Geometry of CR-Slant Warped Products in Nearly Kaehler Manifolds

Abstract

Recently, we studied CR-slant warped products $B_1{\times }_f{M}_{\perp }$, where $B_1=M_T\times M_{\theta }$ is the Riemannian product of holomorphic and proper slant submanifolds and $M_{\perp }$ is a totally real submanifold in a nearly Kaehler manifold. In the continuation, in this paper, we study $B_2{\times }_f{M}_{\theta }$, where $B_2=M_T\times M_{\perp }$ is a CR-product of a nearly Kaehler manifold and establish Chen’s inequality for the squared norm of the second fundamental form. Some special cases of Chen’s inequality are given.

1. Introduction

A submanifold M of an almost Hermitian manifold $\tilde{M}$ is called a complex submanifold of $\tilde{M}$ if its tangent space remains the same under the action of almost complex structure J. On contrary, M is called a totally real submanifold if J carries each tangent space of M into the corresponding normal space (see [1]). A submanifold M of $\tilde{M}$ is called a CR-submanifold (or Cauchy–Riemann submanifold) [2] if there exists a complex distribution $\mathfrak{D}$ on M whose orthogonal complementary distribution ${\mathfrak{D}}^{\perp }$ is a totally real distribution, i.e., $J{\mathfrak{D}}_p^{\perp }\subset T_p^{\perp }N,\forall p\in M$.

A CR-submanifold is called a CR-product [3] if it is a Riemannian product of a complex submanifold $M_T$ and a totally real submanifold $M_{\perp }$. For basic properties of CR-products in Käher manifolds, see, e.g., [2,3,4,5]. In [6,7], the second author introduced and investigated fundamental properties of a much larger class of CR-submanifolds; namely, the class of CR-warped product submanifolds. It was proved in [6] that there are no CR-warped product submanifolds in a Kaehler manifold $\tilde{M}$ which are of the form $M_{\perp }{\times }_f{M}_T$, where $M_{\perp }$ is totally real and $M_T$ is complex in $\tilde{M}$. On the other hand, a CR-submanifold M is called a CR-warped product [6] if it is the warped product $M_T{\times }_f{M}_{\perp }$ of a complex submanifold $M_T$ and a totally real submanifold $M_{\perp }$, where f is the warping function.

The second author proved in [6] that every CR-warped product $M_T{\times }_f{M}_{\perp }$ in an arbitrary Kaehler manifold satisfies the basic inequality,

$${\parallel h\parallel }^2\ge 2p{\parallel \nabla (lnf)\parallel }^2,$$

where p is the dimension of $M_{\perp }$, ${\parallel h\parallel }^2$ is the squared norm of the second fundamental form h, and $\nabla (lnf)$ is the gradient of $lnf$. The second author also classified all CR-warped products in complex space form satisfying the equality of the inequality in [6,7]. For further results in this respect, see [4,5,8,9,10,11,12,13,14].

CR-slant warped product submanifolds of the form $B_1{\times }_f{M}_{\perp }$ in a nearly Kaehler manifold $\tilde{M}$ were studied in [14], where $B_1=M_T\times M_{\theta }$ is the Riemannian product of a complex submanifold and a proper slant submanifold of $\tilde{M}$. In fact, the following Chen type inequality was established in [14].

Theorem 1 

([14]). Let $M=B_1{\times }_f{M}_{\perp }$ be a CR-slant warped product submanifold of a nearly Kaehler manifold $\tilde{M}$, where $B_1=M_T\times M_{\theta }$ is the Riemannian product of complex and proper slant submanifolds of $\tilde{M}$. If M is ${\mathfrak{D}}^{\perp }\oplus {\mathfrak{D}}^{\theta }$-mixed totally geodesic in $\tilde{M}$, then:

(i) 

The second fundamental form h satisfies

$$\begin{array}{c}\hfill {\parallel h\parallel }^2\ge 2s\parallel {\overrightarrow{\nabla }}^T{(lnf)\parallel }^2+s{cot}^2\theta {\parallel {\overrightarrow{\nabla }}^{\theta }(lnf)\parallel }^2\end{array}$$

(1) 

where $s=dim{M}_{\perp }$ and ${\overrightarrow{\nabla }}^T(lnf)$ and ${\overrightarrow{\nabla }}^{\theta }(lnf)$ denote the gradient components of $lnf$ along $M_T$ and $M_{\theta }$, respectively.

(ii) 

If the equality sign in (1) holds identically, then $M_T$ and $M_{\theta }$ are totally geodesic, $B_1$ is mixed totally geodesic in $\tilde{M}$ and $M_{\perp }$ is totally umbilical in $\tilde{M}$.

In the sequel, we study in this paper CR-slant warped product submanifolds of the form $M=B_2{\times }_f{M}_{\theta }^{n_3}$, where $B_2=M_T^{n_1}\times M_{\perp }^{n_2}$ is a CR-product and $M_{\theta }^{n_3}$ is an $n_3$-dimensional proper $\theta$-slant submanifold in a nearly Kaehler manifold ${\tilde{M}}^{2m}$. We prove that the second fundamental form h of M satisfies the following inequality

$$\begin{array}{c}\hfill {\parallel h\parallel }^2\ge \frac{1}{9}{n}_3{cos}^2\theta \parallel {\overrightarrow{\nabla }}^{\perp }{(lnf)\parallel }^2+2n_3\left(1+\frac{10}{9}{cot}^2\theta \right){\parallel {\overrightarrow{\nabla }}^T(lnf)\parallel }^2\end{array}$$

where ${\overrightarrow{\nabla }}^{\perp }(lnf)$ and ${\overrightarrow{\nabla }}^T(lnf)$ are the gradients of $lnf$ along $M_{\perp }$ and $M_T$, respectively. In this paper, we also discuss the equality case of this inequality. Several immediate consequences of this inequality are also given.

