**2. Preliminaries**

Let *S* ¯ be an A-H manifold with an almost complex structure *J* and a Hermitian metric ,, i.e., *J*2 = −*I* and *JU*1, *JU*2 = *<sup>U</sup>*1, *<sup>U</sup>*2, for all vector fields *U*1, *U*2 on *S*¯. If *J* is parallel with respect to the Levi-Civita connection *D* ¯ on *S* ¯ , that is

$$(\bar{D}\_{\mathcal{U}\_1}f)\mathcal{U}\_2 = 0,\tag{2}$$

for all *U*1, *U*2 ∈ *TS* ¯ , then (*S*¯, *J*,,, *D*¯ ) is called a *Kaehler manifold* (K-M).

.

A K-M *S* ¯ is called a *CSF* if it has constant holomorphic sectional curvature *c* denoted by *S* ¯(*c*). The curvature tensor of the CSF *S* ¯(*c*) is given by

$$\begin{aligned} \mathcal{R}\left(\mathcal{U}\_{1}, \mathcal{U}\_{2}, \mathcal{U}\_{3}, \mathcal{U}\_{4}\right) &= \frac{c}{4} \left[ \langle \mathcal{U}\_{2}, \mathcal{U}\_{3} \rangle \langle \mathcal{U}\_{1}, \mathcal{U}\_{4} \rangle - \langle \mathcal{U}\_{1}, \mathcal{U}\_{3} \rangle \langle \mathcal{U}\_{2}, \mathcal{U}\_{4} \rangle + \langle \mathcal{U}\_{1}, \mathcal{U}\_{3} \rangle \langle \mathcal{U}\_{2}, \mathcal{U}\_{4} \rangle \right. \\ &\left. - \left\langle \mathcal{U}\_{2}, \mathcal{U}\_{3} \right\rangle \langle \mathcal{U}\_{1}, \mathcal{U}\_{4} \rangle + 2 \langle \mathcal{U}\_{1}, \mathcal{U}\_{2} \rangle \langle \mathcal{U}\_{3}, \mathcal{U}\_{4} \rangle \right]. \end{aligned} \tag{3}$$

for any *U*1, *U*2, *U*3, *U*4 ∈ *TS* ¯ 

Let *S* be a *<sup>n</sup>*−dimensional Riemannian manifold isometrically immersed in a *m*− dimensional Riemannian manifold *S* ¯ . Then, the Gauss and Weingarten formulas are *D* ¯ *<sup>U</sup>*1*U*<sup>2</sup> = *DU*1*U*<sup>2</sup> + <sup>Γ</sup>(*<sup>U</sup>*1, *<sup>U</sup>*2) and *D* ¯ *U*1 *ξ* = −*AξU*<sup>1</sup> + *<sup>D</sup>*⊥*U*1 *ξ* respectively, for all *U*1, *U*2 ∈ *TS* and *ξ* ∈ *<sup>T</sup>*⊥*S*, where *D* is the induced Levi-Civita connection on *S*, *ξ* is a vector field normal to *S*, Γ is the second fundamental form of *S*, *D*<sup>⊥</sup> is the normal connection in the normal bundle *T*⊥*S* and *<sup>A</sup>ξ* is the shape operator of the second fundamental form. The second fundamental form Γ and the shape operator are related by the following formula

$$
\langle \Gamma(lI\_1, lI\_2), \xi \rangle = \langle A\_{\xi} lI\_1, lI\_2 \rangle. \tag{4}
$$

The Gauss equation is given by

$$R(lL\_1, lL\_2, lL\_3, lL\_4) = \mathcal{R}(lL\_1, lL\_2, lL\_3, lL\_4) + \langle \Gamma(lL\_1, lL\_4), \Gamma(lL\_2, lL\_3) \rangle - \langle \Gamma(lL\_1, lL\_3), \Gamma(lL\_2, lL\_4) \rangle,\tag{5}$$

for all *U*1, *U*2, *U*3, *U*4 ∈ *TS*, where *R* ¯ and *R* are the curvature tensors of *S* ¯ and *S* respectively. Forany*U*1∈ *TS*and*ξ*∈ *<sup>T</sup>*⊥*S*,*JU*1and*Jξ* canbedecomposedasfollows.

$$J\,U\_1 = P\,U\_1 + F\,U\_1\tag{6}$$

$$J\,U\_1 = P\,U\_1 + F\,U\_1\tag{7}$$

and 
$$J\mathfrak{g} = t\mathfrak{F} + f\mathfrak{F},\tag{7}$$

where *PU*1 (resp. *tξ*) is the tangential and *FU*1 (resp. *f ξ*) is the normal component of *JU*1 ( resp. *Jξ*).

It is evident that *JU*1, *<sup>U</sup>*2 = *PU*1, *<sup>U</sup>*2 for any *U*1, *U*2 ∈ *TxS*; this implies that *PU*1,*Y*2 + *<sup>U</sup>*1, *PU*2 = 0. Thus, *P*<sup>2</sup> is a symmetric operator on the tangent space *TxS*, for any *x* ∈ *S*. The eigenvalues of *P*<sup>2</sup> are real and diagonalizable. Moreover, for each *x* ∈ *S*, one can observe

$$L\_x^{\lambda} = \operatorname{Ker}\{P^2 + \lambda^2(x)I\}\_{x\star}$$

where *I* denotes the identity transformation on *TxS*, and *<sup>λ</sup>*(*x*) ∈ [0, 1] such that −*λ*<sup>2</sup>(*x*) is an eigenvalue of *<sup>P</sup>*<sup>2</sup>(*x*). Further, it is easy to observe that *KerF* = *L*1*x* and *KerP* = *<sup>L</sup>*0*x*, where *L*1*x* is the maximal holomorphic sub space of *TxS* and *L*0*x* is the maximal totally real subspace of *TxS*; these distributions are denoted by *L* and *L*<sup>⊥</sup> respectively. If <sup>−</sup>*<sup>λ</sup>*21(*x*), ... , <sup>−</sup>*λ*2*k* (*x*) are the eigenvalues of *P*<sup>2</sup> at *x*, then *TxS* can be decomposed as

$$T\_{\mathbf{x}}\mathbb{S} = L\_{\mathbf{x}}^{\lambda\_1} \oplus L\_{\mathbf{x}}^{\lambda\_2} \oplus \dots \sqcup L\_{\mathbf{x}}^{\lambda\_k}.$$

Every *Lλ<sup>i</sup> x* , 1 ≤ *i* ≤ *k* is a *P*−invariant subspace of *TxS*. Moreover, if *λi* = 0, then *Lλ<sup>i</sup> x* is even dimensional the submanifold *S* of a Kaehler manifold *S* ¯ is a generic submanifold if there exists an integer *k* and functions *λi* 1 ≤ *i* ≤ *k* defined on *S* with *λi* ∈ (0, 1) such that

(i) Each <sup>−</sup>*λ*2*i*(*x*), 1 ≤ *i* ≤ *k*, is a distinct eigenvalue of *P*<sup>2</sup> with

$$T\_{\mathbf{x}}\mathcal{S} = L\_{\mathbf{x}}^T \oplus L\_{\mathbf{x}}^\perp \oplus L\_{\mathbf{x}}^{\lambda\_1} \oplus \dots \oplus L\_{\mathbf{x}}^{\lambda\_k}$$

for any *x* ∈ *S*.

(ii) The distributions of *LTx* , *L*⊥*x* and *Lλ<sup>i</sup> x* , 1 ≤ *i* ≤ *k* are independent of *x* ∈ *S*.

If in addition, each *λi* is constant on *S*, then *S* is called a skew CR-submanifold [7]. It is significant to recount that CR-submanifolds are a particular class of skew CR-submanifold for which *k* = 1, *L<sup>T</sup>* = {0}, *L*<sup>⊥</sup> = {0} and *λ*1 is constant. If *L*<sup>⊥</sup> = {0}, *L*1 = {0} and *k* = 1, then *S* is a semi-slant submanifold, whereas if *L* = {0}, *L*<sup>⊥</sup> = {0} and *k* = 1, then *S* is a hemi-slant submanifold.

**Definition 1.** *A submanifold S of an A-H manifold S* ¯ *is said to be a "skew CR-submanifold of order 1" if S is a skew CR-submanifold with k* = 1 *and λ*1 *is constant.*

We have the following characterization

**Theorem 2.** *Reference [3] let S be a submanifold of an A-H manifold S* ¯ *. Then S is a slant if and only if there exists a constant λ* ∈ [0, 1] *such that*

$$P^2 = -\lambda I.$$

*Furthermore, if θ is a slant angle, then λ* = cos<sup>2</sup> *θ*.

For any orthonormal basis {*<sup>e</sup>*1,*e*2, ... ,*et*} of the tangent space *TxS*, the mean curvature vector <sup>Π</sup>(*x*) and its squared norm are defined as follows.

$$\Pi(\mathbf{x}) = \frac{1}{t} \sum\_{i=1}^{t} \Gamma(e\_i, e\_i), \quad \|\Pi\|^2 = \frac{1}{t^2} \sum\_{i,j=1}^{t} \langle \Gamma(e\_i, e\_i), \Gamma(e\_j, e\_j) \rangle,\tag{8}$$

where *t* is the dimension of *S*. If Γ = 0 then the submanifold is said to be totally geodesic and minimal if Π = 0. If <sup>Γ</sup>(*<sup>U</sup>*1, *<sup>U</sup>*2) = *<sup>U</sup>*1, *<sup>U</sup>*2<sup>Π</sup> for all *U*1, *U*2 ∈ *TS*, then *S* is called totally umbilical (T-U).

