Some Basic Inequalities on (ϵ)-Para Sasakian Manifold

Abstract

We propose fundamental inequalities for contact pseudo-slant submanifolds of $(ϵ)$-para Sasakian space form employing generalized normalized $\delta$-Casorati curvature. We characterize submanifolds for which equality cases hold and illustrate the main result with some applications. Further, we have considered a certain type of submanifold for a Ricci soliton and after computing its scalar curvature, developed an inequality to find correlations between intrinsic or extrinsic invariants.

1. Introduction

It is very interesting to investigate the link between intrinsic and extrinsic invariants with the help of sharp inequality involving $\delta$-invariants. A number of scenarios have been applied to Chen’s invariants since their invention in [1] (refer to [2,3,4,5,6,7], etc.). The study of optimal inequalities turned out to be more appealing to the geometers with the introduction of Casorati curvature due to F. Casorati [8], and this event provided them a new tool to derive optimal inequalities with Casorati curvatures. This notion of Casorati curvature has been applied by several investigators in different ambient spaces ([9,10,11,12], etc.).

On the other hand, Sato [13] began studying almost-paracontact structures on a differentiable manifold in 1976. A structure was devised by Tripathi et al. [14] termed as the $ϵ$-almost-paracontact structure having vector fields $\xi$ that are spacelike ($ϵ=1$) (resp. timelike ($ϵ=-1$)). They further explained an $\left(ϵ\right)$-almost-paracontact manifold and $\left(ϵ\right)$-Sasakian manifold. Recently, Dirik et al. [15] studied $\left(ϵ\right)$-para Sasakian manifold (briefly, $\left(ϵ\right)$-PSM) and obtained certain results on contact pseudo-slant submanifolds (briefly, CPSS) of $\left(ϵ\right)$-PSM and an $\left(ϵ\right)$-para Sasakian space form ($\left(ϵ\right)$-PSSF).

In addition, as a result of [16] Grigori Perelman’s solution of Poincare conjecture (1904), Ricci solitons have gained popularity. Moreover, they simulate how singularities develop in Ricci flows (for details, [16]).

Hamilton [17] first proposed the Ricci flow in 1982, and the Ricci soliton indicates self-identical solutions of such a flow. A Riemannian manifold $(\tilde{M},g)$ equipped with a smooth vector field V forms a Ricci soliton. According to Hamilton’s formulation,

$$S+\frac{1}{2}{\mathfrak{L}}_{V}g+\Lambda g=0;$$

(1)

in this case, ${\mathfrak{L}}_{V}g$ means a Lie derivative of g with respect to V, and S means the Ricci tensor of $(\tilde{M},g)$. Classify $(\tilde{M},g,V,\Lambda )$ by decreasing ($\Lambda <0$), stabilizing ($\Lambda =0$), or expanding ($\Lambda >0$). Additionally, a gradient Ricci soliton emerges if $V=\nabla \psi$ in (1) and on $\tilde{M}$, a smooth function $\psi$ exists where the Ricci tensor of g is provided with

$$S+Hess\left(\psi \right)=\Lambda g.$$

The authors worked with a Ricci soliton immersed into a Riemannian manifold in 2011 [18]. The Ricci soliton has recently gained interest among some researchers for many types of manifolds, including contact, para-contact, Sasakian, and others [19,20]. For example, in [21], Bejan and Crasmareanu explored the Eisenhart issue of finding parallel tensors for the symmetric situation, which was originally discussed for quasi-constant curvature manifolds and provided some characterizations in terms of Ricci solitons. They also looked at the $(0,2)$-type family of parallel symmetric tensor fields and potential Lorentz Ricci solitons [22].

Chen and Deshmukh developed a criterion for constituting a submanifold as a Ricci soliton in [23], and thorough characterization of Ricci solitons on Euclidean hypersurfaces was also shown in [24].

Due to its growing applications in physics, including Yang–Mills theory, Kaluza–Klein theory, string theory, and Hodge theory, the geometry of the Ricci soliton with Riemannian immersion and its extensions, such as hypersurfaces and various types of submanifolds of Riemannian manifold, have attracted increasing interest in modern geometric analysis. We can create additional structures as examples of locally trivial fiber spaces. Thus, we can analyze the spaces with symmetries using the framework on structure-preserving submanifolds. In particular, black holes in different dimensions, Lagrangians (with symmetries), and basic quantum systems (with symmetrical features) can all be studied directly using this theory.

Motivated by all the above developments, this study deals with the contact pseudo-slant submanifold of $\left(ϵ\right)$-PSSF. The purpose of this work is to present some fundamental inequalities that arise from generalized normalized $\delta$-Casorati for CPSS of an $\left(ϵ\right)$-PSSF, and we also describe the submanifolds for which equality cases hold, as well as write some applications of the main result. We also consider certain types of submanifold M of $(\tilde{M},g,V,\Lambda )$ and compute its scalar curvature. Further, for these Ricci solitons, we developed an inequality to find correlations between intrinsic or extrinsic invariants, such as scalar curvature, sectional curvature, or mean curvature.

2. Preliminaries

In the presence of a $(1,1)$-tensor field $\varphi$, vector field $\xi$, 1-form $\eta$ on the differentiable manifold ${\tilde{M}}^n$, there occurs almost-para-contact structure $(\varphi ,\xi ,\eta )$ as [13]

$$\begin{array}{ccc}\hfill {\varphi }^2& =& {\ell }_1-\eta \left({\ell }_1\right)\xi ,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill \eta \left(\xi \right)& =& 1,g({\ell }_1,\xi )=\eta \left({\ell }_1\right),\hfill \end{array}$$

(3)

$$\eta \left(\varphi \right)=0,\varphi \xi =0,$$

${\ell }_1$ being vector field on $\tilde{M}$.

The semi-Riemannian metric on the manifold is supplied by symmetric non-degenerate $(0,2)$-tensor field g. A semi-Riemannian metric having index 1 is corresponded by a Lorentzian metric [25] in this context.

