Geometric Classification of Warped Products Isometrically Immersed into Conformal Sasakian Space Froms

Abstract

Warped products play important roles in differential geometry, general relativity, and symmetry science. In this paper, we study the warped product pointwise semi-slant submanifolds that are isometrically immersed into conformal Sasakian space form. We show that there does not exist any proper warped product pointwise semi-slant submanifolds in conformal Sasakian manifolds. We derived some geometric inequalities for squared norm of second fundamental form from a warped product pointwise semi-slant submanifold into a conformal Sasakian manifolds.

1. Introduction

In 1969, R. L. Bishop and B. O’Neill introduce the geometry of warped products. Warped product manifolds are generalized by product manifolds. This concept is used to construct Riemannian manifolds with negative sectional curvatures. Warped product manifolds can give a natural framework for time-dependent mechanic systems, and there are many applications in physics and relativity. Some applications of warped product manifolds are that they provide a good setting to model space-time close to black holes or bodies with large gravitational fields. For instance, the best relativistic model of Schwarzschild space-time describes the outer space around a massive star or black hole as a warped product. This makes the research for warped product manifolds significant geometrically as well. Chen in [1] and Papaghiuc in [2] studied slant and semi-slant submanifolds in almost Hermitian manifolds. The slant submanifolds are shown by the natural generalized by the complex (holomorphic) and the totally real submanifolds. In [3], Etayo introduced the notions of pointwise slant submanifolds in almost Hermitian manifolds and studied quasi-slant submanifolds as generalized by slant and semi-slant submanifolds. In recent years, Park [4] presented the notions of pointwise slant and semi-slant submanifolds in almost contact metric manifolds. Park presented some examples and classification results of these notions [4]. Sahin in [5] presented some classification results and classical examples about warped product pointwise semi-slant submanifolds in Kaehler manifolds. In [6], Chen studied the geometric properties of pointwise semi-slant warped products in locally conformal Kaehler manifolds. In [7], a new class of warped products were introduced, which are called as generic warped products in locally produced Riemannian manifolds. Later, Uddin [8] developed notions of geometric of warped product pointwise semi-slant submanifolds in locally produced Riemannian manifolds. Park [4] has done work in cosympletic, Kenmotsu, and Sasakian manifolds. Based on previous research, we consider the concept of warped product pointwise semi-slant submanifolds in the conformal Sasakian space forms as generalized by contact CR-warped products. In this paper, we will try to learn the geometric of the second fundamental form of a compact, oriented Riemannian submanifold M that is isometrically immersed into a conformal Sasakian space form $\overline{M\left(c\right)}$. Symmetry is one of the most basic and important notions in all fields of science, technology, and art. Warped products play important roles in differential geometry, as well as in general relativity and symmetry science. We could find some papers about symmetry properties related with this paper [9,10,11]. Singularity theory and submanifold theory are significant fields of modern mathematical research that are useful tools to study the symmetry properties. Our future work will combine the results in this paper with the methods and techniques of singularity theory and submanifolds theory presented in [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] to explore new results and theorems related with more symmetric properties.

2. Preliminaries

Consider $(2m+1)$-dimentional manifold $\tilde{M}$ along with an almost contact structure $(\varphi ,\xi ,\eta ,g)$, and it is said to be an almost contact metric manifold if the following properties are satisfied:

$$\begin{array}{c}\hfill {\varphi }^2=-I+\eta \otimes \xi ,\eta \left(\xi \right)=1,\varphi \left(\xi \right)=0,\eta \circ \varphi =0,\end{array}$$

(1)

$$\begin{array}{c}\hfill g(\varphi X,\varphi Y)=g(X,Y)-\eta \left(X\right)\eta \left(Y\right),and\eta \left(X\right)=g(X,\xi ),\end{array}$$

(2)

for any vector fields X,Y $\in \Gamma \left(T\tilde{M}\right)$, where $\varphi$ is a tensor field of type (1,1), $\xi$ a vector field, $\eta$ a 1-form, and g a Riemannian metric. Further, an almost contact metric manifold is a Sasakian manifold if

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_X\varphi \right)Y=g(X,Y)\xi -\eta \left(Y\right)X,\end{array}$$

(3)

for vector field $X,Y\in \tilde{M}$, where $\tilde{\nabla }$ denotes covariant derivative with respect to metric g, and we denote Lie algebra of vector fields by $T\tilde{M}$.

The Gauss and Weingarten formulas for the immersion of M in $\tilde{M}$ are given by

$$\begin{array}{c}\hfill \left(i\right){\tilde{\nabla }}_{X}Y={\nabla }_{X}Y+\sigma (X,Y),\left(ii\right){\tilde{\nabla }}_{X}N=-A_{N}X+{\nabla }_X^{\perp }N,\end{array}$$

(4)

where ∇ and ${\nabla }^{\perp }$ are the induced connections on the tangent bundle $TM$ and normal bundle $T^{\perp }M$, $X,Y\in \Gamma \left(TM\right)$ and N $\in \Gamma \left(T{M}^{\perp }\right)$, $\sigma$ the second fundamental form, and A the shape operator. Again we have

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

(5)

Now for $X\in \Gamma \left(TM\right)$ and $N\in \Gamma \left(T^{\perp }M\right)$ we can take

$$\begin{array}{c}\hfill \left(i\right)\varphi X=PX+FX\left(ii\right)\varphi N=tN+fN,\end{array}$$

(6)

where $PX\left(tN\right)$ and $FX\left(fN\right)$ are the tangential and normal parts of $\varphi X\left(\varphi N\right)$, respectively. From (1) and (6), it is easy to observe that for each $X,Y,\in \Gamma \left(TM\right)$, we have

$$\begin{array}{c}\hfill \left(i\right)g(PX,Y)=-{g(X,PY)\left(ii\right)\parallel P\parallel }^2=\sum _{k,s=1}^n{g}^2(P{e}_k,e_s).\end{array}$$

(7)

The Gauss equation for submanifolds is given by

$$\begin{array}{c}\hfill \tilde{R}(X,Y,Z,W)=R(X,Y,Z,W)+g(\sigma (X,Z),\sigma (Y,W))-g(\sigma (X,W),\sigma (Y,Z)),\end{array}$$

(8)

for $X,Y,Z,W\in \Gamma \left(TM\right)$ and $\tilde{R}$ and R are curvature tensors on $\tilde{M}$ and M, respectively. A Sasakian manifold with constant $\varphi$—sectional curvature c is said to be Sasakian space form iff the Riemannian curvature tensor $\tilde{R}$ can be written:

$$\begin{array}{ccc}\hfill \tilde{R}(X,Y,Z,W)& =& \frac{c+3}{4}\{g(Y,Z)g(X,W)-g(X,Z)g(Y,W)\}\hfill \\ & & +\frac{c-1}{4}\{\eta \left(X\right)\eta \left(Z\right)g(Y,W)+\eta \left(W\right)\eta \left(Y\right)g(X,Z)-\eta \left(Y\right)\eta \left(Z\right)g(X,W)\hfill \\ & & -\eta \left(X\right)g(Y,Z)\eta \left(W\right)+g(\varphi Y,Z)g(\varphi X,W)-g(\varphi X,Z)g(\varphi Y,W)\hfill \\ & & +2g(X,\varphi Y)g(\varphi Z,W)\}.\hfill \end{array}$$

(9)

The mean curvature vector H for an orthonormal frame $\{e_1,e_2,...,e_n\}$ of the tangent space $TM$ on M is defined by

$$\begin{array}{c}\hfill H=\frac{1}{n}Trace\left(\sigma \right)=\frac{1}{n}\sum _{k=1}^n\sigma (e_k,e_k),and{\parallel H\parallel }^2=\frac{1}{n^2}{\left(\sum _{k=1}^n\sigma (e_k,e_k)\right)}^2,\end{array}$$

(10)

where $n=dimM$. We also have

$$\begin{array}{c}\hfill {\sigma }_{ks}^r=g\left(\sigma (e_k,e_s)e_r\right)and{\parallel \sigma \parallel }^2=\sum _{k,s}^{n}g(\sigma (e_k,e_s),\sigma (e_k,e_s)).\end{array}$$

(11)

For the submanifold M of an almost Hermitian manifold $\tilde{M}$ the scalar curvature $\tau$ is given by

$$\begin{array}{c}\hfill \tau \left(TM\right)=\sum _{1\le k\ne s\le n}K(e_k\wedge e_s),\end{array}$$

