New Obstructions to Warped Product Immersions in Complex Space Forms

Abstract

In this paper, we obtain a geometric inequality for warped product pointwise semi-slant submanifolds of complex space forms endowed with a semi-symmetric metric connection and discuss the equality case of this inequality. We provide some applications concerning the minimality and compactness of such submanifolds.

1. Introduction

In 1969, R.L. Bishop et al. [1] defined the notion of warped product manifolds, by genera-lizing the Riemannian product manifolds, in order to study manifolds of negative sectional curvature. Since then, it has remained a topic of research due to its usefulness and links to other fields, especially physics. Many research articles have been published in this area [2,3,4,5,6,7,8].

In differential geometry, one of the fundamental problems is the immersibility of a Riemannian manifold in a space form. According to the very famous Nash’s embedding theorem, every Riemannian manifold can be isometrically immersed in some Euclidean spaces with sufficiently high codimensions. Starting from this theorem, B.-Y. Chen discovered a method to study intrinsic and extrinsic invariants of a submanifold and provided many applications. For example, for isometric warped product immersion $\phi :M_1{\times }_f{M}_2\to \overline{M}\left(c\right)$ into a Riemannian space form, the following inequality holds ([2]):

$$\begin{array}{c}\hfill \frac{\Delta f}{f}\le \frac{{\left(n_1+n_2\right)}^2}{4n_2}{\parallel H\parallel }^2+n_{1}c,\end{array}$$

(1)

where $n_i=dim{M}_i,i=1,2,{\parallel H\parallel }^2$ is the squared mean curvature of the immersion $\phi$, and $\Delta$ is the Laplace operator of $M_1$. In the same paper [2], the author discussed the equality case of this inequality.

Motivated by the inequality (1), many researchers proved corresponding inequalities for different classes of submanifolds in various space forms. In these papers, the space forms are endowed with the Levi–Civita connection, which is torsion-free.

An important class of connections with non-vanishing torsion are the semi-symmetric connections. They have many applications in affine differential geometry, information geometry, etc.

In the present paper, we consider complex space forms endowed with semi-symmetric metric connections. We extend the above-mentioned result of Chen and prove a geometric inequality for warped product pointwise semi-slant submanifolds $M=M_1{\times }_f{M}_2$ in a complex space form $\overline{M}\left(c\right)$ endowed with a semi-symmetric metric connection. We also discuss the equality case and provide several applications in the compact case and the minimal case, respectively.

2. Preliminaries

Let $(\overline{M},J,g)$ be an almost Hermitian manifold, where J is an almost complex structure and g a Hermitian metric. Then, $\overline{M}$ is a Kaehler manifold if $\left({\overline{\nabla }}_{X}J\right)Y=0$, for all $X,Y\in T\overline{M}$, where $\overline{\nabla }$ is the Levi–Civita connection of the Riemannian metric g.

A complex space form $\overline{M}\left(c\right)$ is a Kaehler manifold of constant holomorphic sectional curvature c; its Riemannian curvature tensor $\overline{R}$ is given by

$$\begin{array}{cc}\hfill \overline{R}(X,Y)Z& =\frac{c}{4}\{g(Y,Z)X-g(X,Z)Y\}\hfill \\ \hfill & +g(X,JZ)JY-g(Y,JZ)JX+2g(X,JY)JZ\},\hfill \end{array}$$

(2)

for all $X,Y,Z\in T\overline{M}$.

Let $\overline{M}$ be an almost Hermitian manifold and M a submanifold of $\overline{M}$ with induced metric g. Let ∇ be the Levi–Civita connection on the tangent bundle $TM$ and ${\nabla }^{\perp }$ the connection on the normal bundle $T^{\perp }M$ of M. Then, the Gauss and Weingarten formulae are

$$\begin{array}{c}{\overline{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y),\hfill \\ {\overline{\nabla }}_{X}N=-A_{N}X+{\nabla }_X^{\perp }N,\hfill \end{array}$$

where $X,Y\in TM$, $N\in T^{\perp }M$ and h, $A_N$ are the second fundamental form and the shape operator, respectively.

The relationship between the shape operator and the second fundamental form is

$$g(h(X,Y),N)=g(A_{N}X,Y),$$

for vector fields $X,Y\in TM$ and $N\in T^{\perp }M$.