The scalar curvature of *S* ¯ is denoted by *τ*¯(*S*¯) and is defined as

$$\bar{\tau}(\bar{S}) = \sum\_{1 \le p < q \le m} \bar{\kappa}\_{pq} \tag{9}$$

where *κ*¯ *pq* = *κ*¯(*ep* ∧ *eq*) and *m* is the dimension of the Riemannian manifold *S*¯. Throughout this study, we shall use the equivalent version of the above equation, which is given by

$$2\pi(S) = \sum\_{1 \le p < q \le m} \kappa\_{pq}.\tag{10}$$

In a similar way, the scalar curvature *τ*¯(*Lx*) of a *<sup>L</sup>*−plane is given by

$$\mathfrak{T}(L\_x) = \sum\_{1 \le p < q \le m} \mathfrak{k}\_{pq}.\tag{11}$$

Let {*<sup>e</sup>*1, ... ,*et*} be an orthonormal basis of the tangent space *TxS* and if *er* belongs to the orthonormal basis {*en*+1,...*em*} of the normal space *<sup>T</sup>*⊥*S*, then we have

$$
\Gamma^r\_{pq} = \langle \Gamma(\mathfrak{e}\_p, \mathfrak{e}\_q), \mathfrak{e}\_r \rangle \tag{12}
$$

and

$$\|\|\Gamma\|\|^2 = \sum\_{p,q=1}^t \langle \Gamma(\varepsilon\_p, \varepsilon\_q), \Gamma(\varepsilon\_{p\prime}, \varepsilon\_q) \rangle. \tag{13}$$

Let *<sup>κ</sup>pq* and *κ*¯ *pq* be the sectional curvatures of the plane sections spanned by *ep* and *eq* at *x* in the submanifold *S* and in the Riemannian space form *S* ¯ *<sup>m</sup>*(*c*), respectively. Thus by Gauss equation, we have

$$\kappa\_{pq} = \kappa\_{pq} + \sum\_{r=t+1}^{m} \left( \Gamma\_{pp}^{r} \Gamma\_{qq}^{r} - (\Gamma\_{pq}^{r})^2 \right). \tag{14}$$

The global tensor field for orthonormal frame of vector field {*<sup>e</sup>*1,...,*et*} on *S* is defined as

$$
\bar{\mathcal{M}}(\mathcal{U}\_1, \mathcal{U}\_2) = \sum\_{i=1}^t \{ \langle \bar{\mathcal{R}}(\varepsilon\_i, \mathcal{U}\_1) \mathcal{U}\_2, \varepsilon\_i \rangle \},
\tag{15}
$$

for all *U*1, *U*2 ∈ *TxS*. The above tensor is called the Ricci tensor. If we fix a distinct vector *en* from {*<sup>e</sup>*1,...,*et*} on *S*, which is governed by *χ*, then the Ricci curvature is defined by

$$R^S(\chi) = \sum\_{\substack{p=1\\p\neq n}}^t \kappa(\mathfrak{e}\_p \wedge \mathfrak{e}\_n). \tag{16}$$

For a smooth function *g* on a Riemannian manifold *S* with Riemannian metric ,, the gradient of *g* is denoted by ∇*g* and is defined as

$$
\langle \nabla \mathcal{g}\_{"\prime} \mathcal{U}\_{1} \rangle = \mathcal{U}\_{1} \mathcal{g}\_{"\prime} \tag{17}
$$

for all *U*1 ∈ *TS*.

> Let the dimension of *S* be *t* and {*<sup>e</sup>*1,*e*2,...,*et*} be a basis of *TS*. Then as a result of (17), we ge<sup>t</sup>

$$\left\|\left\|\nabla\mathcal{g}\right\|\right\|^2 = \sum\_{i=1}^t (e\_i(\mathcal{g}))^2. \tag{18}$$

The Laplacian of *g* is defined by

$$
\Delta \mathbf{g} = \sum\_{i=1}^{t} \{ (\nabla\_{\mathbf{c}\_i} \mathbf{c}\_i) \mathbf{g} - \mathbf{c}\_i \mathbf{c}\_i \boldsymbol{\psi} \}. \tag{19}
$$

For a W-P submanifold *St*11 <sup>×</sup>*g St*22 isometrically immersed in a Riemannian manifold *S*¯, we observe the well known result, which can be described as follows [25]:

$$\sum\_{p=1}^{t\_1} \sum\_{q=1}^{t\_2} \kappa(e\_p \wedge e\_q) = \frac{t\_2 \Delta \text{g}}{\mathcal{S}} = t\_2 (\Delta \ln \text{g} - \|\nabla \ln \text{g}\|^2),\tag{20}$$

where *t*1 and *t*2 are the dimensions of the submanifolds *St*11and *St*22respectively.

## **3. Skew CR-Warped Product Submanifolds**

Recently, B. Sahin [17] demonstrated the existence of SCR W-P of the type *S* = *S*1 <sup>×</sup>*f S*⊥, where *S*1 is a semi-slant submanifold as defined by N. Papaghuic [4] and *S*⊥ is a totally real

submanifold. Throughout this section we consider the SCR W-P *S* = *S*1 <sup>×</sup>*f S*⊥ in a Kaehler manifold *S* ¯ . Then it is evident that *S* is a proper SCR W-P of order 1. Moreover, the tangent space *TS* of *S* can be decomposed as follows.

$$TS = L^{\theta} \oplus L^{T} \oplus L^{\perp},\tag{21}$$

.

where *Lθx* = *Lλ*<sup>1</sup> *x* . If *Lθ* = {0}, then *S* becomes a CR-warped product submanifold defined in [26]. If *L<sup>T</sup>* = {0}, then *S* is reduced to a warped product hemi-slant submanifold [6]. Thus, skew CR-warped product submanifold presents a single platform to study the CR W-P submanifolds and W-P hemi-slant submanifold.

Now, we have an example of SCR W-P submanifold in an A-H manifold

**Example 1.** *Let S be a submanifold in R*<sup>12</sup> *defined by x*1 = *u*, *x*2 = *v sech<sup>α</sup>*, *x*3 = *k tanhβ*, *x*4 = *k sechβ*, *x*5 = *u sechβ*, *x*6 = *u tanhβ*, *y*1 = <sup>−</sup>*v*, *y*2 = *v tanh<sup>α</sup>*, *y*3 = −*r tanhβ*, *y*4 = −*r sechβ*, *y*5 = 0, *y*6 = 0. *Then, we have the following basis of TS*

$$\begin{split} l\boldsymbol{l}\_{1} &= \operatorname{sech}\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{x}\_{5}} + \tanh\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{x}\_{6}} + \frac{\partial}{\partial\mathbf{x}\_{1}}, \quad l\boldsymbol{l}\_{2} = \operatorname{sech}\boldsymbol{\alpha}\frac{\partial}{\partial\mathbf{x}\_{2}} - \frac{\partial}{\partial\mathbf{y}\_{1}} + \tanh\boldsymbol{\alpha}\frac{\partial}{\partial\mathbf{y}\_{2}}, \\ l\boldsymbol{l}\_{3} &= \tanh\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{x}\_{3}} + \operatorname{sech}\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{x}\_{4}}, \; l\boldsymbol{l}\_{4} = -\tanh\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{y}\_{3}} - \operatorname{sech}\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{y}\_{4}}, \\ l\boldsymbol{l}\_{5} &= -\operatorname{k}\operatorname{sech}\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{x}\_{3}} + \operatorname{k}\tanh\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{x}\_{4}} + \operatorname{u\tanh\boldsymbol{\beta}}\frac{\partial}{\partial\mathbf{x}\_{5}} - \operatorname{u\sch}\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{x}\_{6}} + \operatorname{r\sch}\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{y}\_{3}} - \operatorname{r\tanh}\boldsymbol{\beta}\frac{\partial}{\partial\mathbf{y}\_{4}}. \end{split}$$

*It is straightforward to identify that Lθ* = *span*{*<sup>U</sup>*1, *<sup>U</sup>*2} *is a slant distribution with slant angle* 60◦, *L* = *span*{*<sup>U</sup>*3, *<sup>U</sup>*4} *is a holomorphic distribution and JU*5 *is orthogonal to S*. *Thus L*<sup>⊥</sup> = *span*{*<sup>U</sup>*5} *is a totally real distribution. Moreover, it is easy to observe that <sup>L</sup>θ, L and L*<sup>⊥</sup> *are integrable. If Sθ, ST and S*⊥ *are the integral manifolds of the distributions <sup>L</sup>θ, L and L*<sup>⊥</sup> *respectively. Then the induced metric tensor of S is given by*

$$ds^2 = \langle \, \rangle\_{\mathcal{S}\_\theta} + \langle \, \rangle\_{\mathcal{S}\_T} + (k^2 + \mu^2 + r^2) \langle \, \rangle\_{\mathcal{S}\_\perp}$$

