Estimation of Ricci Curvature for Hemi-Slant Warped Product Submanifolds of Generalized Complex Space Forms and Their Applications

Abstract

In this paper, we estimate Ricci curvature inequalities for a hemi-slant warped product submanifold immersed isometrically in a generalized complex space form with a nearly Kaehler structure, and the equality cases are also discussed. Moreover, we also gave the equivalent version of these inequalities. In a later study, we will exhibit the application of differential equations to the acquired results. In fact, we prove that the base manifold is isometric to Euclidean space under a specific condition.

1. Introduction

It is widely acknowledged that the warped product manifold can be found in the generalization of the Riemannian product manifold. As a result of the remarkable work of Chen [1], warped product submanifolds were used as an analytical tool to study Riemannian geometry. Basically, Chen derived the inequality for second fundamental forms by identifying CR-warped product submanifolds on Kaehler manifolds. Indeed, different classes of warped product submanifolds with diverse structures have studied by several researchers. For example, Al-Solamy et al. [2], Sahin [3], Khan et al. [4,5], and Bonanzinga and Matsumoto [6] investigated various geometric properties of submanifolds related to warped products in Kahler, nearly Kaehler, and locally conformal Kaehler manifolds. In [7], Sahin proposed the idea of warped product hemi-slant (pseudo-slant or anti-slant) submanifolds in the Kaehler manifolds. He studied various existence and non existence results. Further, Uddin and Chi [8] explored these submanifolds with the name pseudo-slant warped product submanifolds for nearly Kaehler manifolds. However, Al-Solamy and Khan [9] considered the hemi-slant warped product submanifolds of nearly Kaehler manifolds and established an inequality for squared norms of second fundamental forms.

One of the major tasks in the theory of submanifolds is to compute a simple correlation connecting the extrinsic invariables and the extrinsic invariables. For instance, the sectional curvature, scalar curvature, and Ricci curvature are the intirinsic invariables. However, the main extrinsic invariable is the squared mean curvature. Chen established the following basic link uniting mean curvature and the Ricci curvature for Lagrangian submanifolds in the background of complex space forms [10]

$$Ric\le (m-1)c+\frac{m^2}{4}{\parallel \Omega \parallel }^2,$$

Here, c denotes the constant holomorphic curvature of the complex space form, whereas m and $\Omega$ denote the dimension and mean curvature vector of the submanifold, respectively. In addition, he also explained the equality cases and provided some applications. This inequality has several geometric properties and fascinated several researchers [11,12,13,14,15]. For a differentiable manifold, Chen [16] computed a relation joining the squared mean curvature and Ricci curvature. In 2001, Matsumoto et al. [11] computed the Ricci curvature of submanifolds in complex space forms. Apart from this, Mihai [15] studied the submanifolds of Sasakian space forms and estimae the Ricci curvature. Deng [17] generalized the Chen–Ricci inequality for Lagrangian submanifolds in quaternion space forms. However, Tripathi [11] obtained the Chen–Ricci inequality in terms of curvature-like terms. Later, for spheres with warped product submanifolds, Ali et al. found a relationship between Ricci curvature and squared mean curvature [18] and provided various uses in mechanics.

On the other hand, by using differential equations, the investigations [19,20] give crucial intrinsic geometrical as well as isometric features of the Riemannian manifold. It is generally known that the universal study of Riemannian manifolds is significantly influenced by the categorization of differential equations. In 1978, Tanno [19] investigated several facts of differential equations on Riemannian manifolds. The research carried out in [19,21,22] showed that a non-constant function on a complete Riemannian manifold meet the following differential equation $(\mathcal{E},g)$.

$${\nabla }^{2}f+cg=0$$

iff $(\mathcal{E},g)$ is isometric to Euclidean space $R^n$, where f is a differentiable function on the submanifold, whereas c is constant.

Additionally, Garcia Rio et al. [23] demonstrated that the Riemannian manifold under certain limitations is isometric to warped product $U{\times }_{\psi }R$, where U is complete Riemannian manifold, $\psi$ is the warping function, and R is the real line. Furthermore, the warping function $\psi$ satisfies following differential equation.

$$\frac{d^2\psi }{d{t}^2}+{\mu }_1\psi =0$$

necessary and sufficient condition is that the following differential equation can be solved by a non constant real valued function $\varphi$ with a negative eigenvalue ${\mu }_1\le 0$

$$\Delta \varphi +{\mu }_1\varphi =0.$$

The characterization of differential equations on Riemannian manifolds has grown to be an exciting area of study and has been looked into by many scholars, including these in [24,25,26]. Furthermore, we will combine the methods and results in [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45] for future research to obtain more new developments.

Note: Throughout the paper, we used the abbreviations w p for “warped product”, g.c.s.f for “generalized complex space form”, TG for “totally geodesic”, and TU for “totally umbilical”.

2. Preliminaries

Suppose $\overline{\mathcal{E}}$ be a almost Hermitian manifold with a complex structure J and a Riemannian metric g that satisfies the following conditions.

$$J^2=-I,g(J{X}^1,J{Y}^1)=g(X^1,Y^1),$$

(1)

for each vector fields $X^1,Y^1$ on $\overline{\mathcal{E}}.$ Moreover, if

$$({\overline{\nabla }}_{X^1}J)Y^1+({\overline{\nabla }}_{Y^1}J)X^1=0$$

(2)

for all vector fields $X^1,Y^1\in T\overline{\mathcal{E}},$ then $\overline{\mathcal{E}}$ is known the nearly Kaehler manifold. An example of a nearly Kaehler manifold that is not a Kaehler manifold is the six-dimensional sphere $S^6$. Over and beyond nearly Kaehler manifolds, there is a more broad class of almost Hermitian manifolds known as RK-manifolds. An RK-manifold with constant holomorphic sectional curvature c and constant beta curvature $\beta$ is referred to as a generalized complex space form and is represented by the symbol $\overline{\mathcal{E}}(c,\beta )$. One example of a g.c.s.f, that is not a complex space form, is the sphere $S^6$ furnished with the usual Neely Kaehler structure. The curvature tensor $\overline{R}$ of a g.c.s.f $\overline{\mathcal{E}}(c,\beta )$ is formulated as

$$\begin{array}{cc}\hfill {\overline{R}}^1(X^1,Y^1,Z^1,W^1)& =\frac{c+3\beta }{4}[g(Y^1,Z^1)g(X^1,W^1)-g(X^1,Z^1)g(Y^1,W^1)]\hfill \\ \hfill & +\frac{c-\beta }{4}[g(X^1,J{Z}^1)g(J{Y}^1,W^1)-g(Y^1,J{Z}^1)g(J{X}^1,W^1)\hfill \\ \hfill & +2g(X^1,J{Y}^1)g(J{Z}^1,W^1)],\hfill \end{array}$$

(3)

for any $X^1,Y^1,Z^1,W^1\in T\overline{\mathcal{E}}.$

Assume that $\mathcal{E}$ is an isometrically immersed m-dimensional Riemannian manifold in n-dimensional $\overline{\mathcal{E}}$. Then, the Weingarten and Gauss formulas are ${\overline{\nabla }}_{X^1}\xi =-A_{\xi }{X}^1+D_{X^1}^{\perp }\xi$ and ${\overline{D}}_{X^1}{Y}^1=D_{X^1}{Y}^1+\omega (X^1,Y^1)$, respectively, for all $X^1,Y^1\in T\mathcal{E}$ and $\xi \in T^{\perp }\mathcal{E}$, where D is the induced Levi–Civita connection on $\mathcal{E},$$\xi$ is a vector field normal to $\mathcal{E}$, $\omega$ is the second fundamental form of $\mathcal{E}$, $D^{\perp }$ is the normal connection in the normal bundle $T^{\perp }\mathcal{E}$, and $A_{\xi }$ is the shape operator. The second fundamental form $\omega$ and the shape operator are connected by the subsequent expression.

$$g(\omega (X^1,Y^1),\xi )=g(A_{\xi }{X}^1,Y^1).$$

(4)

The equation of Gauss is given by

$$\begin{array}{ccc}\hfill R^1(X^1,Y^1,Z^1,W^1)& =& {\overline{R}}^1(X^1,Y^1,Z^1,W^1)+g(\omega (X^1,W^1),\omega (Y^1,Z^1))\hfill \\ \hfill & -& g(\omega (X^1,Z^1),\omega (Y^1,W^1)),\hfill \end{array}$$

(5)

for all $X^1,Y^1,Z^1,W^1\in T\mathcal{E}$, where ${\overline{R}}^1$ and $R^1$ are the curvature tensors of $\overline{\mathcal{E}}$ and $\mathcal{E}$, respectively.