(12)

such that $K(e_k\wedge e_s)$ is the sectional curvature of the plane section, which is spanned by $e_k$ and $e_s$. Suppose $G_r$ is an r-plane section on M and suppose $\{e_1,e_2,...,e_r\}$ is any orthonormal basis of $G_r$. The scalar curvature $\tau \left(G\right)$ of $G_r$ is given by

$$\begin{array}{c}\hfill \tau \left(G_r\right)=\sum _{1\le k\ne s\le r}K(e_k\wedge e_s).\end{array}$$

(13)

Let $\tilde{M}$ be an alomost contact metric manifold. A submanifold M of $\tilde{M}$ is said to be totally umbilical and totally geodesic if $\sigma (X,Y)=g(X,Y)H$ and $\sigma (X,Y)=0$, respectively, for all $X,Y\in \Gamma \left(TM\right)$ such that H is the mean curvature vector of M. Furthermore, if $H=0$, then M is minimal submanifold in $\tilde{M}$. We can define the covariant derivative of endomorphism $\varphi$ as

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_X\varphi \right)Y={\tilde{\nabla }}_X\varphi Y-\varphi {\tilde{\nabla }}_{X}Y,\mathrm{for}\mathrm{all}X,Y\in \Gamma \left(T\tilde{M}\right).\end{array}$$

(14)

Definition 1.

A submanifold M of an almost contact metric manifold $\tilde{M}$ is said to be semi-slant submanifold if there exists two orthogonal distributions $D_1$ and $D_2$ on M such that:

(i)

$TM$ admits the orthogonal direct decomposition $TM=D_1\oplus D_2\oplus <\xi >$.

(ii)

The distribution $D_1$ is an invariant distribution, i.e., $\varphi \left(D_1\right)=D_1$.

(iii)

The distribution $D_2$ is slant with slant angle $\theta \ne 0$.

Definition 2.

A $(2n+1)$ dimensional submanifold $M^n$ of an almost contact metric manifold $(\tilde{M},\varphi ,\xi ,\eta ,g)$ is called pointwise slant submanifold if any nonzero vector X tangent to $M^n$ at $x\in M^n$, such that X is not proportional to ${\xi }_x$, the wirtinger angle $\theta =\theta \left(X\right)$ between $\varphi \left(X\right)$ and $T^{*}M=TM-\left\{0\right\}$ is independent of the choice of the nonzero vector $X\in T^{*}M$. The Wirtinger angle becomes a real-valued function, defined on $T^{*}M$ such that $\theta :T^{*}M\to \mathbb{R}$ and said to be a wirtinger function (slant function).

Lemma 1.

[4] Suppose M is a submanifold of an almost contact metric manifold $\tilde{M}$. Then M is said to be pointwise slant iff thers exists $\lambda \in [0,1]$ such that

$$\begin{array}{c}\hfill P^2=-\lambda (I+\eta \otimes \xi )\end{array}$$

(15)

Furthermore, in such case, if θ is the slant angle of M, it satisfies $\lambda ={cos}^2\theta$.

Hence, for a pointwise slant submanifold M of an almost contact metric manifold $\tilde{M}$, the following relations hold:

$$\begin{array}{ccc}& & g(PX,PY)={cos}^2\theta (g(X,Y)-\eta \left(X\right)\eta \left(Y\right)),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & g(FX,FY)={sin}^2\theta (g(X,Y)-\eta \left(X\right)\eta \left(Y\right)),\hfill \end{array}$$

(17)

for any $X,Y\in \Gamma \left(T^{*}M\right)$, these relations are the consequences of Lemma 1.

In [5], Sahin studied the pointwise semi-slant submanifolds as the natural generalization of CR-submanifolds of almost Hermitian manifolds in terms of slant function, and later these results were extended to the settings of contact manifolds by Park [4]. These submanifolds are defined as follows.

Definition 3.

A submanifold M of an almost contact metric manifold $\tilde{M}$ is said to be a pointside semi-slant submanifold iff there exists two orthogonal distributions $D_1$ and $D_2$ such that:

(i)

$TM=D_1\oplus D_2\oplus \xi$, where $\xi \left(x\right)$ is a 1-dimentional distribution spanned for each point $x\in M$,

(ii)

$D_1$ is invariant, i.e., $\varphi \left(D_1\right)\subseteq D_1$,

(iii)

$D_2$ is pointwise slant distribution with slant function $\theta :M\to \mathbb{R}$.

Let p and q be the dimentions of the invariant distribution $D_1$ and the pointwise slant distribution $D_2$ of the pointwise semi-slant submanifold of an almost contact metric manifold $\tilde{M}$, respectively. Then we have the following remarks.

Remark 1.

M is invariant if p = 0 and pointwise slant if q = 0.

Remark 2.

if the slant function $\theta :M\to \mathbb{R}$ is globally constant on M and $\theta =\frac{\pi }{2}$, then M is called a contact CR-submanifold.

Remark 3.

if the slant function $\theta :M\to \left(0,\frac{\pi }{2}\right)$ and $p=q\ne 0$, then M is called a proper pointwise semi-slant submanifold.

Remark 4.

Let μ be an invariant subspace under ϕ of the normal bundle $T^{\perp }M$, then in case of a pointwise semi-slant submanifold of the normal bundle $T^{\perp }M$ can be decomposed as $T^{\perp }M=F{D}_2\oplus \mu$.

Now, let f be a differential function defined on M. Thus, the gradient $\nabla f$ is defined as

$$\begin{array}{c}\hfill \left(i\right)g(\nabla f,X)=Xf,\left(ii\right)\nabla f=\sum _{k=1}^n{e}_k\left(f\right)e_k.\end{array}$$

(18)

Hence, from the above equation we can define the Hamiltonian in a local orthonormal frame as

$$\begin{array}{c}\hfill H(df,x)=\frac{1}{2}\sum _{s=1}^{n}df{\left(e_s\right)}^2=\frac{1}{2}\sum _{s=1}^n{e}_s{\left(f\right)}^2=\frac{1}{2}{\parallel \nabla f\parallel }^2.\end{array}$$

(19)

Moreover, we can define Laplacian $\Delta f$ of f as

$$\begin{array}{c}\hfill \Delta f=\sum _{k=1}^n\left\{\left(\nabla e_k{e}_s\right)-e_k\left(e_k\left(f\right)\right)\right\}=-\sum _{k=1}^{n}g({\nabla }_{e_k}gradf,e_k).\end{array}$$

(20)

In the similar fashion we can define the Hessian tensor of the function f as

$$\begin{array}{c}\hfill \Delta f=-Trace{H}^f=-\sum _{k=1}^n{H}^f(e_k,e_k),\end{array}$$

(21)

where $H^f$ is the Hessian of the function f. Suppose, we can consider the compact manifold M without boundary i.e., $\partial M=0$. Thus we can define the following lemma:

Lemma 2.

[37] (Hopf’s lemma): Let M be a compact Riemannian manifold and let f be a smooth function on M such that $\Delta f\ge 0(\Delta f\le 0)$. Then f is a constant function on M.

Moreover, for a compact oriented Riemannian manifold M without boundary condition, we have the following formula:

$$\begin{array}{c}\hfill {\int }_M\Delta fdV=0.\end{array}$$

(22)

where dV denotes the volume of M.

Theorem 1.

[37] The Euler–Lagrange equation for the Lagrangian $L=\frac{1}{2}{\parallel \nabla f\parallel }^2$ is

$$\Delta f=0.$$

In this case when M is a manifold with boundary, Hopf’s lemma becomes the uniqueness theorem for the Dirichlet problem. Thus we define the following:

Theorem 2.

[37] Let M be a connected, compact manifold and suppose f be a positive differentiable function on M such that $\Delta f=0$ on M and $\frac{f}{\partial M}=0$, where $\partial M$ denotes the boundary of M. Then f = 0.

Again, let M be a compact Riemannian manifold and let f be a positive differentiable function defined on M. Then the kinetic energy function is defined as follows:

$$\begin{array}{c}\hfill E\left(F\right)=\frac{1}{2}{\int }_M{\parallel \nabla f\parallel }^{2}dV,\end{array}$$

(23)

where dV denotes the volume element of M.