2. Basic Definitions and Formulas

Let ${\tilde{M}}^{2m}$ be an almost Hermitian manifold endowed with an almost complex structure J and a Riemannian metric $\tilde{g}$, such that

$$\begin{array}{c}\hfill J^2\left(X\right)=-X,\tilde{g}(JX,JY)=\tilde{g}(X,Y)\end{array}$$

(2) 

for any $X,Y\in \Gamma \left(T{\tilde{M}}^{2m}\right)$, where $\Gamma \left(T{\tilde{M}}^{2m}\right)$ denotes the Lie algebra of vector fields on ${\tilde{M}}^{2m}$. In addition, an almost Hermitian manifold is called Kaehler manifold if

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_{X}J\right)Y=0,\forall X,Y\in \Gamma \left(T{\tilde{M}}^{2m}\right),\end{array}$$

where $\tilde{\nabla }$ is the Levi–Civita connection on ${\tilde{M}}^{2m}$. Furthermore, an almost Hermitian manifold ${\tilde{M}}^{2m}$ is nearly Kaehler if $\left({\tilde{\nabla }}_{X}J\right)X=0,\forall X\in \Gamma \left(T{\tilde{M}}^{2m}\right)$, equivalently

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_{X}J\right)Y+\left({\tilde{\nabla }}_{Y}J\right)X=0,\forall X,Y\in \Gamma \left(T{\tilde{M}}^{2m}\right).\end{array}$$

(3) 

Clearly, every Kaehler manifold is nearly Kaehler but the converse is not true in general. The best known example of a nearly Kaehler non-Kaehlerian manifold is 6-dimensional sphere ${\mathbb{S}}^6$. For further results on nearly Kaehler manifolds, see, e.g., [15,16,17,18,19].

Let M be a Riemannian manifold isometrically immersed in ${\tilde{M}}^{2m}$. We denote the metric $\tilde{g}$ and the induced metric g on M by the same symbol g. The Gauss and Weingarten formulas are, respectively, given by (see, e.g., [4,5])

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y),\hfill \end{array}$$

(4) 

$$\begin{array}{c}\hfill {\tilde{\nabla }}_X\xi =-A_{\xi }X+{\nabla }_X^{\perp }\xi ,\end{array}$$

(5) 

for vector fields $X,Y\in \Gamma \left(TM\right)$ and $\xi \in \Gamma \left(T^{\perp }M\right)$, where $\Gamma \left(T^{\perp }M\right)$ denotes the set of all vector fields normal to M and ∇ and ${\nabla }^{\perp }$ denote the induced connections on the tangent and normal bundles of M, respectively, and h is the second fundamental form A is the shape operator of M; and they are related by

$$\begin{array}{c}\hfill g(A_{\xi }X,Y)=g(h(X,Y),\xi )\end{array}$$

(6) 

for any vector fields $X,Y\in \Gamma \left(TM\right)$ and any normal vector $\xi \in \Gamma \left(T^{\perp }M\right)$. A submanifold M in ${\tilde{M}}^{2m}$ is called totally geodesic if the second fundamental form h vanishes identically on M. Furthermore, M is called totally umbilical if h satisfies

$$\begin{array}{c}\hfill h(X,Y)=g(X,Y)H,\end{array}$$

(7) 

where H is the mean curvature vector M defined by $H=\frac{1}{n}\mathrm{trace}h$, $n=dimM$.

For each vector field X tangent to M, we write

$$\begin{array}{c}\hfill JX=PX+FX,\end{array}$$

(8) 

where $PX$ and $FX$ are the tangential and normal components of $JX$.

Definition 1 

([20,21]). A submanifold M of an almost Hermitian manifold $\tilde{M}$ is called slant if for each $p\in M$, the Wirtinger angle $\theta \left(X\right)$ between $JX$ and $T_{p}M$ is constant on M, i.e., it does not depend on the choice of $X\in T_{p}M$ and $p\in M$. In this case, θ is called the slant angle of M.

Complex and totally real submanifolds are slant submanifolds with slant angle $\theta =0$ and $\theta =\frac{\pi }{2}$, respectively. A slant submanifold is called proper if it is neither complex nor totally real.

More generally, a distribution $\mathfrak{D}$ on M is called a slant distribution if the angle $\theta \left(X\right)$ between $JX$ and ${\mathfrak{D}}_p$ is independent of the choice of $p\in M$ for any $0\ne X\in {\mathfrak{D}}_p$. The second author shown that a submanifold M of $\tilde{M}$ is slant if, and only if, we have [20]

$$\begin{array}{c}\hfill P^{2}X=-\left({cos}^2\theta \right)X,X\in \Gamma \left(TM\right).\end{array}$$

(9) 

Clearly, it follows from (8) and (9) that

$$\begin{array}{c}\hfill g(PX,PY)=\left({cos}^2\theta \right)g(X,Y),g(FX,FY)=\left({sin}^2\theta \right)g(X,Y),\end{array}$$

(10) 

for any vector fields $X,Y$ tangent to M.

Definition 2. 

A submanifold M of an almost Hermitian manifold $\tilde{M}$ is called CR-slant if there exist mutually orthogonal distributions $\mathfrak{D},{\mathfrak{D}}^{\perp }$ and ${\mathfrak{D}}^{\theta }$, such that the tangent bundle is spanned by

$$\begin{array}{c}\hfill TM=\mathfrak{D}\oplus {\mathfrak{D}}^{\perp }\oplus {\mathfrak{D}}^{\theta },\end{array}$$

(11) 

where $\mathfrak{D},{\mathfrak{D}}^{\perp }$ and ${\mathfrak{D}}^{\theta }$ are complex, totally real, and proper slant distributions.