*or*

$$ds^2 = \langle \swarrow \rangle\_{S\_1} + (k^2 + \mathfrak{u}^2 + r^2)\langle \swarrow \rangle\_{S\_\perp}.$$

**Definition 2.** *The warped product S*1 <sup>×</sup>*f S*2 *isometrically immersed in a Riemannian manifold S* ¯ *is called Si totally geodesic if the partial second fundamental form* Γ*i is zero identically. It is called Si-minimal if the partial mean curvature vector* Π*<sup>i</sup> becomes zero for i* = 1, 2*.*

Let {*<sup>e</sup>*1, ... ,*ep*,*ep*+<sup>1</sup> = *Je*1, ... ,*et*1=2*<sup>p</sup>* = *Jep*,*<sup>e</sup>*1, ... ,*eq*,*eq*+<sup>1</sup> = sec *θPe*1, ... ,*<sup>e</sup>*(*<sup>t</sup>*2<sup>=</sup>2*q*) = sec *<sup>θ</sup>Peq*,*et*2<sup>+</sup>1, ... ,*et*<sup>3</sup> } be a local orthonormal frame of vector fields such that {*<sup>e</sup>*1, ... ,*ep*,*ep*+<sup>1</sup> = *Je*1, ... ,*et*1=2*<sup>p</sup>* = *Jep*} is an orthonormal basis of *L*, {*e*1, ... ,*eq*,*eq*+<sup>1</sup> = sec *θPe*1, ... ,*<sup>e</sup>*(*<sup>t</sup>*2<sup>=</sup>2*q*) = sec *θPeq*} is an orthonormal basis of *Lθ*and {*et*2<sup>+</sup>1,...,*et*<sup>3</sup> } is an orthonormal basis of *L*⊥.

Throughout this paper we consider that the SCR W-P submanifold *S*1 <sup>×</sup>*f S*⊥ is *L*−minimal. Presently we have the following outcome for further applications

**Lemma 1.** *Let St* = *S<sup>t</sup>*1+*t*2 1 <sup>×</sup>*f St*3⊥ *be a L*−*minimal SCR W-P submanifold isometrically immersed in a Kaehler manifold; then*

$$\left\|\left\|\Pi\right\|\right\|^2 = \frac{1}{t^2} \sum\_{r=t+1}^{m} \left(\Gamma\_{t\_1+1t\_1+1}^{r} + \dots + \Gamma\_{t\_2t\_2}^{r} + \dots + \Gamma\_{tt}^{r}\right)^2,\tag{22}$$

*where* -Π-2 *represents squared mean curvature.*

#### **4. Ricci Curvature for Skew CR-Warped Product Submanifold**

In this section, we investigate Ricci curvature in terms of the squared norm of mean curvature and the warping functions as follows:

**Theorem 3.** *Let St* = *S<sup>t</sup>*1+*t*2 1 <sup>×</sup>*f Ss*3⊥ *be a L*−*minimal SCR W-P submanifold isometrically immersed in a Complex space form S* ¯ *<sup>m</sup>*(*c*)*. If the holomorphic and slant distributions L and Lθ are integrable with integral submanifolds St*1*T and St*2*θ respectively, then for each orthogonal unit vector field χ* ∈ *TxS, the tangent to St*1*T , St*2*θ or <sup>S</sup><sup>t</sup>*3⊥*, we have that*

	- *(i) If χ* ∈ *TS<sup>t</sup>*1*T, then*

$$\frac{1}{4}t^2||\Pi||^2 \ge R^S(\chi) + \frac{t\_3 \Delta f}{f} + \frac{c}{4}(t - t\_1 t\_2 - t\_2 t\_3 - t\_1 t\_3 - \frac{1}{2}).\tag{23}$$

*(ii) χ* ∈ *TS<sup>t</sup>*2*θ, then*

$$\frac{1}{4}t^2||\Pi||^2 \ge R^S(\chi) + \frac{t\_3\Delta f}{f} + \frac{c}{4}(t - t\_1t\_2 - t\_2t\_3 - t\_1t\_3 + 1 - \frac{3}{2}\cos^2\theta). \tag{24}$$

*(iii) If χ* ∈ *TS<sup>t</sup>*2⊥*, then*

$$\frac{1}{4}t^2||\Pi||^2 \ge R^S(\chi) + \frac{t\_3 \Delta f}{f} + \frac{c}{4}(t - t\_1 t\_2 - t\_2 t\_3 - t\_1 t\_3 + 1). \tag{25}$$

	- *(a) The equality of (23) holds identically for all unit vector fields tangential to St*1*T at each x* ∈ *St iff St is mixed TG and <sup>L</sup>*−*totally geodesic SCR W-P submanifold in S* ¯ *<sup>m</sup>*(*c*)*.*
	- *(b) The equality of (24) holds identically for all unit vector fields tangential to Sθ at each x* ∈ *St iff S is mixed totally geodesic and either St is Lθ- totally geodesic SCR W-P submanifold or St is a Lθ totally umbilical in S* ¯ *<sup>m</sup>*(*c*) *with dim Lθ* = 2*.*
	- *(c) The equality of (25) holds identically for all unit vector fields tangential to St*2⊥ *at each x* ∈ *St iff S is mixed totally geodesic and either St is L*⊥*- totally geodesic SCR W-P or St is a L*<sup>⊥</sup> *totally umbilical in S* ¯ *<sup>m</sup>*(*c*) *with dim L*<sup>⊥</sup> = 2*.*
	- *(d) The equality case of (1) holds identically for all unit tangent vectors to St at each x* ∈ *St iff either St is totally geodesic submanifold or M<sup>t</sup> is a mixed totally geodesic totally umbilical and L totally geodesic submanifold with dim St*2*θ* = 2 *and dim St*3⊥ = 2.

*where t*1, *t*2 *and t*3 *are the dimensions of St*1*T*, *St*2*θand St*3⊥*respectively.*

**Proof.** Suppose that *St* = *S<sup>t</sup>*1+*t*2 1 <sup>×</sup>*f St*3⊥ be a SCR W-P submanifold of a CSF. From Gauss equation, we have

$$\|t^2\|\|\Pi\|\|^2 = 2\pi(S^t) + \|\Gamma\|^2 - 2\pi(S^t). \tag{26}$$

Let {*<sup>e</sup>*1, ... ,*et*1 ,*et*1+1, ... ,*et*2 , ...*et*} be a local orthonormal frame of vector fields on *St* such that {*<sup>e</sup>*1, ... ,*et*1 } is tangential to *St*1*T* , {*et*1+1, ... ,*et*2 } is tangential to *St*2*θ* and {*et*2+1, ... ,*et*} is the tangent to *<sup>S</sup><sup>t</sup>*3⊥. Thus, the unit tangent vector *χ* = *eA* ∈ {*<sup>e</sup>*1,...,*et*} can be expanded (26) as follows.

$$\begin{array}{c} \|\mathbf{t}^2\|\|\Gamma\|\|^2 = 2\tau(S^t) + \frac{1}{2} \sum\_{r=t+1}^m \left\{ (\Gamma\_{11}^r + \dots \,\Gamma\_{t2t}^r + \dots + \Gamma\_{tt}^r - \Gamma\_{AA}^r)^2 + (\Gamma\_{AA}^r)^2 \right\} \\ \qquad - \sum\_{r=t+1}^m \sum\_{1 \le i \ne j \le t} \Gamma\_{ii}^r \Gamma\_{jj}^r - 2\tau(S^t). \end{array} \tag{27}$$

The above expression can be represented as

$$\begin{split} \|t^2\|\|\Gamma\|\|^2 &= 2\pi (\mathcal{S}^t) + \frac{1}{2} \sum\_{r=t+1}^m \left\{ (\Gamma\_{11}^r + \dots \,\Gamma\_{t\_2 t\_2}^r + \dots + \Gamma\_{tt}^r)^2 \right. \\ &\left. + \left( 2\Gamma\_{AA}^r - (\Gamma\_{11}^r + \dots + \Gamma\_{tt}^r) \right)^2 \right\} + 2 \sum\_{r=t+1}^m \sum\_{1 \le i < j \le t} (\Gamma\_{ij}^r)^2 \\ &- 2 \sum\_{r=t+1}^m \sum\_{1 \le i < j \le t} \Gamma\_{ii}^r \Gamma\_{jj}^r - 2\pi (\mathcal{S}^t) . \end{split}$$