3. Conformal Sasakian Manifold

A (2n + 1) dimensional Riemannian manifold $\overline{M}$ endowed with an almost contact metric structure $(\overline{\varphi },\overline{\eta },\overline{\xi },\overline{g})$ is called a conformal Sasakian manifold if for a ${\mathbb{C}}^{\infty }$ function $f:\overline{M}\to \mathbb{R}$, we have $\tilde{g}=exp\left(f\right)\overline{g}$, $\tilde{\xi }={\left(exp(-f)\right)}^{\frac{1}{2}}\overline{\xi }$, $\tilde{\eta }={\left(exp\left(f\right)\right)}^{\frac{1}{2}}\overline{\eta }$$\tilde{\varphi }=\overline{\varphi }$, such that $(\overline{M},\tilde{\varphi },\tilde{\eta },\tilde{\xi },\tilde{g})$ is a Sasakian manifold.

Let $\tilde{\nabla }$ and $\overline{\nabla }$ be the Riemannian connections on $\overline{M}$ with respect to the metrics $\tilde{g}$ and $\overline{g}$, respectively. Using the Koszul formula, the relation between the connections $\tilde{\nabla }$ and $\overline{\nabla }$ is given by

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{X}Y={\overline{\nabla }}_{X}Y+\frac{1}{2}\{w\left(X\right)Y+w\left(Y\right)X-\overline{g}(X,Y)w^{\#}\}\end{array}$$

(24)

for any $X,YinT\overline{M}$, where $w\left(x\right)=X\left(f\right)$ and $\overline{g}(w^{\#},X)=w\left(X\right)$.

Using Equation (24), we derive the relation between the curvature tensors of $(\overline{M},\overline{\varphi },\overline{\eta },\overline{g})$ and $(\overline{M},\tilde{\varphi },\tilde{\eta },\tilde{\xi },\tilde{g})$ as follows:

$$\begin{array}{ccc}\hfill exp(-f)\tilde{R}(X,Y,Z,W)& =& \overline{R}(X,Y,Z,W)+\frac{1}{2}\{B(X,Z)\overline{g}(Y,W)-B(Y,Z)\overline{g}(X,W)\hfill \\ & & +B(Y,W)\overline{g}(X,Z)-B(X,W)\overline{g}(Y,W)\}\hfill \\ & & +\frac{1}{4}{\parallel w^{\#}\parallel }^2\{\overline{g}(X,Z)\overline{g}(Y,W)-\overline{g}(Y,Z)\overline{g}(X,W)\}\hfill \end{array}$$

(25)

for any $X,Y,Z,W\in T\overline{M}$, such that $B=\overline{\nabla }w-\frac{1}{2}w\otimes w$ and $\overline{R}$, $\tilde{R}$ are the curvature tensors of $(\overline{M},\overline{\varphi },\overline{\eta },\overline{\xi },\overline{g})$ and $(\overline{M},\tilde{\varphi },\tilde{\eta },\tilde{\xi },\tilde{g})$, respectively.

From Equation (24) it follows that

$$\begin{array}{c}\hfill {\overline{\nabla }}_X\overline{\xi }=-{\left(exp\left(f\right)\right)}^{\frac{1}{2}}\overline{\varphi }X+\frac{1}{2}\{\overline{\eta }\left(X\right)w^{\#}-w\left(\overline{\xi }\right)X\},\end{array}$$

(26)

$$\begin{array}{ccc}\hfill \left({\overline{\nabla }}_X\overline{\varphi }\right)Y& =& {\left(exp\left(f\right)\right)}^{\frac{1}{2}}\{\overline{g}(X,Y)\overline{\xi }-\overline{\eta }\left(Y\right)X\}\hfill \\ & & -\frac{1}{2}\{w\left(\overline{\varphi }Y\right)X-w\left(Y\right)\overline{\varphi }X+\overline{g}(X,Y)\overline{\varphi }{w}^{\#}\hfill \\ & & -\overline{g}(X,\overline{\varphi }Y)w^{\#}\}.\hfill \end{array}$$

(27)

From (27), we have

$$\begin{array}{ccc}\hfill \left({\nabla }_{X}T\right)Y& =& {\left(exp\left(f\right)\right)}^{\frac{1}{2}}\{g(X,Y)\xi -\eta \left(Y\right)X\}+A_{FY}X+t\sigma (X,Y)\hfill \\ & & -\frac{1}{2}\{w\left(TY\right)X-w\left(Y\right)TX+g(X,Y)T{w}^{\#}-g(X,TY)w^{\#}\}\hfill \end{array}$$

(28)

and,

$$\begin{array}{ccc}\hfill \left({\nabla }_{X}N\right)Y& =& n\sigma (X,Y)-\sigma (X,TY)\hfill \\ & & +\frac{1}{2}\{w\left(Y\right)NX-g(X,Y)N{w}^{\#}+g(X,TY)w^{\#\perp }\}.\hfill \end{array}$$

(29)

The curvature tensor of conformal Sasakian manifold with constant sectional curvature c, denoted by $\overline{M}\left(c\right)$, is given by

$$\begin{array}{ccc}\hfill R(X,Y,Z,W)& =& exp\left(f\right)\{\frac{c+3}{4}(\overline{g}(Y,Z)\overline{g}(X,W)-\overline{g}(X,Z)\overline{g}(Y,W))+\hfill \\ & & \frac{c-1}{4}(\overline{\eta }\left(X\right)\overline{\eta }\left(Z\right)\overline{g}(Y,W)-\overline{\eta }\left(Y\right)\overline{\eta }\left(Z\right)\overline{g}(X,W)\hfill \\ & & +\overline{g}(X,Z)\overline{g}(\overline{\xi },W)\overline{\eta }\left(Y\right)-\overline{g}(Y,Z)\overline{g}(\overline{\xi },W)\overline{\eta }\left(X\right)\hfill \\ & & +\overline{g}(\overline{\varphi }Y,Z)\overline{g}(\overline{\varphi }X,W)-\overline{g}(\overline{\varphi }X,Z)\overline{g}(\overline{\varphi }Y,W)\hfill \\ & & -2\overline{g}(\overline{\varphi }X,Y)\overline{g}(\overline{\varphi }Z,W)\}-\frac{1}{2}\{B(X,Z)\overline{g}(Y,W)\hfill \\ & & -B(Y,Z)\overline{g}(X,W)+B(Y,W)\overline{g}(X,Z)-B(X,W)\overline{g}(Y,Z)\}\hfill \\ & & -\frac{1}{4}{\parallel w^{\#}\parallel }^2\{\overline{g}(X,Z)\overline{g}(Y,W)-\overline{g}(Y,Z)\overline{g}(X,W)\}\hfill \end{array}$$

(30)

for all $X,Y,Z,W$ tangent to $\overline{M}\left(c\right)$.

Example 1.

[12] Let ${\mathbb{R}}^{2n+1}$ be the $(2n+1)$-dimensional Euclidean space endowed with the almost contact metric structure $(\overline{\varphi },\overline{\xi },\overline{\eta },\overline{g})$ defined by

$$\begin{array}{ccc}\hfill \overline{\varphi }\left(\sum _{i=1}^n(X_i\frac{\partial }{\partial x^i}+Y_i\frac{\partial }{\partial y^i})+Z\frac{\partial }{\partial z}\right)& =& \sum _{i=1}^n(Y_i\frac{\partial }{\partial x^i}-X_i\frac{\partial }{\partial y^i})+\sum _{i=1}^n{Y}_i{y}^i\frac{\partial }{\partial z},\hfill \\ \hfill \overline{g}& =& exp(-f)\left\{\overline{\eta }\otimes \overline{\eta }+\frac{1}{4}\sum _{i=1}^n\left\{{\left(d{x}^i\right)}^2+{\left(d{y}^i\right)}^2\right\}\right\},\hfill \\ \hfill \overline{\eta }& =& {\left(exp(-f)\right)}^{\frac{1}{2}}\left\{\frac{1}{2}\left(dz-\sum _{i=1}^n{y}^{i}d{x}^i\right)\right\},\hfill \\ \hfill \overline{\xi }& =& {\left(exp\left(f\right)\right)}^{\frac{1}{2}}\left\{2\frac{\partial }{\partial z}\right\},\hfill \end{array}$$

where $f={\sum }_{i=1}^n{\left(x^i\right)}^2+{\left(y^i\right)}^2+z^2$.