The normal bundle of a CR-slant submanifold M is decomposed by

$$\begin{array}{c}\hfill T^{\perp }M=J{\mathfrak{D}}^{\perp }\oplus F{\mathfrak{D}}^{\theta }\oplus \nu ,\end{array}$$

(12) 

where $\nu$ is an invariant normal sub-bundle of the normal bundle $T^{\perp }M$. A CR-slant product submanifold M is called semi-slant mixed-totally geodesic (resp., hemi-slant mixed-totally geodesic) if its second fundamental form satisfies

$$\begin{array}{cc}\hfill & h(X_1,X_2)=0\forall X_1\in \Gamma \left(\mathfrak{D}\right),\forall X_2\in \Gamma \left({\mathfrak{D}}^{\theta }\right)\hfill \\ & (resp.,h(X_2,X_3)=0\forall X_2\in \Gamma \left({\mathfrak{D}}^{\theta }\right),\forall X_3\in \Gamma \left({\mathfrak{D}}^{\perp }\right)).\hfill \end{array}$$

3. CR-Slant Warped Products $\left({\mathit{M}}_{\mathit{T}}\times {\mathit{M}}_{\perp }\right){\times }_{\mathit{f}}{\mathit{M}}_{\mathit{\theta }}$

In this section, first we recall the definition of warped product manifolds which are the generalizations of Riemannian products. In 1969, Bishop and O’Neill [22] introduced the notion of warped product manifolds as follows:

Definition 3. 

A warped product $B{\times }_{f}F$ of two Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ is the product manifold $M=B\times F$ equipped with the product structure

$$\begin{array}{c}\hfill g_M(X,Y)=g_B({\pi }_{1*}X,{\pi }_{1*}Y)+{(f\circ {\pi }_1)}^2{g}_F({\pi }_{2*}X,{\pi }_{2*}Y),\end{array}$$

where $f:B\to (0,\infty )$ and ${\pi }_1:M\to B,{\pi }_2:M\to F$ are projection maps given by ${\pi }_1(p,q)=p$ and ${\pi }_2(p,q)=q$ for any $(p,q)\in B\times F$ and ∗ denotes the symbol for tangent map.

The function f is called warping function, if f is constant, then M is simply a Riemannian product. It is known that, for any vector field X on B and a vector field Z on F, we have [22,23]

$$\begin{array}{c}\hfill {\nabla }_{X}Z={\nabla }_{Z}X=X(lnf)Z,\end{array}$$

(13) 

where ∇ is the Levi–Civita connection on M. Further, it is well known that the base manifold B is totally geodesic and the fiber F is totally umbilical in M.

Next, we define CR-slant warped products $\left(M_T\times M_{\perp }\right){\times }_f{M}_{\theta }$ as follows.

Definition 4. 

A submanifold M of an almost Hermitian manifold $\tilde{M}$ is said to be CR-slant warped product submanifold if it is a warped product of CR-product $M_T\times M_{\perp }$ and a proper θ-slant submanifold $M_{\theta }$ of $\tilde{M}$.

In [14], we studied CR-slant warped product submanifolds of the form $B_1{\times }_f{M}_{\perp }$, where $B_1=M_T\times M_{\theta }$. In this section, we study CR-slant warped products of the form $B_2{\times }_f{M}_{\theta }$, where $B_2=M_T\times M_{\perp }$. For this, we use the following conventions, $X_1,Y_1,\cdots$ are vector fields on $\mathfrak{D}$ and $X_2,Y_2,\cdots$ are vector fields on ${\mathfrak{D}}^{\theta }$, while $X_3,Y_3,\cdots$ are vector fields on ${\mathfrak{D}}^{\perp }$.

First, we have the following preparatory lemmas.

Lemma 1. 

On a CR-slant warped product submanifold $M=B_2{\times }_f{M}_{\theta }$ of a nearly Kaehler manifold $\tilde{M}$, we have

(i) 

$g(h(X_1,Y_1),F{X}_2)=0,$

(ii) 

$2g(h(X_3,Y_3),F{X}_2)=g(h(X_2,X_3),J{Y}_3)+g(h(X_2,Y_3),J{X}_3).$

for any $X_1,Y_1\in \Gamma \left(T{M}_T\right),X_2\in \Gamma \left(T{M}_{\theta }\right)$ and $X_3,Y_3\in \Gamma \left(T{M}_{\perp }\right)$, where $B_2=M_T\times M_{\perp }$ is the CR-product submanifold in $\tilde{M}$.

Proof. 

The first part is easy to prove by using (3), (4) and (13). For the second part, we have

$$\begin{array}{cc}\hfill g(h(X_3,Y_3),F{X}_2)=& g({\tilde{\nabla }}_{X_3}{Y}_3,J{X}_2)+g({\tilde{\nabla }}_{X_3}P{X}_2,Y_3)\hfill \\ & -g(J{\nabla }_{X_3}{Y}_3,X_2)+g({\nabla }_{X_3}{Y}_3,P{X}_2)\hfill \end{array}$$

for any $X_2\in \Gamma \left(T{M}_{\theta }\right)$ and $X_3,Y_3\in \Gamma \left(T{M}_{\perp }\right)$. Since ${\nabla }_{X_3}{Y}_3\in \Gamma \left(T{M}_{\perp }\right)$, then using orthogonality of vector fields and covariant derivative property of J with (13), we find

$$\begin{array}{cc}\hfill g(h(X_3,Y_3),F{X}_2)& =g(\left({\tilde{\nabla }}_{X_3}J\right)Y_3,X_2)-g({\tilde{\nabla }}_{X_3}J{Y}_3,X_2)+X_3(lnf)g(P{X}_2,Y_3)\hfill \\ \hfill & =g(\left({\tilde{\nabla }}_{X_3}J\right)Y_3,X_2)+g(h(X_2,X_3),J{Y}_3).\hfill \end{array}$$

(14) 

Similarly, by interchanging $X_3$ with $Y_3$ in (14), we brain

$$\begin{array}{c}\hfill g(h(X_3,Y_3),F{X}_2)=g(\left({\tilde{\nabla }}_{Y_3}J\right)X_3,X_2)+g(h(X_2,Y_3),J{X}_3).\end{array}$$

(15) 

Hence, the second part immediately follows from (14) and (15). □

Lemma 2. 

