality situations occur. Lastly, we provided an example derived from this research.

1. Introduction

“The worst form in inequality is to try make things equal.”

Aristotle.

The second fundamental form in differential geometry is a quadratic form on the tangent plane of a smooth surface in three-dimensional Euclidean space $E^3$, often known as the shape tensor. Quadratic forms are crucial in many fields, including number theory, linear algebra, group theory (orthogonal groups), differential geometry (Riemannian metric), and topology (intersection forms of four manifolds).

The coefficients in the arithmetic theory of quadratic forms typically correspond to the p-adic integers ${\mathbb{Z}}_p$ or integers $\mathbb{Z}$, which belong to a specified commutative ring. Furthermore, the theory of quadratic fields, in particular, has examined Binary quadratic forms in great detail.

The second fundamental form defines the extrinsic invariants of the surface and its principal curvatures in conjunction with the first fundamental form. In a Riemannian manifold, such a quadratic form is defined for a smooth immersed submanifold.

How to link intrinsic and extrinsic invariants, which are particularly useful tools for researching Riemannian manifold submanifolds, is one of the most pressing problems in submanifolds theory.

In the early 90s, B. Y. Chen [1] opened a new era by studying the relationship between extrinsic and intrinsic invariants with the help of a sharp inequality and introduced a new tool known as $\delta$-invariants (or Chen’s invariants). This conception was used by many researchers who studied Chen-type inequalities and Chen invariants in different ambient spaces (for more information see [2,3,4,5]).

Furthermore, F. Casorati [6] altered the standard Gauss curvature by the Casorati Curvature, which made it possible to build optimal inequalities in terms of Casorati Curvature. F. Casorati [6] favored this curvature over the conventional Gauss curvature because it better reflects the intuitive understanding of curvature when both principal curvatures of a surface in $E^3$ are zero simultaneously if and only if the Casorati Curvature disappears.

Inequalities using Casorati Curvatures provide the solution to the challenge of determining the relationship between the primary extrinsic and intrinsic invariants. The Casorati Curvature was first defined as a normalized sum of squared primary curvatures for the surfaces in Euclidean three-space [6]. Decu et al. [7,8] expanded it to the broader situation of a submanifold of a Riemannian manifold as the normalized square of the length of the submanifold’s second fundamental form. For several submanifolds, the inequalities involving this curvature have been examined. Vilcu et al. [9,10,11] discovered the optimum inequality for Casorati Curvature of Lagrangian submanifolds in complex space forms and slant submanifolds of quaternionic space forms. The geometric interpretation of the Cauchy–Schwarz inequality in terms of Casorati Curvature was obtained by Brubaker and Suceava [12].

Decu et al. [13] studied Casorati Curvature on holomorphic statistical manifolds having constant holomorphic curvature. Chaudhary and Blaga established the inequality on slant submanifolds in metallic Riemannian space forms [14]. The notion of Casorati Curvature provides a new tool to study optimal inequalities for different submanifolds and submersions by other researchers ([15,16,17,18,19]). There have been a lot of intriguing articles published recently on topics including submanifolds theory [20,21] and singularity theory [22,23].

Moreover, from the view points of some applications, some mathematical models involving Casorati Curvature were first researched in the current era, for example, in computer vision [24]. Verstraelen also reported qualitatively geometrical models of early human eyesight [25]. From this angle, the surfaces $M^2$ provided by the Casorati curves $\mathcal{C}$ can be considered the matching (visual) sensations. The isotropical Casorati Curvature of production surfaces was studied in [26] in the field of economics. The study of $\delta$-Casorati Curvatures was initially conducted in [7,8] using optimum inequalities for submanifolds in real space forms. This subject was later the subject of much research (see, for example, [27]). Suceava and Vajiac have recently established inequalities for strictly convex Euclidean hypersurfaces that involve mean curvature, Casorati Curvature, and several Chen invariants [28].

On the other hand, Mihai and Ozgur [29] studied Chen inequalities on Riemannian manifolds with semi-symmetric metric connections while Zhang et al. [30] studied geometric inequalities for the submanifolds of Riemannian manifolds of quasi-constant curvature with semi-symmetric metric connections. Inequalities for the submanifolds of real space forms with Ricci quarter-symmetric metric connections were studied by Nergis and Halil [31]. Lee et al. [32] established a Casorati Curvature for the submanifolds in generalized space forms endowed with semi-symmetric metric connections and Kenmotso space forms. Majid et al. [33] also derived sharp geometric inequalities that involve generalized normalized $\delta$-Casorati Curvatures for submanifolds of golden Lorentzian manifolds equipped with generalized symmetric metric U-connection.

Trans-Sasakian manifolds, however, in contrast, were created as a result of Chinae and Gonzales’ [34] classification of almost contact metric structures ($a.c.m$ structures). Additionally, a different classification of nearly Hermitian manifolds by Gray-Hervella [35]. Brought into focus class $W_4$ of the Hermitian manifolds, which is strongly connected to locally conformal Kaehler manifolds. We call an $a.c.m$ structure on a manifold M trans-Sasakian ([36,37]) or a $(\alpha ,\beta )$-type almost contact manifold [38], when the product manifold $M\times R$ is a member of the class $W_4$. S. Tanno [39] studied some curvature identities and sectional curvatures for trans-Sasakian manifolds. The trans-Sasakian structures of type $(\alpha ,0)$, $(0,\beta )$, and $(0,0)$ can be used to produce $\alpha$-Sasakian, $\beta$-Kenmotsu, and cosymplectic manifolds, respectively [37]. For dimension $n\ge 5$, J. C. Marrero [40] obtained a complete characterization of trans-Sasakian manifolds. He showed a trans-Sasakian manifold or $(\alpha ,\beta )$-type almost contact manifolds [38].

Extensively, when a differential manifold is equipped with an affine connection, the $SVK$-connection is one of the most natural connections adapted to a pair of complementary distributions. The study of the $SVK$-connection was initiated thanks to J. Schouten and E. van Kampen [41]. Ianus [42] and Olszak [43] also followed this study of the $SVK$-conncetion and discussed several results.

These days, one of the most essential issues in submanifolds theory is how to construct links between intrinsic and extrinsic invariants, which are particularly effective tools for studying Riemannian manifold submanifolds with respect to a certain connection.

Therefore, motivated by the above studies, in the current research note, we investigate and develop optimal inequalities for submanifolds in $(\alpha ,\beta )$-type almost contact manifolds or trans-Sasakian manifolds with the $SVK$-connection that contains generalized normalized $\delta$-Casorati Curvatures (in short $\delta$-CC). The equality cases are also covered.

2. Preliminaries

Let $\mathrm{\Theta }$ represent a connected $a.c.m$ manifold equipped with an $a.c.m$ structure $(\psi ,\zeta ,\gamma ,g)$ consisting of a $(1,1)$ tensor field $\psi$, a vector field $\zeta$, a 1-form $\gamma$, and $\mathit{g}$ denotes the Riemannian metric such that

$$\begin{array}{c}\hfill {\psi }^2=-I+\gamma \otimes \zeta ,\gamma \left(\zeta \right)=1,\gamma \circ \psi =0,\psi \zeta =0,\end{array}$$

(1)

$$\begin{array}{c}\hfill g(\psi p,\psi q)=g(p,q)-\gamma \left(p\right)\gamma \left(q\right),\gamma \left(p\right)=g(p,\zeta )\end{array}$$

(2)

$\forall p,q\in \chi \left(\mathrm{\Theta }\right)$, where $\chi \left(\mathrm{\Theta }\right)$ the Lie algebra of $C^{\infty }$ vector fields on $\mathrm{\Theta }$.

Define the fundamental two-form $\mathrm{\Phi }$ of the manifold $\mathrm{\Theta }$ by

$$\mathrm{\Phi }(p,q)=g(p,\psi q).$$

(3)

$\forall p,q\in \chi \left(\mathrm{\Theta }\right)$, and this form satisfies $\gamma \wedge {\mathrm{\Phi }}^n\ne 0$. This means that every almost contact metric manifold is orientable. The existence of an almost contact structure on $\mathrm{\Theta }$ is equivalent to the existence of a reduction in the structural group to $U\left(n\right)\times 1$.