It is easy to show that $({\mathbb{R}}^{2n+1},\overline{\varphi },\overline{\xi },\overline{\eta },\overline{g})$ is not a Sasakian manifold but ${\mathbb{R}}^{2n+1}$ with the structure $(\tilde{\varphi },\tilde{\xi },\tilde{\eta },\tilde{g})$ given by

$$\begin{array}{ccc}\hfill \tilde{\varphi }& =& \overline{\varphi },\hfill \\ \hfill \tilde{g}& =& \overline{\eta }\otimes \overline{\eta }+\frac{1}{4}\sum _{i=1}^n\left\{{\left(d{x}^i\right)}^2+{\left(d{y}^i\right)}^2\right\},\hfill \\ \hfill \tilde{\eta }& =& \frac{1}{2}\left(dz-\sum _{i=1}^n{y}^{i}d{x}^i\right),\hfill \\ \hfill \tilde{\xi }& =& 2\frac{\partial }{\partial z},\hfill \end{array}$$

is a Sasakian space form with $\tilde{\varphi }$-sectional curvature equal to −3.

4. Characterization of Warped Product Pointwise Semi-Slant Submanifolds

Let $(M_1,g_1)$ and $(M_2,g_2)$ be the Riemannian manifolds and f a positive differentiable function on $M_1$. The warped product of $M_1$ and $M_2$ is the Riemannian manifold $M_1{\times }_f{M}_2=(M_1\times M_2,g)$, where $g=g_1+f^2{g}_2$. In this case the function f is called the warping function on M. We can state the following lemma as consequence of warped product manifolds.

Lemma 3.

Let $M_1{\times }_f{M}_2$ ba a warped product manifold. Then for any $X,Y\in \Gamma \left(T{M}_1\right)$ and $Z,W\in \Gamma \left(T{M}_2\right)$, we have

(i)

${\nabla }_{X}Y\in \Gamma \left(T{M}_1\right)$,

(ii)

${\nabla }_{Z}X={\nabla }_{X}Z=\left(Xlnf\right)Z$,

(iii)

${\nabla }_{Z}W={\nabla }_Z^{\prime }W-g(Z,W)\nabla lnf$.

where ∇ and ${\nabla }^{\prime }$ denote levi-civita connections on $M_1$ and $M_2$, and $\nabla lnf$ is gradient of $lnf$ which is defined by $g(\nabla lnf,X)=Xlnf$.

Remark 5.

A warped product manifold $M=M_1{\times }_f{M}_2$ is said to be trival or simply Riemannian product if the warping function f is constant.

Remark 6.

On a warped product manifold $M=M_1{\times }_f{M}_2$, $M_1$ is totally geodesic and $M_2$ is totally umbilical submanifolds of M, respectively.

Suppose $\kappa :M=M_1{\times }_f{M}_2\to \tilde{M}$ is an isometric immersion from a warped product $M_1{\times }_f{M}_2$ into a Riemannian manifold $\tilde{M}$ with a constant sectional curvature c; let p, q, and n be the dimentions of $M_1$, $M_2$, and $M_1{\times }_f{M}_2$, respectively. Then unit vector fields X and Z are tangent to $M_1$ and $M_2$, respectively, and we have

$$\begin{array}{c}\hfill K(X\wedge Z)=g({\nabla }_Z{\nabla }_{X}X-{\nabla }_X{\nabla }_{Z}X,Z)=\frac{1}{f}\{\left({\nabla }_{X}X\right)f-X^{2}f\}.\end{array}$$

(31)

If we consider the local orthonormal frame $\{e_1.e_2,...e_n\}$ such that $\{e_1,e_2,...,e_p\}$ are tangents to $M_1$ and $\{e_{p+1},...,e_n\}$ are tangents to $M_2$, then we have

$$\begin{array}{c}\hfill \frac{\Delta f}{f}=\sum _{k=1}^{p}K(e_k\wedge e_s)or\sum _{k+1}^p\sum _{s=1}^{q}K(e_k\wedge e_s)=\frac{q\Delta f}{f}=q(\Delta \left(lnf\right)-{\parallel \nabla \left(lnf\right)\parallel }^2)\end{array}$$

(32)

for each $s=p+1,...,n$.

5. Main Results

Theorem 3.

There does not exist any proper warped product pointwise semi-slant submanifold $M=M_{\theta }{\times }_f{M}_T$ in a conformal Sasakian manifold $\overline{M}$ such that $M_{\theta }$ is a proper pointwise slant submanifold tangent to the structure vector field ξ and $M_T$ is an invariant submanifold of M.

Proof. 

Suppose there exists a proper pointwise semi-slant submanifold $M=M_{\theta }{\times }_f{M}_T$ of $\overline{M}$ such that $D_1=T{M}_T$ and $D_2=T{M}_{\theta }$. Now we will show contradiction.

Since $X,Y\in \Gamma \left(T{M}_T\right)$ and $Z\in \Gamma \left(T{M}_{\theta }\right)$, using (3), Lemma 3, we have

$$\begin{array}{c}\hfill Z\left(lnf\right)g(X,Y)=g({\overline{\nabla }}_{X}Z,Y).\end{array}$$

Using (2), (6), and (14) in the above equation we get

$$\begin{array}{ccc}\hfill Z\left(lnf\right)g(X,Y)& =& g({\overline{\nabla }}_X(PZ+FZ)-{\left(exp\left(f\right)\right)}^{\frac{1}{2}}\{\overline{g}(X,Z)\overline{\xi }-\overline{\eta }\left(Z\right)X\}+\frac{1}{2}\{w\left(\overline{\varphi }Z\right)X\hfill \\ & & -w\left(Z\right)\overline{\varphi }X+g(X,Z)\overline{\varphi }{w}^{\#}-g(X,\overline{\varphi }Z)w^{\#}\},\varphi Y)+Z\left(lnf\right)\eta \left(X\right)\eta \left(Y\right).\hfill \end{array}$$

On further simplification, this gives

$$\begin{array}{ccc}\hfill Z\left(lnf\right)g(X,Y)& =& {cos}^2\theta \left(Zlnf\right)g(X,Y)+g(\sigma (X,Y),FPZ)-g(\sigma (X,\varphi Y),FZ)\hfill \\ & & +Z\left(lnf\right)\eta \left(X\right)\eta \left(Y\right),\hfill \end{array}$$

so that,

$$\begin{array}{c}\hfill {sin}^2\theta Z\left(lnf\right)g(X,Y)=g(\sigma (X,Y),FPZ)-g(\sigma (X,\varphi Y),FZ)+Z\left(lnf\right)\eta \left(X\right)\eta \left(Y\right).\end{array}$$

Replacing X and Y by $\xi$ in the above equation and using ${\overline{\nabla }}_X\xi =-\varphi X$

$$\begin{array}{ccc}\hfill {cos}^2\theta Z\left(lnf\right)& =& -g\left(\sigma \right(\xi ,\xi ),FPZ)\hfill \\ & =& -g({\overline{\nabla }}_{\xi }\xi ,FPZ)\hfill \\ & =& -g(\varphi \xi ,FPZ)\hfill \\ & =& 0.\hfill \end{array}$$

The above equation must imply $Z\left(lnf\right)=0$, hence f must be constant, which is a contradiction. □

Theorem 4.

Let $M=M_T{\times }_f{M}_{\theta }$ be a warped product semi-slant submanifold of a conformal Sasakian manifold M such that $M_T$ is invariant submanifold tangent to ξ and $M_{\theta }$ slant submanifold of M, with slant angle θ$\ne 0$, then

$$\begin{array}{ccc}\hfill g\left(\sigma \right(X,Y),FZ)& =& \frac{1}{2}\{g(X,TY)g(w^{\#},Z)-g(X,Y)g(T{w}^{\#},Z)\}\hfill \\ \hfill g\left(\sigma \right(X.W),FZ)& =& g(\sigma (X.Z),FW)+\frac{1}{2}w\left(TZ\right)\left\{g(X,W)\right\}\hfill \\ \hfill g\left(\sigma \right(\varphi X,Z),FW)& =& \left(Xlnf\right)g(Z,W)+\frac{1}{2}\left\{w\left(TX\right)g(Z,W)\right\}\hfill \end{array}$$

for any X, Y $\in \Gamma \left(T{M}_T\right)$ and Z, W $\in \Gamma \left(T{M}_{\theta }\right)$.

Proof. 