Let $M=B_2{\times }_f{M}_{\theta }$ be a CR-slant warped product submanifold of a nearly Kaehler manifold $\tilde{M}$, such that $B_2=M_T\times M_{\perp }$ is the CR-product submanifold in $\tilde{M}$. Then, we have

$$\begin{array}{c}\hfill g(h(X_1,X_3),F{X}_2)=\frac{1}{2}g(h(X_1,X_2),J{X}_3),\end{array}$$

(16) 

for any $X_1\in \Gamma \left(T{M}_T\right),X_2\in \Gamma \left(T{M}_{\theta }\right)$ and $X_3\in \Gamma \left(T{M}_{\perp }\right)$.

Proof. 

For any $X_1\in \Gamma \left(T{M}_T\right),X_2\in \Gamma \left(T{M}_{\theta }\right)$ and $X_3\in \Gamma \left(T{M}_{\perp }\right)$, we have

$$\begin{array}{c}\hfill g(h(X_1,X_3),F{X}_2)=g(\left({\tilde{\nabla }}_{X_3}J\right)X_1,X_2)-g({\tilde{\nabla }}_{X_3}J{X}_1,X_2)=g(\left({\tilde{\nabla }}_{X_3}J\right)X_1,X_2).\end{array}$$

(17) 

On the other hand, we know that

$$\begin{array}{c}\hfill g(h(X_1,X_3),F{X}_2)=g(\left({\tilde{\nabla }}_{X_1}J\right)X_3,X_2)-g({\tilde{\nabla }}_{X_1}J{X}_3,X_2)+g(X_3,{\tilde{\nabla }}_{X_1}P{X}_2).\end{array}$$

(18) 

Then, the lemma follows from (17) and (18) with the help of (3) and (13). □

Lemma 3. 

For a proper CR-slant warped product $M=B_2{\times }_f{M}_{\theta }$, such that $B_2=M_T\times M_{\perp }$ in a nearly Kaehler manifold $\tilde{M}$, we have

$$\begin{array}{c}\hfill g(h(J{X}_1,X_2),F{Y}_2)=X_1(lnf)g(X_2,Y_2)+\frac{1}{3}J{X}_1(lnf)g(X_2,P{Y}_2),\end{array}$$

(19) 

for any $X_1\in \Gamma \left(T{M}_T\right),X_2,Y_2\in \Gamma \left(T{M}_{\theta }\right)$.

Proof. 

From (4) and (13), we have

$$\begin{array}{c}\hfill g(h(X_1,X_2),F{Y}_2)=g(\left({\tilde{\nabla }}_{X_2}J\right)X_1,Y_2)-J{X}_1(lnf)g(X_2,Y_2),\end{array}$$

(20) 

for any orthogonal vector fields $X_1\in \Gamma \left(T{M}_T\right),X_2,Y_2\in \Gamma \left(T{M}_{\theta }\right)$. On the other hand, we derive

$$\begin{array}{c}\hfill g(h(X_1,X_2),F{Y}_2)=g(\left({\tilde{\nabla }}_{X_1}J\right)X_2,Y_2)-X_1(lnf)g(P{X}_2,Y_2)+g(h(X_1,Y_2),F{X}_2).\end{array}$$

(21) 

Then, from (20) and (21), we find

$$\begin{array}{c}\hfill 2g(h(X_1,X_2),F{Y}_2)=X_1(lnf)g(X_2,P{Y}_2)-J{X}_1(lnf)g(X_2,Y_2)+g(h(X_1,Y_2),F{X}_2).\end{array}$$

(22) 

Interchanging $X_2$ by $Y_2$, we obtain

$$\begin{array}{c}\hfill 2g(h(X_1,Y_2),F{X}_2)=X_1(lnf)g(P{X}_2,Y_2)-J{X}_1(lnf)g(X_2,Y_2)+g(h(X_1,X_2),F{Y}_2).\end{array}$$

(23) 

Then, from (22) and (23), we derive

$$\begin{array}{c}\hfill g(h(X_1,X_2),F{Y}_2)=-J{X}_1(lnf)g(X_2,Y_2)+\frac{1}{3}{X}_1(lnf)g(X_2,P{Y}_2).\end{array}$$

(24) 

Hence, (19) follows immediately by interchanging $X_1$ with $J{X}_1$ in (24), which proves the lemma completely. □

The following relations are immediate consequences of (19).

$$\begin{array}{cc}\hfill & g(h(J{X}_1,P{X}_2),F{Y}_2)=X_1(lnf)g(P{X}_2,Y_2)+\frac{1}{3}{cos}^2\theta J{X}_1(lnf)g(X_2,Y_2),\hfill \end{array}$$

(25) 

$$\begin{array}{cc}\hfill & g(h(J{X}_1,P{X}_2),FP{Y}_2)={cos}^2\theta X_1(lnf)g(X_2,Y_2)+\frac{1}{3}{cos}^2\theta J{X}_1(lnf)g(X_2,P{Y}_2),\hfill \end{array}$$

(26) 

$$\begin{array}{cc}\hfill & g(h(J{X}_1,X_2),FP{Y}_2)=X_1(lnf)g(X_2,P{Y}_2)-\frac{1}{3}{cos}^2\theta J{X}_1(lnf)g(X_2,Y_2).\hfill \end{array}$$

(27) 

Lemma 4. 