Let $\overline{R}$ and R be the Riemannian curvature tensors of $\overline{M}$ and M, respectively. We use the notation $R(X,Y,Z,W)=g\left(R\right(X,Y)Z,W),$ for any $X,Y,Z,W\in TM$. Then, the Gauss equation is given by

$$\begin{array}{cc}\hfill \overline{R}(X,Y,Z,W)& =R(X,Y,Z,W)\hfill \\ \hfill & +g\left(h\right(X,Z),h(Y,W\left)\right)-g\left(h\right(X,W),h(Y,Z\left)\right),\hfill \end{array}$$

(3)

for any $X,Y,Z,W\in TM$.

The notion of a semi-symmetric linear connection was introduced by Friedmann and Schouten [9]. Let $(\overline{M},g)$ be a Riemannian manifold with a Riemannian metric g. A linear connection $\tilde{\nabla }$ on $\overline{M}$ is called a semi-symmetric connection if its torsion tensor T

$$T(X,Y)={\tilde{\nabla }}_{X}Y-{\tilde{\nabla }}_{Y}X-[X,Y]$$

satisfies

$$T(X,Y)=\omega \left(Y\right)X-\omega \left(X\right)Y,$$

for any vector fields $X,Y$ on $\overline{M}$, where $\omega$ is a 1-form. Denote by P its dual vector field, i.e., $\omega \left(X\right)=g(X,P)$. If a semi-symmetric connection satisfies

$$\tilde{\nabla }g=0,$$

then it is said to be a semi-symmetric metric connection $\tilde{\nabla }$.

Further, with respect to a semi-symmetric metric connection $\tilde{\nabla }$ on $\overline{M}$, the curvature tensor $\tilde{R}$ is given by

$$\begin{array}{cc}\hfill \tilde{R}(X,Y,Z,W)& =\overline{R}(X,Y,Z,W)-\alpha (Y,Z)g(X,W)\hfill \\ \hfill & +\alpha (X,Z)g(Y,W)-\alpha (X,W)g(Y,W)+\alpha (Y,W)g(X,Z),\hfill \end{array}$$

(4)

for any $X,Y,Z,W\in T\overline{M}$, where $\alpha$ is the $(0,2)$-tensor field defined by

$$\begin{array}{c}\hfill \alpha (X,Y)=\left({\overline{\nabla }}_X\omega \right)Y-\omega \left(X\right)\omega \left(Y\right)+\frac{1}{2}\omega \left(P\right)g(X,Y),\end{array}$$

(5)

for all $X,Y\in T\overline{M}$.

Let M be an n-dimensional submanifold of a complex space form $\overline{M}\left(c\right)$ of complex dimension m. Then, we decompose

$$JX=TX+FX,$$

where $TX$ and $FX$ are the tangential and normal components of $JX$, respectively, for any $X\in TM$.

The submanifold is called anti-invariant if $T=0$.

The submanifold is called invariant if $F=0$.

Let $\{e_1,\cdots ,e_n\}$ be an orthonormal basis of $T_{p}M$, $p\in M$. One is denoted by

$$\begin{array}{c}\hfill {\parallel T\parallel }^2=\sum _{i,j=1}^n{g}^2(J{e}_i,e_j).\end{array}$$

(6)

Let $p\in M$ and $\pi \subset T_{p}M$ be a plane section.

If $\{e_1,\cdots ,e_n\}$ is an orthonormal basis of $T_{p}M$ and $\{e_{n+1},\cdots ,e_{2m}\}$ an orthonormal basis of $T_p^{\perp }M$, then the sectional curvature $K\left(\pi \right)$ is defined by $K\left(\pi \right)=g\left(R\right(X,Y)X,Y)$, where $X,Y\in \pi$ are orthonormal, and the scalar curvature $\tau$ at p by

$$\begin{array}{c}\hfill \tau \left(p\right)=\sum _{1\le i<j\le n}K(e_i\wedge e_j).\end{array}$$

(7)

The mean curvature vector field H of M is

$$\begin{array}{c}\hfill H=\frac{1}{n}\sum _{i=1}^{n}h(e_i,e_i).\end{array}$$

A submanifold is minimal if the mean curvature vector H vanishes identically, that is, $H=0$.

We recall the definition of pointwise slant submanifolds.