For every orthonormal basis of the tangent space $T_x\overline{\mathcal{E}}$, the mean curvature vector $\Omega (x)$ and its squared norm are determined as follows.

$$\Omega (x)=\frac{1}{k}\sum _{i=1}^k\omega (u_i,u_i),{\parallel \Omega \parallel }^2=\frac{1}{k^2}\sum _{i,j=1}^{k}g(\omega (u_i,u_i),\omega (u_j,u_j)),$$

(6)

where k is the dimension of $\mathcal{E}$. If $\omega =0$, then the submanifold is said to be TG and minimal if $\Omega =0.$ If $\omega (X^1,Y^1)=g(X^1,Y^1)\Omega$ for every $X^1,Y^1\in T\mathcal{E},$ then $\mathcal{E}$ is known as TU.

The scalar curvature of $\overline{\mathcal{E}}$ is denoted by $\overline{\alpha }(\overline{\mathcal{E}})$ and is given by

$$\overline{\alpha }({\mathcal{E}}^t)=\sum _{1\le r<s\le m}{\overline{K}}_{rs},$$

(7)

where ${\overline{K}}_{rs}=\overline{K}(e_r\wedge e_s)$ and m are the dimension of the Riemannian manifold $\overline{\mathcal{E}}$. In this paper, we refer to the equivalent form of the previously mentioned equation, which is defined by

$$2\overline{\alpha }({\mathcal{E}}^t)=\sum _{1\le r,s\le m}{\overline{K}}_{rs}.$$

(8)

Similar to this, the scalar curvature of a L-plane is represented by the expression

$$\overline{\alpha }(L_x)=\sum _{1\le r<s\le m}{\overline{K}}_{rs}.$$

(9)

Assume $\{u_1,\cdots ,u_k\}$ be an orthonormal basis of the tangent space $T_x\mathcal{E}$, and if $u_r$ belongs to the orthonormal basis $\{u_{k+1},\cdots ,u_m\}$ of the normal space $T^{\perp }\mathcal{E}$, we get

$${\omega }_{pq}^r=g(\omega (u_p,u_q),u_r))$$

(10)

and

$${\parallel \omega \parallel }^2=\sum _{p,q=1}^{k}g(\omega (u_p,u_q),\omega (u_p,u_q)).$$

(11)

Let $K_{rs}$ and ${\overline{K}}_{rs}$ be the sectional curvatures of the plane sections spanned by $u_r$ and $u_s$ at x in the submanifold ${\mathcal{E}}^t$ and in the Riemannian space form ${\overline{\mathcal{E}}}^m(c)$, respectively. Thus through the Gauss equation, we get

$$K_{rs}={\overline{K}}_{rs}+\sum _{p=k+1}^m({\omega }_{rr}^p{\omega }_{ss}^p-{({\omega }_{rs}^r)}^2).$$

(12)

The orthonormal frame of the vector field $\{u_1,\cdots ,u_k\}$ on ${\mathcal{E}}^k$ is described by the global tensor field as

$$\overline{S}(X^1,Y^1)=\sum _{i=1}^k\left\{g(\overline{R}(u_i,X^1)Y^1,u_i)\right\},$$

(13)

for all $X^1,Y^1\in T_x{\mathcal{E}}^k.$ The tensor $\overline{S}(X^1,Y^1)$ is known as Ricci tensor. If we fix a unique vector, $u_e$, from $\{u_1,\cdots ,u_k\}$ on ${\mathcal{E}}^k$ on ${\mathcal{E}}^k$, which is symbolized by P. Then, the Ricci curvature is described by

$$Ric(P)=\sum _{\begin{array}{c}p=1\\ p\ne e\end{array}}^{k}K(u_p\wedge u_e).$$

(14)

for the wp submanifold ${\mathcal{E}}_1{\times }_{\psi }{\mathcal{E}}_2$ in a Riemannian manifold. Let $X^{1}T{\mathcal{E}}_1$ and $Z^{1}T{\mathcal{E}}_2$. Then, from Lemma 7.3 of [46], we obtain

$$D_{X^1}{Z}^1=D_{Z^1}{X}^1=(\frac{X^1\psi }{\psi })Z^1$$

(15)

where D denotes the Levi–Civita connection on $\mathcal{E}$. For a wp $\mathcal{E}={\mathcal{E}}_1{\times }_{\psi }{\mathcal{E}}_2$, it is easy to observe that

$$D_{X^1}{Z}^1=D_{Z^1}{X}^1=(X^{1}ln\psi )Z^1$$

(16)

for $X^1\in T{\mathcal{E}}_1$ and $Z^1\in T{\mathcal{E}}_2.$ The gradient of $\psi$, $\nabla \psi$, is described as

$$g(\nabla \psi ,X^1)=X^1\psi ,$$

(17)

for all $X^1\in T\mathcal{E}.$

Let $\mathcal{E}$ be an k-dimensional Riemannian manifold with the Riemannian metric g and let $\{u_1,u_2,\cdots ,u_k\}$ be an orthogonal basis of $T\mathcal{E}.$ Then, as a conclusion of (17), we obtain

$${\parallel \nabla \psi \parallel }^2=\sum _{i=1}^k{(u_i(\psi ))}^2.$$

(18)

The Laplacian of $\psi$ is given by

$$\Delta \psi =\sum _{i=1}^k\{({\nabla }_{u_i}{u}_i)\psi -u_i{u}_i\psi \}.$$

(19)

The mathematical description of the symmetric covariant tensor $\psi$, which is the Hessian tensor for a differentiable function, is

$$\Delta \psi =-traceH\psi$$

For a wp submanifold ${\mathcal{E}}_1^{t_1}{\times }_{\psi }{\mathcal{E}}_2^{t_2}$ of an arbitrary Riemannian manifold, what follows is the well-known result [47]

$$\sum _{p=1}^{t_1}\sum _{q=1}^{t_2}K(e_p\wedge e_q)=\frac{t_2\Delta \psi }{\psi }=t_2{(\Delta ln\psi -\parallel \nabla ln\psi \parallel }^2).$$

(20)

where $t_1$ and $t_2$ represent, respectively, the dimensions of the submanifolds ${\mathcal{E}}_1^{t_1}$ and ${\mathcal{E}}_2^{t_2}.$

As a result of the integration theory of manifolds, with a compact orientable Riemannian manifold $\mathcal{E}$ with or without border, we obtain [46]

$${\int }_{\mathcal{E}}\Delta \psi dV=0,$$

(21)

where $dV$ is the volume element of $\mathcal{E},$ and $\psi$ is a function on $\mathcal{E}.$

For a differential function h on a Riemannian manifold ${\overline{\mathcal{E}}}^t$, the Bochner formula [48] is given by

$$\frac{1}{2}{\Delta \parallel \nabla h\parallel }^2=Ric(\nabla h,\nabla h)+{\parallel H_{ess}h\parallel }^2+g(\nabla \Delta h,\nabla h),$$

(22)

where $H_{ess}\psi$ is the Hessian of the function $\psi .$

3. Hemi-Slant Wp Submanifolds of a Nearly Kaehler Manifold

The concept of hemi-slant submanifolds of the Kaehler manifold is an innovative and captivating topic in geometry. Indeed, this represents the investigation of the differential geometry of the hemi-slant in a Riemannian metric as a generic version of CR-submanifolds and slant submanifolds. Uddin and Chi [8] and Al-solamy and Khan [9] studied pseudo-slant wp submanifolds with the name of hemi-slant wp submanifolds in the setting of nearly Kaehler manifold. Suppose ${\mathcal{E}}_{\perp }$ and ${\mathcal{E}}_{\theta }$ are the totally real and slant submanifolds. However, two possible choices are available of wp submanifolds of $\overline{\mathcal{E}}$; these wp are ${\mathcal{E}}_{\theta }{\times }_{\psi }{\mathcal{E}}_{\perp }$ and ${\mathcal{E}}_{\perp }{\times }_{\psi }{\mathcal{E}}_{\theta }$. The wp of the type ${\mathcal{E}}_{\theta }{\times }_{\psi }{\mathcal{E}}_{\perp }$ were explored by Al-Solamy and Khan [9], and they proved an estimation for the squared norm of second fundamental forms. However, the intrinsic study of the wp of the type ${\mathcal{E}}_{\perp }{\times }_{\psi }{\mathcal{E}}_{\theta }$ is not available in the literature. Therefore, in this study, we consider the wp of the type ${\mathcal{E}}_{\perp }{\times }_{\psi }{\mathcal{E}}_{\theta },$ and obtained an inequality for the Ricci curvature for these types of submanifolds. Across this study, we assume the hemi-slant wp submanifolds $\mathcal{E}={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ of a g.c.s.f, where $t_1$ and $t_2$ are the dimensions of the totally real and slant submanifolds, respectively.

Definition 1.

If If the partial second fundamental form ${\omega }_i$ disappear identically, the wp isometrically immersed in a Riemannian manifold is said to be ${\mathcal{E}}_i$ TG. If $i=1,2$ and the partial mean curvature vector $H_i$ reaches zero, it is referred to as being ${\mathcal{E}}_i$-minimal.