For a semi-Riemannian metric g on an almost-para-contact manifold, if [14]

$$g(\varphi {\ell }_1,\varphi {\ell }_2)=g({\ell }_1,{\ell }_2)-ϵ\eta \left({\ell }_1\right)\eta \left({\ell }_2\right),$$

(4)

$\tilde{M}(\varphi ,\xi ,\eta ,g,ϵ)$ becomes $\left(ϵ\right)$-almost-para-contact metric manifold. Here, $ϵ=+1$ or $ϵ=-1$.

We define an $(ϵ)$-almost-para-contact metric manifold as a Lorentzian almost-para-contact metric manifold provided the index of g is 1. In addition, the $(ϵ)$-almost-para-contact metric manifold is the usual almost-para contact metric manifold having positive definite metric g.

In light of (2), (3), and (4), the obtained equations are

$$g(\varphi {\ell }_1,{\ell }_2)=g({\ell }_1,\varphi {\ell }_2),$$

$$g({\ell }_1,\xi )=ϵ\eta \left({\ell }_1\right).$$

(5)

Again, (3) and (5) provide

$$g(\xi ,\xi )=ϵ.$$

When $(\tilde{M},\varphi ,\xi ,\eta ,g)$ stands for an almost-para-contact metric manifold and $\tilde{\nabla }$ is a Levi–Civita connection on it, $\tilde{M}$ represents $\left(ϵ\right)$-PSM iff

$$({\tilde{\nabla }}_{{\ell }_1}\varphi ){\ell }_2=-g(\varphi {\ell }_1,\varphi {\ell }_2)\xi -ϵ\eta \left({\ell }_2\right){\varphi }^2{\ell }_1.$$

(6)

By putting $\xi$ for ${\ell }_2$ in (6), we have

$${\nabla }_{{\ell }_1}\xi =ϵ\varphi {\ell }_1.$$

When the $(0,2)$-type Ricci-tensor of an $\eta$-Einstein holds,

$$S({\ell }_1,{\ell }_2)=ag({\ell }_1,{\ell }_2)+b\eta \left({\ell }_1\right)\eta \left({\ell }_2\right),$$

it becomes an $\left(ϵ\right)$-PSM $\tilde{M}$. In this scenario, $a,b$ indicate smooth functions on $\tilde{M}$. Further, $\tilde{M}$ becomes an Einstein manifold for $b=0$.

The $\left(ϵ\right)$-PSSF $\tilde{M}\left(k\right)$ having constant $\varphi$-para holomorphic sectional curvature k satisfies

$$\begin{array}{ccc}\hfill g(\tilde{R}({\ell }_1,{\ell }_2){\ell }_3,{\ell }_4)& =& \left(\frac{k-3}{4}\right)\{g({\ell }_2,{\ell }_3)g({\ell }_1,{\ell }_4)-g({\ell }_1,{\ell }_3)g({\ell }_2,{\ell }_4)\}\hfill \\ & -& \left(\frac{k+1}{4}\right)\{\eta \left({\ell }_2\right)\eta \left({\ell }_3\right)g({\ell }_1,{\ell }_4)-\eta \left({\ell }_1\right)\eta \left({\ell }_2\right)g({\ell }_2,{\ell }_4)\hfill \\ & +& g({\ell }_1,{\ell }_3)\eta \left({\ell }_2\right)\eta \left({\ell }_4\right)-g({\ell }_2,{\ell }_3)\eta \left({\ell }_1\right)\eta \left({\ell }_4\right)\hfill \\ & +& g({\ell }_2,\varphi {\ell }_3)g(\varphi {\ell }_1,{\ell }_4)-g({\ell }_1,\varphi {\ell }_3)g(\varphi {\ell }_2,{\ell }_4)\hfill \\ & +& 2g(\varphi {\ell }_1,{\ell }_2)g(\varphi {\ell }_1,{\ell }_4)\}.\hfill \end{array}$$

(7)

Let ∇ and ${\nabla }^{\perp }$ be the induced connections of tangent bundle and normal bundle of submanifold M of $\left(ϵ\right)$-PSM $\tilde{M}$. For $V\in \mathrm{\Gamma }\left(T^{\perp }M\right)$, write the Gauss and Weingarten formulae as

$$\begin{array}{ccc}\hfill {\tilde{\nabla }}_{{\ell }_1}{\ell }_2& =& {\nabla }_{{\ell }_1}{\ell }_2+\sigma ({\ell }_1,{\ell }_2),\hfill \\ \hfill {\tilde{\nabla }}_{{\ell }_1}V& =& -A_V{\ell }_1+{\nabla }_{{\ell }_1}^{\perp }V,\forall {\ell }_1,{\ell }_2\in \mathrm{\Gamma }\left(TM\right),\hfill \end{array}$$

$\sigma$ is second fundamental form, and $A_V$ refers to a shape operator. $A_V$ and $\sigma$ have a close relationship because

$$g(A_V{\ell }_1,{\ell }_2)=g(\sigma ({\ell }_1,{\ell }_2),V)).$$

Let $\tilde{R}$ and R represent Riemannian curvature tensors of $\tilde{M}$ and M. Then,

$$\tilde{R}({\ell }_1,{\ell }_2){\ell }_3=R({\ell }_1,{\ell }_2){\ell }_3-A_{\sigma ({\ell }_2,{\ell }_3)}{\ell }_1+A_{\sigma ({\ell }_1,{\ell }_3)}{\ell }_2-({\tilde{\nabla }}_{{\ell }_2}\sigma )({\ell }_1,{\ell }_3)+({\tilde{\nabla }}_{{\ell }_1}\sigma )({\ell }_2,{\ell }_3),$$

(8)

$\forall {\ell }_i\in \mathrm{\Gamma }\left(TM\right),i=1,2,3$.