(i) Since we know that in a warped product $M_T$ is totally geodesic in M and therefore $\left({\overline{\nabla }}_{X}T\right)Y\in \Gamma \left(T_{M}T\right)$ for all $X,Y\in \Gamma \left(T_{M}T\right)$

Now, taking the inner product of (28) with $Z\in \Gamma \left(T{M}_{\theta }\right)$ we get

$$\begin{array}{c}\hfill g(t\sigma (X,Y),Z)=\frac{1}{2}\{g(X,Y)g(T{w}^{\#},Z)-g(X,TY)g(w^{\#},Z)\},\end{array}$$

(33)

On simplifying (33) we get

$$\begin{array}{c}\hfill g(\sigma (X,Y),FZ)=\frac{1}{2}\{g(X,TY)g(w^{\#},Z)-g(X,Y)g(T{w}^{\#},Z)\}.\end{array}$$

(34)

(ii) We know that

$$\begin{array}{c}\hfill {\nabla }_{X}Z={\nabla }_{Z}X=\left(Xlnf\right)Z.\end{array}$$

(35)

Again, we have

$$\begin{array}{c}\hfill \left({\nabla }_{X}T\right)Z={\nabla }_{X}TZ-T{\nabla }_{X}Z.\end{array}$$

(36)

Using (28) and (35) in (36) we get

$$\begin{array}{c}\hfill A_{FZ}X+t\sigma (X,Z)-\frac{1}{2}\{w\left(TZ\right)X-w\left(Z\right)TX\}=0.\end{array}$$

Now, taking the inner product of the above equation with $W\in \Gamma \left(TM\theta \right)$, we get

$$\begin{array}{c}\hfill g(\sigma (X,W),FZ)=g(\sigma (X,Z),FW)+\frac{1}{2}w\left(TZ\right)\left\{g(X,W)\right\}.\end{array}$$

(37)

which follows (ii).

(iii) We have

$$\begin{array}{c}\hfill \left({\nabla }_{Z}T\right)X=\left(TXlnf\right)Z-\left(Xlnf\right)TZ.\end{array}$$

(38)

Using (28) we have

$$\begin{array}{c}\hfill \left({\nabla }_{Z}T\right)X=t\sigma (X,Z)-\frac{1}{2}\{w\left(TX\right)Z-w\left(X\right)TZ\}.\end{array}$$

(39)

Combining (38), (39) and then taking inner product with $W\in \Gamma \left(T{M}_{\theta }\right)$, we get

$$\begin{array}{c}\hfill g(t\sigma (\varphi X,Z),W)-\frac{1}{2}\left\{w\left(TX\right)g(Z,W)\right\}=-\left(Xlnf\right)g(Z,W)-\left(TXlnf\right)g(Z,W).\end{array}$$

(40)

On making use of (37) in (40), this yields

$$\begin{array}{c}\hfill g(\sigma (\varphi X,Z),FW)=\left(Xlnf\right)g(Z,W)+\frac{1}{2}\left\{w\left(TX\right)g(Z,W)\right\}.\end{array}$$

(41)

This completes the proof. □

6. Inequality for Warped Product Pointwise Semi-Slant Submanifold of Conformal Sasakian Manifold and Its Applications

Proposition 1.

Consider a warped product pointwise semi-slant submanifold $M^n$ in a conformal Sasakian manifold ${\tilde{M}}^{2n+1}$. Then, we have the following:

$$\begin{array}{ccc}\hfill \left(g\right(\sigma (X,X),FZ)=g(\sigma (X,X),FPZ)& =& \frac{1}{2}\left(\overline{\varphi }PZ\right)\parallel X\parallel -\frac{1}{2}w\left(PZ\right)g(\overline{\varphi }X,X)\hfill \\ & +& \frac{1}{2}\overline{g}(X,PZ)\overline{g}(\overline{\varphi }{w}^{\#},X)-\frac{1}{2}\overline{g}(X,\overline{\varphi }PZ)g(w^{\#},X)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill g\left(\sigma \right(\varphi X,\varphi X),FZ)=g\left(\sigma \right(\varphi X,\varphi X),FPZ)& =& \frac{1}{2}\left(\varphi P^{2}Z\right)\parallel \varphi X\parallel +\frac{1}{2}w\left(P^{2}Z\right)g(X,\varphi X)\hfill \\ & +& \frac{1}{2}\overline{g}(\varphi X,P^{2}Z)\overline{g}(\overline{\varphi }{w}^{\#},\varphi X)\hfill \\ & -& \frac{1}{2}\overline{g}(\varphi X,\varphi P^{2}Z)g(w^{\#},\varphi X)\hfill \end{array}$$

$$\begin{array}{c}\hfill g\left(\sigma \right(X,X),\rho )=-g\left(\sigma \right(\varphi X,\varphi X\left)\rho \right),\end{array}$$

for any $X\in \Gamma \left(T{M}_T\right)$ and $Z\in \Gamma \left(T{M}_{\theta }\right)$ and $\rho \in \Gamma \left(\mu \right)$.

Proof. 

From (4) (i), we have

$$\begin{array}{c}\hfill g(\sigma (X,X),FPZ)=g({\tilde{\nabla }}_{X}X,FPZ)=-g({\tilde{\nabla }}_{X}FPZ,X).\end{array}$$

(42)

Then, using (5) in (42) and from the covariant derivative of endomorphism $\varphi$ we may get

$$\begin{array}{c}\hfill g(\sigma (X,X),FPZ)=g({\tilde{\nabla }}_{X}PZ,\varphi X)-g(\left({\tilde{\nabla }}_X\varphi \right)PZ,X)+g({\tilde{\nabla }}_X{P}^{2}Z,X).\end{array}$$

(43)

Using the structural equation of conformal Sasakian manifold (27) in (43) and than using Lemma (1) for pointwise slant manifold, we may obtain

$$\begin{array}{ccc}\hfill g\left(\sigma \right(X,X),FPZ)& =& g({\tilde{\nabla }}_X\varphi X,PZ)+sin2\theta X\left(\theta \right)g(Z,X)-{cos}^2\theta g({\tilde{\nabla }}_{X}X,Z)\hfill \\ & +& \frac{1}{2}\left(\overline{\varphi }PZ\right)\parallel X\parallel -\frac{1}{2}w\left(PZ\right)g(\overline{\varphi }X,X)+\frac{1}{2}\overline{g}(X,PZ)g(\overline{\varphi }{w}^{\#},X)\hfill \\ & -& \overline{g}(X,\overline{\varphi }PZ)g(w^{\#},X).\hfill \end{array}$$

(44)

Now using the fact that $M_{\theta }$ is totally geodesic in M, (44) reduces to

$$\begin{array}{ccc}\hfill g\left(\sigma \right(X,X),FZ)=g\left(\sigma \right(X,X),FPZ)& =& \frac{1}{2}\left(\overline{\varphi }PZ\right)\parallel X\parallel -\frac{1}{2}w\left(PZ\right)g(\overline{\varphi }X,X)\hfill \\ & +& \frac{1}{2}\overline{g}(X,PZ)\overline{g}(\overline{\varphi }{w}^{\#},X)\hfill \\ & -& \frac{1}{2}\overline{g}(X,\overline{\varphi }PZ)g(w^{\#},X).\hfill \end{array}$$

(45)

Interchanging Z with $PZ$ and X with $PX$ in (45), we obtain (ii).

Now, for (iii), we can use for conformal Sasakian manifold

$$\begin{array}{ccc}\hfill {\tilde{\nabla }}_X\varphi X& =& \varphi {\tilde{\nabla }}_{X}X+{\left(exp\left(f\right)\right)}^{\frac{1}{2}}\{\overline{g}(X,X)\overline{\xi }-\overline{\eta }\left(X\right)X\}\hfill \\ & -& \frac{1}{2}\{w\left(\overline{\varphi }X\right)X-w\left(X\right)\overline{\varphi }X+\overline{g}(X,X)\overline{\varphi }{w}^{\#}-\overline{g}(X,\overline{\varphi }X)w^{\#}\}.\hfill \end{array}$$

(46)

Using the Gauss formula, (46) can be further reduced as

$$\begin{array}{ccc}\hfill {\nabla }_X\varphi X+\sigma (\varphi X,X)& =& \varphi {\nabla }_{X}X+\varphi \sigma (X,X)+{\left(exp\left(f\right)\right)}^{\frac{1}{2}}\{\parallel X\parallel \overline{\xi }-\overline{\eta }\left(X\right)X\}\hfill \\ & -& \frac{1}{2}\{w\left(\overline{\varphi }X\right)X-w\left(X\right)\overline{\varphi }X+\parallel X\parallel \overline{\varphi }{w}^{\#}-\overline{g}(X,\overline{\varphi }X)w^{\#}\}.\hfill \end{array}$$

(47)

Taking the inner product with ${\varphi }_{\rho }$ in (47) for any $\rho \in \Gamma \left(\mathsf{\mu }\right)$ we have

$$\begin{array}{c}\hfill g(\sigma (\varphi X,X),{\varphi }_{\rho })=g(\sigma (X,X),\rho ).\end{array}$$

(48)

Interchanging X with $\varphi X$ in (48) and using (1) (i), also using the fact $\mathsf{\mu }$ is invariant normal sub bundle of $T^{\perp }M$ under the action of $\varphi$, we have

$$\begin{array}{c}\hfill -g(\sigma (X,X),{\varphi }_p)=g(\sigma (\varphi X,\varphi X),\rho ).\end{array}$$

(49)

comparing (48) and (49) we get the result. □

Theorem 5.