Definition 1

([10]). Let $\overline{M}$ be an almost Hermitian manifold. Then a submanifold M of $\overline{M}$ is called a pointwise slant submanifold if for each point $p\in M$ and any non-zero vector $X\in T_{p}M$ the angle $\theta \left(X\right)$ between $JX$ and $T_{p}M$ is independent of the choice of X.

In [10], Chen and Garay obtained a necessary and sufficient condition for a submanifold to be a pointwise slant submanifold. They proved that a submanifold M of an almost Hermitian manifold $\tilde{M}$ is pointwise slant if and only if

$$\begin{array}{c}\hfill T^2=-\left({cos}^2\theta \right)I,\end{array}$$

(8)

for a real-valued function $\theta$ defined on M, where I is the identity transformation of the tangent bundle $TM$ of M. On the other hand, Chen and the forth author generalized the above concept and defined pointwise semi-slant submanifolds as follows [6].

Definition 2.

Let $\overline{M}$ be an almost Hermitian manifold. Then a submanifold M of $\overline{M}$ is called a pointwise semi-slant submanifold if a pair of orthogonal distributions ${\mathfrak{D}}_1$ and ${\mathfrak{D}}_2$ exist such that

(i) 

$TM$ admits the orthogonal direct decomposition $TM={\mathfrak{D}}_1\oplus {\mathfrak{D}}_2$;

(ii) 

${\mathfrak{D}}_1$ is invariant;

(iii) 

${\mathfrak{D}}_2$ is pointwise slant with a slant function θ.

The submanifold M is called a proper pointwise semi-slant submanifold if both distributions are non-trivial. Denote their dimensions by $2d_1$ and $2d_2$.

Let M be a proper pointwise semi-slant submanifold of a complex space form $\overline{M}\left(c\right)$. We set the following

$$\begin{array}{cc}\hfill & e_1,e_2=T{e}_1,\cdots ,e_{2d_1-1},e_{2d_1}=T{e}_{2d_1-1},\hfill \\ \hfill & e_{2d_1+1},e_{2d_1+2}=sec\theta T{e}_{2d_1+1},\cdots ,e_{2d_1+2d_2-1},e_{2d_1+2d_2}=sec\theta T{e}_{d_1-1}.\hfill \end{array}$$

Then, we obtain

$$\begin{array}{c}\hfill g^2(J{e}_i,e_{i+1})=\left\{\begin{array}{c}1,\forall i=1,3,\cdots ,2d_1-1,\hfill \\ {cos}^2\theta ,\forall i=2d_1+1,\cdots ,2d_1+2d_2-1.\hfill \end{array}\right.\end{array}$$

Hence, we have

$$\begin{array}{c}\hfill {\parallel T\parallel }^2=\sum _{i,j=1}^n{g}^2(J{e}_i,e_j)=n_1+n_2{cos}^2\theta ,\end{array}$$

(9)

where $n_1=2d_1$ and $n_2=2d_2$.

Further, we state an algebraic lemma due to Chen.

Lemma 1

([11]). Let $n\ge 2$ and $a_1,\cdots ,a_n,b$ real numbers such that

$$\begin{array}{c}\hfill {\left(\sum _{i=1}^n{a}_i\right)}^2=(n-1)\left(\sum _{i=1}^n{a}_i^2+b\right).\end{array}$$

Then $2a_1{a}_2\ge b$ and the equality holds if and only if

$$\begin{array}{c}\hfill a_1+a_2=a_3=\cdots =a_n.\end{array}$$

Finally, we conclude this section with the following relation between sectional curvature and the Laplacian of the warping function for warped products. Let $\{e_1,e_2,\cdots ,e_n,e_{n+1},\cdots ,e_{2m}\}$ be a local orthonormal frame such that $e_1,e_2,\cdots ,e_{n_1}$ are tangent to $M_1$, $e_{n_1+1},\cdots ,e_n$ are tangent to $M_2$ and $e_{n+1}$ is parallel to the mean curvature vector H. Then,

$$\begin{array}{cc}\hfill & \sum _{1\le i\le n_1}\sum _{n_1+1\le j\le n}K(e_i\wedge e_j)=n_2\frac{\Delta f}{f}=n_2\left({\parallel \nabla (lnf)\parallel }^2-\Delta (lnf)\right),\hfill \end{array}$$

(10)

where $\nabla (lnf)$ is the gradient vector on $M_1$.