Describe the local orthonormal tangent frame of $TM$ as $\{{\vartheta }_1,\cdots ,{\vartheta }_n\}$ and the local orthonormal normal frame of $T^{\perp }M$ in $\tilde{M}$ as $\{{\vartheta }_{n+1},\cdots ,{\vartheta }_m\}$. M becomes

• totally umbilical when $\sigma ({\ell }_1,{\ell }_2)=g({\ell }_1,{\ell }_2)H$, $H=\frac{1}{n}{\sum }_{i=1}^n\sigma ({\vartheta }_i,{\vartheta }_i)$ is the mean curvature of M,
• totally geodesic if $\sigma =0$,
• minimal if $H=0$.

Write the scalar curvature by

$$\begin{array}{c}\hfill \tau =\sum _{1\le i<j\le n}R({\vartheta }_i,{\vartheta }_j,{\vartheta }_j,{\vartheta }_i),\end{array}$$

and the normalized scalar curvature with

$$\begin{array}{c}\hfill \rho =\frac{2\tau }{n(n-1)}.\end{array}$$

By setting ${\sigma }_{ij}^r=g(\sigma ({\vartheta }_i,{\vartheta }_j),{\vartheta }_r),$ $1\le i,j\le n$, one obtains

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

(9)

The divergence of vector field ℓ on $\mathrm{\Gamma }\left(TM\right)$ is determined with

$$Div(\ell )=\sum _{i=1}^{n}g({\nabla }_{{\vartheta }_i}\ell ,{\vartheta }_i).$$

(10)

The simple formula for the Casorati curvature of M is

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

(11)

With the t-dimensional subspace $\mathcal{L}$ of $TM,t\ge 2$, let $\{{\vartheta }_1,\cdots ,{\vartheta }_t\}$ be its orthonormal basis. The t-plane section’s scalar curvature may thus be expressed as follows:

$$\begin{array}{c}\hfill \tau \left(\mathcal{L}\right)=\sum _{1\le i<j\le t}R({\vartheta }_i,{\vartheta }_j,{\vartheta }_j,{\vartheta }_i)\end{array}$$

and $\mathcal{L}$’s Casorati curvature is determined by

$$\begin{array}{c}\hfill \mathcal{C}\left(\mathcal{L}\right)=\frac{1}{t}\sum _{r=n+1}^m\sum _{i,j=1}^t{\left({\sigma }_{ij}^r\right)}^2.\end{array}$$

Let $\mathcal{L}$ be hyperplane of $T_{p}M$; then, specify the normalized δ-Casorati curvatures as

$$\begin{array}{c}\hfill {\left[{\delta }_c(n-1)\right]}_p=\frac{1}{2}{\mathcal{C}}_p+\frac{1}{2}(1+\frac{1}{n})\inf \left\{\mathcal{C}\left(\mathcal{L}\right)\right\},\end{array}$$

$$\begin{array}{c}\hfill {\left[{\widehat{\delta }}_c(n-1)\right]}_p=2{\mathcal{C}}_p-(1-\frac{1}{2n})\sup \left\{\mathcal{C}\left(\mathcal{L}\right)\right\}.\end{array}$$

Let ${\mathcal{N}}_1=\frac{(n^2+nr-n-r)(n^2-n-r)}{rn}$. Set the generalized normalized δ-Casorati curvatures of $M^n$ as

$$\begin{array}{ccc}\hfill {\left[{\delta }_c(r;n-1)\right]}_p& =& r{\mathcal{C}}_p+{\mathcal{N}}_1.\inf \left\{\mathcal{C}\left(\mathcal{L}\right)\right\},\hfill \end{array}$$

when $0<r<n^2-n$, and

$$\begin{array}{ccc}\hfill {\left[{\widehat{\delta }}_c(r;n-1)\right]}_p& =& r{\mathcal{C}}_p+{\mathcal{N}}_1.\sup \left\{\mathcal{C}\left(\mathcal{L}\right)\right\},\hfill \end{array}$$

provided $r>n(n-1).$

The Riemannian curvature tensor of an immersed submanifold M of $\left(ϵ\right)$-para Sasakian space of the form $\tilde{M}\left(k\right)$ is provided by using (7) and (8) as

$$\begin{array}{ccc}\hfill R({\ell }_1,{\ell }_2){\ell }_3& =& \left(\frac{k-3}{4}\right)\{g({\ell }_2,{\ell }_3){\ell }_1-g({\ell }_1,{\ell }_3){\ell }_2\}+\left(\frac{k+1}{4}\right)\{\eta \left({\ell }_1\right)\eta \left({\ell }_3\right){\ell }_2\hfill \\ & -& \eta \left({\ell }_2\right)\eta \left({\ell }_3\right){\ell }_1+\eta \left({\ell }_2\right)g({\ell }_1,{\ell }_3)\xi -\eta \left({\ell }_1\right)g({\ell }_2,{\ell }_3)\xi \hfill \\ & -& g({\ell }_1,\varphi {\ell }_3)\varphi {\ell }_2+g({\ell }_2,\varphi {\ell }_3)\varphi {\ell }_1+2g(\varphi {\ell }_1,{\ell }_2)\varphi {\ell }_3\}\hfill \\ & +& A_{\sigma ({\ell }_2,{\ell }_3)}{\ell }_1-A_{\sigma ({\ell }_1,{\ell }_3)}{\ell }_2+({\tilde{\nabla }}_{{\ell }_2}\sigma )({\ell }_1,{\ell }_3)-({\tilde{\nabla }}_{{\ell }_1}\sigma )({\ell }_2,{\ell }_3).\hfill \end{array}$$

3. Contact Pseudo-Slant-Submanifold of an $\left(\mathbf{ϵ}\right)$-PSM

Definition 1. 

There must be two orthogonal distributions $D^{\perp }$ and $D_{\theta }$ on any submanifold M of $\left(ϵ\right)$-PSM $\tilde{M}$ for it to be a contact pseudo-slant submanifold, such that [15]:

(i)

$D_{\theta }$ is slant.