Let $M=B_2{\times }_f{M}_{\theta }$ be a CR-slant warped product submanifold of a nearly Kaehler manifold $\tilde{M}$ such that $B_2=M_T\times M_{\perp }$ is the CR-product submanifold in $\tilde{M}$. Then, we have

$$\begin{array}{c}\hfill g(h(X_2,Y_2),J{X}_3)=g(h(X_2,X_3),F{Y}_2)+\frac{1}{3}{X}_3(lnf)g(X_2,P{Y}_2)\end{array}$$

(28) 

for any $X_2,Y_2\in \Gamma \left(T{M}_{\theta }\right)$ and $X_3\in \Gamma \left(T{M}_{\perp }\right)$.

Proof. 

From the definition of covariant derivative with (4) and (8), we have

$$\begin{array}{c}\hfill g(h(X_2,X_3),F{Y}_2)=g(\left({\tilde{\nabla }}_{X_3}J\right)X_2,Y_2)-g({\tilde{\nabla }}_{X_3}P{X}_2,Y_2)-g({\tilde{\nabla }}_{X_3}F{X}_2,Y_2)-g({\tilde{\nabla }}_{X_3}{X}_2,P{Y}_2).\end{array}$$

Again, using (4), (5), and (13), we find

$$\begin{array}{c}\hfill g(h(X_2,X_3),F{Y}_2)=g(\left({\tilde{\nabla }}_{X_3}J\right)X_2,Y_2)+g(h(Y_2,X_3),F{X}_2).\end{array}$$

(29) 

On the other hand, we derive

$$\begin{array}{cc}\hfill g(h(X_2,X_3),F{Y}_2)& =g(\left({\tilde{\nabla }}_{X_2}J\right)X_3,Y_2)-g({\tilde{\nabla }}_{X_2}J{X}_3,Y_2)-g({\tilde{\nabla }}_{X_2}{X}_3,P{Y}_2)\hfill \\ \hfill & =g(\left({\tilde{\nabla }}_{X_2}J\right)X_3,Y_2)+g(h(X_2,Y_2),J{X}_3)-X_3(lnf)g(X_2,P{Y}_2).\hfill \end{array}$$

(30) 

Then, from (29) and (30), we obtain

$$\begin{array}{c}\hfill 2g(h(X_2,X_3),F{Y}_2)=g(h(X_2,Y_2),J{X}_3)+g(h(Y_2,X_3),F{X}_2)-X_3(lnf)g(X_2,P{Y}_2).\end{array}$$

(31) 

Interchanging $X_2$ by $Y_2$, we obtain

$$\begin{array}{c}\hfill 2g(h(Y_2,X_3),F{X}_2)=g(h(X_2,Y_2),J{X}_3)+g(h(X_2,X_3),F{Y}_2)+X_3(lnf)g(X_2,P{Y}_2).\end{array}$$

(32) 

Then, from (31) and (32), we obtain (28); which proves the Lemma completely. □

4. Chen’s Inequality and Its Consequences

In this section, first we prove the following main result by using Lemma 3.

Theorem 2. 

Let $M=B_2{\times }_f{M}_{\theta }$ be a proper CR-slant warped product submanifold of a nearly Kaehler manifold $\tilde{M}$. Then, M is a Riemannian product if, and only if, either M is semi-slant mixed-totally geodesic, (i.e., $h(X_1,X_2)=0,\forall X_1\in \Gamma \left(\mathfrak{D}\right),X_2\in \Gamma \left({\mathfrak{D}}^{\theta }\right))$ or $h(\mathfrak{D},{\mathfrak{D}}^{\theta })$ is orthogonal to $F{\mathfrak{D}}^{\theta }$.

Proof. 

From Lemma 3, we find

$$\begin{array}{c}\hfill g(h(J{X}_1,X_2),F{Y}_2)=\frac{1}{3}J{X}_1(lnf)g(X_2,P{Y}_2)+X_1(lnf)g(X_2,Y_2),\end{array}$$

(33) 

for any $X_1\in \Gamma \left(\mathfrak{D}\right),X_2,Y_2\in \Gamma \left({\mathfrak{D}}^{\theta }\right)$. Then, from (27) and (33), we derive

$$\begin{array}{c}\hfill g(h(J{X}_1,X_2),F{Y}_2)+\frac{1}{3}g(h(X_1,X_2),FP{Y}_2)=\left(1-\frac{1}{9}{cos}^2\theta \right)X_1(lnf)g(X_2,Y_2).\end{array}$$

(34) 

If M is semi-slant mixed totally geodesic or $h(\mathfrak{D},{\mathfrak{D}}^{\theta })$ is orthogonal to $F{\mathfrak{D}}^{\theta }$ then from (34), we find

$$\begin{array}{c}\hfill \left(1-\frac{1}{9}{cos}^2\theta \right)X_1(lnf)g(X_2,Y_2)=0.\end{array}$$

Since g is a Riemannian metric and $-1\le cos\theta \le 1$, then from above equation we obtain $X_1(lnf)=0$, i.e., f is constant along $M_T$.

Conversely, if f is constant then again from (34), we obtain

$$\begin{array}{c}\hfill g(h(J{X}_1,X_2),F{Y}_2)+\frac{1}{3}g(h(X_1,X_2),FP{Y}_2)=0.\end{array}$$

(35) 

Interchanging $X_1$ by $J{X}_1$ and $Y_2$ by $P{Y}_2$ in (35), we derive

$$\begin{array}{c}\hfill g(h(X_1,X_2),FP{Y}_2)+\frac{1}{3}{cos}^2\theta g(h(J{X}_1,X_2),F{Y}_2)=0.\end{array}$$

(36) 

Then, from (35) and (36), we obtain

$$\begin{array}{c}\hfill \left(1-\frac{1}{9}{cos}^2\theta \right)g(h(J{X}_1,X_2),F{Y}_2)=0.\end{array}$$

(37) 

Since $-1\le cos\theta \le 1$ for any value of $\theta \in \mathbb{R}$, thus we find either $h(\mathfrak{D},{\mathfrak{D}}^{\theta })=\left\{0\right\}$ or $h(\mathfrak{D},{\mathfrak{D}}^{\theta })$ is orthogonal to $F{\mathfrak{D}}^{\theta }$, which completes the proof. □

Next, we derive the Chen’s inequality for CR-slant wanted products $M=B_2{\times }_f{M}_{\theta }$, where $B_2=M_T\times M_{\perp }$ is a CR-product in a nearly Kaehler manifold.