Let $\{u_1,\cdots ,u_p=u_{t_1},u_{t_1+1}=u^1,\cdots ,u_{t_1+q}=u^q,u_{t_1+q+1}=u^{q+1}=sec\theta P{u}^1,\cdots ,u_{(t_2=2q)}=u^{t_2}=sec\theta P{u}^q\}$ is a basis of orthonormal vector fields on a hemi-slant wp submanifold ${\mathcal{E}}^t={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ and the set $\{u_1,\cdots ,u_p,\}$ is tangent to ${\mathcal{E}}_{\perp }^{t_1}$, and the set $\{u^1,\cdots ,u^q,\cdots u^{t_2}\}$ is tangent to ${\mathcal{E}}_{\theta }^{t_2}$. Moreover, $\{u_{t+1}=J{u}_1,\cdots ,u_{n+t_1}=J{u}_{t_1},u_{n+t_1+1}=csc\theta F{u}^1,\cdots ,u_{n+t_1+t_2}=csc\theta F{u}^q,u_{n+t_1+t_2+1}={\overline{u}}^1,\cdots ,u_m={\overline{u}}^k\}$ is a basis for the normal bundle $T^{\perp }{\mathcal{E}}^n$ such that the set $\{J{u}_1,\cdots ,J{u}_{t_1}\}$ is tangent to $J{D}^{\perp }$. The set $\{u_{t+1}=csc\theta F{u}^1,\cdots ,u_{n+t_2}=csc\theta F{u}^q\}$ is tangent to $F{D}^{\theta }$, and $\{{\overline{u}}^1,\cdots ,{\overline{u}}^k\}$ has a tangent to complementary invariant subspace μ.

The hemi-slant wp submanifold ${\mathcal{E}}_{\perp }\times {\mathcal{E}}_{\theta }$ is taken into consideration to be $D^{\perp }$-minimal throughout this paper. We now have the following lemma for further use.

Lemma 1.

Let ${\mathcal{E}}^t={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ be a hemi-slant wp submanifold immersed isometrically in a nearly Kaehler manifold. If ${\mathcal{E}}^t$ is $D^{\perp }$-minimal, then

$${\parallel \Omega \parallel }^2=\frac{1}{t^2}\sum _{r=t+1}^m{({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{nn}^r)}^2,$$

(23)

where ${\parallel \Omega \parallel }^2$ is the squared mean curvature.

4. Estimation of Ricci Curvature

Under this section, the squared norm of mean curvature and warping function are used to investigate the Ricci curvature.

Theorem 1.

Let $\mathcal{E}={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ be a hemi-slant wp submanifold immersed isometrically in a g.c.s.f $\overline{\mathcal{E}}(c,\beta )$ having a nearly Kaehler structure. Thus, we have for each orthogonal unit vector field an $P\in T_x\mathcal{E}$ that is tangent to either ${\mathcal{E}}_{\perp }$ or ${\mathcal{E}}_{\theta }$.

(1) 

The estimation of Ricci curvature is given by

(i) 

If $P\in T{\mathcal{E}}_{\perp }^{t_1}$, then

$$Ric(P)\le \frac{1}{4}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }+\frac{c+3\beta }{4}(t+t_1{t}_2-1).$$

(24)

(ii) 

If $P\in {\mathcal{E}}_{\theta }^{t_2}$, then

$$Ric(P)\le \frac{1}{4}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }+\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}{cos}^2\theta .$$

(25)

(2) 

If the mean curvature vector $\Omega (x)$ is zero at each point x on ${\mathcal{E}}^t$, there exits a vector field P of unit length, equality holds in (1) iff ${\mathcal{E}}^t$ is mixed TG and P belongs to the relative null space $N_x$.

(3) 

For situaltion of equality,

(a) 

Equality holds in (24) for all unit vector fields belong to $T{\mathcal{E}}_{\perp }^{t_1}$, for all $x\in {\mathcal{E}}^t$ iff ${\mathcal{E}}_{\perp }^t$ is mixed TG and $D^{\perp }$ is TG hemi-slant wp submanifold of ${\overline{\mathcal{E}}}^m(c,\beta )$.

(b) 

Equality holds in (25) for all unit vector fields belong to $T{\mathcal{E}}_{\theta }^{t_2}$, for all $x\in {\mathcal{E}}^t$ iff ${\mathcal{E}}_{\perp }^t$ is mixed TG and either ${\mathcal{E}}_{\perp }^t$ is $D^{\theta }$ TG or $D^{\theta }$ TU in ${\overline{\mathcal{E}}}^m(c,\beta )$ and dimension of slant submanifold is 2.

(c) 

Equality holds in (1) identically, for all unit vector fields belong to $T{\mathcal{E}}^t$ at every point $x\in {\mathcal{E}}^t$ iff ${\mathcal{E}}^t$ is TG or ${\mathcal{E}}^t$ is mixed TG-TU, and $T{\mathcal{E}}_{\perp }^{t_1}$ is TG and the dimension of slant submanifold $T{\mathcal{E}}_{\theta }^{t_2}$ is 2.

Here, $t_1$ and $t_2$ denote the dimensions of ${\mathcal{E}}_{\perp }$ and ${\mathcal{E}}_{\theta }$, respectively.

Proof. 

Let $\mathcal{E}={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ be a hemi-slant wp submanifold of a g. c. s. f. Through the Gauss equation, we find

$$t^2{\parallel \Omega \parallel }^2=2\alpha ({\mathcal{E}}^t)+{\parallel \omega \parallel }^2-2\overline{\alpha }({\mathcal{E}}^t).$$

(26)

Suppose $\{u_1,\cdots ,u_{t_1},u_{t_1+1},\cdots ,u_t\}$ is a basis of orthonormal vector fields on ${\mathcal{E}}^t$ like that $\{u_1,\cdots ,u_{t_1}\}\in T{\mathcal{E}}_{\perp }$, and $\{u_{t_1+1},\cdots ,u_t\}\in T{\mathcal{E}}_{\theta }$. So, the unit tangent vector $P=u_A\in \{u_1,\cdots ,u_t\}$ can be modified (28) as below.

$$\begin{array}{cc}\hfill t^2{\parallel \Omega \parallel }^2=2\alpha ({\mathcal{E}}^t)& +\frac{1}{2}\sum _{r=t+1}^m\{{({\omega }_{11}^r+\cdots +{\omega }_{tt}^r-{\omega }_{AA}^r)}^2+{({\omega }_{AA}^r)}^2\}-\hfill \\ \hfill & \sum _{r=t+1}^m\sum _{1\le p\ne q\le t}{\omega }_{pp}^r{\omega }_{qq}^r-2\overline{\alpha }({\mathcal{E}}^t).\hfill \end{array}$$

(27)

The preceding relation can be expressed as follows:

$$\begin{array}{cc}\hfill t^2{\parallel \Omega \parallel }^2=& 2\alpha ({\mathcal{E}}^t)+\frac{1}{2}\sum _{r=t+1}^m\{{({\omega }_{11}^r+\cdots +{\omega }_{tt}^r)}^2+{(2{\omega }_{AA}^r-({\omega }_{11}^r+\cdots +{\omega }_{tt}^r))}^2\}\hfill \\ & +2\sum _{r=t+1}^m\sum _{1\le p<q\le t}{({\omega }_{pq}^r)}^2-2\sum _{r=t+1}^m\sum _{1\le p<q\le t}{\omega }_{pp}^r{\omega }_{qq}^r-2\overline{\alpha }({\mathcal{E}}^t).\hfill \end{array}$$

According to Lemma 1, the previous relation has the following form:

$$\begin{array}{cc}\hfill t^2{\parallel \Omega \parallel }^2=& \sum _{r=t+1}^m\{{({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{tt}^r)}^2++{(2{\omega }_{AA}^r-({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{tt}^r))}^2\}\hfill \\ \hfill & +2\alpha ({\mathcal{E}}^t)+\sum _{r=t+1}^m\sum _{1\le p<q\le t}{({\omega }_{pq}^r)}^2-\sum _{r=t+1}^m\sum _{1\le p<q\le t}{\omega }_{pp}^r{\omega }_{qq}^r+\sum _{r=t+1}^m\sum _{\begin{array}{c}a=1\\ a\ne A\end{array}}{({\omega }_{aA}^r)}^2\hfill \\ \hfill & +\sum _{r=t+1}^m\sum _{\begin{array}{c}1\le p<q\le t\\ p,q\ne A\end{array}}{({\omega }_{pq}^r)}^2-\sum _{r=t+1}^m\sum _{\begin{array}{c}1\le p<q\le t\\ p,q\ne A\end{array}}{\omega }_{pp}^r{\omega }_{qq}^r-2\overline{\alpha }({\mathcal{E}}^t).\hfill \end{array}$$

(28)