(ii)

([26]) $TM=D^{\perp }\oplus D_{\theta },\xi \in \mathrm{\Gamma }\left(D_{\theta }\right)$.

(iii)

$\varphi \left(D^{\perp }\right)\subset T^{\perp }M$.

M is a semi-invariant submanifold with $\theta =0$.

Suppose $d_1=dim\left(D^{\perp }\right)$ and $d_2=dim\left(D_{\theta }\right)$. We have [15]:

(i)

$d_2=0$ ⇒ M is anti-invariant submanifold.

(ii)

$\theta =0$, $d_1=0$ ⇒ M is invariant submanifold.

(iii)

$\theta \in (0,\frac{\pi }{2})$, $d_1=0$ ⇒ M is a properly slanted submanifold.

(iv)

$\theta =\frac{\pi }{2}$ ⇒ M is an anti-invariant submanifold.

(v)

$\theta =0$, $d_2{d}_1\ne 0$ ⇒ M is a semi-invariant submanifold.

(vi)

$\theta \in (0,\frac{\pi }{2})$, $d_2{d}_1\ne 0$ ⇒ M is a contact pseudo-slant submanifold.

Let $\mu$ be an invariant subspace of $T^{\perp }M$; then, for contact pseudo-slant submanifolds, we have:

$$T^{\perp }M=F\left(D_{\theta }\right)\oplus F\left(D^{\perp }\right)\oplus \mu .$$

Below are the contact pseudo-slant submanifolds’ bases.

${\vartheta }_1,{\vartheta }_2,...,{\vartheta }_p,{\vartheta }_{p+1}=sec\theta T{\vartheta }_1,{\vartheta }_{p+2}=sec\theta T{\vartheta }_2,...,{\vartheta }_{2p}=sec\theta T{\vartheta }_p,{\vartheta }_{2p+1}=\xi ,$

${\vartheta }_{2p+2},{\vartheta }_{2p+3},\dots ,{\vartheta }_{2p+q+1}$ are the orthonormal basis of $\mathrm{\Gamma }\left(TM\right)$,

${\vartheta }_1,{\vartheta }_2,...,{\vartheta }_p,{\vartheta }_{p+1}=sec\theta T{\vartheta }_1,{\vartheta }_{p+2}=sec\theta T{\vartheta }_2,...,{\vartheta }_{2p}=sec\theta T{\vartheta }_p,{\vartheta }_{2p+1}=\xi$ are tangent to $\mathrm{\Gamma }\left(D_{\theta }\right)$ and ${\vartheta }_{2p+2},{\vartheta }_{2p+3},...,{\vartheta }_{2p+q+1}$ are tangent to $\mathrm{\Gamma }\left(D^{\perp }\right)$. In this case, $dim\left(M\right)=2p+q+1=n$ [27].

Observe the outcomes shown below [15]:

Theorem 1. 

Assume that M is a contact pseudo-slant submanifold of an $\left(ϵ\right)$-PSSF $\tilde{M}\left(k\right)$. We observe

$$\begin{array}{ccc}\hfill \tau & =& \{\frac{1}{2}{\mathcal{K}}_1(n-1)+\frac{1}{2}{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\left(n\right)-{\parallel \sigma \parallel }^2\hfill \\ & +& \frac{1}{2}{\mathcal{K}}_2(5-n+2{ϵ}^2-ϵ-ϵco{s}^2\theta )+{\left(n\right)}^2{\parallel H\parallel }^2,\hfill \end{array}$$

(12)

here ${\mathcal{K}}_1=\frac{k-3}{2},{\mathcal{K}}_2=\frac{k+1}{2}$.

Theorem 2. 

Let M be totally umbilical contact pseudo-slant submanifold of an $\left(ϵ\right)$-PSSF $\tilde{M}\left(c\right)$. Following that, we have

$$\begin{array}{ccc}\hfill \tau & =& \{\frac{1}{2}{\mathcal{K}}_1(n-1)+\frac{1}{2}{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\left(n\right)\hfill \\ & +& \frac{1}{2}{\mathcal{K}}_2(5-n+2{ϵ}^2-ϵ-ϵco{s}^2\theta ).\hfill \end{array}$$

4. Main Results

Here, it is discussed how to derive a sharp inequality involving generalized normalized $\delta$-Casorati curvatures for a pseudo-slant submanifold of $\left(ϵ\right)$-PSSF $\tilde{M}\left(k\right)$.

Theorem 3. 

Assume M stands for a contact pseudo-slant submanifold of an $\left(ϵ\right)$-PSSF $\tilde{M}\left(k\right)$. One obtains that

(i)

the generalized normalized δ-Casorati curvature holds

$$\begin{array}{ccc}\hfill \rho & \le & \frac{{\delta }_c(r;n-1)}{n^2-1}\hfill \\ & & +\{{\mathcal{K}}_1(n-1)+{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\frac{1}{(n-1)}\hfill \\ & & +{\mathcal{K}}_2\left(\frac{1}{n^2-n}\right)(ϵ(2ϵ-1-co{s}^2\theta )+5-n),\hfill \end{array}$$

(13)

$0<r<(n^2-n)$, r is a real number;

(ii)

the generalized normalized δ-Casorati curvature verifies

$$\begin{array}{ccc}\hfill \rho & \le & \frac{{\widehat{\delta }}_c(r;n-1)}{n^2-1}\hfill \\ & & +\{{\mathcal{K}}_1(n-1)+{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\frac{1}{(n-1)}\hfill \\ & & +\frac{1}{n^2-n}(ϵ(2ϵ-1-co{s}^2\theta )+5-n){\mathcal{K}}_2,\hfill \end{array}$$

(14)

$r>(n^2-n).$

The relationships (13) and (14) also hold for equality iff M is invariantly quasi-umbilical for trivial normal connection in $\tilde{M}$ and $A_r$, $r\in \{n+1,\cdots ,m\}$ for orthonormal tangent frame $\{{\vartheta }_1,\cdots ,{\vartheta }_n\}$, and orthonormal normal frame $\{{\vartheta }_{n+1},\cdots ,\vartheta m\}$ can be expressed as:

$$\begin{array}{c}\hfill A_{n+1}=\left(\begin{array}{cccccc}\mathbb{g}& 0& 0& \cdots & 0& 0\\ 0& \mathbb{g}& 0& \cdots & 0& 0\\ 0& 0& \mathbb{g}& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & \mathbb{g}& 0\\ 0& 0& 0& \cdots & 0& \frac{n^2-n}{r}\mathbb{g}\end{array}\right),A_{n+2}=\cdots =A_m=0.\end{array}$$

(15)

Proof. 