Theorem 3. 

Let $M=\left(M_T^{n_1}\times M_{\perp }^{n_2}\right){\times }_f{M}_{\theta }^{n_3}$ be a CR-slant warped product submanifold of a nearly Kaehler manifold $\tilde{M}$, such that M is hemi-slant mixed-totally geodesic. Then, the squared norm of the second fundamental form satisfies

$$\begin{array}{c}\hfill {\parallel h\parallel }^2\ge \frac{1}{9}{n}_3{cos}^2\theta \parallel {\overrightarrow{\nabla }}^{\perp }{(lnf)\parallel }^2+2n_3\left(1+\frac{10}{9}{cot}^2\theta \right){\parallel {\overrightarrow{\nabla }}^T(lnf)\parallel }^2\end{array}$$

(38) 

where ${\overrightarrow{\nabla }}^T(lnf)$ and ${\overrightarrow{\nabla }}^{\perp }(lnf)$ denote the gradient components of $lnf$ along $M_T$ and $M_{\perp }$, respectively.

Furthermore, if the equality holds in (38), then $M_T\times M_{\perp }$ is totally geodesic and $M_{\theta }$ is totally umbilical in $\tilde{M}$. Moreover, M is not a semi-slant mixed totally geodesic submanifold of $\tilde{M}$.

Proof. 

If we denote the tangent bundles of $M_T,M_{\perp }$ and $M_{\theta }$ by $\mathfrak{D},{\mathfrak{D}}^{\perp }$ and ${\mathfrak{D}}^{\theta }$, respectively; then we use the following frame fields for the CR-slant warped product

$$\begin{array}{cc}\hfill & \mathfrak{D}=\mathrm{Span}\{e_1,\cdots ,e_p,e_{p+1}=J{e}_1,\cdots ,e_{n_1}=e_{2p}=J{e}_p\},\hfill \\ \hfill & {\mathfrak{D}}^{\perp }=\mathrm{Span}\{e_{n_1+1}={\widehat{e}}_1,\cdots ,e_{n_1+n_2}={\widehat{e}}_{n_2}\},\hfill \\ \hfill & {\mathfrak{D}}^{\theta }=\mathrm{Span}\{e_{n_1+n_2+1}=e_1^{*},\cdots ,e_{n_1+n_2+q}=e_q^{*},e_{n_1+n_2+q+1}=sec\theta P{e}_1^{*},\cdots ,\hfill \\ \hfill & e_n=e_{2q}^{*}=sec\theta P{e}_q^{*}\}.\hfill \end{array}$$

Additionally, the normal bundle frame will be

$$\begin{array}{cc}\hfill & J{\mathfrak{D}}^{\perp }=\mathrm{Span}\{e_{n+1}={\tilde{e}}_1=J{\widehat{e}}_1,\cdots ,e_{n+n_2}={\tilde{e}}_{n_2}=J{\widehat{e}}_{n_2}\},\hfill \\ \hfill & F{\mathfrak{D}}^{\theta }=\mathrm{Span}\{e_{n+n_2+1}={\tilde{e}}_{n_2+1}=E_1^{*}=csc\theta F{e}_1^{*},\cdots ,e_{n+n_2+q}={\tilde{e}}_{n_2+q}=E_q^{*}=csc\theta F{e}_q^{*},\hfill \\ \hfill & e_{n+n_2+q+1}={\tilde{e}}_{n_2+q+1}=E_{q+1}^{*}=csc\theta sec\theta FP{e}_1^{*},\cdots ,\hfill \\ \hfill & e_{n+n_2+n_3}={\tilde{e}}_{n_2+n_3}=E_{n_3}^{*}=csc\theta sec\theta FP{e}_q^{*}\},\hfill \\ \hfill & \nu =\mathrm{Span}\{e_{n+n_2+n_3+1}={\tilde{e}}_{n_2+n_3+1},\cdots ,e_{2m}={\tilde{e}}_{2m-n-n_2-n_3}\}.\hfill \end{array}$$

From the definition of h, we find

$$\begin{array}{cc}\hfill {\parallel h\parallel }^2& ={\parallel h(\mathfrak{D},\mathfrak{D})\parallel }^2+\parallel h({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\perp }){\parallel }^2+{\parallel h({\mathfrak{D}}^{\theta },{\mathfrak{D}}^{\theta })\parallel }^2\hfill \\ \hfill & +2\left(\parallel h(\mathfrak{D},{\mathfrak{D}}^{\perp }){\parallel }^2+\parallel h(\mathfrak{D},{\mathfrak{D}}^{\theta }){\parallel }^2+{\parallel h({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\theta })\parallel }^2\right).\hfill \end{array}$$

(39) 

Using the frame fields and preparatory lemmas, we expand each term of (39) as follows:

$$\begin{array}{cc}\hfill {\parallel h(\mathfrak{D},\mathfrak{D})\parallel }^2=& \sum _{k=1}^{n_2}\sum _{i,j=1}^{n_1}{\left(g(h(e_i,e_j),J{\widehat{e}}_k)\right)}^2+\sum _{k=1}^{n_3}\sum _{i,j=1}^{n_1}{\left(g(h(e_i,e_j),E_k^{*})\right)}^2\hfill \\ \hfill & +\sum _{k=1}^{2m-n-n_2-n_3}\sum _{i,j=1}^{n_1}{\left(g(h(e_i,e_j),{\tilde{e}}_k)\right)}^2.\hfill \end{array}$$