By Equation (12), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{r=t+1}^m\sum _{\begin{array}{c}1\le p<q\le t\\ p,q\ne A\end{array}}{({\omega }_{pq}^r)}^2& -\sum _{r=t+1}^m\sum _{\begin{array}{c}1\le p<q\le t\\ p,q\ne A\end{array}}{\omega }_{pp}^r{\omega }_{qq}^r=\sum _{\begin{array}{c}1\le p<q\le t\\ p,q\ne A\end{array}}{\overline{K}}_{pq}-\sum _{\begin{array}{c}1\le p<q\le t\\ p,q\ne A\end{array}}{K}_{pq}.\hfill \end{array}\end{array}$$

(29)

Upon entering the values of Equation (29) into (28), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill t^2{\parallel \Omega \parallel }^2=& 2\alpha ({\mathcal{E}}^t)+\frac{1}{2}\sum _{r=t+1}^m{(2{\omega }_{AA}^r-({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{tt}^r))}^2+\sum _{r=t+1}^m\sum _{1\le p<q\le t}{({\omega }_{pq}^r)}^2\hfill \\ \hfill & -\sum _{r=t+1}^m\sum _{1\le p<q\le t}{\omega }_{pp}^r{\omega }_{qq}^r-2\overline{\alpha }({\mathcal{E}}^t)+\sum _{r=t+1}^m\sum _{\begin{array}{c}a=1\\ a\ne A\end{array}}{({\omega }_{aA}^r)}^2+\sum _{\begin{array}{c}1\le p<q\le t\\ p,q\ne A\end{array}}{\overline{K}}_{p,q}-\sum _{\begin{array}{c}1\le p<q\le t\\ p,q\ne A\end{array}}{K}_{pq}.\hfill \end{array}\end{array}$$

(30)

Since, ${\mathcal{E}}^t={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$, by then using (9), the scalar curvature of ${\mathcal{E}}^t$ is written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \alpha ({\mathcal{E}}^t)=\sum _{1\le p<q\le t}K(e_p\wedge e_q)=\sum _{i=1}^{t_1}\sum _{j=t_1+1}^{t}K(e_i\wedge e_j)& +\sum _{1\le r<k\le t_1}K(e_r\wedge e_k)\hfill \\ \hfill +\sum _{t_1+1\le l<o\le t}K(e_l\wedge e_o).\end{array}\end{array}$$

(31)

With the help of (9) and (20), we deduce

$$\alpha ({\mathcal{E}}^t)=\frac{t_2\Delta \psi }{\psi }+\alpha ({\mathcal{E}}_{\perp }^{t_1})+\alpha ({\mathcal{E}}_{\theta }^{t_2}).$$

(32)

Utilizing (32) together with (12) and (5) in (30), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{1}{2}{t}^2{\parallel \Omega \parallel }^2=& \frac{t_2\Delta \psi }{\psi }+\sum _{\begin{array}{c}1\le p<q\le t\\ p,q\ne A\end{array}}{\overline{K}}_{p,q}+\overline{\alpha }({\mathcal{E}}_{\perp }^{t_1})+\overline{\alpha }({\mathcal{E}}_{\theta }^{t_2})+\sum _{r=t+1}^m\{\sum _{1\le p<q\le t}{({\omega }_{pq}^r)}^2-\sum _{\begin{array}{c}1\le p<q\le t\\ p,q\ne A\end{array}}{\omega }_{pp}^r{\omega }_{qq}^r\}\hfill \\ \hfill & +\sum _{r=t+1}^m\sum _{\begin{array}{c}a=1\\ a\ne A\end{array}}{({\omega }_{aA}^r)}^2+\sum _{r=t+1}^m\sum _{1\le i\ne j\le t_1}({\omega }_{ii}^r{\omega }_{jj}^r-{({\omega }_{ij}^r)}^2)+\sum _{r=t+1}^m\sum _{t_1+1\le s\ne t\le t}({\omega }_{ss}^r{\omega }_{tt}^r-{({\omega }_{st}^r)}^2)\hfill \\ \hfill & +\frac{1}{2}\sum _{r=t+1}^m{(2{\omega }_{AA}^r-({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{tt}^r))}^2-\frac{c+3\beta }{4}(t(t-1))-\frac{(c-\beta )}{4}(3t_2{cos}^2\theta ).\hfill \end{array}\end{array}$$

(33)

By taking into account the unit tangent vector $P=e_a$, there are two possibilities: $P\in T{\mathcal{E}}_{\theta }^{t_2}$ or $P\in T{\mathcal{E}}_{\perp }^{t_1}$ or.

Case i: taking a unit tangent vector from $\{u_1,\cdots ,u_{t_1}\}$ and assume $P=u_a=u_1$, if $u_a$ is tangent to ${\mathcal{E}}_{\perp }^{t_1}$. Next, based on (33) and (14), we discover

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Ric(P)\le & \frac{1}{2}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }-\frac{1}{2}\sum _{r=t+1}^m{(2{\omega }_{11}^r-({\omega }_{t_1+1t_1+1}^r+\cdots {\omega }_{tt}^r))}^2\hfill \\ \hfill & -\sum _{r=t+1}^m\sum _{1\le p<q\le t_1}{({\omega }_{pq}^r)}^2+\sum _{r=t+1}^m[\sum _{1\le i<j\le t_1}{({\omega }_{ij}^r)}^2-\sum _{1\le i<j\le t_1}{\omega }_{ii}^r{\omega }_{jj}^r]\hfill \\ \hfill & +\sum _{r=t+1}^m\sum _{t_1+1\le s<t\le t}{({\omega }_{st}^r)}^2+\sum _{r=t+1}^m[\sum _{t_1+1\le s<t\le n}{({\omega }_{ij}^r)}^2-\sum _{t_1+1\le s<t\le n}{\omega }_{ss}^r{\omega }_{tt}^r]\hfill \\ \hfill & +\sum _{r=t+1}^m\sum _{2\le p<q\le t}{\omega }_{pp}^r{\omega }_{qq}^r+\frac{c+3\beta }{4}(t(t-1))+\frac{(c-\beta )}{4}(3t_2{cos}^2\theta )\hfill \\ \hfill & -\sum _{2\le p<q\le t}{\overline{K}}_{pq}-\overline{\alpha }({\mathcal{E}}_{\perp }^{t_1})-\overline{\alpha }({\mathcal{E}}_{\theta }^{t_2}).\hfill \end{array}\end{array}$$

(34)

By the application of (5), (9), and (10), we deduce

$$\sum _{2\le p<q\le t}{\overline{K}}_{p,q}=\frac{c+3\beta }{8}(t-1)(t-2)+\frac{c-\beta }{8}(3t_2{cos}^2\theta ),$$

(35)

$$\overline{\alpha }({\mathcal{E}}_{\perp }^{t_1})=\frac{c+3\beta }{8}{t}_1(t_1-1),$$

(36)

$$\overline{\alpha }({\mathcal{E}}_{\theta }^{t_2})=\frac{c+3\beta }{8}{t}_2(t_2-1)+\frac{c-\beta }{8}3t_2{cos}^2\theta .$$

(37)

Using (42)–(44) in (34), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Ric(P)\le & \frac{1}{2}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }+\frac{c+3\beta }{4}(t+t_1{t}_2-1)-\frac{1}{2}\sum _{r=t+1}^m(2{\omega }_{11}^r-({\omega }_{t_1+1t_1+1}^r+\cdots \hfill \\ \hfill & +{\omega }_{tt}^r{))}^2-\sum _{r=t+1}^m\sum _{1\le p<q\le t}{({\omega }_{pq}^r)}^2+\sum _{r=t+1}^m[\sum _{1\le i<j\le t_1}{({\omega }_{ij}^r)}^2+\sum _{r=t+1}^m\sum _{t_1+1\le s<t\le n}{({\omega }_{st}^r)}^2]\hfill \\ \hfill & -\sum _{r=t+1}^m[\sum _{1\le i<j\le t_1}{\omega }_{ii}^r{\omega }_{jj}^r+\sum _{t_1+1\le s<t\le n}{\omega }_{ss}^r{\omega }_{tt}^r]+\sum _{r=t+1}^m\sum _{2\le p<q\le t}{\omega }_{pp}^r{\omega }_{qq}^r.\hfill \end{array}\end{array}$$

(38)