An $a.c.m$ structure [44] on $\mathrm{\Theta }$ is known as trans-Sasakian manifold or $(\alpha ,\beta )$-type almost contact manifold (see [36,37,38]) if $(\mathrm{\Theta }\times R,J,\mathbb{G})$ is a member of the class $W_4$ [39], where J represents the almost complex structure.

$$\begin{array}{c}\hfill J\left(p,f\frac{d}{dt}\right)=\left(\psi \left(p\right)-f\zeta ,\gamma \left(p\right)\frac{d}{dt}\right)\end{array}$$

(4)

where f is a smooth function on $\mathrm{\Theta }\times R$ and $\mathbb{G}$ denotes the product type metric on $\mathrm{\Theta }\times R$, is integrable, which is equivalent to the condition $[\psi ,\psi ]+2d\gamma \otimes \zeta =0$ where $[\psi ,\psi ]$ denotes the Nijenhuis torsion of $\psi$.

An almost contact metric structure $(\psi ,\zeta ,\gamma ,g)$ in $\mathrm{\Theta }$ is said to be:

Almost cosymplectic if $d\mathrm{\Phi }=O$ and $d\gamma =0.$ Cosymplectic if it is almost cosymplectic and normal.

Quasi Sasakian if $d\mathrm{\Phi }=0$ and $(\psi ,\zeta ,\gamma )$ is normal.

Almost $\alpha$-Kenmotsu if $d\gamma =0$ and $d\mathrm{\Phi }(p,q,t)=\frac{2}{3}\mathcal{G}\left[\gamma \left(p\right)\mathrm{\Phi }(q,t)\right]$, being a differentiable function $\alpha$ on $\mathrm{\Theta }$.

For $\alpha =constant$ our definition of almost $\alpha$-Kenmotsu and almost $\alpha$-Sasakian structures coincides with the structures introduced in [37].

Moreover, $(\psi ,\zeta ,\gamma ,g)$ is said to be Kenmotsu if it is 1-Kenmotsu, contact if it is almost 1-Sasakian, and Sasakian if it is 1-Sasakian. For an extensive study of these structures we refer to ([36,37,38]). On the other hand, J. Oubina [37] defined other classes of almost contact metric structure through the almost Hermitian structure $(\mathrm{\Theta }\times R,J)$.

The following equation expresses it:

$$\begin{array}{c}\hfill \left({\nabla }_p\psi \right)q=\alpha (g(p,q)\zeta -\gamma \left(q\right)p)+\beta (g(\psi p,q)\zeta -\gamma \left(q\right)\psi p)\end{array}$$

(5)

for any vector fields p and q on $\mathrm{\Theta }$, here ∇ is used for the Levi–Civita connection with respect to g. In this case, we call it the trans-Sasakian manifold of type $(\alpha ,\beta )$ or $(\alpha ,\beta )$-type almost contact manifold [38].

Thanks to (5), we write

$$\begin{array}{c}\hfill {\nabla }_p\zeta =-\alpha \left(\psi p\right)+\beta (p-\gamma \left(p\right)\zeta ,\end{array}$$

(6)

and

$$\begin{array}{c}\hfill \left({\nabla }_p\gamma \right)q=-\alpha g(\psi p,q)+\beta g(\psi p,\psi q).\end{array}$$

(7)

Let $\mathrm{\Theta }$ represent a $(\alpha ,\beta )$-type almost contact manifolds of dimension s, then we have [45]

$${\Re }_{im}(p,q)\zeta =({\alpha }^2-{\beta }^2)[\gamma \left(q\right)p-\gamma \left(p\right)q]+2\alpha \beta [\gamma \left(q\right)\psi p-\gamma \left(p\right)\psi q]$$

(8)

$$+[\left(q\alpha \right)\psi p-\left(p\alpha \right)\psi q+\left(q\beta \right){\psi }^{2}p-\left(p\beta \right){\psi }^{2}q]$$

$${\Re }_{ic}(p,\zeta )=[((n-1)({\alpha }^2-{\beta }^2)-\left(\zeta \beta \right)]\gamma \left(p\right)+\left(\left(\psi p\right)\alpha \right)+(n-2)\left(p\beta \right),$$

(9)

$$Q\zeta =(n-1)({\alpha }^2-{\beta }^2)-\left(\zeta \beta \right))\zeta +\varphi \left(grad\alpha \right)-(n-2)\left(grad\beta \right),$$

(10)

here, ${\Re }_{im}$ and ${\Re }_{ic}$ are used to symbolize the curvature tensor and Ricci curvature tensor, respectively, and Q defines the Ricci operator by the relation ${\Re }_{ic}(p,q)=g(Qp,q)$.

Further, for any $(\alpha ,\beta )$-type almost contact manifolds or trans-Sasakian manifolds, we write

$$\psi \left(grad\alpha \right)=(n-2)\left(grad\beta \right),$$

(11)

and

$$2\alpha \beta +\left(\zeta \alpha \right)=0.$$

(12)

Taking into use (11) and (12), we write [45]

$${\Re }_{im}(\zeta ,p)q=({\alpha }^2-{\beta }^2)[g(p,q)\zeta -\gamma \left(q\right)p]$$

(13)

$${\Re }_{im}(p,q)\zeta =({\alpha }^2-{\beta }^2)[\gamma \left(q\right)p-\gamma \left(p\right)q]$$

(14)

$$\gamma \left({\Re }_{im}(p,q)r\right)=({\alpha }^2-{\beta }^2)[g(q,r)\gamma \left(p\right)-g(p,r)\gamma \left(q\right)]$$

(15)

$${\Re }_{ic}(P,\zeta )=[\left((n-1)({\alpha }^2-{\beta }^2)\right]\gamma \left(p\right)$$

(16)

$$Q\zeta =\left[(n-1)({\alpha }^2-{\beta }^2)\right]\zeta .$$

(17)

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

(18)

where ℜ is a scalar curvature.

Let p be an arbitrary vector field, then

$$d\gamma (\zeta ,p)=0.$$

(19)

The sectional curvature of the plane spanned by $\zeta$ and a unit vector field U is used to calculate the $\zeta$-sectional curvature ${\mathcal{K}}_{\zeta }$ of $\mathrm{\Theta }$. Equation (14) results

$${\mathcal{K}}_{\zeta }=g({\Re }_{im}(\zeta ,p)\zeta ,p)=({\alpha }^2-{\beta }^2).$$

(20)

(20) shows that the $\zeta$-sectional curvature is independent of U.

3. $\mathit{SVK}$-Connection

Let us suppose that $\mathrm{\Theta }$ represents a Riemannian manifold with signature $(s,t-s)$, $0\le s\le t$, $dim\left(M\right)=t\ge 2$. When $\mathtt{H}$ and $\mathtt{V}$ are two orthogonal complementary distributions on $\mathrm{\Theta }$ with $dim\mathtt{H}=t-1$ and $dim\mathtt{V}=1$, and $\mathtt{V}$ is a non-null distribution, then

$$T\mathrm{\Theta }=\mathtt{H}\oplus \mathtt{V},\mathtt{H}\cap \mathtt{V}=\left\{0\right\}and\mathtt{H}\perp \mathtt{V}.$$

Let a unit vector field $\zeta$ and a linear form $\gamma$ such that $\gamma \left(\zeta \right)=1$, $g(\zeta ,\zeta )=1$ and

$$\mathtt{H}=ker\gamma ,\mathtt{V}=span\left\{\zeta \right\}.$$

We can at least locally adopt $\zeta$ and $\gamma$. Such that One can note $\gamma \left(p\right)=g(p,\zeta )$. Furthermore, it is evident that ${\nabla }_p\zeta \in \mathtt{H}$.