3. An Inequality for Warped Product Pointwise Semi-Slant Submanifolds

The following theorem is the main result of this article; it gives an estimate of the squared mean curvature in terms of the warping function.

Theorem 1.

Let $\overline{M}\left(c\right)$ be a complex space form endowed with a semi-symmetric metric connection $\tilde{\nabla }$ and $M=M_1{\times }_f{M}_2$ a warped product pointwise semi-slant submanifold of $\overline{M}\left(c\right)$. Then we have

$$n_2\frac{\Delta f}{f}\le \frac{{\left(n_1+n_2\right)}^2}{4}{\parallel H\parallel }^2+\frac{c}{4}{n}_1{n}_2+\varpi -\frac{3c}{8}\left(n_1+n_2{cos}^2\theta \right),$$

(11)

where ϖ denotes the trace of α, θ is the slant function on $M_2$ and $dim{M}_i=n_i$, $i=1,2$.

The equality case holds if and only if M is a mixed totally geodesic submanifold and $n_1{H}_1=n_2{H}_2$, where $H_1$ and $H_2$ are the partial mean curvature vectors corresponding to $M_1$ and $M_2$, respectively.

Proof. 

From (2)–(4) we have

$$\begin{array}{cc}\hfill R(X,Y,& Z,W)+g(h(X,Z),h(Y,W))-g(h(X,W),h(Y,Z))\hfill \\ \hfill & =\frac{c}{4}\{g(Y,Z)g(X,W)-g(X,Z)g(Y,W)\}\hfill \\ \hfill & +\frac{c}{4}\{g(X,JZ)g(JY,W)-g(Y,JZ)g(JX,W)+2g(X,JY)g(JZ,W)\}\hfill \\ \hfill & -\alpha (Y,Z)g(X,W)+\alpha (X,Z)g(Y,W)-\alpha (X,W)g(Y,W)+\alpha (Y,W)g(X,Z).\hfill \end{array}$$

(12)

We consider the orthonormal frame defined in the previous section. For $X=W=e_i$, $Y=Z=e_j$, $i,j=1,...,n$, summing after $1\le i,j\le n$, one obtains

$$\begin{array}{cc}\hfill 2\tau & =\frac{c}{4}n\left(n-1\right)-{\parallel h\parallel }^2\hfill \\ \hfill & +\frac{c}{4}\left(3n_1+3n_2{cos}^2\theta \right)+2(n-1)\varpi +n^2{\parallel H\parallel }^2.\hfill \end{array}$$

(13)

We denote

$$\delta =2\tau -\frac{c}{4}n\left(n-1\right)-\frac{c}{4}\left(3n_1+3n_2{cos}^2\theta \right)-2(n-1)\varpi -\frac{n^2}{2}{\parallel H\parallel }^2.$$

(14)

Then, from (13) and (14) we derive that

$$\begin{array}{c}\hfill n^2{\parallel H\parallel }^2={2(\delta +\parallel h\parallel }^2),\end{array}$$

(15)

which can be written as

$$\begin{array}{c}\hfill {\left(\sum _{i=1}^n{h}_{ii}^{n+1}\right)}^2=2\left\{\delta +\sum _{i=1}^n{\left(h_{ii}^{n+1}\right)}^2+\sum _{i\ne j}{\left(h_{ij}^{n+1}\right)}^2+\sum _{r=n+2}^{2m}\sum _{i,j=1}^n{\left(h_{ij}^r\right)}^2\right\},\end{array}$$

(16)

where $h_{ij}^r=g(h(e_i,e_j),e_r),i,j=1,...,n;r=n+1,...,2m.$

If we put $a_1=h_{11}^{n+1}$, $a_2={\sum }_{i=2}^{n_1}{h}_{ii}^{n+1}$ and $a_3={\sum }_{t=n_1+1}^n{h}_{tt}^{n+1}$, the above equation is reduced to

$$\begin{array}{cc}\hfill {\left(\sum _{i=1}^3{a}_i\right)}^2& =2\{\delta +\sum _{i=1}^3{a}_i^2+\sum _{1\le i\ne j\le n}{\left(h_{ij}^{n+1}\right)}^2+\sum _{r=n+2}^{2m}\sum _{i,j=1}^n{\left(h_{ij}^r\right)}^2\hfill \\ \hfill & -\sum _{2\le j\ne k_{\le }{n}_1}{h}_{jj}^{n+1}{h}_{kk}^{n+1}-\sum _{n_1+1\le s\ne t_{\le }n}{h}_{ss}^{n+1}{h}_{tt}^{n+1}\}.\hfill \end{array}$$