(i) Assume that $\tilde{M}$ is $\left(ϵ\right)$-PSSF. Based on (11) and (12), we obtain

$$\begin{array}{ccc}\hfill 2\tau \left(p\right)& =& \{{\mathcal{K}}_1(n-1)+{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\left(n\right)\hfill \\ & +& n^2{\left|\right|H\left|\right|}^2+{\mathcal{K}}_2(ϵ(2ϵ-1-co{s}^2\theta )+5-n)-\left(n\right)\mathcal{C}.\hfill \end{array}$$

(16)

Let $\mathcal{Q}$ be a quadratic polynomial and $\mathcal{L}$ represents a hyperplane of $TpM$:

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& r\mathcal{C}+{\mathcal{N}}_1+{\mathcal{K}}_2(ϵ(2ϵ-1-co{s}^2\theta )+5-n)\hfill \\ & & -2\tau \left(p\right)+\{{\mathcal{K}}_1(n-1)+{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\left(n\right)\hfill \end{array}$$

Without sacrificing generality, it may be assumed that $\{{\vartheta }_1,\cdots ,{\vartheta }_{n-1}\}$ are used to span $\mathcal{L}$. Several calculations later, we obtained

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \frac{r}{n}{\mathcal{B}}_1+\frac{(n^2-n-r)}{r}{\mathcal{B}}_2{\mathcal{N}}_3\hfill \\ & & -2\tau \left(p\right)+\{\left(\frac{n-1}{2}\right)(k-3)+{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\left(n\right)\hfill \\ & & +{\mathcal{K}}_2(ϵ(2ϵ-1-co{s}^2\theta )+5-n).\hfill \end{array}$$

(17)

${\mathcal{B}}_1={\sum }_{\alpha =n+1}^m{\sum }_{i,j=1}^n{({\sigma }_{ij}^{\alpha })}^2,{\mathcal{B}}_2={\sum }_{\alpha =n+1}^m{\sum }_{i,j=1}^{n-1}{\left({\sigma }_{ij}^{\alpha }\right)}^2,{\mathcal{N}}_3=\frac{(n+r)}{n}.$

With (16) and (17), one derives

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& (1+\frac{r}{n}){\mathcal{B}}_1+\frac{(n^2-n-r)}{r}{\mathcal{B}}_2{\mathcal{N}}_3-\sum _{\alpha =n+1}^m{\left(\sum _{i=1}^n{\sigma }_{ii}^{\alpha }\right)}^2.\hfill \end{array}$$

Simple steps can decrease it to

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \sum _{\alpha =n+1}^m\sum _{i=1}^{n-1}\left[\frac{n^2+n(r-1)-2r}{r}{\left({\sigma }_{ii}^{\alpha }\right)}^2+2{\mathcal{N}}_3{\left({\sigma }_{i.n}^{\alpha }\right)}^2\right]\hfill \\ & & +\sum _{\alpha =n+1}^m\left[2{\mathcal{N}}_2\sum _{i<j=1}^{n-1}{\left({\sigma }_{ij}^{\alpha }\right)}^2-2\sum _{i<j=1}^n{\sigma }_{ii}^{\alpha }{\sigma }_{jj}^{\alpha }+\frac{r}{n}{\left({\sigma }_{n.n}^{\alpha }\right)}^2\right];\hfill \end{array}$$

(18)

here ${\mathcal{N}}_2=\frac{(n+r)(n-1)}{r}$.

In light of (18), the solutions to the system of linear homogeneous equations

$$\begin{array}{ccc}\hfill \frac{\partial \mathcal{Q}}{\partial {\sigma }_{ii}^{\alpha }}& =& 2{\mathcal{N}}_2{\sigma }_{ii}^{\alpha }-2\sum _{l=1}^n{\sigma }_{ll}^{\alpha }=0,\hfill \\ \hfill \frac{\partial \mathcal{Q}}{\partial {\sigma }_{n.n}^{\alpha }}& =& \frac{2r}{n}{\sigma }_{n.n}^{\alpha }-2\sum _{l=1}^{n-1}{\sigma }_{ll}^{\alpha }=0,\hfill \\ \hfill \frac{\partial \mathcal{Q}}{\partial {\sigma }_{ij}^{\alpha }}& =& 4{\mathcal{N}}_2{\sigma }_{ij}^{\alpha }=0,\hfill \\ \hfill \frac{\partial \mathcal{Q}}{\partial {\sigma }_{i.n}^{\alpha }}& =& 4{\mathcal{N}}_3{\sigma }_{i.n}^{\alpha }=0,\hfill \end{array}$$

(19)

are the critical points

$$\begin{array}{c}\hfill {\sigma }^c=({\sigma }_{11}^{n+1},{\sigma }_{12}^{n+1},\cdots ,{\sigma }_{n.n}^{n+1},\cdots ,{\sigma }_{11}^m,\cdots ,{\sigma }_{n.n}^m)\end{array}$$

of $\mathcal{Q}$, $\forall i,j=\{1,2,\cdots ,n-1\}$, $i\ne j$, $\alpha \in \{n+1,\cdots ,m\}$.