Let us suppose, $p\in T\mathrm{\Theta }$ and by $p^h$ and $p^v$, we use projections as a symbol of p onto $\mathtt{H}$ and $\mathtt{V}$, respectively. So, we turn up $p=p^h\oplus h^v$ and

$$p^h=p-\gamma \left(p\right)\zeta ,p^v=\gamma \left(p\right)\zeta .$$

(21)

The $SVK$-connection $\overline{\nabla }$ associated to the Levi–Civita connection ∇ and in accordance with the distribution’s pair $(\mathtt{H},\mathtt{V})$ is characterized by [46]

$${\overline{\nabla }}_{p}q={\left({\nabla }_p{q}^h\right)}^h+{\left({\nabla }_p{q}^v\right)}^v,$$

(22)

also specify the second fundamental form $\mathrm{\Pi }$ by $\mathrm{\Pi }=\nabla -\overline{\nabla }$. Thanks to (22), we achieve parallelism of the distributions $\mathtt{H}$ and $\mathtt{V}$ with respect to the $SVK$-connection $\overline{\nabla }$. In light of (21), we gain

$${\left({\nabla }_p{q}^h\right)}^h=\left({\nabla }_{p}q\right)-\gamma \left({\nabla }_{p}q\right)\zeta -\gamma \left(q\right){\nabla }_p\zeta ,$$

(23)

$${\left({\nabla }_p{q}^v\right)}^v=\left({\nabla }_p\gamma \right)\left(q\right)\zeta +\gamma \left(q\right){\nabla }_p\zeta ,$$

(24)

which can describe the $SVK$-connection in terms of the Levi–Civita connection by [47]

$$\left({\overline{\nabla }}_{p}q\right)=\left({\nabla }_{p}q\right)-\gamma \left(q\right){\nabla }_p\zeta +\left({\nabla }_p\gamma \right)\left(q\right)\zeta .$$

(25)

Hence, the torsion $\overline{T}$ of $\overline{\nabla }$ and the second fundamental form $\mathrm{\Pi }$ a are [47]

$$\overline{T}(p,q)=\gamma \left(p\right){\nabla }_q\zeta -\gamma \left(q\right){\nabla }_p\zeta +2d\gamma (p,q)\zeta .$$

(26)

$$\mathrm{\Pi }(p,q)=\gamma \left(q\right){\nabla }_p\zeta -\left({\nabla }_p\zeta \right)\left(q\right)\zeta ,$$

(27)

with respect to the $SVK$-connection (25) and $g,\zeta$ and $\gamma$ are parallel, that is,

$$\overline{\nabla }g=0,\overline{\nabla }\zeta =0,\overline{\nabla }\gamma =0$$

4. $(\mathit{\alpha },\mathit{\beta })$-Type Almost Contact Manifold with the $\mathit{SVK}$-Connection

In this section, we turn up some curvatures’ restriction on $(\alpha ,\beta )$-type almost contact manifolds with respect to the $SVK$-connection.

Let $\mathrm{\Theta }$ represents a $2t+1$-dimensional trans-Sasakian manifolds of type $(\alpha ,\beta )$. The Riemannian curvature tensor $\overline{{\Re }_{im}}$ of $\mathrm{\Theta }$ with respect to the $SVK$-connection $\overline{\nabla }$ can be defined by

$$\overline{{\Re }_{im}}(p,q)r={\overline{\nabla }}_p{\overline{\nabla }}_{q}r-{\overline{\nabla }}_q{\overline{\nabla }}_{p}r-{\overline{\nabla }}_{[p,q]}r,\forall p,q,r\in \chi \left(\mathrm{\Theta }\right).$$

(28)

In view of (6), (28) and (25), the Riemannian curvature $\overline{{\Re }_{im}}$ of $\mathrm{\Theta }$ with respect to the $SVK$-connection $\overline{\nabla }$ is turned to [48]

$$\overline{{\Re }_{im}}(p,q)r={\Re }_{im}(p,q)r-\left(q\alpha \right)[\gamma \left(r\right)\psi p-g(\psi p,r)\zeta ]+\left(q\beta \right)[\gamma \left(r\right)p-g(p,r)\zeta ]$$

(29)

$$+\left(p\alpha \right)\left[\gamma \right(r)\psi q-g(\psi q,r\left)\zeta \right]-\left(p\beta \right)\left[\gamma \right(r)q-g(q,r\left)\zeta \right]$$

$$+{\alpha }^2[g(p,r)\gamma \left(q\right)\zeta -g(q,r)\gamma \left(p\right)\zeta -g(r,\psi p)\psi q$$

$$+g(r,\psi q)\psi p+\gamma \left(p\right)\gamma \left(r\right)q-\gamma \left(q\right)\gamma \left(r\right)p]+{\beta }^2[g(q,r)p-g(p,r)q]$$

$$+\alpha \beta \left[g\right(\psi p,r)q-g(\psi q,r)p+g(p,r)\psi q-g(q,r)\psi p+g(\psi p,r\left)\gamma \right(q)\zeta$$

$$-g(\psi q,r)\gamma \left(p\right)\zeta +\gamma \left(p\right)\gamma \left(r\right)\psi q-\gamma \left(q\right)\gamma \left(q\right)\gamma \left(r\right)\psi p].$$

Here, ${\Re }_{im}$ is used to denote the Riemannian curvature tesnor of $\mathrm{\Theta }$ with respect to the Levi–Civita connection ∇. Now, we also have

Theorem 1. 

Let Θ represent a $(\alpha ,\beta )$-type almost contact manifold of dimension $(2t+1)$ equipped with the $SVK$-connection $\overline{\nabla }$. Then, the Ricci tensor $\overline{{\Re }_{ic}}$ and the scalar curvature $\overline{\Re }$ of Θ with respect to the $SVK$-connection are [48]

$$\overline{{\Re }_{ic}}(p,q)={\Re }_{ic}(p,q)-(2t-2)\alpha \beta g(\psi p,q)+[2t{\beta }^2+\left(\zeta \beta \right)]g(p,q)$$

(30)

$$-2t{\alpha }^2\gamma \left(p\right)\gamma \left(q\right)+[(2t-1)\left(p\beta \right)+\left(\psi p\right)\alpha ]\gamma \left(q\right),$$

$$\overline{\Re }=\Re +2t[(2t+1){\beta }^2-{\alpha }^2+2\left(\zeta \beta \right)],$$

(31)

for all $p,q$$\in \chi \left(\mathrm{\Theta }\right)$.

Now, we investigate the Casorati Curvature ($CC$) of s-dimensional submanifold ℵ of $(2t+1)$-dimensional $(\alpha ,\beta )$-type almost contact manifolds $\mathrm{\Theta }$ equipped with the $SVK$-connection and based on the generalized normalized $\delta$-Casorati Curvatures (briefly, $\delta -CC)$ for the submanifold, deduce two major inequalities for the submanifold ℵ in $\mathrm{\Theta }$.

Let $\{I_1,\cdots ,I_s\}$ represent a local tangent frame of the tangent bundle $T\aleph$ of ℵ and $\{I_{s+1},\cdots ,I_{2t+1}\}$ be a local normal frame of the normal bundle $T^{\perp }\aleph$ of ℵ in $\mathrm{\Theta }$. Then, the scalar curvature denoted by $\overline{\Re }$ is given by

$$\begin{array}{c}\hfill \overline{\Re }=\sum _{1\le i<j\le s}\Re (I_i,I_j,I_j,I_i).\end{array}$$

The normalized scalar curvature $\sigma$ can be recalled in the following manner:

$$\begin{array}{c}\hfill \sigma =\frac{2\overline{\Re }}{s(s-1)}\end{array}$$

and the mean curvature using the formula

$$\begin{array}{c}\hfill \mathrm{\Omega }=\sum _{i=1}^s\frac{1}{s}\mathrm{\Pi }(I_i,I_i).\end{array}$$

Let us consider, ${\mathrm{\Pi }}_{ij}^a=g(\mathrm{\Pi }(I_i,I_j),I_a),$ for $i,j=\{1,\cdots ,s\}$ and $a=\{s+1,\cdots ,2t+1\}$. Then, the squared norm of the mean curvature vector is given by

$$\begin{array}{c}\hfill {\left|\right|\mathrm{\Omega }\left|\right|}^2=\frac{1}{s^2}\sum _{a=s+1}^{2t+1}{\left(\sum _{i=1}^s{\mathrm{\Omega }}_{ii}^a\right)}^2\end{array}$$

and the second fundamental form $\mathrm{\Pi }$ is written as

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

Represent the Casorati Curvature $\mathcal{C}$ of $\alpha$ with the help of second fundamental form $\mathrm{\Pi }$ as follows:

$$\begin{array}{c}\hfill \mathcal{C}=\frac{1}{s}{\left|\right|\mathrm{\Pi }\left|\right|}^2.\end{array}$$

(32)

Let us consider $\{I_1,\cdots ,I_q\}$ represents an orthonormal basis of q-dimensional subspace $\mathfrak{L}$ of $TN,q\ge 2$. Then, we express the scalar curvature of the q-plane section $\mathfrak{L}$ by

$$\begin{array}{c}\hfill \overline{\Re }\left(\mathfrak{L}\right)=\sum _{1\le i<j\le q}\Re (I_i,I_j,I_j,I_i)\end{array}$$

and the following is the Casorati Curvature of $\mathfrak{L}$

$$\begin{array}{c}\hfill \mathcal{C}\left(\mathfrak{L}\right)=\frac{1}{q}\sum _{a=s+1}^{2t+1}\sum _{i,j=1}^q{\left({\mathrm{\Pi }}_{ij}^a\right)}^2.\end{array}$$