In view of the assumption that SCR W-P submanifold *S*1 <sup>×</sup>*f S*⊥ is *L*−minimal submanifold, the preceding expression takes the form

$$\begin{split} t^2 \|\Pi\|^2 &= 2\pi (S^t) + \frac{1}{2} \sum\_{r=t+1}^m \left\{ (\Gamma\_{t\_1+1t\_1+1}^r + \dots \Gamma\_{t\_2t\_2}^r + \dots + \Gamma\_{tt}^r)^2 \right. \\ &\left. + \frac{1}{2} \sum\_{r=t+1}^m \left( 2\Gamma\_{AA}^r - (\Gamma\_{t\_1+1t\_1+1}^r + \dots \Gamma\_{t\_2t\_2}^r + \dots + \Gamma\_{tt}^r) \right)^2 \right. \\ &\left. + \sum\_{r=t+1}^m \sum\_{1 \le i < j \le t} (\Gamma\_{ij}^r)^2 - \sum\_{r=t+1}^m \sum\_{\substack{1 \le i < j \le t \\ i, j \ne A}} \Gamma\_{ii}^r \Gamma\_{jj}^r - 2\pi (S^t) \right. \\ &\left. + \sum\_{r=t+1}^m \sum\_{\substack{a=1 \\ a \ne A}}^t (\Gamma\_{aA}^r)^2 + \sum\_{r=t+1}^m \sum\_{\substack{1 \le i < j \le t \\ i, j \ne A}} (\Gamma\_{ij}^r)^2 - \sum\_{r=t+1}^m \sum\_{\substack{1 \le i < j \le t \\ i, j \ne A}} \Gamma\_{ii}^r \Gamma\_{jj}^t. \right. \end{split} \tag{28}$$

Equation (14) can be written as

$$\sum\_{\substack{1 \le p < q \le t \\ p, q \ne A}} \mathbb{K}\_{pq} - \sum\_{\substack{1 \le p < q \le t \\ p, q \ne A}} \kappa\_{pq} = \sum\_{r=t+1}^{m} \sum\_{\substack{1 \le p < q \le t \\ p, q \ne A}} (\Gamma\_r^{pq})^2 - \sum\_{r=t+1}^{m} \sum\_{\substack{1 \le p < q \le t \\ p, q \ne A}} \Gamma\_{pp}^r \Gamma\_{qq}^r$$

Substituting this value in (28), we derive

$$\begin{split} 2^2 \|\Pi\|^2 &= 2\pi (S^t) + \frac{1}{2} \sum\_{r=t+1}^m \left\{ (\Gamma\_{t\_1+1t\_1+1}^r + \dots \,\Gamma\_{t\_2t\_2}^r + \dots + \Gamma\_{tt}^r)^2 \right. \\ &\left. + \frac{1}{2} \sum\_{r=t+1}^m \left( 2\Gamma\_{AA}^r - (\Gamma\_{t\_1+1t\_1+1}^r + \dots \,\Gamma\_{t\_2t\_2}^r + \dots + \Gamma\_{tt}^r) \right)^2 \right. \\ &\left. + \sum\_{r=t+1}^m \sum\_{1 \le i < j \le t} (\Gamma\_{ij}^r)^2 - \sum\_{r=t+1}^m \sum\_{\substack{1 \le i < j \le t \\ i, j \ne A}} \Gamma\_{ii}^r \Gamma\_{jj}^r - 2\pi (S^t) \right. \\ &\left. + \sum\_{r=t+1}^m \sum\_{\substack{a=1 \\ a \ne A}}^t (\Gamma\_{aA}^r)^2 + \sum\_{\substack{1 \le i < j \le t \\ i, j \ne A}} \kappa\_{ij} - \sum\_{1 \le i < j \le t} \kappa\_{ij} \right. \end{split} \tag{29}$$

On the other hand, from (9) we have

$$\begin{split} \pi(\mathcal{S}^{\dagger}) &= \sum\_{1 \le i < j \le t} \kappa(\varepsilon\_{i} \wedge \varepsilon\_{j}) = \sum\_{a=1}^{t\_{1}+t\_{2}} \sum\_{\substack{\beta = t\_{1}+t\_{2}+1 \\ t\_{1}+1 \le l < 0 \le t\_{2}}}^{t} \kappa(\varepsilon\_{a} \wedge \varepsilon\_{\beta}) + \sum\_{1 \le a < \gamma \le t\_{1}} \kappa(\varepsilon\_{a} \wedge \varepsilon\_{\gamma}) \\ &+ \sum\_{t\_{1}+1 \le l < 0 \le t\_{2}} \kappa(\varepsilon\_{l} \wedge \varepsilon\_{0}) + \sum\_{t\_{2}+1 \le u < \gamma \le t} \kappa(\varepsilon\_{u} \wedge \varepsilon\_{\overline{v}}). \end{split} \tag{30}$$

Using (9) and (20), we derive

$$\tau(S^t) = \frac{t\_3 \Delta f}{f} + \tau(S^{t\_1}\_T) + \tau(S^{t\_2}\_\theta) + \tau(S^{t\_3}\_\perp).$$

Using this in (29), we ge<sup>t</sup>

$$\begin{split} 2\|\Pi\|^2 &= \frac{t\_3\Delta f}{f} + \frac{1}{2} \sum\_{r=t+1}^m (\Upsilon'\_{t\_1+1t\_1+1} + \dots \Upsilon'\_{t\_2t\_2} + \dots + \Gamma'\_{tt})^2 \\ &+ \frac{1}{2} \sum\_{r=t+1}^m (2\Gamma'\_{AA} - (\Gamma'\_{t\_1+1t\_1+1} + \dots \Gamma'\_{t\_2t\_2} + \dots + \Gamma'\_{tt}))^2 \\ &+ \sum\_{r=t+1}^m \sum\_{1 \le a < \delta\_1 \le t\_1} (\Gamma'\_{aa}\Gamma'\_{\beta\delta} - (\Gamma'\_{a\beta})^2) \\ &+ \sum\_{r=t+1}^m \sum\_{\substack{1 \le a < \delta\_1 \le t \le n \\ r \ne t+1 \; i\_1 \le j \le n}} (\Gamma'\_{x\delta}\Gamma'\_{\delta\theta} - (\Gamma'\_{pq})^2) \\ &+ \sum\_{r=t+1 \; i\_2 \le r \le t}^m (\Gamma'\_{x\delta}\Gamma'\_{\delta\theta} - (\Gamma'\_{\delta\theta})^2) \\ &+ \sum\_{r=t+1 \; 1 \le i < j \le t}^m (\Gamma'\_{ij})^2 - \sum\_{r=t+1 \; 1 \le i < j \le t}^m (\Gamma'\_{ii}\Gamma'\_{jj}) \\ &- 2\tau(S^t) + \sum\_{\substack{1 \le i < j \le t \\ i, j \ne A}} \pounds\_{ij} + \tau(S^{t\_1}\_T) + \tau(S^{t\_2}\_\theta) + \tau(S^{t\_3}\_\bot) . \end{split} \tag{31}$$

Considering unit tangent vector *χ* = *eA*, we have three choices: *χ* is the tangent to the base manifold *St*1*T* or *St*2*θ* , or to the fiber *St*3⊥ .

**Case 1:** If *χ* ∈ *St*1*T* , then we need to choose a unit vector field from {*<sup>e</sup>*1, ... ,*et*1 }. Let *χ* = *e*1; then by (15) and the assumption that the submanifolds is *L*−minimal, we have

*t*2-Π-2 <sup>≥</sup>*R<sup>S</sup>*(*χ*) + 12 *m*∑ *<sup>r</sup>*=*t*+1 (Γ*rt*1+1*t*1+<sup>1</sup> + ... <sup>Γ</sup>*rt*2*t*2 + ··· + Γ*rtt*)<sup>2</sup> + *<sup>t</sup>*3<sup>Δ</sup>*f f* + 1 2 *m* ∑ *<sup>r</sup>*=*t*+1 (2Γ*<sup>r</sup>*11 − (Γ*rt*1+1*t*1+<sup>1</sup> + ... <sup>Γ</sup>*rt*2*t*2 + ··· + Γ*rtt*))<sup>2</sup> + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ 1≤*α*<sup>&</sup>lt;*β*≤*t*1 (Γ*<sup>r</sup>αα*Γ*<sup>r</sup>ββ* − (Γ*<sup>r</sup>αβ*)<sup>2</sup>) + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ *<sup>t</sup>*1+1≤*p*<*q*≤*t*2 (Γ*rpp*Γ*rqq* − (Γ*rpq*)<sup>2</sup>) + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ *t*2+1≤*s*<*n*≤*t* (Γ*rss*Γ*rnn* − (Γ*rsn*)<sup>2</sup>) + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ 1≤*i*<*j*≤*<sup>t</sup>* (Γ*rij*)<sup>2</sup> − *m* ∑ *<sup>r</sup>*=*t*+1 ∑ 2≤*i*<*j*≤*<sup>t</sup>* (Γ*rii*Γ*rjj*) − <sup>2</sup>*τ*¯(*S<sup>t</sup>*) + ∑ 2≤*i*<*j*≤*<sup>t</sup> κ* ¯(*ei*,*ej*) + *<sup>τ</sup>*¯(*S<sup>t</sup>*1*T* ) + *<sup>τ</sup>*¯(*S<sup>t</sup>*2*θ* ) + *<sup>τ</sup>*¯(*S<sup>t</sup>*3⊥). (32)