With reference to (19), each solution ${\sigma }^c$ has ${\sigma }_{ij}^{\alpha }=0$, $i\ne j$, and the associated determinant to the first two sets of equations in (19) is zero. Additionally, discover the Hessian matrix $H\left(\mathcal{Q}\right)$ of $\mathcal{Q}$, by

$$\begin{array}{c}\hfill H\left(\mathcal{Q}\right)=\left(\begin{array}{ccc}{H}_1& 0& 0\\ 0& H_2& 0\\ 0& 0& H_3\end{array}\right),\end{array}$$

where

$$\begin{array}{c}\hfill H_1=\left(\begin{array}{cccc}2{\mathcal{N}}_2-2& -2& \cdots & -2\\ -2& 2{\mathcal{N}}_2-2& \cdots & -2\\ ⋮& ⋮& \ddots & ⋮\\ -2& -2& \cdots & \frac{2r}{n}\end{array}\right),\end{array}$$

0 represents the null matrices of corresponding sizes, and the diagonal matrices $H_2$ and $H_3$ are

$$\begin{array}{c}\hfill H_2=\mathrm{diag}(2{\mathcal{N}}_2,\cdots ,2{\mathcal{N}}_2),\end{array}$$

$$\begin{array}{c}\hfill H_3=\mathrm{diag}(2{\mathcal{N}}_3,\cdots ,2{\mathcal{N}}_3).\end{array}$$

Therefore, $H\left(\mathcal{Q}\right)$ has the following eigenvalues:

$$\begin{array}{c}\hfill {\lambda }_{11}=0,{\lambda }_{22}=\frac{2(n^3-n^2+r^2)}{rn},\\ \hfill {\lambda }_{33}=\cdots ={\lambda }_{n.n}=2{\mathcal{N}}_2,\end{array}$$

$$\begin{array}{c}\hfill {\lambda }_{ij}=2{\mathcal{N}}_2,{\lambda }_{i.n}=2{\mathcal{N}}_3,\end{array}$$

$\forall i,j\in \{1,2,\cdots ,n-1\}$, $i\ne j$.

We conclude that $\mathcal{Q}$ is parabolic and approaches a minimum $\mathcal{Q}\left({\sigma }^c\right)$ at any solution ${\sigma }^c$ of (19). Expressions (18) and (19) produce $\mathcal{Q}\left({\sigma }^c\right)=0$, implying $\mathcal{Q}\ge 0$, and we conclude that

$$\begin{array}{ccc}\hfill 2\tau \left(p\right)& \le & r\mathcal{C}+{\mathcal{N}}_1.\mathcal{C}\left(\mathcal{L}\right)++{\mathcal{K}}_2(ϵ(2ϵ-1-co{s}^2\theta )+5-n)\hfill \\ & & +\{{\mathcal{K}}_1(n-1)+{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\left(n\right),\hfill \end{array}$$

providing

$$\begin{array}{ccc}\hfill \rho & \le & \frac{r}{(n^2-n)}\mathcal{C}+\frac{(n^2-n-r)}{rn}{\mathcal{N}}_3.\mathcal{C}\left(\mathcal{L}\right)\hfill \\ & & +\{{\mathcal{K}}_1(n-1)+{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\frac{1}{(n-1)}\hfill \\ & & +{\mathcal{K}}_2\left(\frac{1}{n^2-n}\right)(ϵ(2ϵ-1-co{s}^2\theta )+5-n),\hfill \end{array}$$

for any tangent hyperplane $\mathcal{L}$ of M, and (13) easily follows from the previous equation. Additionally, the equality is valid in (13) iff

$$\begin{array}{c}\hfill {\sigma }_{ij}^{\alpha }=0,\forall i,j\in \{1,\cdots ,n\},i\ne j\end{array}$$

(20)

and

$$\begin{array}{ccc}\hfill {\sigma }_{n.n}^{\alpha }& =& \frac{n^2-n}{r}{\sigma }_{11}^{\alpha }=\frac{n^2-n}{r}{\sigma }_{22}^{\alpha }\hfill \\ & =& \cdots =\frac{n^2-n}{r}{\sigma }_{n-1.n-1}^{\alpha },\hfill \end{array}$$

(21)

$\forall \alpha \in \{n+1,\cdots ,m\}.$

Consequently, (20) and (21) imply equality in (13) iff M is invariantly quasi-umbilical for trivial normal connection in $\tilde{M}$, and for local orthonormal tangent and orthonormal normal frames, the shape operators satisfy (15).

(ii) (14) can be demonstrated similarly. □

Corollary 1. 

Let M be a contact pseudo-slant submanifold of an $\left(ϵ\right)$-PSSF form $\tilde{M}\left(k\right)$. We have

(i)

for normalized δ-Casorati curvature

$$\begin{array}{ccc}\hfill \rho & \le & {\delta }_c(n-1)\hfill \\ & & +\{{\mathcal{K}}_1(n-1)+{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\frac{1}{(n-1)}\hfill \\ & & +{\mathcal{K}}_2\left(\frac{1}{n^2-n}\right)(ϵ(2ϵ-1-co{s}^2\theta )+5-n);\hfill \end{array}$$

(22)

(ii)

for normalized δ-Casorati curvature

$$\begin{array}{ccc}\hfill \rho & \le & {\widehat{\delta }}_c(n-1)\hfill \\ & & +\{{\mathcal{K}}_1(n-1)+{\mathcal{K}}_2(-2ϵ-{ϵ}^{2}co{s}^2\theta )\}\frac{1}{(n-1)}\hfill \\ & & +{\mathcal{K}}_2\left(\frac{1}{n^2-n}\right)(ϵ(2ϵ-1-co{s}^2\theta )+5-n).\hfill \end{array}$$

(23)