Consider κ: $M^n=M_T^p{\times }_f{M}_{\theta }^q$→${\tilde{M}}^{2m+1}$ to be an isometrically immersion from a warped product pointwise semi-slant submanifold $M_T^p{\times }_f{M}_{\theta }^q$ into a conformal Sasakian manifold ${\tilde{M}}^{2m+1}$. Then we have

(i)

The squared norm of the second fundamental form of M is given by

$$\begin{array}{ccc}\hfill {\parallel \sigma \parallel }^2& \ge & 2exp\left(f\right)\left\{{q\parallel \nabla \lambda \parallel }^2+\tilde{\tau }\left(TM\right)-\tilde{\tau }\left(T{M}_T\right)-\tilde{\tau }\left(T{M}_{\theta }\right)-q\Delta \lambda \right\}\hfill \\ & +& nTraceB+\frac{1}{4}{\parallel w^{\#}\parallel }^{2}n(n-1).\hfill \end{array}$$

(50)

where q is the dimension of a pointwise slant submanifold $M_{\theta }$, and $\lambda =lnf$.

(ii)

Equality holds in (50) iff $M_T^p$ is totally geodesic and $M_{\theta }^q$ is totally umbilical submanifolds in ${\tilde{M}}^{2m+1}$. Moreover, $M^n$ is minimal submanifold in ${\tilde{M}}^{2m+1}$.

Proof. 

The proof follows from Theorem 5.1 [17] and for contact version see [38]. Here we can rewrite Equation (4.10) of [38] for conformal Sasakian manifold as

$$\begin{array}{ccc}\hfill {\parallel \sigma \parallel }^2& =& -2exp\left(f\right)\{\sum _{i=1}^p\sum _{j=p+1}^{n}K(e_i\wedge e_j)-\tilde{\tau }\left(T{M}_T\right)-\tilde{\tau }\left(T{M}_{\theta }\right)+\tilde{\tau }\left(T{M}^n\right)\hfill \\ & +& \sum _{r=n+1}^{2m+1}\sum _{1\le i\ne k\le n_1}{\left({\sigma }_{ik}^r\right)}^2+n^2{\parallel H\parallel }^2-n^2{\parallel H\parallel }^2+\sum _{r=n+1}^{2m+1}\sum _{n_1+1\le j\ne t\le n}{\left({\sigma }_{jt}^r\right)}^2\}\hfill \\ & +& \frac{1}{4}{\parallel w^{\#}\parallel }^{2}n(n-1)+nTraceB.\hfill \end{array}$$

(51)

Using (32) in (51) we have,

$$\begin{array}{ccc}\hfill {\parallel \sigma \parallel }^2& \ge & 2exp\left(f\right)\{q\parallel \nabla \lambda {\parallel }^2+\tilde{\tau }\left(T{M}^n\right)-\tilde{\tau }\left(T{M}_T\right)-\tilde{\tau }\left(T{M}_{\theta }\right)-q\Delta \lambda \}\hfill \\ & +& nTraceB+\frac{1}{4}{\parallel w^{\#}\parallel }^{2}n(n-1).\hfill \end{array}$$

(52)

which completes the proof of (i).

(ii) the proof is the same as in [38]. □

Theorem 6.

Consider κ: $M^n=M_T^p{\times }_f{M}_{\theta }^q\to {\tilde{M}}^{2m+1}\left(c\right)$ is an isometric immersion from an n- dimensional warped product pointwise semi-slant submanifold $M_T^p{\times }_f{M}_{\theta }^q$ into a conformal Sasakian space form ${\tilde{M}}^{2m+1}\left(c\right)$. Then,

(i)

The squared norm of the second fundamental form of $M^n$ is defined by

$$\begin{array}{c}\hfill {\parallel \sigma |}^2\ge 2exp\left(f\right)q\left\{{\parallel \nabla \lambda \parallel }^2+\frac{c+3}{4}p-\frac{c-1}{4}-\Delta \lambda \right\}+\frac{1}{4}{\parallel w^{\$}\parallel }^2+nTraceB.\end{array}$$

(53)

where p and q are dimensions of the invarient $M_T^p$ and pointwise slant submanifold $M_{\theta }^q$, respectively, and $\lambda =lnf$.

(ii)

Equality holds in (53) iff $M_T^p$ is totally geodesic and $M_{\theta }^q$ is totally umbilical submanifolds in ${\tilde{M}}^{2m+1}\left(c\right)$, also $M^n$ is minimal in ${\tilde{M}}^{2m+1}\left(c\right)$.

Proof. 

Putting $X=W=$$e_k$, $Y=Z=e_s$ in (30) we can obtain

$$\begin{array}{ccc}\hfill R(e_k,e_s,e_s,e_k)& =& exp\left(f\right)\{\frac{c+3}{4}(g(e_k,e_k)g(e_s,e_s)-g(e_k,e_s)g(e_s,e_k))\hfill \\ & +& \frac{c-1}{4}(\eta \left(e_k\right)\eta \left(e_s\right)g(e_k,e_s)-\eta \left(e_s\right)\eta \left(e_s\right)g(e_k,e_k)+\eta \left(e_k\right)\eta \left(e_s\right)g(e_k,e_s)\hfill \\ & -& \eta \left(e_k\right)\eta \left(e_k\right)g(e_s,e_s)+\overline{g}(\varphi e_s,e_s)\overline{g}(\varphi e_k,e_k)-\overline{g}(\varphi e_s,e_s)\overline{g}(\varphi e_s,e_k)\hfill \\ & -& 2\overline{g}(\overline{\varphi }{e}_k,e_s)\overline{g}(\varphi e_s,e_k)\left)\right\}-\frac{1}{2}\{B(e_k,e_s)\overline{g}(e_s,e_k)-B(e_s,e_s)\overline{g}(e_k,e_k)\hfill \\ & +& B(e_k,e_s)\overline{g}(e_k,e_s)-B(e_k,e_k)\overline{g}(e_s,e_s)\}-\frac{1}{4}\parallel w^{\#}{\parallel }^2\{\overline{g}(e_k,e_s)\overline{g}(e_s,e_k)\hfill \\ & -& \overline{g}(e_s,e_s)\overline{g}(e_k,e_k)\}.\hfill \end{array}$$

(54)

Taking the sum over the vector fields on $T{M}_T$ in (54) and than using (7) (ii) we get

$$\begin{array}{ccc}\hfill 2\tilde{\rho }\left(T{M}_T\right)& =& exp\left(f\right)\left\{\left(\frac{c+3}{4}\right)p(p-1)+\left(\frac{c-1}{4}\right){(3\parallel P\parallel }^2-2p(p-1)\right\}\hfill \\ & +& \frac{1}{4}{\parallel w^{\#}\parallel }^{2}p(p-1)+pTraceB.\hfill \end{array}$$

(55)

As $\xi \left(p\right)$ is tangent to $T{M}_T$ for the p-dimensional Invariant submanifold, we may take ${\parallel P\parallel }^2=p-1$, in (55) then we have

$$\begin{array}{ccc}\hfill 2\tilde{\rho }\left(T{M}_T\right)& =& exp\left(f\right)\left\{\left(\frac{c+3}{4}\right)p(p-1)+\left(\frac{c-1}{4}\right)(p-1)\right\}+\frac{1}{4}{\parallel w^{\#}\parallel }^{2}p(p-1)\hfill \\ & +& pTraceB.\hfill \end{array}$$

(56)