The normalized δ-CC ${\delta }_c(s-1)$ and ${\widehat{\delta }}_c(s-1)$ are expressed as

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

and

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

The generalized normalized δ-CC ${\delta }_C(\lambda ;s-1)$ and ${\widehat{\delta }}_C(\lambda ;s-1)$ of the submanifold ℵ are expressed for any positive real number $\mu \ne s(s-1)$ as

$$\begin{array}{ccc}\hfill {\left[{\delta }_c(\mu ;s-1)\right]}_p& =& \mu {\mathcal{C}}_p\hfill \\ & +& \frac{(s-1)(s+\lambda )(s^2-s-\mu )}{\mu s}\inf \left\{\mathcal{C}\left(\mathfrak{L}\right)\right|\mathfrak{L}:\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_p\aleph \}\hfill \end{array}$$

when $0<\mu <s(s-1),$ and

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

when $\mu >s(s-1).$

Any s-dimensional submanifold ℵ of $(2t+1)$-dimensional Riemannian manifold $\mathrm{\Theta }$ is known as invariantly quasi-umbilical [49] if there exist $(2t+1-s)$ orthogonal unit normal vectors $\{I_{s+1},\cdots ,I_{2t+1}\}$ such that the distinguishing eigendirection for each $I_a$ is the same and the shape operators $\mathrm{\Lambda }$ with respect to all directions $I_a$ have an eigenvalue of multiplicity $s-1$.

Now we have gained the following outcomes:

Theorem 2. 

Let ℵ be an s-dimensional submanifold of a $(\alpha ,\beta )$-type almost contact manifold Θ equipped with the $SVK$-connection $\overline{\nabla }$. Then,

(i)

the δ-CC ${\delta }_c(\mu ;s-1)$ obeys

$$\begin{array}{ccc}\hfill \sigma & \le & \frac{{\delta }_c(\mu ;s-1)}{s(s-1)}+\frac{4t}{s}(2s+1)({\beta }^2-{\alpha }^2)\hfill \\ & & +\frac{4t}{s(s-1)}[(2s+1){\beta }^2-{\alpha }^2+2\left(\zeta \beta \right)],\hfill \end{array}$$

(33)

for every real number μ such that $0<\mu <s(s-1)$;

(ii)

the δ-CC ${\widehat{\delta }}_c(\mu ;s-1)$ obeys

$$\begin{array}{ccc}\hfill \sigma & \le & \frac{{\widehat{\delta }}_c(\mu ;s-1)}{s(s-1)}+\frac{4t}{s(s-1)}(2t+1)({\beta }^2-{\alpha }^2)\hfill \\ & & +\frac{4t}{s(s-1)}[(2t+1){\beta }^2-{\alpha }^2+2\left(\zeta \beta \right)],\hfill \end{array}$$

(34)

for every real number $\mu >s(s-1).$

Moreover, (33) and (34) exist for equality if and only if the submanifold ℵ has a trivial normal connection in Θ and is invariantly quasi-umbilical, such that the shape operators ${\mathrm{\Lambda }}_a$, $a\in \{s+1,\cdots ,2t+1\}$ with respect to some orthonormal tangent frame $\{I_1,\cdots ,I_s\}$ and orthonormal normal frame $\{I_{s+1},\cdots ,I_{2t+1}\}$ take the following forms:

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}b& 0& 0& \cdots & 0& 0\\ 0& b& 0& \cdots & 0& 0\\ 0& 0& b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b& 0\\ 0& 0& 0& \cdots & 0& \frac{s(s-1)}{\mu }b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0.\end{array}$$

(35)

Proof. 

(i) Let us suppose that $\mathrm{\Theta }$ is a $(2t+1)$-dimensional $(\alpha ,\beta )$-type almost contact manifold $\mathrm{\Theta }$ equipped with the $SVK$-connection $\overline{\nabla }$. Further, let a local orthonormal tangent frame of the tangent bundle $T\aleph$ of ℵ is $\{I_1,\cdots ,I_s\}$ and let a local orthonormal normal frame of the normal bundle $T^{\perp }\aleph$ of ℵ is $\{I_{s+1},\cdots ,I_{2t+1}\}$ in $\mathrm{\Theta }$. Then, in light of Gauss equation and (18), (31), we turn up

$$\begin{array}{ccc}\hfill 2\overline{\Re }& =& 4t(2t+1)({\beta }^2-{\alpha }^2)\hfill \\ & & +4t[(2t+1){\beta }^2-{\alpha }^2+2\left(\zeta \beta \right)]+s^2{\left|\right|\mathrm{\Pi }\left|\right|}^2-s\mathcal{C},\hfill \end{array}$$

(36)

where we have used (32).

Let a hyperplane of $T_p\aleph$ is $\mathfrak{L}$ and use for a quadratic polynomial $\mathcal{Q}$ in the second fundamental form components as

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \mu \mathcal{C}+\frac{(s-1)(s+\mu )(s^2-s-\mu )}{\mu s}\mathcal{C}\left(\mathfrak{L}\right)-2\overline{r}\left(p\right)\hfill \\ & & +4t(2t+1)({\beta }^2-{\alpha }^2)+4t[(2t+1){\beta }^2-{\alpha }^2+2\left(\zeta \beta \right)].\hfill \end{array}$$

(37)

In order to maintain generality, suppose that $\mathfrak{L}$ is spanned by $\{I_1,\cdots ,I_{s-1}\}$. After, some easy computations yield

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \frac{\mu }{s}\sum _{a=s+1}^{2t+1}\sum _{i,j=1}^s{\left({\mathrm{\Pi }}_{ij}^a\right)}^2+\frac{(s+\mu )(s^2-s-\mu )}{\mu s}\sum _{a=s+1}^{2t+1}\sum _{i,j=1}^{s-1}{\left({\mathrm{\Pi }}_{ij}^a\right)}^2-2\overline{\Re }\left(p\right)\hfill \\ & & +4t(2t+1)({\beta }^2-{\alpha }^2)+4t[(2t+1){\beta }^2-{\alpha }^2+2\left(\zeta \beta \right)].\hfill \end{array}$$

(38)

Taking into use (36) and (38), we turn up

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \frac{s+\mu }{n}\sum _{a=s+1}^{2t+1}\sum _{i,j=1}^n{\left({\mathrm{\Pi }}_{ij}^a\right)}^2+\frac{(s+\mu )(s^2-s-\mu )}{\mu s}\sum _{a=s+1}^{2t+1}\sum _{i,j=1}^{s-1}{\left({\mathrm{\Pi }}_{ij}^a\right)}^2\hfill \\ & & -\sum _{a=s+1}^{2t+1}{\left(\sum _{i=1}^s{\mathrm{\Pi }}_{ii}^a\right)}^2.\hfill \end{array}$$

Simple calculations turn the above equation into

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \sum _{a=s+1}^{2t+1}\sum _{i=1}^{s-1}\left[\frac{s^2+s(\mu -1)-2\mu }{\mu }{\left({\mathrm{\Pi }}_{ii}^a\right)}^2+\frac{2(s+\mu )}{s}{\left({\mathrm{\Pi }}_{ij}^a\right)}^2\right]\hfill \\ & & +\sum _{a=s+1}^{2t+1}[\frac{2(s+\mu )(s-1)}{\mu }\sum _{i<j=1}^{n-1}{\left({\mathrm{\Pi }}_{ij}^a\right)}^2-2\sum _{i<j=1}^s{B}_{ii}^a{\mathrm{\Pi }}_{jj}^a\hfill \\ & & +\frac{\mu }{s}{\left({\mathrm{\Pi }}_{ss}^a\right)}^2].\hfill \end{array}$$

(39)

Moreover, with the help of (39), the critical points

$$\begin{array}{c}\hfill {\mathrm{\Pi }}^c=({\mathrm{\Pi }}_{11}^{n+1},{\mathrm{\Pi }}_{12}^{s+1},\cdots ,{\mathrm{\Pi }}_{ss}^{s+1},\cdots ,{\mathrm{\Pi }}_{11}^{2t+1},\cdots ,{\mathrm{\Pi }}_{ss}^{2t+1})\end{array}$$