(17)

It follows that $a_1,a_2,a_3$ satisfy Lemma 1 for $n=3$; then, $2a_1{a}_2\ge b$, with equality holding if and only if $a_1+a_2=a_3$. This means

$$\begin{array}{cc}\hfill \sum _{1\le j<k_{\le }{n}_1}{h}_{jj}^{n+1}{h}_{kk}^{n+1}& +\sum _{n_1+1\le s<t_{\le }n}{h}_{ss}^{n+1}{h}_{tt}^{n+1}\hfill \\ \hfill & \ge \frac{\delta }{2}+\sum _{1\le \alpha <\beta \le n}{\left(h_{\alpha \beta }^{n+1}\right)}^2+\frac{1}{2}\sum _{r=n+2}^{2m}\sum _{\alpha ,\beta =1}^n{\left(h_{\alpha \beta }^r\right)}^2.\hfill \end{array}$$

(18)

We have the equality if and only if

$$\begin{array}{c}\hfill \sum _{i=1}^{n_1}{h}_{ii}^{n+1}=\sum _{t=n_1+1}^n{h}_{tt}^{n+1}.\end{array}$$

(19)

From the Gauss equation we obtain

$$\begin{array}{cc}\hfill n_2\frac{\Delta f}{f}& =\tau -\sum _{1\le j<k\le n_1}K(e_j\wedge e_k)-\sum _{n_1+1\le s<t\le n}K(e_s\wedge e_t).\hfill \end{array}$$

(20)

Combining (10), (18) and (20), we derive

$$\begin{array}{cc}\hfill n_2\frac{\Delta f}{f}& =\tau -\frac{c}{8}{n}_1\left(n_1-1\right)-\frac{3c}{4}{n}_1-(n_1-1)\varpi \hfill \\ \hfill & -\sum _{r=n+1}^{2m}\sum _{1\le j<k\le n_1}\left(h_{jj}^r{h}_{kk}^r-{\left(h_{jk}^r\right)}^2\right)\hfill \\ \hfill & -\frac{c}{8}{n}_2\left(n_2-1\right)-\frac{3c}{4}{n}_2{cos}^2\theta -(n_2-1)\varpi \hfill \\ \hfill & -\sum _{r=n+1}^{2m}\sum _{n_1+1\le s<t\le n}\left(h_{ss}^r{h}_{tt}^r-{\left(h_{st}^r\right)}^2\right).\hfill \end{array}$$

(21)

Taking into account (18) and (21), we find

$$\begin{array}{cc}\hfill n_2\frac{\Delta f}{f}& \le \tau -\frac{c}{8}n\left(n-1\right)+\frac{c}{4}{n}_1{n}_2\hfill \\ \hfill & -(n-2)\varpi -\frac{3c}{4}\left(n_1+n_2{cos}^2\theta \right)-\frac{\delta }{2}.\hfill \end{array}$$

(22)

Using (14) in the previous inequality, we derive

$$\begin{array}{cc}\hfill n_2\frac{\Delta f}{f}& \le \frac{n^2}{4}{\parallel H\parallel }^2+\frac{c}{4}{n}_1{n}_2+\varpi -\frac{3c}{8}\left(n_1+n_2{cos}^2\theta \right),\hfill \end{array}$$

(23)

which represents the inequality to prove.

For the equality case, from (19) it follows that $n_1{H}_1=n_2{H}_2$.

Moreover, from (18) we obtain

$$\begin{array}{cc}\hfill h_{\alpha \beta }^r=0,\forall 1\le & \alpha \le n_1,n_1+1\le \beta \le n,n+1\le r\le 2m,\hfill \end{array}$$

(24)

i.e., M is a mixed totally geodesic submanifold.

The converse is trivial, and the proof is achieved. □

In the following subsections, we derive certain consequences of Theorem 1.

3.1. The Compact Case

It is well-known that if M is a compact oriented Riemannian manifold without boundary, one has

$${\int }_M\Delta fdV=0,$$

(25)

where f is a smooth function on M and $dV$ denotes the volume element of M.