On the right side of the last inequality, the sixth and seventh terms can be stated as follows:

$$\begin{array}{cc}\hfill \sum _{r=t+1}^m[\sum _{1\le i<j\le t_1}{({\omega }_{ij}^r)}^2+\sum _{t_1+1\le s<t\le t}{({\omega }_{st}^r)}^2]& -\sum _{r=t+1}^m\sum _{1\le p<q\le t}{({\omega }_{pq}^r)}^2=-\sum _{r=t+1}^m\sum _{p=1}^{t_1}\sum _{q=t_1+1}^t{({\omega }_{pq}^r)}^2.\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}\hfill \sum _{r=t+1}^m[\sum _{1\le i<j\le t_1}{\omega }_{ii}^r{\omega }_{jj}^r& +\sum _{t_1+1\le s\ne t\le t}{\omega }_{ss}^r{\omega }_{tt}^r-\sum _{2\le p<q\le t}{\omega }_{pp}^r{\omega }_{qq}^r]=\sum _{r=t+1}^m[\sum _{p=2}^{t_1}\sum _{q=t_1+1}^t{\omega }_{pp}^r{\omega }_{qq}^r-\sum _{j=2}^{t_1}{\omega }_{11}^r{\omega }_{jj}^r].\hfill \end{array}$$

Using the two values listed above in (38), we obtain

$$\begin{array}{cc}\hfill Ric(P)\le & \frac{1}{2}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }+\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}-\frac{1}{2}\sum _{r=t+1}^m(2{\omega }_{11}^r\hfill \\ \hfill & -({\omega }_{t_1+1t_1+1}^r+\cdots {\omega }_{tt}^r){)}^2-\sum _{r=t+1}^m\sum _{p=1}^{t_1}\sum _{q=t_1+1}^t{({\omega }_{pq}^r)}^2+\sum _{b=2}^{t_1}{\omega }_{11}^r{\omega }_{bb}^r\hfill \\ \hfill & -\sum _{p=2}^{t_1}\sum _{q=t_1+1}^t{\omega }_{pp}^r{\omega }_{qq}^r.\hfill \end{array}$$

(39)

Through the assumption, ${\mathcal{E}}^t={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ is ${\mathcal{E}}_{\perp }^{t_1}$ minimal, the following can be seen

$$\sum _{r=t+1}^m\sum _{p=2}^{t_1}\sum _{q=t_1+1}^t{\omega }_{pp}^r{\omega }_{qq}^r=-\sum _{r=t+1}^m\sum _{q=t_1+1}^t{\omega }_{11}^r{\omega }_{qq}^r$$

(40)

and

$$\sum _{r=t+1}^m\sum _{b=2}^{t_1}{\omega }_{11}^r{\omega }_{bb}^r=-\sum _{r=t+1}^m{({\omega }_{11}^r)}^2.$$

(41)

Hence, we can state simultaneously

$$\begin{array}{cc}\hfill \frac{1}{2}\sum _{r=t+1}^m(2{\omega }_{11}^r& -({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{tt}^r){)}^2+\sum _{r=t+1}^m\sum _{q=t_1+1}^t{\omega }_{11}^r{\omega }_{qq}^r\hfill \\ \hfill & =2\sum _{r=t+1}^m{({\omega }_{11}^r)}^2+\frac{1}{2}{t}^2{\parallel \Omega \parallel }^2.\hfill \end{array}$$

(42)

Using (40) and (41) in (39), in the light of (42), we obtain

$$\begin{array}{cc}\hfill Ric(P)\le & \frac{1}{2}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }+\frac{c+3\beta }{4}(n+t_1{t}_2-1)+\frac{3(c-\beta )}{8}\hfill \\ \hfill & -\frac{1}{4}\sum _{r=t+1}^m\sum _{q=t_1+1}^t{({\omega }_{qq}^r)}^2-\sum _{r=t+1}^m\{{({\omega }_{11}^r)}^2-\sum _{q=t_1+1}^t{\omega }_{11}^r{\omega }_{qq}^r\hfill \\ \hfill & +\frac{1}{4}{({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{tt}^r)}^2\}.\hfill \end{array}$$

(43)

Next, using the condition ${\sum }_{r=t+1}^m({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{tt}^r)=t^2{\parallel \Omega \parallel }^2,$ we find

$$\begin{array}{ccc}\hfill Ric(P)& \le & \frac{1}{4}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }+\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}\hfill \\ \hfill & -& \frac{1}{4}\sum _{r=t+1}^m{(2{\omega }_{11}^r-\sum _{q=t_1+1}^t{\omega }_{qq}^r)}^2.\hfill \end{array}$$

(44)

We can deduce the inequality (24) from the preceding inequality.

Case ii: If $u_a$ is tangent to ${\mathcal{E}}_{\theta }^{t_2}$, the unit vector is chosen from $\{u_{t_1+1},\cdots ,u_n\}$. Assume that the unit vector is $u_t$, which means that $P=u_t.$ Then, using (5), (9), and (10), we get

$$\sum _{1\le p<q\le t-1}{\overline{K}}_{pq}=\frac{c+3\beta }{8}(t-1)(t-2)-\frac{c-\beta }{8}(3(t_2-1){cos}^2\theta ),$$

(45)

$$\overline{\alpha }({\mathcal{E}}_{\perp }^{t_1})=\frac{c+3\beta }{8}{t}_1(t_1-1),$$

(46)

$$\overline{\alpha }({\mathcal{E}}_{\theta }^{t_2})=\frac{c+3\beta }{8}{t}_2(t_2-1)+\frac{c-\beta }{8}(3t_2{cos}^2\theta ).$$

(47)

Again, in a similar manner as in Case i, by applying (45), (46), and (47), we obtain

$$\begin{array}{cc}\hfill Ric(P)\le & \frac{1}{2}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }-\frac{1}{2}\sum _{r=t+1}^m{(({\omega }_{t_1+1t_1+1}^r+\cdots {\omega }_{tt}^r)-2{\omega }_{tt}^r)}^2\hfill \\ \hfill & -\sum _{r=t+1}^m\sum _{1\le p<q\le t_1}{({\omega }_{pq}^r)}^2+\sum _{r=t+1}^m[\sum _{1\le i<j\le t_1}{({\omega }_{ij}^r)}^2-\sum _{1\le i<j\le t_1}{\omega }_{ii}^r{\omega }_{jj}^r]\hfill \\ \hfill & +\sum _{r=t+1}^m\sum _{t_1+1\le s<t\le t}{({\omega }_{st}^r)}^2+\sum _{r=t+1}^m[\sum _{t_1+1\le s<t\le t}{({\omega }_{ij}^r)}^2-\sum _{t_1+1\le s<t\le t}{\omega }_{ss}^r{\omega }_{tt}^r]\hfill \\ \hfill & +\sum _{r=t+1}^m\sum _{1\le p<q\le t-1}{\omega }_{pp}^r{\omega }_{qq}^r+\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}{cos}^2\theta .\hfill \end{array}$$

(48)

The aforementioned inequality takes the form by following the same procedures as in Case i.

$$\begin{array}{cc}\hfill Ric(P)\le & \frac{1}{2}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }+\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}{cos}^2\theta -\frac{1}{2}\sum _{r=t+1}^m\{({\omega }_{t_1+1t_1+1}^r+\cdots \hfill \\ \hfill & +{\omega }_{tt}^r{)-2{\omega }_{tt}^r\}}^2-\sum _{r=t+1}^m[\sum _{p=1}^{t_1}\sum _{q=t_1+1}^t{({\omega }_{pq}^r)}^2+\sum _{b=t_1+1}^{t-1}{\omega }_{tt}^r{\omega }_{bb}^r-\sum _{p=1}^{t_1}\sum _{q=t_1+1}^{t-1}{\omega }_{pp}^r{\omega }_{qq}^r].\hfill \end{array}$$

(49)

One can see, by the Lemma 1, that

$$\sum _{r=t+1}^m\sum _{p=1}^{t_1}\sum _{q=t_1+1}^{t-1}{\omega }_{pp}^r{\omega }_{qq}^r=0.$$

(50)

Making use of this in (49), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Ric(P)\le & \frac{1}{2}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }+\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}{cos}^2\theta \hfill \\ \hfill & -\frac{1}{2}\sum _{r=t+1}^m{(({\omega }_{t_1+1t_1+1}^r+\cdots {\omega }_{tt}^r)-2{\omega }_{tt}^r)}^2-\sum _{r=t+1}^m\sum _{p=1}^{t_1}\sum _{q=t_1+1}^t{({\omega }_{pq}^r)}^2-\sum _{r=t+1}^m\sum _{b=t_1+1}^{n-1}{\omega }_{tt}^r{\omega }_{bb}^r.\hfill \end{array}\end{array}$$

(51)

Last term of the inequality mentioned above can be expressed as

$$-\sum _{r=t+1}^m\sum _{b=t_1+1}^{t-1}{\omega }_{tt}^r{\omega }_{bb}^r=-\sum _{r=t+1}^m\sum _{b=t_1+1}^t{\omega }_{tt}^r{\omega }_{bb}^r+\sum _{r=t+1}^m{({\omega }_{tt}^r)}^2.$$