Furthermore, for $\mathcal{Q}$ are the solutions of the system of linear homogeneous equations:

$$\begin{array}{ccc}\hfill \frac{\partial \mathcal{Q}}{\partial {\mathrm{\Pi }}_{ii}^a}& =& \frac{2(s+\mu )(s-1)}{\mu }{\mathrm{\Pi }}_{ii}^a-2\sum _{l=1}^s{\mathrm{\Pi }}_{ll}^a=0,\hfill \\ \hfill \frac{\partial \mathcal{Q}}{\partial {\mathrm{\Pi }}_{ss}^a}& =& \frac{2\mu }{n}{\mathrm{\Pi }}_{ss}^a-2\sum _{l=1}^{s-1}{\mathrm{\Pi }}_{ll}^a=0,\hfill \\ \hfill \frac{\partial \mathcal{Q}}{\partial {\mathrm{\Pi }}_{ij}^a}& =& \frac{4(s+\mu )(s-1)}{\mu }{\mathrm{\Pi }}_{ij}^a=0,\hfill \\ \hfill \frac{\partial \mathcal{Q}}{\partial {\mathrm{\Pi }}_{is}^a}& =& \frac{4(s+\mu )}{s}{\mathrm{\Pi }}_{is}^a=0,\hfill \end{array}$$

(40)

for all $i,j=\{1,2,\cdots ,s-1\},i\ne j,a\in \{s+1,\cdots ,2t+1\}$.

The corresponding determinant to the first two sets of equations in (40) is zero, demonstrating the possibility of solutions for non-totally geodesic submanifolds when $i\ne j$ and every solution $B^c$ shows $B_{ij}^a=0$. Additionally, the Hessian matrix ${\hslash }^{\prime }\left(\mathcal{Q}\right)$ of $\mathcal{Q}$ is provided by

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

wherein

$$\begin{array}{c}\hfill {\hslash }_1=\left(\begin{array}{ccccc}\frac{2(s+\mu )(s-1)}{\mu }-2& -2& \cdots & -2& -2\\ -2& \frac{2(s+\mu )(n-1)}{\mu }-2& \cdots & -2& -2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -2& -2& \cdots & \frac{2(s+\mu )(s-1)}{\mu }-2& -2\\ -2& -2& \cdots & -2& \frac{2\mu }{s}\end{array}\right),\end{array}$$

The diagonal matrices ${\hslash }_2$ and ${\hslash }_3$ are presented as follows, with 0 signifying the null matrices of the corresponding sizes.

$$\begin{array}{c}\hfill {\hslash }_2=\mathrm{diag}(\frac{2(s+\mu )(s-1)}{\mu },\frac{2(s+\mu )(s-1)}{\mu },\cdots ,\frac{2(s+\mu )(s-1)}{\lambda }),\end{array}$$

$$\begin{array}{c}\hfill {\hslash }_3=\mathrm{diag}(\frac{2(s+\mu )}{s},\frac{2(s+\mu )}{s},\cdots ,\frac{2(s+\mu )}{s}).\end{array}$$

The eigenvalues of $\hslash \left(\mathcal{Q}\right)$ are as follows.

$$\begin{array}{c}\hfill {\omega }_{11}=0,{\omega }_{22}=\frac{2(s^3-s^2+{\mu }^2)}{\mu n},{\omega }_{33}=\cdots ={\omega }_{ss}=\frac{2(s+\mu )(s-1)}{\mu },\end{array}$$

$$\begin{array}{c}\hfill {\omega }_{ij}=\frac{2(s+\mu )(s-1)}{\mu },{\omega }_{is}=\frac{2(s+\mu )}{s},\end{array}$$

for all $i,j\in \{1,2,\cdots ,s-1\},i\ne j$.

Therefore, one concludes that $\mathcal{Q}$ is parabolic at any solution ${\mathrm{\Pi }}^c$ of (40), reaches a minimum $\mathcal{Q}\left(B^c\right)$. With the help of (39) and (40), we find $\mathcal{Q}\left({\mathrm{\Pi }}^c\right)=0$ and that shows $\mathcal{Q}\ge 0$ and hence

$$\begin{array}{ccc}\hfill 2\overline{\Re }\left(p\right)& \le & \mu \mathcal{C}+\frac{(s-1)(s+\mu )(s^2-s-\mu )}{\mu s}\mathcal{C}\left(\mathfrak{L}\right)\hfill \\ & & +4t(2t+1)({\beta }^2-{\alpha }^2)+4t[(2t+1){\beta }^2-{\alpha }^2+2\left(\zeta \beta \right)],\hfill \end{array}$$

whereby, we obtain

$$\begin{array}{ccc}\hfill \sigma & \le & \frac{\mu }{s(s-1)}\mathcal{C}+\frac{(s+\mu )(s^2-s-\mu )}{\mu s^2}\mathcal{C}\left(\mathfrak{L}\right)\hfill \\ & & +\frac{4t}{s(s-1)}(2t+1)({\beta }^2-{\alpha }^2)+\frac{4t}{s(s-1)}[(2t+1){\beta }^2-{\alpha }^2+2\left(\zeta \beta \right)],\hfill \end{array}$$

for any tangent hyperplane $\mathfrak{L}$ of $T_p\aleph$. Therefore, (33) easily holds in the light of the above equation. In addition, a sign of the equality holds in (33) if and only if

$$\begin{array}{c}\hfill {\mathrm{\Pi }}_{ij}^a=0,\forall i,j\in \{1,\cdots ,s\},i\ne j\end{array}$$

(41)

and

$$\begin{array}{c}\hfill {\mathrm{\Pi }}_{ss}^a=\frac{s(s-1)}{\mu }{\mathrm{\Pi }}_{11}^a=\frac{s(s-1)}{\mu }{\mathrm{\Pi }}_{22}^a\cdots =\frac{s(s-1)}{\mu }{\mathrm{\Pi }}_{s-1s-1}^a,\\ \hfill \forall a\in \{s+1,\cdots ,2t+1\}.\end{array}$$

(42)

Thus, thanks to (41) and (42), one obtains the equality sign in (33) if and only if the submanifold ℵ is invariantly quasi-umbilical with a trivial normal connection in $\mathrm{\Theta }$. Therefore, with respect to suitable local orthonormal normal and orthonormal tangent a frames, the shape operators satisfy (35). (ii) Similarly one can prove (34).

□

Next, we gain sharp inequalities involving the normalized $\delta$-CC for submanifolds of a $(2t+1)$-dimensional $(\alpha ,\beta )$-type almost contact manifold $\mathrm{\Theta }$ equipped with the $SVK$-connection $\overline{\nabla }$.

Corollary 1. 

Let ℵ be an s-dimensional submanifold of a $(2t+1)$-dimensional $(\alpha ,\beta )$-type almost contact manifolds Θ equipped with the $SVK$-connection $\overline{\nabla }$. Then

(i)

the normalized δ-CC ${\delta }_c(s-1)$ holds

$$\begin{array}{ccc}\hfill \sigma & \le & {\delta }_c(s-1)+\frac{4t}{s(s-1)}(2t+1)({\beta }^2-{\alpha }^2)\hfill \\ & & +\frac{4t}{s(s-1)}[(2t+1){\beta }^2-{\alpha }^2+2\left(\zeta \beta \right)].\hfill \end{array}$$

(43)

(ii)

the normalized δ-CC ${\widehat{\delta }}_c(s-1)$ satisfies

$$\begin{array}{ccc}\hfill \sigma & \le & {\widehat{\delta }}_c(s-1)+\frac{4t}{s(s-1)}(2t+1)({\beta }^2-{\alpha }^2)\hfill \\ & & +\frac{4t}{s(s-1)}[(2t+1){\beta }^2-{\alpha }^2+2\left(\zeta \beta \right)].\hfill \end{array}$$

(44)

Furthermore, the equalities obey in (43) and (44) if and only if submnaifold ℵ is an invariantly quasi-umbilical with trivial normal connection in Θ such that with some orthonormal tangent frame $\{I_1,\cdots ,I_s\}$ of the tangent bundle $T\aleph$ of ℵ and orthonormal normal frame $\{I_{s+1},\cdots ,I_{2t+1}\}$ of the normal bundle $T^{\perp }\aleph$ of ℵ in $\mathrm{\Theta },$ the shape operators ${\mathrm{\Lambda }}_a$ satisfy

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}b& 0& 0& \cdots & 0& 0\\ 0& b& 0& \cdots & 0& 0\\ 0& 0& b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b& 0\\ 0& 0& 0& \cdots & 0& 2b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0\end{array}$$

(45)

and

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}2b& 0& 0& \cdots & 0& 0\\ 0& 2b& 0& \cdots & 0& 0\\ 0& 0& 2b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 2b& 0\\ 0& 0& 0& \cdots & 0& b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0.\end{array}$$

(46)

5. Applications of Theorem 2

We write the results below as an application of Theorem 2.