Putting *U*1, *U*3 = *ei*, *U*2, *U*4 = *ej* in the formula (3), we have

$$2\overline{\tau}(S) = \frac{c}{4}[t(t-1) + 3t\_1 + 3t\_2\cos^2\theta] \tag{33}$$

$$\sum\_{2 \le i < j \le t} \overline{\kappa}(e\_i, e\_j) = \frac{c}{8}[(t-1)(t-2) + 3(t\_1 - 1) + 3t\_2\cos^2\theta]$$

$$\overline{\tau}(S\_T^{t\_1}) = \frac{c}{8}[t\_1(t\_1 - 1) + 3t\_1]$$

$$\overline{\tau}(S\_\theta^{t\_2}) = \frac{c}{8}[t\_2(t\_2 - 1) + 3t\_2\cos^2\theta]$$

$$\overline{\tau}(S\_\perp^{t\_3}) = \frac{c}{8}[t\_3(t\_3 - 1)].$$

Using these values in (32), we ge<sup>t</sup>

$$\begin{split} 2^{l} \|\Pi\|^{2} &\geq \mathcal{R}^{S}(\chi) + \frac{1}{2}t^{2} \|\Pi\|^{2} + \frac{1}{2} \sum\_{r=t+1}^{m} \left( 2\Gamma\_{11}' - (\Gamma\_{t\_{1}+1t\_{1}+1}^{r} + \dots + \Gamma\_{tt}') \right)^{2} \\ &+ \frac{t\_{3}\Delta f}{f} + \sum\_{r=t+1}^{m} \sum\_{i=1}^{t\_{1}} \sum\_{j=t\_{1}+1}^{t\_{2}} (\Gamma\_{ij}^{r})^{2} \\ &+ \sum\_{r=t+1}^{m} \sum\_{i=1}^{t\_{1}} \sum\_{k=t\_{2}+1}^{t} (\Gamma\_{ik}^{r})^{2} + \sum\_{r=t+1}^{m} \sum\_{j=2}^{t\_{1}} \Gamma\_{11}^{r} \Gamma\_{j\beta}^{r} \\ &- \sum\_{r=t+1}^{m} \sum\_{i=2}^{t\_{1}} \sum\_{j=t\_{1}+1}^{t\_{2}} \Gamma\_{i1}^{r} \Gamma\_{j\overline{i}}^{r} - \sum\_{r=t+1}^{m} \sum\_{i=2}^{t\_{1}} \sum\_{k=t\_{2}+1}^{t\_{1}} \Gamma\_{i\overline{i}}^{r} \Gamma\_{k\overline{k}}^{r} \\ &+ \frac{c}{4} (t - t\_{1}t\_{2} - t\_{2}t\_{3} - t\_{3}t\_{1} - \frac{1}{2}). \end{split} \tag{34}$$

In view of the assumption that the submanifold is *L*−minimal, then

$$\sum\_{r=t+1}^{m} \sum\_{\beta=2}^{t\_1} \Gamma\_{11}^{r} \Gamma\_{\beta\beta}^{r} = \sum\_{r=t+1}^{m} \left(\Gamma\_{11}^{r}\right)^{2}$$

$$= \sum\_{r=t+1}^{m} \sum\_{i=2}^{t\_1} \left[\sum\_{j=t\_1+1}^{t\_2} \Gamma\_{ii}^{r} \Gamma\_{jj}^{r} + \sum\_{k=t\_2+1}^{t} \Gamma\_{ii}^{r} \Gamma\_{kk}^{r}\right] = \sum\_{r=t+1}^{m} \sum\_{j=t\_1+1}^{t} \Gamma\_{11}^{r} \Gamma\_{jj}^{r} \Gamma\_{jj}^{r}$$

Utilizing that in (34), we have

$$\begin{split} t^2 \|\Pi\|^2 &\ge \mathcal{R}^S(\chi) + \frac{1}{2}t^2 \|\Pi\|^2 + \frac{1}{2} \sum\_{r=t+1}^m \left( 2\Gamma\_{11}^r - (\Gamma\_{t\_1+1t\_1+1}^r + \dots + \Gamma\_{mn}^r) \right)^2 \\ &+ \frac{t\_3 \Delta f}{f} + \sum\_{r=t+1}^m \sum\_{i=1}^{t\_1} \sum\_{j=t\_1+1}^{t\_2} (\Gamma\_{ij}^r)^2 \\ &+ \sum\_{r=t+1}^m \sum\_{i=1}^{t\_1} \sum\_{k=t\_2+1}^t (\Gamma\_{ik}^r)^2 - \sum\_{r=t+1}^m (\Gamma\_{11}^r)^2 + \sum\_{i=1}^{t\_1} \sum\_{j=t\_1+1}^t \Gamma\_{ii}^r \Gamma\_{jj}^r \\ &+ \frac{c}{4}(t - t\_1t\_2 - t\_2t\_3 - t\_3t\_1 - \frac{1}{2}). \end{split} \tag{35}$$

The third term on the right hand side can be written as

$$\begin{split} \frac{1}{2} \sum\_{r=t+1}^{m} \left( 2\Gamma\_{11}^{r} - (\Gamma\_{t\_{1}+1t\_{1}+1}^{r} + \dots + \Gamma\_{t\_{2}t\_{2}}^{r} + \dots + \Gamma\_{mn}^{r}) \right)^{2} \\ = 2 \sum\_{r=t+1}^{m} (\Gamma\_{11}^{r})^{2} + \frac{1}{2}t^{2}||\Pi||^{2} - 2\sum\_{r=t+1}^{m} \left[ \sum\_{j=t\_{1}+1}^{t\_{2}} \Gamma\_{11}^{r} \Gamma\_{jj}^{r} \\ + \sum\_{k=t\_{2}+1}^{t} \Gamma\_{11}^{r} \Gamma\_{kk}^{r} \right]. \end{split} \tag{36}$$

Combining above two expressions, we have

$$\begin{aligned} \frac{1}{2}t^2||\Pi||^2 &\geq \mathcal{R}^S(\chi) + \sum\_{r=t+1}^m (\Gamma\_{11}')^2 - \sum\_{r=t+1}^m \sum\_{j=t\_1+1}^t \Gamma\_{11}' \Gamma\_{jj}' \\ &+ \frac{1}{2} \sum\_{r=t+1}^m (\Gamma\_{t\_1+1t\_1+1}^{\prime} + \dots + \Gamma\_{t\_2t\_2}^{\prime} + \dots + \Gamma\_{mn}^{\prime})^2 \\ &+ \sum\_{r=t+1}^m \sum\_{i=1}^{t\_1} \sum\_{j=t\_1+1}^t (\Gamma\_{ij}^{r})^2 + \frac{t\_3 \Delta f}{f} \\ &+ \frac{c}{4} (t - t\_1 t\_2 - t\_2 t\_3 - t\_3 t\_1 - \frac{1}{2}), \end{aligned} \tag{37}$$

or equivalently

$$\begin{split} \frac{1}{4}t^2||\Pi||^2 &\geq \mathcal{R}^S(\chi) + \frac{1}{4} \sum\_{r=t+1}^m (2\Gamma\_{11}^r - (\Gamma\_{t\_1+1t\_1+1}^r + \dots + \Gamma\_{t\_2t\_2}^r + \dots + \Gamma\_{mn}^r))^2 \\ &+ \sum\_{r=t+1}^m \sum\_{i=1}^{t\_1} \sum\_{j=t\_1+1}^t (\Gamma\_{ij}^r)^2 + \frac{t\_3\Delta f}{f} \\ &+ \frac{c}{4}(t - t\_1t\_2 - t\_2t\_3 - t\_3t\_1 - \frac{1}{2}), \end{split} \tag{38}$$

which proves the inequality (*i*) of (1).

**Case 2.** If *χ* is tangential to *St*2*θ* , we choose the unit vector from {*et*1+1, ... ,*et*2 }. Suppose *χ* = *et*2 ; then from (28), we deduce

*t*2-Π-2 <sup>≥</sup>*R<sup>S</sup>*(*χ*) + 12 *m*∑ *<sup>r</sup>*=*t*+1 (Γ*rt*1+1*t*1+<sup>1</sup> + ... <sup>Γ</sup>*rt*2*t*2 + ··· + Γ*rtt*)<sup>2</sup> + *<sup>t</sup>*3<sup>Δ</sup>*f f* + 1 2 *m* ∑ *<sup>r</sup>*=*t*+1 ((Γ*rt*1+1*t*1+<sup>1</sup> + ... <sup>Γ</sup>*rt*2*t*2 + ··· + Γ*rtt*) − <sup>2</sup><sup>Γ</sup>*rt*2*t*2 )2 + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ 1≤*α*<sup>&</sup>lt;*β*≤*t*1 (Γ*<sup>r</sup>αα*Γ*<sup>r</sup>ββ* − (Γ*<sup>r</sup>αβ*)<sup>2</sup>) + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ *t*1+1≤*s*<*n*≤*t*2 (Γ*rss*Γ*rnn* − (Γ*rsn*)<sup>2</sup>) + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ *<sup>t</sup>*2+1≤*p*<*q*≤*<sup>t</sup>* (Γ*rpp*Γ*rqq* − (Γ*rpq*)<sup>2</sup>) + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ 1≤*i*<*j*≤*<sup>t</sup>* (Γ*rij*)<sup>2</sup> − *m* ∑ *<sup>r</sup>*=*t*+1 ∑ 1≤*i*<*j*≤*n i*,*j* <sup>=</sup>*t*<sup>2</sup> (Γ*rii*Γ*rjj*) − <sup>2</sup>*τ*¯(*S<sup>t</sup>*) + ∑ 1≤*i*<*j*≤*<sup>t</sup> i*,*j* <sup>=</sup>*t*<sup>2</sup> *κ* ¯(*ei*,*ej*) + *<sup>τ</sup>*¯(*S<sup>t</sup>*1*T* ) + *<sup>τ</sup>*¯(*S<sup>t</sup>*2*θ* + *<sup>τ</sup>*¯(*S<sup>t</sup>*3⊥)). (39)