As a consequence of Theorem 1, we prove the following result.

Theorem 2.

Let $M=M_1{\times }_f{M}_2$ be a compact oriented warped product pointwise semi-slant submanifold in a complex space form $\overline{M}\left(c\right)$ endowed with a semi-symmetric metric connection. Then M is a Riemannian product if and only if

$$\begin{array}{cc}\hfill {\parallel H\parallel }^2& \ge \frac{3c}{2{\left(n_1+n_2\right)}^2}\left(n_1+n_2{cos}^2\theta \right)-\frac{n_1{n}_2}{{\left(n_1+n_2\right)}^2}c-\frac{4\varpi }{{\left(n_1+n_2\right)}^2}.\hfill \end{array}$$

(26)

Proof. 

By using (10), the inequality (11) reduces to

$$\begin{array}{cc}\hfill n_2(\Delta (lnf)& -{\parallel \nabla (lnf)\parallel }^2)\hfill \\ \hfill & \le \frac{n^2}{4}{\parallel H\parallel }^2+\frac{c}{4}{n}_1{n}_2+\varpi -\frac{3c}{8}\left(n_1+n_2{cos}^2\theta \right).\hfill \end{array}$$

(27)

Let assume that M is a Riemannian product, i.e., f is constant on M. Then, we obtain (26).

Conversely, suppose that the inequality (26) holds; then, integrating (27) and using (25), we obtain

$$\begin{array}{c}\hfill 0\le {\int }_M(n_2{\parallel \nabla (lnf)\parallel }^2)dV\le 0,\end{array}$$

from where $\nabla (lnf)=0$, which implies that f is a constant function on M. □

3.2. The Minimal Case

In this subsection, we obtain obstructions to the minimality of warped product pointwise semi-slant submanifolds in a complex space form endowed with a semi-symmetric metric connection.

An immediate consequence of Theorem 1 is the following:

Theorem 3.

Let $M=M_1{\times }_f{M}_2$ be a warped product pointwise semi-slant submanifold in a complex space form $\overline{M}\left(c\right)$ endowed with a semi-symmetric metric connection. If there exists a point $p\in M$ such that

$$\begin{array}{c}\hfill n_2\frac{\Delta f}{f}>\frac{c}{4}{n}_1{n}_2+\varpi -\frac{3c}{8}\left(n_1+n_2{cos}^2\theta \right),\end{array}$$

at p, then M cannot be minimal.

As with special cases of Theorem 3, we state the following corollaries.

Corollary 1.

There does not exist any minimal warped product pointwise semi-slant submanifold $M=M_1{\times }_f{M}_2$ in a complex space form $\overline{M}\left(c\right)$ endowed with a semi-symmetric metric connection if

$$\begin{array}{c}\hfill n_2\frac{\Delta f}{f}>\frac{c}{4}{n}_1{n}_2+\varpi -\frac{3c}{8}\left(n_1+n_2{cos}^2\theta \right).\end{array}$$

Corollary 2.

Let $M=M_T{\times }_f{M}_{\perp }$ be a warped product CR-submanifold of a complex space form $\overline{M}\left(c\right)$ endowed with a semi-symmetric metric connection, where $M_T$ and $M_{\perp }$ are holomorphic and totally real submanifolds of $\overline{M}$, respectively. If the following inequality

$$\begin{array}{c}\hfill n_2\frac{\Delta f}{f}>\frac{c}{4}{n}_1{n}_2+\varpi -\frac{3c{n}_1}{8}\end{array}$$

holds at a point $p\in M$, then M cannot be a minimal submanifold.

Also from Theorem 2, we obtain

Corollary 3.

Let $M=M_1{\times }_f{M}_2$ be a compact oriented warped product pointwise semi-slant submanifold in a complex space form $\overline{M}\left(c\right)$ endowed with a semi-symmetric metric connection. If

$$\begin{array}{cc}\hfill c& <\frac{3\left(n_1+n_2{cos}^2\theta \right)}{n_1{n}_2}-\frac{{\left(n_1+n_2\right)}^2}{n_1{n}_2}\varpi ,\hfill \end{array}$$

then M cannot be a minimal submanifold of $\overline{M}$.

4. Conclusions

By using the above methods, one can obtain corresponding inequalities for other classes of warped product submanifolds in complex space forms endowed with semi-symmetric connections.