Corollary 2. 

If ℵ be an s-dimensional submanifold of a $(2t+1)$-dimensional β-Kenmotsu manifold Θ equipped with the $SVK$ connection $\overline{\nabla }$, then,

(i)

the δ-CC ${\delta }_c(\mu ;s-1)$ satisfies

$$\begin{array}{ccc}\hfill \sigma & \le & \frac{{\delta }_c(\mu ;s-1)}{s(s-1)}+\frac{4t}{s(s-1)}(2t+1){\beta }^2\hfill \\ & & +\frac{4t}{s(s-1)}[(2t+1){\beta }^2+2\left(\zeta \beta \right)],\hfill \end{array}$$

(47)

for some real number μ such that $0<\mu <s(s-1)$;

(ii)

the δ-CC ${\widehat{\delta }}_c(\mu ;s-1)$ satisfies

$$\begin{array}{ccc}\hfill \sigma & \le & \frac{{\widehat{\delta }}_c(\mu ;s-1)}{s(s-1)}+\frac{4t}{s(s-1)}(2t+1){\beta }^2\hfill \\ & & +\frac{4t}{s(s-1)}[(2t+1){\beta }^2+2\left(\zeta \beta \right)],\hfill \end{array}$$

(48)

for some real number $\mu >s(s-1).$

In addition, (47) and (48) obey for equality if and only if the submanifold ℵ is invariantly quasi-umbilical with a trivial normal connection in Θ. Moreover, the shape operators ${\mathrm{\Lambda }}_a$, $a\in \{s+1,\cdots ,2t+1\}$ with a orthonormal tangent frame $\{I_1,\cdots ,I_s\}$ and orthonormal normal frame $\{I_{s+1},\cdots ,I_{2t+1}\}$ take the following forms:

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}b& 0& 0& \cdots & 0& 0\\ 0& b& 0& \cdots & 0& 0\\ 0& 0& b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b& 0\\ 0& 0& 0& \cdots & 0& \frac{s(s-1)}{\mu }b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2r+1}=0.\end{array}$$

(49)

Corollary 3. 

Let ℵ be an s-dimensional submanifold of a $(2t+1)$-dimensional α-Sasakian manifold Θ equipped with the $SVK$-connection $\overline{\nabla }$. Then

(i)

the δ-CC ${\delta }_c(\mu ;s-1)$ holds

$$\begin{array}{ccc}\hfill \sigma & \le & \frac{{\delta }_c(\mu ;s-1)}{s(s-1)}-\frac{4t}{s(s-1)}(2t+1){\alpha }^2-\frac{4t}{s(s-1)}{\alpha }^2,\hfill \end{array}$$

(50)

for some real number μ such that $0<\mu <s(s-1)$;

(ii)

the δ-CC ${\widehat{\delta }}_c(\mu ;s-1)$ satisfies

$$\begin{array}{ccc}\hfill \sigma & \le & \frac{{\widehat{\delta }}_c(\mu ;s-1)}{s(s-1)}-\frac{4t}{s(s-1)}(2t+1){\alpha }^2-\frac{4t}{s(s-1)}{\alpha }^2,\hfill \end{array}$$

(51)

for some real number $\mu >s(s-1).$

Additionally, (50) and (51) maintain for equality if and only if the submanifold ℵ is invariantly quasi-umbilical with a trivial normal connection in Θ, such that the shape operators ${\mathrm{\Lambda }}_a$, $a\in \{s+1,\cdots ,2t+1\}$ with respect to some orthonormal tangent frame $\{I_{s+1},\cdots ,I_{2t+1}\}.$

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}b& 0& 0& \cdots & 0& 0\\ 0& b& 0& \cdots & 0& 0\\ 0& 0& b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b& 0\\ 0& 0& 0& \cdots & 0& \frac{s(s-1)}{\mu }b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0.\end{array}$$

(52)

Corollary 4. 

If N is an s-dimensional submanifold of a $(2t+1)$-dimensional Kenmotsu manifold Θ equipped with the $SVK$-connection $\overline{\nabla }$, then,

(i)

the δ-CC ${\delta }_c(\mu ;s-1)$ satisfies

$$\begin{array}{ccc}\hfill \sigma & \le & \frac{{\delta }_c(\mu ;s-1)}{s(s-1)}+\frac{4t}{s(s-1)}(2t+1)+\frac{4t}{s(s-1)}[(2t+1)+2\zeta ]\hfill \end{array}$$

(53)

such that $0<\mu <s(s-1)$ for any real number μ;

(ii)

the δ-CC ${\widehat{\delta }}_c(\mu ;s-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & \frac{{\widehat{\delta }}_c(\mu ;s-1)}{s(s-1)}+\frac{4t}{s(s-1)}(2t+1)+\frac{4t}{s(s-1)}[(2t+1)+2\zeta ]\hfill \end{array}$$

(54)

for some real number $\mu >s(s-1).$

Additionally, (53) and (54) maintain for equality if and only if the submanifold ℵ is invariantly quasi-umbilical with a trivial normal connection in Θ, such that the shape operators ${\mathrm{\Lambda }}_a$, $a\in \{s+1,\cdots ,2t+1\}$ with respect to some orthonormal tangent frame $\{I_{s+1},\cdots ,I_{tr+1}\}.$

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}b& 0& 0& \cdots & 0& 0\\ 0& b& 0& \cdots & 0& 0\\ 0& 0& b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b& 0\\ 0& 0& 0& \cdots & 0& \frac{s(s-1)}{\mu }b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0.\end{array}$$

(55)

Corollary 5. 

If ℵ is an s-dimensional submanifold of a $(2t+1)$-dimensional Sasakian manifold Θ equipped with the $SVK$-connection $\overline{\nabla }$, then,

(i)

the δ-CC ${\delta }_c(\mu ;s-1)$ satisfies

$$\begin{array}{ccc}\hfill \sigma & \le & \frac{{\delta }_c(\mu ;s-1)}{s(s-1)}-\frac{4t}{s(s-1)}(2t+1)-\frac{4t}{s(s-1)}\hfill \end{array}$$

(56)

such that $0<\mu <s(s-1)$ for any real number μ;

(ii)

the δ-CC ${\widehat{\delta }}_c(\mu ;s-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & \frac{{\widehat{\delta }}_c(\mu ;s-1)}{s(s-1)}-\frac{4t}{s(s-1)}(2t+1)-\frac{4t}{s(s-1)}\hfill \end{array}$$

(57)

for some real number $\mu >s(s-1).$

Moreover, (56) and (57) maintain for equality if and only if the submanifold ℵ is invariantly quasi-umbilical with trivial normal connection in Θ, such that the shape operators ${\mathrm{\Lambda }}_a$, $a\in \{s+1,\cdots ,2t+1\}$ with respect to some orthonormal tangent frame $\{I_{s+1},\cdots ,I_{2t+1}\}.$

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}b& 0& 0& \cdots & 0& 0\\ 0& b& 0& \cdots & 0& 0\\ 0& 0& b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b& 0\\ 0& 0& 0& \cdots & 0& \frac{s(s-1)}{\mu }b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0.\end{array}$$

(58)

In view of Corollary 1, we gain the following results.

Corollary 6. 