From (3) by putting *U*1, *U*3 = *ei*, *U*2, *U*3 = *ej*, one can compute

$$\sum\_{\substack{1 \le i < j \le t \\ i, j \ne \ell\_2}} \mathbb{F}(e\_i, e\_j) = \frac{c}{8} [(t - 1)(t - 2) + 3t\_1 + 3t\_2 \cos^2 \theta]$$

$$\mathbb{F}(S\_T^{t\_1}) = \frac{c}{8} [t\_1(t\_1 - 1) + 3t\_1]$$

$$\mathbb{F}(S\_\theta^{t\_3}) = \frac{c}{8} [t\_2(t\_2 - 1) + 3t\_2 \cos^2 \theta]$$

$$\mathbb{F}(S\_\perp^{t\_3}) = \frac{c}{8} [t\_3(t\_3 - 1)].$$

 Using these values together with (33) in (39) and applying similar techniques as in Case 1, we obtain

$$\begin{split} \ell^{2} \|\Pi\|^{2} &\geq \mathcal{R}^{S}(\chi) + \frac{1}{2} \sum\_{r=t+1}^{m} ((\Gamma\_{t\_{1}+1t\_{1}+1}^{r} + \dots \Gamma\_{t\_{2t}}^{r}) + \dots + \Gamma\_{tt}^{r}) - 2\Gamma\_{tt2}^{r}))^{2} \\ &+ \frac{1}{2}t^{2} \|\Pi\|^{2} + \frac{t\_{3}\Delta f}{f} + \sum\_{r=t+1}^{m} \sum\_{1 \leq i < j \leq n} (\Gamma\_{ij}^{r})^{2} \\ &+ \sum\_{r=t+1}^{m} \left[ \sum\_{m=t\_{1}+1}^{t\_{2}-1} \Gamma\_{t\_{2}t\_{2}}^{r} \Gamma\_{mn}^{r} + \sum\_{l=t\_{2}+1}^{t} \Gamma\_{t\_{2}t\_{2}}^{r} \Gamma\_{ll}^{r} \right] \\ &\sum\_{r=1}^{m} \sum\_{i=1}^{t\_{1}} \left[ \sum\_{j=t\_{1}+1}^{t\_{2}-1} \Gamma\_{i1}^{r} \Gamma\_{jt}^{r} + \sum\_{k=t\_{2}+1}^{t} \Gamma\_{i1}^{r} \Gamma\_{k1}^{r} \right] \\ &+ \frac{c}{4}(t - t\_{1}t\_{2} - t\_{2}t\_{3} - t\_{3}t\_{1} + 1). \end{split} \tag{40}$$

By the assumption that the submanifold *St* is *L*−minimal, one can conclude

$$\sum\_{r=1}^{m} \sum\_{i=1}^{t\_1} \left[ \sum\_{j=t\_1+1}^{t\_2-1} \Gamma^r\_{ii} \Gamma^r\_{jj} + \sum\_{k=t\_2+1}^{t} \Gamma^r\_{ii} \Gamma^r\_{kk} \right] = 0.$$

The second and seventh terms on right hand side of (40) can be solved as follows:

$$\begin{split} \frac{1}{2} \sum\_{r=t+1}^{m} ((\Gamma\_{t+1t\_1+1}^{r} + \dots + \Gamma\_{tt}^{r}) - 2\Gamma\_{t2t\_2}^{r})^2 &+ \sum\_{r=t+1}^{m} \left[ \sum\_{n=t\_1+1}^{t\_2} \Gamma\_{t2t\_2}^{r} \Gamma\_{mt}^{r} + \sum\_{l=t\_2+1}^{t} \Gamma\_{t2t\_2}^{r} \Gamma\_{l1}^{r} \right] \\ &= \frac{1}{2} \sum\_{r=t+1}^{m} (\Gamma\_{t+1t\_1+1}^{r} + \dots + \Gamma\_{mt}^{r})^2 + 2 \sum\_{r=t+1}^{m} (\Gamma\_{t2t\_2}^{r})^2 \\ &- 2 \sum\_{r=t+1}^{m} \sum\_{j=t\_1+1}^{t} \Gamma\_{t2t\_2}^{r} \Gamma\_{jt}^{r} + \sum\_{r=t+1}^{m} \sum\_{n=t\_1+1}^{t} \Gamma\_{t2t\_2}^{r} \Gamma\_{mn}^{r} - \sum\_{r=t+1}^{m} (\Gamma\_{t2t\_2}^{r})^2 \\ &= \frac{1}{2} \sum\_{r=t+1}^{m} (\Gamma\_{t\_1+1t\_1+1}^{r} + \dots + \Gamma\_{mt}^{r})^2 + \sum\_{r=t+1}^{m} (\Gamma\_{t\_2t\_2}^{r})^2 \\ &- \sum\_{r=t+1}^{m} \sum\_{j=t\_1+1}^{t} \Gamma\_{mt}^{r} \Gamma\_{jt}^{r} .\end{split} \tag{41}$$

By utilizing those two values in (40), we arrive at

$$\begin{split} \frac{1}{2}t^{2}||\Pi||^{2} &\geq \mathsf{R}^{S}(\mathsf{x}) + \sum\_{r=t+1}^{m} (\Gamma\_{t2t}^{r})^{2} - \sum\_{r=t+1}^{m} \sum\_{i=t\_{1}+1}^{t} \Gamma\_{mt}^{r} \Gamma\_{\overline{j}j}^{r} \\ &+ \frac{1}{2} \sum\_{r=t+1}^{m} (\Gamma\_{t\_{1}+1t\_{1}+1}^{r} + \dots + \Gamma\_{n}^{r}n)^{2} + \frac{1}{2}t^{2}||\Pi||^{2} + \frac{t\_{3}\Delta f}{f} \\ &+ \sum\_{r=t+1}^{m} \sum\_{i=1}^{t\_{1}} \sum\_{j=t\_{1}+1}^{t} (\Gamma\_{ij}^{r})^{2} + \frac{c}{4}(t - t\_{1}t\_{2} - t\_{2}t\_{3} - t\_{3}t\_{1} + 1). \end{split} \tag{42}$$

By using similar steps as in Case 1, the above inequality can be written as

$$\begin{split} \frac{1}{4}t^2||\Pi||^2 &\geq R^S(\chi) + \frac{1}{4} \sum\_{r=t+1}^m (2\Gamma\_{t\_2t\_2}^r - (\Gamma\_{t\_1+1t\_1+1}^r + \dots + \Gamma\_{mn}^r))^2 \\ &+ \frac{t\_3\Delta f}{f} + \frac{c}{4}(t - t\_1t\_2 - t\_2t\_3 - t\_1t\_3 + 1). \end{split} \tag{43}$$

The last inequality leads to inequality (*ii*) of (1).

**Case 3.** If *χ* is tangential to *<sup>S</sup><sup>t</sup>*3⊥, then we choose the unit vector field from {*et*2+1, ... ,*en*}. Suppose the vector *χ* is *en*. Then from (28)

*t*2-Π-2 <sup>≥</sup>*R<sup>S</sup>*(*χ*) + 12 *m*∑ *<sup>r</sup>*=*t*+1 (Γ*rt*1+1*t*1+<sup>1</sup> + ... <sup>Γ</sup>*rt*2*t*2 + ··· + Γ*rtt*)<sup>2</sup> + *<sup>t</sup>*3<sup>Δ</sup>*f f* + 1 2 *m* ∑ *<sup>r</sup>*=*t*+1 ((Γ*rt*1+1*t*1+<sup>1</sup> + ... <sup>Γ</sup>*rt*2*t*2 + ··· + Γ*rtt*) − 2Γ*rtt*)<sup>2</sup> + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ 1≤*α*<sup>&</sup>lt;*β*≤*t*1 (Γ*<sup>r</sup>αα*Γ*<sup>r</sup>ββ* − (Γ*<sup>r</sup>αβ*)<sup>2</sup>) + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ *t*1+1≤*s*<*n*≤*t*2 (Γ*rss*Γ*rnn* − (Γ*rsn*)<sup>2</sup>) + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ *<sup>t</sup>*2+1≤*p*<*q*≤*<sup>t</sup>* (Γ*rpp*Γ*rqq* − (Γ*rpq*)<sup>2</sup>) + *m* ∑ *<sup>r</sup>*=*t*+1 ∑ 1≤*i*<*j*≤*n* (Γ*rij*)<sup>2</sup> − *m* ∑ *<sup>r</sup>*=*t*+1 ∑ 1≤*i*<*j*≤*t*−1 Γ*rii*Γ*rjj* − <sup>2</sup>*τ*¯(*S<sup>t</sup>*) + ∑ 1≤*i*<*j*≤*t*−1 *κ* ¯(*ei*,*ej*) + *<sup>τ</sup>*¯(*S<sup>t</sup>*1*T* ) + *<sup>τ</sup>*¯(*S<sup>t</sup>*2*θ* ) + *<sup>τ</sup>*¯(*S<sup>t</sup>*3⊥). (44)