Leaving the $\nu$-components terms and the is no warped product relation for the first term, then from Lemma 1 (i), we obtain

$$\begin{array}{c}\hfill {\parallel h(\mathfrak{D},\mathfrak{D})\parallel }^2\ge 0.\end{array}$$

(40) 

Similarly, for the second term of (39), we derive

$$\begin{array}{cc}\hfill \parallel h({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\perp }){\parallel }^2=& \sum _{k=1}^{n_2}\sum _{i,j=1}^{n_2}{\left(g(h({\widehat{e}}_i,{\widehat{e}}_j),J{\widehat{e}}_k)\right)}^2+\sum _{k=1}^{n_3}\sum _{i,j=1}^{n_2}{\left(g(h({\widehat{e}}_i,{\widehat{e}}_j),E_k^{*})\right)}^2\hfill \\ \hfill & +\sum _{k=1}^{2m-n-n_2-n_3}\sum _{i,j=1}^{n_2}{\left(g(h({\widehat{e}}_i,{\widehat{e}}_j),{\tilde{e}}_k)\right)}^2.\hfill \end{array}$$

Using Lemma 1 (ii) with the given hemi-slant totally geodesic condition and leaving the first and last positive terms, we find

$$\begin{array}{c}\hfill \parallel h({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\perp }){\parallel }^2\ge 0.\end{array}$$

(41) 

For the third term of (39), we find

$$\begin{array}{cc}\hfill \parallel h({\mathfrak{D}}^{\theta },{\mathfrak{D}}^{\theta }){\parallel }^2=& \sum _{k=1}^{n_2}\sum _{i,j=1}^{n_3}{\left(g(h(e_i^{*},e_j^{*}),J{\widehat{e}}_k)\right)}^2+\sum _{k=1}^{n_3}\sum _{i,j=1}^{n_3}{\left(g(h(e_i^{*},e_j^{*}),E_k^{*})\right)}^2\hfill \\ \hfill & +\sum _{k=1}^{2n-n_2-n_3}\sum _{i,j=1}^{n_3}{\left(g(h(e_i^{*},e_j^{*}),{\tilde{e}}_k)\right)}^2.\hfill \end{array}$$

Leaving the last two positive terms and using Lemma 4 with mixed totally geodesic condition, we obtain

$$\begin{array}{c}\hfill \parallel h({\mathfrak{D}}^{\theta },{\mathfrak{D}}^{\theta }){\parallel }^2\ge \frac{2q}{9}{cos}^2\theta \sum _{k=1}^{n_2}{\left(e_k(lnf)\right)}^2=\frac{1}{9}{n}_3{cos}^2\theta {\parallel {\overrightarrow{\nabla }}^{\perp }(lnf)\parallel }^2.\end{array}$$

(42) 

Similarly, we derive the other terms of (39) as follows

$$\begin{array}{cc}\hfill \parallel h(\mathfrak{D},{\mathfrak{D}}^{\perp }){\parallel }^2=& \sum _{k,j=1}^{n_2}\sum _{i=1}^{n_1}{\left(g(h(e_i,{\widehat{e}}_j),J{\widehat{e}}_k)\right)}^2+\sum _{k=1}^{n_3}\sum _{i=1}^{n_1}\sum _{j=1}^{n_2}{\left(g(h(e_i,{\widehat{e}}_j),E_k^{*})\right)}^2\hfill \\ \hfill & +\sum _{k=1}^{2m-n-n_2-n_3}\sum _{i=1}^{n_1}\sum _{j=1}^{n_2}{\left(g(h(e_i,{\widehat{e}}_j),{\tilde{e}}_k)\right)}^2.\hfill \end{array}$$

There is no relation for the first positive term in terms of warped products and leaving the last $\nu$-components term. Then, using Lemma 2, we derive

$$\begin{array}{cc}\hfill \parallel h(\mathfrak{D},{\mathfrak{D}}^{\perp }){\parallel }^2& \ge \frac{1}{4}\sum _{j=1}^{n_2}\sum _{i=1}^{n_1}\sum _{k=1}^{n_3}{\left(g(h(e_i,e_k^{*}),J{\widehat{e}}_j)\right)}^2\ge 0.\hfill \end{array}$$

(43) 

On the other hand, we also have

$$\begin{array}{cc}\hfill \parallel h(\mathfrak{D},{\mathfrak{D}}^{\theta }){\parallel }^2=& \sum _{k=1}^{n_2}\sum _{j=1}^{n_3}\sum _{i=1}^{n_1}{\left(g(h(e_i,e_j^{*}),J{\widehat{e}}_k)\right)}^2+\sum _{k,j=1}^{n_3}\sum _{i=1}^{n_1}{\left(g(h(e_i,e_j^{*}),E_k^{*})\right)}^2\hfill \\ \hfill & +\sum _{k=1}^{2m-n-n_2-n_3}\sum _{i=1}^{n_1}\sum _{j=1}^{n_3}{\left(g(h(e_i,e_j^{*}),{\tilde{e}}_k)\right)}^2.\hfill \end{array}$$

For the first term we use (43) and omit the $\nu$-components terms and using frame fields of ${\mathfrak{D}}^{\theta }$ and $F{\mathfrak{D}}^{\theta }$, we derive

$$\begin{array}{cc}\hfill \parallel h(\mathfrak{D},{\mathfrak{D}}^{\theta }){\parallel }^2\ge & {csc}^2\theta \sum _{k,j=1}^q\sum _{i=1}^{n_1}{\left(g(h(e_i,e_j^{*}),F{e}_k^{*})\right)}^2\hfill \\ & +{csc}^2\theta {sec}^2\theta \sum _{k,j=1}^q\sum _{i=1}^{n_1}{\left(g(h(e_i,T{e}_j^{*}),F{e}_k^{*})\right)}^2\hfill \\ \hfill & +{csc}^2\theta {sec}^2\theta \sum _{k,j=1}^q\sum _{i=1}^{n_1}{\left(g(h(e_i,e_j^{*}),FT{e}_k^{*})\right)}^2\hfill \\ \hfill & +{csc}^2\theta {sec}^4\theta \sum _{k,j=1}^q\sum _{i=1}^{n_1}{\left(g(h(e_i,T{e}_j^{*}),FT{e}_k^{*})\right)}^2.\hfill \end{array}$$