If ℵ is an s-dimensional submanifold of a $(2t+1)$-dimensional β-Kenmotsu manifold Θ equipped with the $SVK$-connection $\overline{\nabla }$, then,

(i)

the δ-CC ${\delta }_c(s-1)$ holds

$$\begin{array}{ccc}\hfill \sigma & \le & {\delta }_c(s-1)+\frac{4t}{s(s-1)}(2t+1){\beta }^2\hfill \\ & & +\frac{4t}{s(s-1)}[(2t+1){\beta }^2+2\left(\zeta \beta \right)].\hfill \end{array}$$

(59)

(ii)

the δ-CC ${\widehat{\delta }}_c(s-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\widehat{\delta }}_c(s-1)+\frac{4t}{s(s-1)}(2t+1){\beta }^2\hfill \\ & & +\frac{4t}{s(s-1)}[(2t+1){\beta }^2+2\left(\zeta \beta \right)].\hfill \end{array}$$

(60)

Furthermore, the equalities maintain in the expression (59) and (60) if and only if ℵ is an invariantly quasi-umbilical submanifold with a trivial normal connection in Θ. Therefore, with an orthonormal tangent set $\{I_1,\cdots ,I_s\}$ of the tangent bundle $T\aleph$ of ℵ and orthonormal normal set $\{I_{s+1},\cdots ,I_{2t+1}\}$ of the normal bundle $T^{\perp }\aleph$ of ℵ in $\mathrm{\Theta },$ the shape operators ${\mathrm{\Lambda }}_a$ satisfy

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}b& 0& 0& \cdots & 0& 0\\ 0& b& 0& \cdots & 0& 0\\ 0& 0& b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b& 0\\ 0& 0& 0& \cdots & 0& 2b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0\end{array}$$

(61)

and

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}2b& 0& 0& \cdots & 0& 0\\ 0& 2b& 0& \cdots & 0& 0\\ 0& 0& 2b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 2b& 0\\ 0& 0& 0& \cdots & 0& b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0.\end{array}$$

(62)

Corollary 7. 

Let ℵ be an s-dimensional submanifold of a $(2t+1)$-dimensional α-Sasakian manifold Θ equipped with the $SVK$-connection $\overline{\nabla }$. Then,

(i)

the δ-CC ${\delta }_c(s-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\delta }_c(s-1)-\frac{4t}{s(s-1)}(2t+1){\alpha }^2-\frac{4t}{s(s-1)}{\alpha }^2.\hfill \end{array}$$

(63)

(ii)

the δ-CC ${\widehat{\delta }}_c(s-1)$ holds

$$\begin{array}{ccc}\hfill \sigma & \le & {\widehat{\delta }}_c(s-1)-\frac{4t}{s(s-1)}(2t+1){\alpha }^2-\frac{4t}{s(s-1)}{\alpha }^2.\hfill \end{array}$$

(64)

Also, the relationships (63) and (64) maintain the equality. if and only if ℵ is an invariantly quasi-umbilical submanifold with trivial normal connection in Θ such that with some orthonormal tangent frame $\{I_1,\cdots ,I_n\}$ of the tangent bundle $T\aleph$ of ℵ and orthonormal normal set $\{I_{s+1},\cdots ,I_{2t+1}\}$ of the normal bundle $T^{\perp }\aleph$ of ℵ in $\mathrm{\Theta },$ the shape operators ${\mathrm{\Lambda }}_a$ holds.

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}b& 0& 0& \cdots & 0& 0\\ 0& b& 0& \cdots & 0& 0\\ 0& 0& b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b& 0\\ 0& 0& 0& \cdots & 0& 2b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0\end{array}$$

(65)

and

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}2b& 0& 0& \cdots & 0& 0\\ 0& 2b& 0& \cdots & 0& 0\\ 0& 0& 2b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 2b& 0\\ 0& 0& 0& \cdots & 0& b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0.\end{array}$$

(66)

Corollary 8. 

If ℵ be an s-dimensional submanifold of a $(2t+1)$-dimensional Kenmotsu manifold Θ equipped with the $SVK$-connection $\overline{\nabla }$, then,

(i)

the δ-CC ${\delta }_c(s-1)$ satisfies

$$\begin{array}{ccc}\hfill \sigma & \le & {\delta }_c(s-1)+\frac{4t}{s(s-1)}(2t+1)+\frac{4t}{s(s-1)}[(2t+1)+2\zeta ].\hfill \end{array}$$

(67)

(ii)

the δ-CC ${\widehat{\delta }}_c(s-1)$ holds

$$\begin{array}{ccc}\hfill \sigma & \le & {\widehat{\delta }}_c(s-1)+\frac{4t}{s(s-1)}(2t+1)+\frac{4t}{s(s-1)}[(2t+1)+2\zeta ].\hfill \end{array}$$

(68)

Furthermore, the equalities exist in the relations (67) and (68) if and only if ℵ is an invariantly quasi-umbilical submanifold with trivial normal connection in Θ such that with a orthonormal tangent set $\{I_1,\cdots ,I_n\}$ of the tangent bundle $T\aleph$ of ℵ and orthonormal normal set $\{I_{n+1},\cdots ,I_{2t+1}\}$ of the normal bundle $T^{\perp }\aleph$ of ℵ in $\mathrm{\Theta },$ the shape operators ${\mathrm{\Lambda }}_a$ obey

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}b& 0& 0& \cdots & 0& 0\\ 0& b& 0& \cdots & 0& 0\\ 0& 0& b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b& 0\\ 0& 0& 0& \cdots & 0& 2b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0\end{array}$$

(69)

and

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}2b& 0& 0& \cdots & 0& 0\\ 0& 2b& 0& \cdots & 0& 0\\ 0& 0& 2b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 2b& 0\\ 0& 0& 0& \cdots & 0& b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0.\end{array}$$

(70)

Corollary 9. 

If ℵ be an s-dimensional submanifold of a $(2t+1)$-dimensional Sasakian manifold Θ equipped with the $SVK$-connection $\overline{\nabla }$, then

(i)

the δ-CC ${\delta }_c(s-1)$ satisfies

$$\begin{array}{ccc}\hfill \sigma & \le & {\delta }_c(s-1)-\frac{4t}{s(s-1)}(2t+1)-\frac{4t}{s(s-1)}.\hfill \end{array}$$

(71)

(ii)

the δ-CC ${\widehat{\delta }}_c(s-1)$ satisfies

$$\begin{array}{ccc}\hfill \sigma & \le & {\widehat{\delta }}_c(s-1)-\frac{4t}{s(s-1)}(2t+1)-\frac{4t}{s(s-1)}.\hfill \end{array}$$

(72)

In addition, the equalities exits in the (71) and (72) if and only if ℵ is an invariantly quasi-umbilical submanifold with a trivial normal connection in Θ such that with an orthonormal tangent frame $\{I_1,\cdots ,I_s\}$ of the tangent bundle $T\aleph$ of ℵ and an orthonormal normal set $\{I_{n+1},\cdots ,I_{2t+1}\}$ of the normal bundle $T^{\perp }\aleph$ of ℵ in $\mathrm{\Theta },$ the shape operators ${\mathrm{\Lambda }}_a$ satisfy

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}b& 0& 0& \cdots & 0& 0\\ 0& b& 0& \cdots & 0& 0\\ 0& 0& b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & b& 0\\ 0& 0& 0& \cdots & 0& 2b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0\end{array}$$

(73)

and

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{s+1}=\left(\begin{array}{cccccc}2b& 0& 0& \cdots & 0& 0\\ 0& 2b& 0& \cdots & 0& 0\\ 0& 0& 2b& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 2b& 0\\ 0& 0& 0& \cdots & 0& b\end{array}\right),{\mathrm{\Lambda }}_{s+2}=\cdots ={\mathrm{\Lambda }}_{2t+1}=0.\end{array}$$

(74)

6. Based Example

Example 1. 

We consider the three-dimensional manifold $\mathrm{\Theta }=[(x,y,z)\in R^3\mid z\ne 0],$ where $(x,y,z)$ are the Cartesian coordinates in $R^3$. Choosing the vector fields

$$e_1=-z\left(\frac{\partial }{\partial x}+y\frac{\partial }{\partial z}\right),e_2=-z\frac{\partial }{\partial y},e_3=\frac{\partial }{\partial z},$$

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

$$g(e_1,e_3)=g(e_2,e_3)=g(e_2,e_2)=0,g(e_1,e_1)=g(e_2,e_2)=g(e_3,e_3)=1.$$

Let γ be the 1-form defined by $\gamma \left(Z\right)=g(Z,e_3)$ for any vector field Z on $T\mathrm{\Theta }$. Let ψ be the $(1,1)$ tensor field defined by

$$\psi \left(e_1\right)=-e_2,\psi \left(e_2\right)=-e_1,\psi \left(e_3\right)=0.$$

Then, by the linearity property of ψ and g, we have

$${\psi }^{2}Z=-Z+\gamma \left(Z\right)e_3,\gamma \left(e_3\right)=1andg(\psi Z,\psi W)=g(Z,W)-\gamma \left(Z\right)\gamma \left(W\right)$$

for any vector fields $Z,W$ on Θ. Thus, $(\psi ,\zeta ,\gamma ,g)$ defines an almost contact metric structure on Θ.