From (3), one can compute

$$\sum\_{1 \le i < j \le t-1} \overline{\pi}(\underline{e}\_i, \underline{e}\_j) = \frac{c}{8} [(t-1)(t-2) + 3t\_1 + 3(t\_2 - 1)\cos^2\theta],$$

$$\overline{\pi}(S\_T^{t\_1}) = \frac{c}{8} [t\_1(t\_1 - 1) + 3t\_1]$$

$$\overline{\pi}(S\_\theta^{t\_2}) = \frac{c}{8} [t\_2(t\_2 - 1) + 3t\_2\cos^2\theta]$$

$$\overline{\pi}(t\_\perp^{t\_3}) = \frac{c}{8} [t\_3(t\_3 - 1)].$$

By usage of those values together with (33) in (44), and analogously to Case 1 and Case 2, we obtain

$$\begin{split} \|\mathbf{r}^2\|\Pi\|^2 &\geq \mathbb{R}^5(\chi) + \frac{1}{2}t^2\|\Pi\|^2 + \frac{1}{2}\sum\_{r=t+1}^m \left( (\Gamma\_{t\_1+1t\_1+1}^{\prime} + \dots \,\Gamma\_{t\_2t\_2}^{\prime} + \dots + \Gamma\_{t\_2t\_2}^{\prime} + \dots + \Gamma\_{tt}^{\prime}) - 2\Gamma\_{tt}^{\prime}\right)^2 \\ &+ \frac{t\_3\Delta f}{f} + \sum\_{r=t+1}^m \sum\_{1\leq i$$

Again, using the assumption that *St* is *L* − *minimal*, it is easy to verify

$$\sum\_{r=t+1}^{m} \sum\_{i=1}^{t\_1} \sum\_{j=t\_1+1}^{t-1} \Gamma\_{ii}^r \Gamma\_{jj}^r = 0. \tag{46}$$

Using in (45), we obtain

$$\begin{split} \|\mathbf{r}^2\|\|\Gamma\|^2 &\geq \mathcal{R}^S(\chi) + \frac{1}{2}t^2\|\|\Gamma\|^2 + \frac{1}{2}\sum\_{r=t+1}^m \left( (\Gamma\_{t\_1+1t\_1+1}^{\prime} + \dots \,\Gamma\_{t\_2t\_2}^{\prime} + \dots + \Gamma\_{tt\_1}^{\prime}) - 2\Gamma\_{tt}^{\prime} \right)^2 \\ &+ \frac{t\_3\Delta f}{f} + \sum\_{r=t+1}^m \sum\_{1 \leq i < j \leq n} (\Gamma\_{ij}^{\prime})^2 + \sum\_{r=t+1}^m \sum\_{q=t\_1+1}^{t-1} \Gamma\_{tt}^{\prime} \Gamma\_{qq}^{\prime} \\ &+ \frac{c}{4}(t - t\_1t\_2 - t\_2t\_3 - t\_1t\_3 + 1 - \frac{3}{2}\cos^2\theta). \end{split} \tag{47}$$

The third and sixth terms on the right hand side of (47) in a similar way as in Case 1 and Case 2 can be simplified as

$$\begin{split} \frac{1}{2} \sum\_{r=t+1}^{m} \left( (\Gamma\_{t\_1+1t\_1+1}' + \dots \Gamma\_{t\_2t\_2}' + \dots + \Gamma\_{tt}') - 2\Gamma\_{tt}' \right)^2 &+ \sum\_{r=t+1}^{m} \sum\_{q=t\_1+1}^{t-1} \Gamma\_{tt}' \Gamma\_{qq}' \\ = \frac{1}{2} \sum\_{r=t+1}^{m} \left( \Gamma\_{t\_1+1t\_1+1}' + \dots \Gamma\_{t\_2t\_2}' + \dots + \Gamma\_{tt}' \right)^2 + \sum\_{r=t+1}^{m} \left( \Gamma\_{tt}' \right)^2 \\ - \sum\_{r=t+1}^{m} \sum\_{j=t+1}^{t} \Gamma\_{tt}' \Gamma\_{jj}'. \end{split} \tag{48}$$

By combining (47) and (48) and using similar techniques as used in Case 1 and Case 2, we can derive

$$\begin{split} \frac{1}{4}t^2||\Pi||^2 &\geq \mathcal{R}^S(\chi) + \frac{1}{4} \sum\_{r=t+1}^m (2\Gamma\_{tt}^r - (\Gamma\_{t\_1+1t\_1+1}^r + \dots + \Gamma\_{tt}^r))^2 \\ &+ \frac{t\_3\Delta f}{f} + \frac{c}{4}(t - t\_1t\_2 - t\_2t\_3 - t\_1t\_3 + 1 - \frac{3}{2}\cos^2\theta). \end{split} \tag{49}$$

The last inequality leads to inequality (*iii*) in (1).

Next, we explore the equality cases of (1). First, we redefine the notion of the relative null space N*x* of the submanifold *St* in the CSF *<sup>S</sup>*¯*m*(*c*) at any point *x* ∈ *S<sup>t</sup>*; the relative null space was defined by B.-Y. Chen [19], as follows:

$$\mathcal{N}\_{\mathbf{x}} = \{ \mathcal{U}\_{\mathbf{l}} \in T\_{\mathbf{x}} \mathcal{S}^{t} \; : \; \Gamma(\mathcal{U}\_{\mathbf{l}}, \mathcal{U}\_{\mathbf{2}}) = 0, \forall \mathcal{U}\_{\mathbf{2}} \in T\_{\mathbf{x}} \mathcal{S}^{t} \}.$$

For *A* ∈ {1, ... , *t*} a unit vector field *eA* tangential to *St* at *x* satisfies the equality sign of (23) identically iff

$$(i)\ \sum\_{p=1}^{t\_1} \sum\_{q=t\_1+1}^{t} \Gamma\_{pq}^r = 0 \text{ (ii)} \ \sum\_{b=1}^t \sum\_{\substack{A=1\\b \neq A}}^t \Gamma\_{bA}^r = 0 \text{ (iii)} \ 2\Gamma\_{AA}^r = \sum\_{q=t\_1+1}^t \Gamma\_{qq}^r \tag{50}$$

such that *r* ∈ {*t* + 1, ... *m*} the condition (*i*) implies that *St* is mixed totally geodesic SCR W-P submanifold. Combining statements (*ii*) and (*iii*) with the fact that *St* is *L*−minimal, we ge<sup>t</sup> that the unit vector field *χ* = *eA* ∈ N*<sup>x</sup>*. The converse is trivial; this proves statement (2).

For a SCR W-P submanifold, the equality sign of (23) holds identically for all unit tangent vector belong to *St*1*T*at *x* iff

$$(i)\sum\_{p=1}^{t\_1} \sum\_{q=t\_1+1}^{t} \Gamma\_{pq}^r = 0 \text{ (ii)}\\ \sum\_{b=1}^{t} \sum\_{\substack{A=1\\b\neq A}}^{t\_1} \Gamma\_{bA}^r = 0 \text{ (iii)}\\ \, 2\Gamma\_{pp}^r = \sum\_{q=t\_1+1}^{t} \Gamma\_{qq}^r \tag{51}$$

where *p* ∈ {1, ... , *<sup>t</sup>*1} and *r* ∈ {*t* + 1, ... , *<sup>m</sup>*}. Since *St* is *L*−minimal SCR W-P submanifold, the third condition implies that Γ*rpp* = 0, *p* ∈ {1, ... , *<sup>t</sup>*1}. Using this in the condition (*ii*), we conclude that *St* is *<sup>L</sup>*−totally geodesic SCR W-P submanifold in *S* ¯ *<sup>m</sup>*(*c*) and totally mixed geodesicness follows from the condition (*i*), which proves (*a*) in the statement (3).

For a SCR W-P submanifold, the equality sign of (24) holds identically for all unit tangent vector fields tangential to *St*2*θ*at *x* if and only if

$$(i)\sum\_{p=1}^{t\_1} \sum\_{q=t\_1+1}^{t} \Gamma\_{pq}^r = 0 \text{ (ii)}\\\sum\_{b=1}^{t} \sum\_{\substack{A=t\_1+1\\b \neq A}}^{t\_2} \Gamma\_{bA}^r = 0 \text{ (iii)}\\\ 2\Gamma\_{KK}^r = \sum\_{q=t\_1+1}^{t} \Gamma\_{qq}^r \tag{52}$$

such that *K* ∈ {*<sup>t</sup>*1 + 1, ... , *<sup>t</sup>*2} and *r* ∈ {*t* + 1, ... , *<sup>m</sup>*}. From the condition (*iii*) two cases emerge; that is,

$$
\Gamma^{r}\_{\mathcal{K}\mathcal{K}} = 0,\ \forall \mathcal{K} \in \{t\_1 + 1, \dots, t\_2\} \text{ and } \ r \in \{t + 1, \dots, m\} \text{ or } \dim S^{t\_2}\_{\theta} = 2. \tag{53}
$$

If the first case of (52) is satisfied, then by virtue of condition (*ii*), it is easy to conclude that *St* is a *Dθ*− totally geodesic SCR W-P submanifold in *S* ¯ *<sup>m</sup>*(*c*). This is the first case of part (*b*) of statement (3).