The fifth term in (51) can also be modified as

$$\begin{array}{cc}\hfill -\frac{1}{2}\sum _{r=t+1}^m{(({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{tt}^r)-2{\omega }_{tt}^r)}^2& =-\frac{1}{2}\sum _{r=t+1}^m{({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{tt}^r)}^2\hfill \\ \hfill & -2\sum _{r=t+1}^m{({\omega }_{tt}^r)}^2+\sum _{r=t+1}^m\sum _{j=t_1+1}^t{\omega }_{tt}^r{\omega }_{jj}^r.\hfill \end{array}$$

(52)

Using last pair of values in (51), we have

$$\begin{array}{cc}\hfill Ric(P)\le & \frac{1}{2}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }+\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}{cos}^2\theta \hfill \\ \hfill & -\frac{1}{2}\sum _{r=t+1}^m{({\omega }_{t_1+1t_1+1}^r+\cdots {\omega }_{tt}^r)}^2-2\sum _{r=t+1}^m{({\omega }_{tt}^r)}^2+2\sum _{r=t+1}^m\sum _{j=t_1+1}^t{\omega }_{tt}^r{\omega }_{jj}^r\hfill \\ \hfill & -\sum _{r=t+1}^m\sum _{p=1}^{t_1}\sum _{q=t_1+1}^n{({\omega }_{pq}^r)}^2-\sum _{r=t+1}^m\sum _{b=t_1+1}^t{\omega }_{tt}^r{\omega }_{bb}^r+\sum _{r=t+1}^m{({\omega }_{tt}^r)}^2,\hfill \end{array}$$

(53)

or equivalently

$$\begin{array}{cc}\hfill Ric(P)\le & \frac{1}{2}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }+\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}{cos}^2\theta \hfill \\ \hfill & -\frac{1}{2}\sum _{r=t+1}^m{({\omega }_{t_1+1t_1+1}^r+\cdots {\omega }_{tt}^r)}^2-\sum _{r=t+1}^m{({\omega }_{tt}^r)}^2+\sum _{r=t+1}^m\sum _{j=t_1+1}^t{\omega }_{tt}^r{\omega }_{jj}^r\hfill \\ \hfill & -\sum _{r=t+1}^m\sum _{p=1}^{t_1}\sum _{q=t_1+1}^t{({\omega }_{pq}^r)}^2.\hfill \end{array}$$

(54)

Using similar methods to those used in the previous case, we find

$$\begin{array}{cc}\hfill Ric(P)\le \frac{1}{4}{t}^2{\parallel \Omega \parallel }^2-\frac{t_2\Delta \psi }{\psi }& +\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}{cos}^2\theta \hfill \\ \hfill & -\frac{1}{4}\sum _{r=t+1}^m{(2{\omega }_{tt}^r-({\omega }_{t_1+1t_1+1}^r+\cdots +{\omega }_{tt}^r))}^2,\hfill \end{array}$$

(55)

This provides the inequality (25).

Now, we examine cases of equality for the inequality (24). First, In the g.c.s.f. ${\overline{\mathcal{E}}}^m(c,\beta )$, we clarify the concept of the relative null space $N_x$ of the submanifold ${\mathcal{E}}^t$ at any point $x\in {\mathcal{E}}^t.$ B. Y. Chen [16] defined the relative null space as follows

$$N_x=\{X^1\in T_x{\mathcal{E}}^t:\omega (X^1,Y^1)=0,\forall Y^1\in T_x{\mathcal{E}}^t\}.$$

If $A\in \{1,\cdots ,t\}$, then a unit vector field $u_A\in T{\mathcal{E}}^t$ at x fulfills the equality symbol of (24) identically iff

$$(i)\sum _{p=1}^{t_1}\sum _{q=t_1+1}^t{\omega }_{pq}^r=0(ii)\sum _{b=1}^t\sum _{\begin{array}{c}A=1\\ b\ne A\end{array}}^t{\omega }_{bA}^r=0(iii)2{\omega }_{AA}^r=\sum _{q=t_1+1}^t{\omega }_{qq}^r,$$

(56)

like that $r\in \{t+1,\cdots m\}$; the case $(i)$ gives that ${\mathcal{E}}^t$ is a mixed TG hemi-slant wp submanifold. The unit vector field $P=u_A\in N_x$ when propositions $(ii)$ and $(iii)$ are combined with the information that ${\mathcal{E}}_t$ is a hemi-slant wp submanifold. The converse is straigtforward, which is shown inassertion (2).

The equality sign of (24) satisfies a hemi-slant wp submanifold identically for all unit tangent vectors belonging to ${\mathcal{E}}_{\perp }^{t_1}$ at x iff

$$(i)\sum _{p=1}^{t_1}\sum _{q=t_1+1}^t{\omega }_{pq}^r=0(ii)\sum _{b=1}^n\sum _{\begin{array}{c}A=1\\ b\ne A\end{array}}^{t_1}{\omega }_{bA}^r=0(iii)2{\omega }_{pp}^r=\sum _{q=t_1+1}^t{\omega }_{qq}^r,$$

(57)

such that $p\in \{1,\cdots ,t_1\}$, and $r\in \{t+1,\cdots ,m\}.$ Since ${\mathcal{E}}^t$ is a hemi-slant wp submanifold, the third condition implies that ${\omega }_{pp}^r=0,p\in \{1,\cdots ,t_1\}$. By applying this to the condition $(ii)$, we obtain a conclusion that ${\mathcal{E}}^t$ is a $D^{\perp }$ TG hemi-slant wp submanifold in ${\overline{\mathcal{E}}}^m(c,\beta )$, and, from the requirement $(i)$, mixed totally geodesicness follows, thereby proving $(a)$ of the statement (3).

For a hemi-slant wp submanifold, the equality sign of (25) holds identically for all unit tangent vector fields tangent to ${\mathcal{E}}_{\theta }^{t_2}$ at x iff

$$(i)\sum _{p=1}^{t_1}\sum _{q=t_1+1}^t{\omega }_{pq}^r=0(ii)\sum _{b=1}^t\sum _{\begin{array}{c}A=t_1+1\\ b\ne A\end{array}}^t{\omega }_{bA}^r=0(iii)2{\omega }_{KK}^r=\sum _{q=t_1+1}^t{\omega }_{qq}^r,$$

(58)

where $K\in \{t_1+1,\cdots ,t\}$, and $r\in \{t+1,\cdots ,m\}.$ Two situations arise from $(iii)$; these are,

$${\omega }_{KK}^r=0,\forall K\in \{t_1+1,\cdots ,t\}\mathrm{and}r\in \{t+1,\cdots ,m\}\mathrm{or}dim{\mathcal{E}}_{\theta }^{t_2}=2.$$

(59)

It is normal to deduce that ${\mathcal{E}}^t$ is a $D_{\theta }$ TG hemi-slant wp submanifold in ${\overline{\mathcal{E}}}^m(c,\beta )$ if the first case of (58) is satisfied; this is the first case of statement $(b)$ of (3).

Regarding the potential conflict, assume that ${\mathcal{E}}^t$ is not a $D^{\theta }$ TG hemi-slant wp submanifold, and dim ${\mathcal{E}}_{\theta }^{t_2}=2.$ As a result, condition $(ii)$ of (58) indicates that ${\mathcal{E}}^t$ is a $D^{\theta }$ TU hemi-slant wp submanifold in $\overline{\mathcal{E}}(c,\beta )$; this is the second situation of this part, which verifies section $(b)$ of (3).

Using sections $(a)$ and $(b)$ of (3), we unify (57) and (58) to demonstrate $(c)$. Assume that $dim{\mathcal{E}}_{\theta }^{t_2}\ne 2$ is true for the first case of this section. from parts $(a)$ and $(b)$ of statement $(3)$, we deduce that ${\mathcal{E}}^t$ is a $D^{\perp }$ TG and $D^{\theta }$ TG submanifold in ${\overline{\mathcal{E}}}^m(c,\beta )$. Therefore, ${\mathcal{E}}^t$ is a submanifold in ${\overline{\mathcal{E}}}^m(c,\beta ).$

Consider a different situation in which the first situation does not hold. Then, parts $(a)$ and $(b)$ give that ${\mathcal{E}}^t$ is a mixed TG and $D^{\perp }$ TG submanifold of ${\overline{\mathcal{E}}}^m(c,\beta )$ with $dim{\mathcal{E}}_{\theta }^{t_2}=2$. By the situation $(b)$, it leads that ${\mathcal{E}}^t$ is a $D^{\theta }$ TU hemi-slant wp submanifold and, from $(a)$, it is a $D^{\perp }$ TG, which is $(c)$. This validates the assertion. □

In light of (20), we have a different formulation of Theorem (1), which is as follows.

Theorem 2.