In the same way, for pointwise slant submanifold $T{M}_{\theta }$, we put ${\parallel P\parallel }^2=q{cos}^2\theta$, and using Lemma (1) and (7) (ii) in (55), we have

$$\begin{array}{ccc}\hfill 2\tilde{\rho }\left(T{M}_{\theta }\right)& =& exp\left(f\right)\left\{\left(\frac{c+3}{4}\right)q(q-1)+\left(\frac{c-1}{4}\right)3q{cos}^2\theta \right\}\hfill \\ & +& \frac{1}{4}{\parallel w^{\#}\parallel }^{2}q(q-1)+qTraceB.\hfill \end{array}$$

(57)

Taking the sum over basic vectors of $T{M}^n$ in (54) such that $1\le k\ne s\le n$, we obtain

$$\begin{array}{ccc}\hfill 2\tilde{\rho }\left(T{M}^n\right)& =& exp\left(f\right)\left\{\left(\frac{c+3}{4}\right)n(n-1)+\left(\frac{c-1}{4}\right)\{3\sum _{1\le k\ne s\le n}{g}^2(\varphi e_k,e_s)-2(n-1)\}\right\}\hfill \\ & +& \frac{1}{4}{\parallel w^{\#}\parallel }^{2}n(n-1)+nTraceB.\hfill \end{array}$$

(58)

Hence, $M^n$ is a proper pointwise semi-slant submanifold of a conformal Sasakian space form $\tilde{M}\left(c\right)$. Thus from [39] we set the following frame, i.e.,

$$\begin{array}{ccc}\hfill e_1,e_2& =& \varphi e_1,....,e_{2\gamma }=\varphi e_{2\gamma -1},\hfill \\ \hfill e_{2\gamma +1},e_{2\gamma +2}& =& sec\theta P_{e_{2\gamma +1}},....\hfill \\ \hfill e_{2\gamma -1},e_{2\gamma }& =& sec\theta P_{e_{2\gamma -1}},......\hfill \\ & ⋮& \\ \hfill e_{2\gamma +2\delta -1},e_{2d\gamma +2\delta }& =& sec\theta P_{e_{2\gamma -1},e_{2\gamma +2\delta ,}{e}_{2\gamma +2\delta +1}}=\xi .\hfill \end{array}$$

Obviously, we can derive

$$\begin{array}{c}\hfill g^2(\varphi e_k,e_{k+1})=\left\{\begin{array}{cc}1,\hfill & \mathrm{for}\mathrm{each},i\in \{1,...,2\gamma -1\}\hfill \\ {cos}^2\theta ,\hfill & \mathrm{for}\mathrm{each},i\in \{2\gamma +1,...,2\gamma +2\delta -1\},\hfill \end{array}\right.\end{array}$$

it can be easily obtained

$$\begin{array}{c}\hfill \sum _{k,s=1}^n{g}^2(P{e}_k,e_s)=2(\gamma +\delta {cos}^2\theta ).\end{array}$$

(59)

From (58) and (59), it follows that

$$\begin{array}{ccc}\hfill 2\tilde{\rho }\left(T{M}^n\right)& =& exp\left(f\right)\left\{\frac{c+3}{4}n(n-1)+\frac{c-1}{4}(6(\gamma +\delta {cos}^2\theta )-2(n-1))\right\}\hfill \\ & +& \frac{1}{4}{\parallel w^{\#}\parallel }^{2}n(n-1)+nTraceB.\hfill \end{array}$$

(60)

Therefore, by using (56), (57), and (60) in (50), we obtain the above result (55). Moreover, the equality holds according to the second part of Theorem (5). Hence the proof of the theorem. □

Theorem 7.

Consider κ: $M^n=M_T^p{\times }_f{M}_{\theta }^q\to {\tilde{M}}^{2m+1}\left(c\right)$ to be an isometric immersion from a compact oriented warped product pointwise semi-slant submanifold into a conformal Sasakian space form ${\tilde{M}}^{2m+1}\left(c\right)$. Then $M^n$ is a trivial warped product iff

$$\begin{array}{c}\hfill {\parallel \sigma \parallel }^2\ge exp\left(f\right)\left\{\left(\frac{c+3}{2}\right)pq-\left(\frac{c-1}{2}\right)q\right\}+\frac{1}{2}q{\parallel w^{\#}\parallel }^2+2nqTraceB.\end{array}$$

(61)

Proof. 

If the inequality follows in (53), than after further simplification gives

$$\begin{array}{ccc}\hfill exp\left(f\right)\left(\frac{c+3}{2}pq-\frac{c-1}{2}q+{\parallel \nabla lnf\parallel }^2\right)& -& \frac{1}{2q}{\parallel \sigma \parallel }^2+\frac{1}{4}{\parallel w^{\#}\parallel }^2\hfill \\ & +& nTraceB\le exp\left(f\right)\left(\Delta \left(lnf\right)\right).\hfill \end{array}$$

(62)

From the theory of integration of manifolds, for a compact Riemannian manifold without boundary on $M^n$ and from (22) and (62), we have

$$\begin{array}{ccc}\hfill {\int }_{M^n}\{exp\left(f\right)\left(\frac{c+3}{2}pq-\frac{c-1}{2}q+{\parallel \nabla lnf\parallel }^2\right)-\frac{1}{2q}{\parallel \sigma \parallel }^2& +& \frac{1}{4}{\parallel w^{\#}\parallel }^2+nTraceB\}dV\hfill \\ & \le & {\int }_{M^n}\Delta \left(lnf\right)dV=0.\hfill \end{array}$$

if

$$\begin{array}{c}\hfill {\parallel \sigma \parallel }^2\ge exp\left(f\right)\left\{\left(\frac{c+3}{2}\right)pq-\left(\frac{c-1}{2}q\right)\right\}+\frac{1}{2}q{\parallel w^{\#}\parallel }^2+2nqTraceB\end{array}$$

then

$$\begin{array}{c}\hfill {\int }_{M^n}{(\parallel \nabla lnf\parallel }^2)dV\le 0.\end{array}$$

We know that integration for positive functions is always positive, we find ${\parallel \nabla lnf\parallel }^2\le 0$, but ${\parallel \nabla lnf|}^2\ge 0$, which means $\nabla lnf=0$. That is, f must be constant function on $M^n$. Thus $M^n$ is a Riemannain product of $M_T^p$ and $M_{\theta }^q$, respectively. The converse follows easily and this completes the proof. □

Theorem 8.

Consider κ: $M^n=M_T^p{\times }_f{M}_{\theta }^q\to {\tilde{M}}^{2m+1}\left(c\right)$ is an isometric immersion from a compact oriented proper warped product pointwise semi-slant submanifold $M_T^p{\times }_f{M}_{\theta }^q$ into a conformal Sasakian space form ${\tilde{M}}^{2m+1}\left(c\right)$. Then $M^n$ is simply Riemannian product iff

$$\begin{array}{ccc}\hfill \sum _{k=1}^p\sum _{s=1}^q{\parallel {\sigma }_{\mu }(e_k,e_s)\parallel }^2& =& \{exp\left(f\right)\left(\frac{c+3}{4}pq-\frac{c-1}{4}q\right)+\frac{1}{8}{\parallel w^{\#}\parallel }^2+\frac{1}{2}nTraceB\hfill \\ & -& q{sin}^2\theta \}.\hfill \end{array}$$

(63)

where θ is a real valued function defined on $T^{*}M$ called the slant function, and $h_{\mu }$ is the second fundamental form component in $\Gamma \left(\mu \right)$.

Proof. 