Furthermore, the equality conditions in (22) and (23) are satisfied iff M is invariantly quasi-umbilical for trivial normal connection in $\tilde{M}$ and for orthonormal tangent frame $\{{\vartheta }_1,\cdots ,{\vartheta }_n\}$ and orthonormal normal frame $\{{\vartheta }_{n+1},\cdots ,{\vartheta }_m\}$. $A_r$ can be written as

$$\begin{array}{c}\hfill A_{n+1}=\left(\begin{array}{cccccc}\mathbb{g}& 0& 0& \cdots & 0& 0\\ 0& \mathbb{g}& 0& \cdots & 0& 0\\ 0& 0& \mathbb{g}& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & \mathbb{g}& 0\\ 0& 0& 0& \cdots & 0& 2\mathbb{g}\end{array}\right),A_{n+2}=\cdots =A_m=0\end{array}$$

and

$$\begin{array}{c}\hfill A_{n+1}=\left(\begin{array}{cccccc}2\mathbb{g}& 0& 0& \cdots & 0& 0\\ 0& 2\mathbb{g}& 0& \cdots & 0& 0\\ 0& 0& 2\mathbb{g}& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 2\mathbb{g}& 0\\ 0& 0& 0& \cdots & 0& \mathbb{g}\end{array}\right),A_{n+2}=\cdots =A_m=0.\end{array}$$

5. Contact Pseudo-Slant Submanifold of Ricci Solitons

Now, we discuss the scalar curvature of the submanifold of the Ricci soliton to infer a connection between the intrinsic and extrinsic invariants. Next, we offer an important inequality for the Ricci soliton and gradient Ricci soliton in order to describe such a submanifold.

Let $({\tilde{M}}^n,g)$ be a Riemannian manifold and $\phi :M\to \tilde{M}$ be an isometric immersion from the Riemannian manifold $(M,g)$ into $(\tilde{M},g)$. The Ricci tensor may therefore be expressed as

$$\tilde{S}({\ell }_1,{\ell }_2)=\tilde{S}{{|}_{T_{p}M}({\ell }_1,{\ell }_2)+\tilde{S}|}_{T_p^{\perp }M}({\ell }_1,{\ell }_2)$$

for any ${\ell }_1,{\ell }_2\in T_{p}M.$

Now, this section begins with the preceding result:

Lemma 1. 

Let $(\tilde{M},g,V,\Lambda )$ be a Ricci soliton and M be a contact pseudo-slant submanifold of an $\left(ϵ\right)$-PSSF $\tilde{M}\left(k\right)$. We get

$$\begin{array}{c}\hfill Div\left(V\right)+\{ϵ\left(k+1\right)(-2-ϵco{s}^2\theta )+(n-1)(k-3)\}\left(n\right)\\ \hfill +{\mathcal{K}}_2(ϵ(2ϵ-1-co{s}^2\theta )+5-n)\\ \hfill +n\Lambda -{\mathcal{A}}_1+{\parallel \sigma \parallel }^2=0.\end{array}$$

(24)

Here, ${\mathcal{A}}_1=n^2{\parallel H\parallel }^2,{\mathcal{A}}_2={\sum }_{\alpha =n+1}^m{\sum }_{i,j\ne 1}^n{\left({\sigma }_{ij}^{\alpha }\right)}^2$.

Proof. 

$(\tilde{M},g,V,\Lambda )$ is a Ricci soliton. From (1), one produces

$$\begin{array}{ccc}\hfill \sum _{i=1}^{n=n}\tilde{S}({\vartheta }_i,{\vartheta }_i)& +& \frac{1}{2}\sum _{i=1}^{n=n}\{g({\nabla }_{{\vartheta }_i}V,{\vartheta }_i)+g({\vartheta }_i,{\nabla }_{{\vartheta }_i}V\}\hfill \\ & +& \sum _{i=1}^{n=n}\Lambda g({\vartheta }_i,{\vartheta }_i)=0,\hfill \end{array}$$

(25)

$\{{\vartheta }_1,\cdots ,{\vartheta }_n\}$ is a local orthonormal tangent frame of $TM$. By adopting (9) and (10) in (25), we get

$$Div\left(V\right)+\tilde{S}{|}_{T_{p}M}({\ell }_1,{\ell }_2)+\Lambda \sum _{i=1}^{n}g({\vartheta }_i,{\vartheta }_i)=0.$$

(26)

Using the Gauss formula in (26) we obtain

$$Div\left(V\right)+2\tau -\sum _{i,j=1}^n\{g\left(h({\vartheta }_i,{\vartheta }_i)h({\vartheta }_j,{\vartheta }_j)\right)-g(h({\vartheta }_i,{\vartheta }_j),h({\vartheta }_i,{\vartheta }_j))\}+\Lambda \sum _{i=1}^{n}g({\vartheta }_i,{\vartheta }_i)=0.$$

Then, the proof is completed. □

At this point, we recall the following lemma [28]:

Lemma 2. 

If ${\omega }_1,{\omega }_2\cdots {\omega }_n$ for $n>1$, are real numbers, then

$$\frac{1}{n}{\{\sum _{i=1}^n{\omega }_i\}}^2\le \sum _{i=1}^n{\omega }_i^2,$$

and the equality is satisfied iff ${\omega }_1={\omega }_2=\cdots {\omega }_n.$

Theorem 4. 

Assume $(\tilde{M},g,V,\Lambda )$ is a Ricci soliton and M is a contact pseudo-slant submanifold of an $\left(ϵ\right)$-PSSF $\tilde{M}\left(k\right)$. We obtain

$$\begin{array}{ccc}\hfill Div\left(V\right)& \le & {n(n-1)\parallel H\parallel }^2-n\Lambda \hfill \\ \hfill & & -\{ϵ\left(k+1\right)(-2-ϵco{s}^2\theta )+(n-1)(k-3)\}\left(n\right)\hfill \\ \hfill & & -{\mathcal{K}}_2(ϵ(2ϵ-1-co{s}^2\theta )+5-n).\hfill \end{array}$$

(27)

Proof. 