Let $\mathcal{E}={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ be a hemi-slant wp submanifold immersed isometrically in a g.c.s.f $\overline{\mathcal{E}}(c,\beta )$ having a nearly Kaehler structure. Then, for every orthogonal unit vector field $P\in T_x\mathcal{E}$, either tangent to ${\mathcal{E}}_{\perp }$ or ${\mathcal{E}}_{\theta }$, the estimation of Ricci curvature is given by:

(i) 

If $P\in T{\mathcal{E}}_{\perp }$, then

$$Ric(P)\le \frac{1}{4}{t}^2{\parallel \Omega \parallel }^2-t_2\Delta ln\psi +t_2{\parallel \nabla ln\psi \parallel }^2+\frac{c+3\beta }{4}(t+t_1{t}_2-1).$$

(60)

(ii) 

If $P\in T{\mathcal{E}}_{\theta }$, then

$$\begin{array}{cc}\hfill Ric(P)\le \frac{1}{4}{t}^2{\parallel \Omega \parallel }^2-t_2\Delta ln\psi & +t_2{\parallel \nabla ln\psi \parallel }^2+\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}{cos}^2\theta .\hfill \end{array}$$

(61)

The equality conditions are analogous to Theorem (1).

5. Characterization via Differential Equations

In this section, we apply the Bochner formula to the Ricci curvature inequalities and obtain some characterizations related to differential equations.

Theorem 3.

For a $D^{\perp }$-minimal t-dimensional complete hemi-slant wp submanifold ${\mathcal{E}}^t={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ in a g.c.s.f $\overline{\mathcal{E}}(c)$, the Ricci curvature is $Ric(P)\ge L,L\ge 0.$ If $X\in D^{\perp }$ and satisfies the following equality,

$$(\frac{{\lambda }_1}{t_2}+1)K=\frac{{\lambda }_1^2}{t_1}+\frac{{\lambda }_1}{t_2}.\frac{c+3\beta }{4}(t+t_1{t}_2-1)-\frac{{\lambda }_1{t}^2}{4t_2}{\parallel \Omega \parallel }^2,$$

(62)

then the base manifold ${\mathcal{E}}_{\perp }^{t_1}$ is isometric to Euclidean space $R^n,$ where $t_1,t_2$ are the dimensions of the submanifolds ${\mathcal{E}}_{\perp }^{t_1},{\mathcal{E}}_{\theta }^{t_2}$, respectively, and ${\lambda }_1$ is the eigenvalue corresponding to the eigen function $ln\psi .$

Proof. 

Since $P\in D^{\perp },$ by Equation (60), we have

$$Ric(P)+t_2\Delta ln\psi \le \frac{1}{4}{t}^2{\parallel \Omega \parallel }^2+t_2{\parallel \nabla ln\psi \parallel }^2+\frac{c+3\beta }{4}(t+t_1{t}_2-1).$$

(63)

By the assumption that $Ric(P)\ge L,$ we obtain

$$L+t_2\Delta ln\psi \le \frac{1}{4}{t}^2{\parallel \Omega \parallel }^2+t_2{\parallel \nabla ln\psi \parallel }^2+\frac{c+3\beta }{4}(t+t_1{t}_2-1).$$

(64)

Since $Ric(P)\ge L$, it follows the theorem of Myer’s [20] that the base manifold ${\mathcal{E}}_{\perp }^{t_1}$ is compact if the Ricci curvature is exceeded by a positive constant. Using Green’s theorem and integrating (64), we obtain

$$Vol({\mathcal{E}}_{\perp }^{t_1})L\le \frac{t^2}{4}{\int }_{{\mathcal{E}}^t}{\parallel \Omega \parallel }^{2}dV+t_2{\int }_{{\mathcal{E}}^t}{\parallel \nabla ln\psi \parallel }^{2}dV+{\int }_{{\mathcal{E}}^t}\frac{c+3\beta }{4}(n+t_1{t}_2-1)dV$$

(65)

or

$${\int }_{{\mathcal{E}}^t}{\parallel \nabla ln\psi \parallel }^{2}dV\ge \frac{L}{t_2}Vol({\mathcal{E}}_{\perp }^{t_1})-\frac{t^2}{4t_2}{\int }_{{\mathcal{E}}^t}{\parallel \Omega \parallel }^{2}dV+\frac{c+3\beta }{4t_2}{\int }_{{\mathcal{E}}^t}(t+t_1{t}_2-1)dV.$$

(66)

If we consider $H_{ess}(ln\psi )$ to be the Hessian of the warping function, we obtain

$$\parallel H_{ess}{(ln\psi )-rI\parallel }^2=\parallel H_{ess}{(ln\psi )\parallel }^2+r^2{\parallel I\parallel }^2-2rg(I,H_{ess}(ln\psi )),$$

(67)

where r is a real number. The above formula gives

$$\parallel H_{ess}{(ln\psi )-rI\parallel }^2=2r\Delta (ln\psi )+r^2{t}_1+{\parallel H_{ess}(ln\psi )\parallel }^2.$$

(68)

Putting $r=\frac{{\lambda }_1}{t_1}$ and integrating the above equation with respect to $dV,$ we obtain

$${\int }_{{\mathcal{E}}^t}\parallel H_{ess}(ln\psi )-\frac{{\lambda }_1}{t_1}{I\parallel }^{2}dV={\int }_{{\mathcal{E}}^t}{\parallel H_{ess}(ln\psi )\parallel }^{2}dV+{\int }_{{\mathcal{E}}^t}\frac{{\lambda }_1^2}{t_1}dV.$$

(69)

Using (22), with the fact that $\Delta ln\psi ={\lambda }_{1}ln\psi ,$ we have

$${\int }_{{\mathcal{E}}^t}\parallel H_{ess}{(ln\psi )\parallel }^{2}dV=-{\lambda }_1{\int }_{{\mathcal{E}}^t}{\parallel \nabla ln\psi \parallel }^{2}dV-{\int }_{{\mathcal{E}}^t}Ric(\nabla ln\psi ,\nabla ln\psi ).$$

(70)

By combining (69) and (70), we derive

$${\int }_{{\mathcal{E}}^t}\parallel H_{ess}(ln\psi )-\frac{{\lambda }_1}{t_1}{I\parallel }^{2}dV={\int }_{{\mathcal{E}}^t}\frac{{\lambda }_1^2}{t_1}dV-{\lambda }_1{\int }_{{\mathcal{E}}^t}{\parallel \nabla ln\psi \parallel }^{2}dV-{\int }_{{\mathcal{E}}^t}Ric(\nabla ln\psi ,\nabla ln\psi )dV.$$

(71)

According to the assumption that $Ric(\nabla f,\nabla f)\ge L$, the equation above becomes

$${\int }_{{\mathcal{E}}^t}\parallel H_{ess}(ln\psi )-\frac{{\lambda }_1}{t_1}{I\parallel }^{2}dV={\int }_{{\mathcal{E}}^t}\frac{{\lambda }_1^2}{t_1}dV-{\lambda }_1{\int }_{{\mathcal{E}}^t}{\parallel \nabla ln\psi \parallel }^{2}dV-LVol({\mathcal{E}}_{\perp }^{t_1}).$$

(72)

Using (74), the last inequality leads to

$$\begin{array}{cc}\hfill {\int }_{{\mathcal{E}}^t}{\parallel H_{ess}(ln\psi )-\frac{{\lambda }_1}{t_1}I\parallel }^{2}dV\le {\int }_{{\mathcal{E}}^t}& \frac{{\lambda }_1^2}{t_1}dV-{\int }_{{\mathcal{E}}^t}(\frac{{\lambda }_{1}K}{t_2}+K)dV\hfill \\ \hfill & +\frac{{\lambda }_1}{t_2}{\int }_{{\mathcal{E}}^t}\frac{c+3\beta }{4}(n+t_1{t}_2-1)dV\hfill \\ \hfill & -\frac{{\lambda }_1{t}^2}{4t_2}{\int }_{{\mathcal{E}}^t}{\parallel \Omega \parallel }^{2}dV.\hfill \end{array}$$

(73)

If (62) holds, then the above inequality gives

$$\parallel H_{ess}(ln\psi )-\frac{{\lambda }_1}{t_1}{I\parallel }^2=0$$

Therefore, we have $H_{ess}(ln\psi )(X,X)=\frac{{\lambda }_1}{t_1}$. Hence, by the aplication of Tashiro’s result [21], the fiber ${\mathcal{E}}_{\perp }^{t_1}$ is isometric to Euclidean space $R^{t_1}$. □

The following conclusions follow if we take into account the unit vector field $P\in T{\mathcal{E}}_{\theta },$ and they may be demonstrated by applying the same procedures as in Theorem 3.

Theorem 4.