Suppose the equality holds in the inequality (53) then, we can obtain

$$\begin{array}{ccc}\hfill {\parallel \sigma (D,D)\parallel }^2+\parallel \sigma (D^{\theta },D^{\theta }){\parallel }^2+2{\parallel \sigma (D,D^{\theta })\parallel }^2& =& exp\left(f\right)\left\{\left(\frac{c+3}{2}\right)pq-\left(\frac{c-1}{2}\right)q\right\}\hfill \\ & +& 2qexp\left(f\right)\left\{{\parallel \nabla lnf\parallel }^2-\Delta \left(lnf\right)\right\}\hfill \\ & +& \frac{1}{4}{\parallel w^{\#}\parallel }^2+nTraceB.\hfill \end{array}$$

(64)

As $M_T^p$ is totally geodesic in $\tilde{M}\left(c\right)$, i.e., $\sigma (D,D)=0$ and $M_{\theta }^q$ is totally umbilical in ${\tilde{M}}^{2m+1}\left(c\right)$, i.e., $h(Z,W)=g(Z,W)H$. In addition, $M^n$ is minimal submanifold in ${\tilde{M}}^{2m+1}\left(c\right)$, then $H=0$, which implies that $\sigma (Z,W)=0$. That means $\parallel \sigma (D^{\theta },D^{\theta }){\parallel }^2=0$. From (64), we have

$$\begin{array}{ccc}\hfill exp\left(f\right)\left\{\frac{c+3}{4}p-\frac{c-1}{4}\right\}-\frac{1}{q}{\parallel \sigma (D,D^{\theta })\parallel }^2& +& {exp\left(f\right)(\parallel \nabla lnf\parallel }^2)+\frac{1}{8q}{\parallel w^{\#}\parallel }^2\hfill \\ & +& \frac{1}{2q}nTraceB=exp\left(f\right)\{\Delta \left(lnf\right)\}\hfill \end{array}$$

(65)

Thus $M^n$ is a compact oriented submanifold. Therefore, $M^n$ is closed and bounded. Hence we can take integration over the volume element dV, by using (22), we have

$$\begin{array}{ccc}\hfill {\int }_{M^n}\left(exp\left(f\right)\right\{\frac{c+3}{4}pq& -& \frac{c-1}{4}q\}+\frac{1}{8}{\parallel w^{\#}\parallel }^2+\frac{1}{2}nTraceB)dV=\hfill \\ & & {\int }_{M^n}\left(\parallel \sigma (D,D^{\theta }){\parallel }^2-exp\left(f\right)q{\parallel \nabla lnf\parallel }^2\right)dV\hfill \end{array}$$

(66)

Now, consider $X=e_k$ and $Z=e_s$ for $1\le k\le p$ and $1\le s\le q$, respectively, and we can define

$$\begin{array}{c}\hfill \sigma (e_k,e_s)=\sum _{r=1}^{q}g(\sigma (e_k,e_s),F{e}_r^{*})F{e}_r^{*}+\sum _{l=1}^{2m+1}g(\sigma (e_k,e_s),\rho e_l)\rho e_l\end{array}$$

for $\rho \in \Gamma \left(\mu \right)$. Now we can take sum over the vector fields $M_T^p$ and $M_{\theta }^q$ and using the adapted frame for pointwise semi-slant submanifolds, we have

$$\begin{array}{ccc}\hfill \sum _{k=1}^p\sum _{s=1}^{q}g(\sigma (e_k,e_s),\sigma (e_k,e_s))& =& cs{c}^2\theta \sum _{k=1}^{\gamma }\sum _{s,m}^{\delta }g{(\sigma (e_k,e_s^{*}),F{e}_k^{*})}^2\hfill \\ & & +cs{c}^2\theta {sec}^2\theta \sum _{k=1}^{\gamma }\sum _{s,m=1}^{\delta }g{(\sigma (e_k,P{e}_s^{*}),F^{*}{e}_m)}^2\hfill \\ & & +{csc}^2\theta {sec}^2\theta \sum _{k=1}^{\gamma }\sum _{s,m=1}^{\delta }g{(\sigma (\varphi e_k,e_s^{*}),FP{e}_m^{*})}^2\hfill \\ & & +{csc}^2\theta {sec}^2\theta \sum _{k=1}^{\gamma }\sum _{s,m=1}^{\delta }g{(\sigma (\varphi e_k,e_s^{*}),FP{e}_m^{*})}^2\hfill \\ & & +{csc}^2\theta {sec}^4\theta \sum _{k=1}^{\gamma }\sum _{s,m=1}^{\delta }g{(\sigma (\varphi e_k,P{e}_s^{*}),FP{e}_m^{*})}^2\hfill \\ & & +{csc}^2\theta {sec}^2\theta \sum _{k=1}^{\gamma }\sum _{s,m=1}^{\delta }g{(\sigma (\varphi e_k,P{e}_s^{*}),F{e}_m^{*})}^2\hfill \\ & & +{csc}^2\theta \sum _{k=1}^{\gamma }\sum _{s,m=1}^{\delta }g{(\sigma (\varphi e_k,e_s^{*}),F{e}_m^{*})}^2\hfill \end{array}$$

$$\begin{array}{ccc}& & +{csc}^2\theta {sec}^4\theta \sum _{k=1}^{\gamma }\sum _{s,m=1}^{\delta }g{(\sigma (e_k,P{e}_s^{*}),FP{e}_m^{*})}^2\hfill \\ & & +\sum _{k=1}^p\sum _{s=1}^q\sum _{r=n+q+1}^{2m+1}g{(\sigma (e_k,e_s),e_r)}^2.\hfill \end{array}$$

Now, we can use Lemma (3.4) [15] in the above equation, and then derive

$$\begin{array}{c}\hfill \parallel \sigma (D,D^{\theta }){\parallel }^2=q(1+2{cot}^2\theta ){\parallel \nabla lnf|}^2+q{sin}^2\theta +\sum _{k=1}^p\sum _{s=1}^q{\parallel \sigma \mu (e_k,e_s)\parallel }^2.\end{array}$$

(67)

We use (67) in (66), from there we get

$$\begin{array}{c}\hfill {\int }_{M^n}\left(exp\left(f\right)\left\{\frac{c+3}{4}pq-\frac{c-1}{4}q\right\}+\frac{1}{8}{\parallel w^{\#}\parallel }^2+\frac{1}{2}nTraceB-q{sin}^2\theta \right)dV=\\ \hfill {\int }_{M^n}\left(\sum _{k=1}^p\sum _{s=1}^q{\parallel \sigma \mu (e_k,e_s)\parallel }^2\right)dV+{\int }_{M^n}\left(1-exp\left(f\right)+2{cot}^2\theta \right){\parallel \nabla lnf\parallel }^{2}dV.\end{array}$$

(68)

If (63) is satisfied, then (68) implies that $lnf$ is a constant function for a proper pointwise semi-slant submanifold $M^n$, and hence $M^n$ becomes a usual Riemannian product of invariant and pointwise submanifolds $M_T^p$ and $M_{\theta }^q$, respectively.

Conversely, suppose that $M^n$ is simply a Riemannain product, then its warping function must be constant, i.e., $\nabla lnf=0$, which means ${\parallel \nabla \left(lnf\right)\parallel }^2=0$. Thus from the Equation (68) we get (63). This completes the proof. □

Theorem 9.

Consider κ: $M^n=M_T^p{\times }_f{M}_{\theta }^q\to {\tilde{M}}^{2m+1}\left(c\right)$ to be an isometric immersion from a warped product pointwise semi-slant submanifold $M_T^p{\times }_f{M}_{\theta }^q$ in a conformal Sasakian space form ${\tilde{M}}^{2m+1}\left(c\right)$. Let Ψ be a nonzero eigenvalue of the on the compact invariant submanifold $M_T^p$. Then,

$$\begin{array}{ccc}\hfill {\int }_{M_T^p}{\parallel \sigma \parallel }^{2}d{V}_T& \ge & {\int }_{M_T^p}\left(exp\left(f\right)\left(\frac{c+3}{2}pq-\frac{c-1}{2}q\right)+\frac{1}{4}{\parallel w^{\#}\parallel }^2+nTraceB\right)d{V}_T\hfill \\ & & +2q\Psi exp\left(f\right){\int }_{M_T^p}{\lambda }^{2}d{V}_T.\hfill \end{array}$$

(69)

where $d{V}_T$ is the volume element on $M_T^p$. The equality holds in (69) iff we have

(i)

$\Delta \lambda =\Psi \lambda$.

(ii)

in the warped product pointwise semi-slant submanifold, both $M_T^p$ and $M_{\theta }^q$ are totally geodesic in ${\tilde{M}}^{2m+1}\left(c\right)$.

Proof. 

Suppose that f is a non-constant function. Using the minimum principle property for eigenvalue $\Psi$, we have

$$\begin{array}{c}\hfill {\int }_{M_T^p}{\parallel \nabla \lambda \parallel }^{2}d{V}_T\ge \Psi {\int }_{M_T^p}{\lambda }^{2}d{V}_T,\end{array}$$

(70)

and the equality holds iff $\Delta lnf=\Psi lnf$ by setting $\lambda =lnf$. From (53) and (70), we can get the result (69). This completes the proof. □