In light of Equation (24), we get

$$\begin{array}{c}\hfill \{ϵ\left(k+1\right)(-2-ϵco{s}^2\theta )+(n-1)(k-3)\}\left(n\right)\\ \hfill +{\mathcal{K}}_2(ϵ(2ϵ-1-co{s}^2\theta )+5-n)=-Div\left(V\right)\\ \hfill +{\mathcal{A}}_1-{\parallel \sigma \parallel }^2-\left(n\right)\Lambda \end{array}$$

$$\begin{array}{c}\hfill \{ϵ\left(k+1\right)(-2-ϵco{s}^2\theta )+(n-1)(k-3)\}\left(n\right)\\ \hfill +{\mathcal{K}}_2(ϵ(2ϵ-1-co{s}^2\theta )+5-n)=-Div\left(V\right)\\ \hfill +{\mathcal{A}}_1-\left(n\right)\Lambda -\sum _{\alpha =n+1}^m\sum _{i,j=1}^n{\left({\sigma }_{ij}^{\alpha }\right)}^2-{\mathcal{A}}_2\end{array}$$

$$\begin{array}{c}\hfill \{ϵ\left(k+1\right)(-2-ϵco{s}^2\theta )+(n-1)(k-3)\}\left(n\right)\\ \hfill +{\mathcal{K}}_2(ϵ(2ϵ-1-co{s}^2\theta )+5-n)\le -Div\left(V\right)\\ \hfill +{\mathcal{A}}_1-\left(n\right)\Lambda -\frac{1}{n}{\mathcal{A}}_1-{\mathcal{A}}_2\end{array}$$

$$\begin{array}{c}\hfill \le -Div\left(V\right)-n\Lambda +{n(n-1)\parallel H\parallel }^2-{\mathcal{A}}_2\\ \hfill \end{array}$$

then, one finds

$$\begin{array}{c}\hfill \{ϵ\left(k+1\right)(-2-ϵco{s}^2\theta )+(n-1)(k-3)\}\left(n\right)\\ \hfill +{\mathcal{K}}_2(ϵ(2ϵ-1-co{s}^2\theta )+5-n)\\ \hfill \le -Div\left(V\right)-n\Lambda +{n(n-1)\parallel H\parallel }^2\end{array}$$

providing us (27). If the equality of (27) is satisfied, then M is totally umbilical. □

Next, in [29] Blaga and Carasmareanu established an inequality for a lower boundary of the geometry of g in terms of gradient Ricci solton for a smooth function $\psi$ on ambient space M, such as

$${\left|\right|S\left|\right|}_g^2\ge {\left|\right|Hess\left|\right|}_g^2-\frac{1}{n}{(\Delta \psi )}^2,$$

(28)

where $Hess$ means the Hessian of the smooth function $\psi$ on M. Now, let that soliton vector field V satisfy $V=\nabla \psi$. Then, (28) helps to articulate:

Theorem 5. 

Let $(\tilde{M},g,\nabla \psi ,\Lambda )$ be a gradient Ricci soliton with a soliton vector field V of gradient type and M be a contact pseudo-slant submanifold of an $\left(ϵ\right)$-PSSF $\tilde{M}\left(k\right)$. We write

$$\begin{array}{ccc}\hfill {\left|\right|S\left|\right|}_g^2& \ge & {\left|\right|Hess\left|\right|}_g^2-n^3{\parallel H\parallel }^4+{\parallel \sigma \parallel }^4+n{\Lambda }^2\hfill \\ \hfill & & +{\{ϵ\left(k+1\right)(-2-ϵco{s}^2\theta )+(n-1)(k-3)\}}^2\hfill \\ \hfill & & +\frac{1}{n}{\left({\mathcal{K}}_2\right)}^2{(ϵ(2ϵ-1-co{s}^2\theta )+5-n)}^2.\hfill \end{array}$$

Theorem 6. 

Let M be a totally umbilical contact pseudo-slant submanifold of an $\left(ϵ\right)$-PSSF $\tilde{M}\left(k\right)$. Further, assume that all the assumptions of Theorem 5 hold; then,

$$\begin{array}{ccc}\hfill {\left|\right|S\left|\right|}_g^2& \ge & {\left|\right|Hess\left|\right|}_g^2+n{\Lambda }^2\hfill \\ \hfill & & +{\{ϵ\left(k+1\right)(-2-ϵco{s}^2\theta )ϵ+(n-1)(k-3)\}}^2\hfill \\ \hfill & & +\frac{1}{n}{\left({\mathcal{K}}_2\right)}^2{(ϵ(2ϵ-1-co{s}^2\theta )+5-n)}^2.\hfill \end{array}$$

6. Conclusions

In [30], algebraic lemmas were used to establish Chen-type inequalities. However, our approach uses an optimization procedure involving a quadratic polynomial, which is shown to be parabolic. Along similar lines of Theorem 3 and with the help of Definition 1 and Theorem 1, one can write normalized scalar curvature in the same ambient space form. Submanifolds for which the equality sign of established inequalities of Casorati curvatures holds are known as Casorati-ideal submanifolds. It is a very typical task to completely classify these submanifolds, although one can check some classifications of such submanifolds in [31,32].

Ricci solitons are some of the most important tools for describing the geometric characteristics of submanifolds of Riemannian manifolds and other ambient space forms. Numerous writers investigated axioms such as rigidity and triviality in terms of Ricci solitons and discovered various inequalities with scalar curvature. See [16,18,21,22,23,24,29] for other applications of Ricci solitons on submanifolds of various ambient spaces. As a result, we have discovered an intriguing inequality in the current article, specifically on pseudo-slant submanifolds of the $ϵ$-para Sasakian manifold, in terms of gradient Ricci soliton with vector field of gradient type. Further, as applications of Theorems 4 and 5, using Definition 1 and Theorem 1, one can write inequalities for invariant, anti-invariant, proper-slant, and semi-invariant submanifolds.

It is remarkable to note it here that some more inequalities have been obtained by different researchers in other settings (see [33,34,35]). It will be interesting to derive such inequalities for a $ϵ$-para Sasakian manifold.