Using Lemma 3 with (24)–(27), we obtain

$$\begin{array}{cc}\hfill \parallel h(\mathfrak{D},{\mathfrak{D}}^{\theta }){\parallel }^2& \ge 2q{csc}^2\theta \sum _{i=1}^p\left[{\left(J{e}_i(lnf)\right)}^2+{\left(e_i(lnf)\right)}^2\right]\hfill \\ \hfill & +\frac{2q}{9}{cot}^2\theta \sum _{i=1}^p\left[{\left(J{e}_i(lnf)\right)}^2+{\left(e_i(lnf)\right)}^2\right]\hfill \\ \hfill & =2q{csc}^2\theta \sum _{i=1}^{n_1}{\left(e_i(lnf)\right)}^2+\frac{2q}{9}{cot}^2\theta \sum _{i=1}^{n_1}{\left(e_i(lnf)\right)}^2\hfill \\ \hfill & =n_3\left({csc}^2\theta +\frac{1}{9}{n}_3{cot}^2\theta \right){\parallel {\overrightarrow{\nabla }}^T(lnf)\parallel }^2.\hfill \end{array}$$

(44) 

Last term of (39) is identically zero by the hemi-slant mixed totally geodesic condition. Then, for all values of h from (40)–(44), finally we obtain the required inequality (38).

For the equality case, since M is ${\mathfrak{D}}^{\perp }\oplus {\mathfrak{D}}^{\theta }$-mixed totally geodesic, i.e.,

$$\begin{array}{c}\hfill h({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\theta })=\left\{0\right\}.\end{array}$$

(45) 

Form the leaving and vanishing terms, we also find

$$\begin{array}{cc}\hfill & h(\mathfrak{D},\mathfrak{D})=\left\{0\right\},h({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\perp })=\left\{0\right\},h(\mathfrak{D},{\mathfrak{D}}^{\perp })=\left\{0\right\},\hfill \\ \hfill & h({\mathfrak{D}}^{\theta },{\mathfrak{D}}^{\theta })\subseteq J{\mathfrak{D}}^{\perp },h(\mathfrak{D},{\mathfrak{D}}^{\theta })\subseteq F{\mathfrak{D}}^{\theta }.\hfill \end{array}$$

(46) 

Then, $M_T\times M_{\perp }$ is totally geodesic and $M_{\theta }$ is totally umbilical in $\tilde{M}$ due to the fact that $M_T\times M_{\perp }$ is totally geodesic and $M_{\theta }$ is totally umbilical in M [6,22] with equality holding case of (46). Furthermore, due to Theorem 2 and Lemma 2, we observe that M is not a $\mathfrak{D}\oplus {\mathfrak{D}}^{\theta }$-mixed totally geodesic submanifold of $\tilde{M}$. Hence, the proof is complete. □

Now, we give the following consequences of Theorem 3.

A warped submanifold of the form $M=M_{\theta }{\times }_f{M}_{\perp }$ in a nearly Kaehler manifold $\tilde{M}$ is called hemi-slant if $M_{\perp }$ is a totally real submanifold and $M_{\theta }$ is a proper slant submanifold.

If $dim{M}_T=0$ in Theorem 3, then we have

Theorem 4. 

Let $M=M_{\perp }^{n_1}{\times }_f{M}_{\theta }^{n_2}$ be a mixed totally geodesic hemi-slant warped product submanifold in a nearly Kaehler manifold $\tilde{M}$. Then

(i) 

The second fundamental form h of M satisfies

$$\begin{array}{c}\hfill {\left|\right|h\left|\right|}^2\ge \frac{1}{9}{n}_2{cos}^2\theta \left|\right|{\overrightarrow{\nabla }}^{\perp }(lnf){\left|\right|}^2,\end{array}$$

(47) 

where ${\overrightarrow{\nabla }}^{\perp }(lnf)$ is the gradient of $lnf$ along $M_{\perp }$.

(ii) 

If the equality sign of (47) holds identically, then $M_{\perp }$ and $M_{\theta }$ are totally geodesic and totally umbilical submanifolds of $\tilde{M}$, respectively.

On the other hand, if $M_{\perp }=\left\{0\right\}$, we have the following special case of Theorem 3.

Theorem 5 

([24]). Let $M=M_T^{n_1}{\times }_f{M}_{\theta }^{n_2}$ be a semi-slant warped product submanifold in a nearly Kaehler manifold $\tilde{M}$. Then, we have

(i) 

The second fundamental form h and the warping function f satisfy

$$\begin{array}{c}\hfill {\parallel h\parallel }^2\ge 2n_2\left(1+\frac{10}{9}{cot}^2\theta \right){\parallel {\overrightarrow{\nabla }}^T(lnf)\parallel }^2.\end{array}$$

(48) 

where ${\overrightarrow{\nabla }}^{T}lnf$ is gradient of $lnf$ along $M_T$.

(ii) 

If the equality sign in (48) holds identically, then $M_T$ is totally geodesic and $M_{\theta }$ is totally umbilical in $\tilde{M}$. Moreover, M is a minimal submanifold in $\tilde{M}$.

Furthermore, if $dim{M}_{\perp }=0$ and $\theta =\frac{\pi }{2}$ in Theorem 3, then $M=M_T^{n_1}{\times }_f{M}_{\perp }^{n_2}$ is a CR-warped product submanifold of a nearly Kaehler manifold $\tilde{M}$ and they were studied in [25] and, hence, the main Theorem 4.2 of [25] is a special case of Theorem 3.