Let ${\mathcal{E}}^t={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ be a $D^{\perp }$-minimal t-dimensional complete hemi-slant wp submanifold in a g.c.s.f $\overline{\mathcal{E}}(c)$, such that the Ricci curvature is $Ric(P)\ge L,L\ge 0.$ If $P\in D^{\perp }$ and satisfies the following equality

$$(\frac{{\lambda }_1}{t_2}+1)L=\frac{{\lambda }_1^2}{t_1}+\frac{{\lambda }_1}{t_2}[\frac{c+3\beta }{4}(t+t_1{t}_2-1)+\frac{3(c-\beta )}{8}{cos}^2\theta ]-\frac{{\lambda }_1{t}^2}{4t_2}{\parallel \Omega \parallel }^2,$$

(74)

then the base manifold ${\mathcal{E}}_{\perp }^{t_1}$ is isometric to Euclidean space $R^n,$ where $t_1,t_2$ are the dimensions of the submanifolds ${\mathcal{E}}_{\perp }^{t_1},{\mathcal{E}}_{\theta }^{t_2}$, respectively, and ${\lambda }_1$ is the eigenvalue corresponding to the eigen function $ln\psi .$

The following result is based on research done by Garcia-Rio et al. [23].

Theorem 5.

Let ${\mathcal{E}}^t={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ be a $D^{\perp }$-minimal t-dimensional complete hemi-slant wp submanifold in a g.c.s.f $\overline{\mathcal{E}}(c),$ such that the Ricci curvature is $Ric(P)>L,L>0.$ If $P\in D^{\perp }$ and satisfies the following expression

$$\frac{{\lambda }_1}{t_1{t}_2}(\frac{t^2}{4}{\parallel \Omega \parallel }^2+\frac{c+3\beta }{4}-L)+{\parallel H_{ess}ln\psi \parallel }^2=0$$

(75)

for ${\lambda }_1<0$, then ${\mathcal{E}}_{\perp }^{t_1}$ is isometric to the wp of the type $R{\times }_{f}U,$ where R is the Euclidean line and U is the complete Riemannian manifold with the warping function f, which is the solution of the differential equation $\frac{d^{2}f}{d{t}^2}+{\lambda }_{1}f=0.$

Proof. 

We define the following relation on ${\mathcal{E}}_{\perp }^{t_1}$ for the warping function $ln\psi$.

$$\parallel Lln\psi I+H_{ess}{ln\psi \parallel }^2=K^2{(ln\psi )}^2{\parallel I\parallel }^2+\parallel H_{ess}ln\psi {\parallel }^2+2Lln\psi g(I,H_{ess}ln\psi )).$$

(76)

Let $ln\psi$ be an eigenfunction correponding to the eigenvalue ${\lambda }_1$ satisfying $\Delta ln\psi ={\lambda }_{1}ln\psi$. Then, we have

$$\parallel Lln\psi I+H_{ess}{ln\psi \parallel }^2={\parallel H_{ess}ln\psi \parallel }^2+(t_1{L}^2-2L{\lambda }_1){(ln\psi )}^2.$$

(77)

Again, using $\Delta ln\psi ={\lambda }_{1}ln\psi ,$ we observe that

$$\nabla \frac{{(ln\psi )}^2}{2}=ln\psi {\lambda }_{1}ln\psi -{\parallel \nabla ln\psi \parallel }^2,$$

(78)

through which integration gives

$${\int }_{{\mathcal{E}}^t}{(ln\psi )}^{2}dV=\frac{1}{{\lambda }_1}{\int }_{{\mathcal{E}}^t}{\parallel \nabla ln\psi \parallel }^{2}dV.$$

(79)

From (77), we have

$${\int }_{{\mathcal{E}}^t}\parallel Lln\psi I+H_{ess}{ln\psi \parallel }^{2}dV={\int }_{{\mathcal{E}}^t}\parallel H_{ess}{ln\psi \parallel }^{2}dV+(\frac{t_1{L}^2}{{\lambda }_1}-2L){\int }_{{\mathcal{E}}^t}{\parallel \nabla ln\psi \parallel }^{2}dV.$$

(80)

Putting $L=\frac{{\lambda }_1}{t_1}$ in (79), we have

$${\int }_{{\mathcal{E}}^t}\parallel H_{ess}ln\psi +\frac{{\lambda }_1}{t_1}{ln\psi I\parallel }^{2}dV={\int }_{{\mathcal{E}}^t}\parallel H_{ess}{ln\psi \parallel }^{2}dV-\frac{{\lambda }_1}{t_1}{\int }_{{\mathcal{E}}^t}{\parallel \nabla ln\psi \parallel }^{2}dV$$

(81)

Furthermore, by integrating (64) and applying Green’s Lemma, we find

$${\int }_{{\mathcal{E}}^t}Ric(P)dV\le \frac{t^2}{4}{\int }_{{\mathcal{E}}^t}{\parallel \Omega \parallel }^{2}dV+t_2{\int }_{{\mathcal{E}}^t}{\parallel \nabla ln\psi \parallel }^{2}dV+\frac{c+3\beta }{4}{\int }_{{\mathcal{E}}^t}(n+t_1{t}_2-1)dV.$$

(82)

From the above two expressions, we obtain

$$\begin{array}{cc}\hfill \frac{1}{t_2}{\int }_{{\mathcal{E}}^t}Ric(P)dV\le & \frac{t^2}{4t_2}{\int }_{{\mathcal{E}}^t}{\parallel \Omega \parallel }^{2}dV+\frac{t_1}{{\lambda }_1}{\int }_{{\mathcal{E}}^t}\parallel H_{ess}ln\psi {\parallel }^{2}dV-\frac{t_1}{{\lambda }_1}{\int }_{{\mathcal{E}}^t}\parallel H_{ess}ln\psi \hfill \\ & +\frac{{\lambda }_1}{t_1}{ln\psi I\parallel }^{2}dV+\frac{c+3\beta }{4t_2}{\int }_{{\mathcal{E}}^t}(t+t_1{t}_2-1)dV.\hfill \end{array}$$

(83)

Upon using the assumption that $Ric(P)>L$ for $L>0$, we obtain

$$\begin{array}{cc}\hfill {\int }_{{\mathcal{E}}^t}\parallel H_{ess}ln\psi +\frac{{\lambda }_1}{t_1}& {ln\psi I\parallel }^{2}dV\le \frac{t^2{\lambda }_1}{4t_1{t}_2}{\int }_{{\mathcal{E}}^t}{\parallel \Omega \parallel }^{2}dV+{\int }_{{\mathcal{E}}^t}{\parallel H_{ess}ln\psi \parallel }^{2}dV\hfill \\ \hfill & -\frac{{\lambda }_1}{t_1{t}_2}{\int }_{{\mathcal{E}}^t}Ric(P)dV+\frac{{\lambda }_1(c+3\beta )}{4t_1{t}_2}{\int }_{{\mathcal{E}}^t}(t+t_1{t}_2-1)dV.\hfill \end{array}$$

(84)

or

$$\begin{array}{cc}\hfill {\int }_{{\mathcal{E}}^t}\parallel H_{ess}ln\psi +\frac{{\lambda }_1}{t_1}& {ln\psi I\parallel }^{2}dV\le \frac{t^2{\lambda }_1}{4t_1{t}_2}{\int }_{{\mathcal{E}}^t}{\parallel \Omega \parallel }^{2}dV+{\int }_{{\mathcal{E}}^t}{\parallel H_{ess}ln\psi \parallel }^{2}dV\hfill \\ \hfill & -\frac{{\lambda }_1}{t_1{t}_2}{\int }_{{\mathcal{E}}^t}KdV+\frac{{\lambda }_1(c+3\beta )}{4t_1{t}_2}{\int }_{{\mathcal{E}}^t}(t+t_1{t}_2-1)dV.\hfill \end{array}$$

(85)

Now, by applying the assumption (75), we find

$$\parallel H_{ess}ln\psi +\frac{{\lambda }_1}{t_1}{ln\psi I\parallel }^2\le 0,$$

or

$$\parallel H_{ess}ln\psi +\frac{{\lambda }_1}{t_1}{ln\psi I\parallel }^2=0.$$

□

Thus, by the application of result obtained in [23], together with the fact ${\mathcal{E}}^t={\mathcal{E}}_{\perp }^{t_1}{\times }_{\psi }{\mathcal{E}}_{\theta }^{t_2}$ is nontrivial, we deduce that ${\mathcal{E}}_{\perp }^{t_1}$ is isometric to a wp of the form $R{\times }_{f}U,$ where U is a complete Riemannian manifold. In addition, the warping function f is the solution of the differential equation $\frac{d^{2}f}{d{t}^2}+{\lambda }_{1}f=0.$

6. Conclusions

Since the Ricci curvature is a symmetric bilinear form and In general relativity, it has a significant impact, the Ricci curvature is one of the key terms in the Einstein field equation and Ricci flow equation. In this paper, we approximated Ricci curvature for hemi-slant warped product submanifolds of a generalized complex space form. We provided the application of Bochner’s formula to the Ricci curvature inequalities via differential equation. Our results can be used in the theory of relativity and mathematical physics. However, in this study, we used the Levi-Civita connection. For further study, one can see the impact of semi-symmetric and quarter symmetric connections on the present type of study.