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

$$[e_1,e_2]=-y{e}_2-z^2{e}_3,[e_1,e_3]=-\frac{1}{z}{e}_1,[e_2,e_3]=-\frac{1}{z}{e}_2.$$

The Riemannian connection ∇ with respect to the metric g is given by

$$2g({\nabla }_{X}Y,Z)=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)+g([X,Y],Z)-g([Y,Z],X)$$

$$+g\left(\right[Z,X],Y).$$

From the above equation, which is known as Koszul’s formula, we have

$${\nabla }_{e_1}{e}_3=-\frac{1}{z}{e}_1+\frac{1}{2}{z}^2{e}_2,{\nabla }_{e_2}{e}_3=-\frac{1}{2}{z}^2{e}_1-\frac{1}{z}{e}_2,{\nabla }_{e_3}{e}_3=0,$$

(75)

$${\nabla }_{e_1}{e}_2=-\frac{1}{2}{z}^2{e}_3,{\nabla }_{e_2}{e}_2=-y{e}_1+\frac{1}{z}{e}_3,{\nabla }_{e_3}{e}_2=-\frac{1}{2}{z}^2{e}_1,$$

$${\nabla }_{e_1}{e}_1=\frac{1}{z}{e}_3,{\nabla }_{e_2}{e}_1=y{e}_2+\frac{1}{2}{z}^2{e}_3,{\nabla }_{e_3}{e}_1=\frac{1}{2}{z}^2{e}_2.$$

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

$${\nabla }_p\zeta =-\alpha \psi p+\left\{\beta (p-\gamma (p)\zeta \right\},$$

for $\zeta \in e_3$, $\alpha =-\frac{1}{2}{z}^2$ and $\beta =-\frac{1}{z}$. Hence, the manifold Θ under consideration is a trans-Sasakian manifold of type $(-\frac{1}{2}{z}^2,-\frac{1}{z})$ manifold of dimension three [48].

Now, we consider at this structure for $SVK$-connection from (75) and (25), we obtain:

$${\overline{\nabla }}_{e_1}{e}_3=0,{\overline{\nabla }}_{e_2}{e}_3=0,{\overline{\nabla }}_{e_3}{e}_3=0,$$

(76)

$${\overline{\nabla }}_{e_1}{e}_2=0,{\overline{\nabla }}_{e_2}{e}_2=-y{e}_1,{\overline{\nabla }}_{e_3}{e}_2=-\frac{1}{2}{z}^2{e}_1,$$

$${\overline{\nabla }}_{e_1}{e}_1=0,{\overline{\nabla }}_{e_2}{e}_1=y{e}_2,{\overline{\nabla }}_{e_3}{e}_1=\frac{1}{2}{z}^2{e}_2.$$

Then, the non-vanishing Riemannian, Ricci curvature, and scalar curvature tensor components with respect to the $SVK$-connection are given by:

$${\overline{\Re }}_{im}(e_1,e_2)e_2=-\left(y^2+\frac{1}{2}{z}^4\right)e_1,,{\overline{\Re }}_{im}(e_1,e_2)e_1=\left(y^2+\frac{1}{2}{z}^4\right)e_2,$$

$${\overline{\Re }}_{im}(e_1,e_3)e_1=-y{z}^2{e}_2,{\overline{\Re }}_{im}(e_1,e_3)e_2=y{z}^2{e}_1,$$

$${\overline{\Re }}_{im}(e_2,e_3)e_1=\frac{y}{z}{e}_2,{\overline{\Re }}_{im}(e_2,e_3)e_2=-\frac{y}{z}{e}_1.$$

$$\begin{array}{c}\hfill {\overline{\Re }}_{ic}{(e_i,e_i)}_{i=1,2,3}=\left(\begin{array}{ccc}-\left(y^2+\frac{1}{2}{z}^4\right)& 0& 0\\ 0& -\left(y^2+\frac{1}{2}{z}^4\right)& 0\\ 0& 0& 0\end{array}\right),\end{array}$$

$$\overline{\Re }=Trace\left({\overline{\Re }}_{ic}\right)=-(2y^2+z^4).$$

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

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

Let ℵ be a two-dimensional submanifold of a 3-dimensional $(\alpha ,\beta )$-type almost contact manifold Θ, which is spanned by $\{e_1,e_2\}$. Since $\overline{\nabla }$ is a $SVK$-connection. Thus, from (76) we get

$${\overline{\nabla }}_{e_1}{e}_1=0,{\overline{\nabla }}_{e_2}{e}_3=0,{\overline{\nabla }}_{e_2}{e}_2=-y{e}_2.$$

(77)

And the second fundamental forms with respect to the $SVK$-connection are given a

$$\begin{array}{c}\hfill \mathrm{\Pi }{(e_i,e_i)}_{i=1,2}=\left(\begin{array}{cc}\frac{1}{2}z{e}_3& 0\\ 0& \frac{1}{2}z{e}_3\end{array}\right).\end{array}$$

(78)

From this, we can easily compute the mean curvature with respect to the $SVK$-connection as

$$\mathrm{\Omega }=\frac{1}{2}\sum _{1=1}^2\mathrm{\Pi }(e_i,e_i)=\frac{1}{2}z{e}_3.$$

(79)

And in light of (32), the Casorati Curvature $\mathcal{C}$ with the $SVK$-connection is given by

$$\mathcal{C}=\frac{1}{s}{\left|\right|\mathrm{\Pi }\left|\right|}^2=\frac{z^2}{4}.$$

(80)

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

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

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

7. Conclusions

Submanifolds in other ambient spaces with a Schouten–Van Kampen connection can be studied using the techniques presented in this study. It is required to establish a suitable sectional curvature in order to examine corresponding problems for submanifolds in trans-Sasakian manifolds with a Schouten–Van Kampen connection. In addition, we pursued the equality cases of these inequalities if and only if the submanifold has a trivial normal connection in trans-Sasakian manifolds with a Schouten–Van Kampen connection and is invariantly quasi-umbilical and deduced some results below as an application of Theorem 2 for the Sasakian manifold, Kenmostu manifolds, and cosymplectic manifolds with respect to the Schouten–Van Kampen connection. Finally, we furnished a non-trivial example to illustrate the results.

For further research, it would be interesting to obtain such type inequalities for various classes of submanifold and submersions (see [19]). In this case, it could be necessary to use not only techniques from submanifold theory, but also from singularity theory [22,23].

In addition, J.C. Marrero proved that, at the Riemannian case, for dimensions greater or equal to five the only existing trans-Sasakian manifolds are Sasakian and Kenmotsu ones. Maybe it is different in the semi-Riemannian case. This would be a really interesting open problem for the geometers.

The optimization techniques have a pivotal role in improving inequalities involving Chen invariants. T. Oprea applied the constrained extremum problem to prove Chen–Ricci inequalities for Lagrangian submanifolds of complex space forms. In the characterization of our main result, is it possible to apply the following lemma to derive such inequalities?

Let $(\overline{\mathrm{\Theta }},\overline{g})$ be a Riemannian manifold, B be a Riemannian submanifold of it, g be the metric induced on B by $\overline{g}$, and $f:(\overline{\mathrm{\Theta }},\overline{g})⟶(\mathbb{R},<·,·>)$ be a differentiable function. Consider the constrained extremum problem $\underset{x\in \mathrm{\Theta }}{min}f\left(x\right),$ then we have the following:

Lemma 1. 

If $x_0\in \mathrm{\Theta }$ is the solution of the above problem, then

1.

$\left(gradf\right)\left(x_0\right)\in T_{x_0}^{\perp }\mathrm{\Theta }$,

2.

the bilinear form

$\mathbf{A}:T_{x_0}\mathrm{\Theta }\times T_{x_0}\mathrm{\Theta }⟶\mathbb{R}$, $\mathbf{A}(p,q)=Hes{s}_f(p,q)+\overline{g}(h(p,q),\left(gradf\right)\left(x_0\right))$ is positive semi-definite, where h is the second fundamental form of B in $\overline{\mathrm{\Theta }}$, and $gradf$ denotes gradient of f.