For a SCR W-P submanifold, the equality sign of (25) holds identically for all unit tangent vector fields tangent to *St*3⊥at *x* if and only if

$$(i)\sum\_{p=1}^{t\_1} \sum\_{q=t\_1+1}^{t} \Gamma\_{pq}^r = 0 \text{ (ii)} \sum\_{b=1}^{t} \sum\_{\substack{A=t\_2+1\\b\neq A}}^{t\_3} \Gamma\_{bA}^r = 0 \text{ (iii)} \ 2\Gamma\_{LL}^r = \sum\_{q=t\_1+1}^{t} \Gamma\_{qq}^r \tag{54}$$

such that *L* ∈ {*<sup>t</sup>*2 + 1, . . . , *t*} and *r* ∈ {*t* + 1, . . . , *<sup>m</sup>*}. From the condition (*iii*) two cases arise; that is,

$$
\Gamma\_{LL}^r = 0, \; \forall L \in \{t\_2 + 1, \ldots, t\} \text{ and } r \in \{t + 1, \ldots, m\} \text{ or } \dim \mathcal{S}\_{\perp}^{t\_3} = 2. \tag{55}
$$

If the first case of (54) is satisfied, then by virtue of condition (*ii*), it is easy to conclude that *St* is a *L*⊥− totally geodesic SCR W-P submanifold in *S* ¯ *<sup>m</sup>*(*c*). This is the first case of part (*c*) of statement (3).

For the other case, assume that *St* is not *<sup>L</sup>*⊥−totally geodesic SCR W-P submanifold and dim *St*3⊥ = 2. Then condition (*ii*) of (54) implies that *St* is *L*⊥− totally umbilical SCR W-P submanifold in *S* ¯(*c*), which is second case of this part. This verifies part (*c*) of (3).

To prove (*d*) using parts (*a*),(*b*) and (*c*) of (3), we combine (51), (52) and (54). For the first case of this part, assume that *dimS<sup>t</sup>*2*θ* = 2 and *dimS<sup>t</sup>*3⊥ = 2. From parts (*a*), (*b*) and (*c*) of statement (3) we concluded that *M<sup>t</sup>* is *<sup>L</sup>*−totally geodesic, *Lθ*− is totally geodesic and *D*⊥− is a totally geodesic submanifold in *S* ¯ *<sup>m</sup>*(*c*). Hence *St* is a totally geodesic submanifold in *<sup>S</sup>*¯*m*(*c*).

For another case, suppose that first case is not satisfied. Then parts (*a*), (*b*) and (*c*) provide that *St* is mixed totally geodesic and *L*− totally geodesic submanifold of *<sup>S</sup>*¯*m*(*c*) with *dimS<sup>t</sup>*2*θ* = 2 and *dimS<sup>t</sup>*3⊥ = 2. From the conditions (*b*) and (*c*) it follows that *St* is *Lθ*− and *<sup>L</sup>*⊥−totally umbilical SCR W-P submanifolds and from (*a*) it is *<sup>L</sup>*−totally geodesic, which is part (*d*). This proves the theorem.

If, *St*2*θ* = {0} then the SCR W-P submanifold becomes the CR W-P submanifold. In this case we have the following corollary

**Corollary 1.** *Let St* = *St*1*T* <sup>×</sup>*f St*3⊥ *be a CR W-P submanifold isometrically immersed in a CSF <sup>S</sup>*¯*m*(*c*)*. Then for each orthogonal unit vector field χ* ∈ *TxSt, either tangent to St*1*Tor St*3⊥*, we have*

	- *(i) If χ* ∈ *St*1*T, then*

$$\frac{1}{4}t^2||\Pi||^2 \ge R^S(\chi) + \frac{t\_3\Delta f}{f} + \frac{c}{4}(t - t\_1t\_3 - \frac{1}{2}).\tag{56}$$

*(ii) If χ* ∈ *<sup>S</sup><sup>t</sup>*3⊥*, then*

$$\frac{1}{4}t^2||\Pi||^2 \ge R^S(\chi) + \frac{t\_3\Delta f}{f} + \frac{c}{4}(t - t\_1t\_3 + 1). \tag{57}$$

	- *(a) The equality case of (56) holds identically for all unit vector fields tangent to St*1*T at each x* ∈ *St iff St is mixed totally geodesic and <sup>L</sup>*−*totally geodesic CR W-P submanifold in <sup>S</sup>*¯*m*(*c*)*.*
	- *(b) The equality case of (57) holds identically for all unit vector fields tangent to St*3⊥ *at each x* ∈ *M<sup>t</sup> iff St is mixed totally geodesic and either St is L*⊥*- totally geodesic CR-warped product or St is a L*⊥ *totally umbilical in S* ¯ *<sup>m</sup>*(*c*) *with dim L*⊥ = 2*.*
	- *(c) The equality case of (1) holds identically for all unit tangent vectors to St at each x* ∈ *St if and only if either St is totally geodesic submanifold or St is a mixed totally geodesic totally umbilical and L*− *totally geodesic submanifold with dim St*3⊥ = 2

*where t*1 *and t*3 *are the dimensions of St*1*T and St*3⊥ *respectively.*

In view of (20) we have the another version of the Theorem 2 as follows:

**Theorem 4.** *Let St* = *S<sup>t</sup>*1+*t*2 1 <sup>×</sup>*f St*3⊥ *be a L*−*minimal SCR W-P submanifold isometrically immersed in a CSF M* ¯ (*c*)*. If the holomorphic and slant distributions L and Lθ are integrable with integral submanifolds St*1*T and St*2*θ respectively. Then for each orthogonal unit vector field χ* ∈ *TxS, either tangent to St*1*T , St*2*θ or <sup>S</sup><sup>t</sup>*3⊥*, we have*

	- *(i) If χ* ∈ *TS<sup>t</sup>*1*T , then*

$$\begin{split} \frac{1}{4}t^2||\Pi||^2 \geq R^S(\chi) + t\_3(\Delta \ln f - ||\nabla \ln f||^2) + \frac{c}{4}(t - t\_1t\_2 - t\_2t\_3 \\ & -t\_1t\_3 - \frac{1}{2}). \end{split} \tag{58}$$

*(ii) χ* ∈ *TS<sup>t</sup>*2*θ, then*

$$\begin{split} \frac{1}{4}t^2||\Pi||^2 \geq R^S(\chi) + t\_3(\Delta ln f - ||\nabla ln f||^2) + \frac{c}{4}(t - t\_1 t\_2 - t\_2 t\_3 \\ & -t\_1 t\_3 + 1 - \frac{3}{2} \cos^2 \theta). \end{split} \tag{59}$$

*(iii) If χ* ∈ *TS<sup>t</sup>*2⊥*, then*

$$\frac{1}{4}t^2\|\Pi\|^2 \ge R^S(\chi) + t\_3(\Delta \ln f - \|\nabla \ln f\|^2) + \frac{c}{4}(t - t\_1t\_2 - t\_2t\_3 - t\_1t\_3 + 1). \tag{60}$$

	- *(a) The equality of (58) holds identically for all unit vector fields tangent to St*1*T at each x* ∈ *St iff St is mixed TG and <sup>L</sup>*−*totally geodesic SCR W-P submanifold in S* ¯ *<sup>m</sup>*(*c*)*.*
	- *(b) The equality of (59) holds identically for all unit vector fields tangent to Sθ at each x* ∈ *St iff S is mixed totally geodesic and either St is Lθ- totally geodesic SCR W-P submanifold or St is a Lθ totally umbilical in S* ¯ *<sup>m</sup>*(*c*) *with dim Lθ* = 2*.*
	- *(c) The equality of (60) holds identically for all unit vector fields tangent to St*2⊥ *at each x* ∈ *St iff S is mixed totally geodesic and either St is L*⊥*- totally geodesic SCR W-P or St is a L*<sup>⊥</sup> *totally umbilical in S* ¯ *<sup>m</sup>*(*c*) *with dim L*<sup>⊥</sup> = 2*.*
	- *(d) The equality case of (1) holds identically for all unit tangent vectors to St at each x* ∈ *St iff either St is totally geodesic submanifold or M<sup>t</sup> is a mixed totally geodesic totally umbilical and L totally geodesic submanifold with dim St*2*θ*= 2 *and dim St*3⊥= 2.

*Where t*1, *t*2 *and t*3 *are the dimensions of St*1*T*, *St*2*θand St*3⊥*respectively.*
