# Warped product pointwise bi-slant submanifolds of locally product Riemannian
manifolds

Prince Majeed  Department of Mathematics,
National Institute of Technology Srinagar,
190006, Kashmir, India<EMAIL_ADDRESS>Mehraj Ahmad Lone
Department of Mathematics,
National Institute of Technology Srinagar,
190006, Kashmir, India<EMAIL_ADDRESS>
###### Abstract.

In this paper we introduce the concept of pointwise bi-slant submanifolds of
locally product Riemannian manifolds and studied warped product pointwise bi-
slant submanifolds of locally product Riemannian manifolds. We obtain some
characterization results for warped products pointwise bi-slant submanifolds.
Also, we provide some non-trivial examples of such warped product
submanifolds.

###### Key words and phrases:

Locally product Riemannian manifold, Pointwise bi-slant submanifolds, Warped
products.

###### 1991 Mathematics Subject Classification:

53C15, 53C25, 53C40, 53C42

##  1\.
Introduction

In [7], Chen introduced the notion of slant submanifolds. It includes totally
real as well as holomorphic submanifolds. Numerous geometer groups continue to
study and conduct research on this idea of submanifolds. Recently, the related
literature of slant submanifolds has been compiled in the form of two books by
Chen, Shahid and Solamy (see [16, 17]). Since the introduction of slant
submanifolds, many generalizations and extensions of slant submanifolds have
been introduced, like: semi-slant, pointwise slant, hemi-slant, pointwise
hemislant and many more. The related literature of these kind of
generalizations can be be found in (see, [12, 18, 22, 24, 25]). A more generic
class of submanifolds in the form of bi-slant submanifolds was introduced by
Cabrerizo and Cariazo [6]. This class of submanifolds acts as a natural
generalization of CR, semi-slant, slant, hemi-slant submanifolds [22, 24, 26].
In connection to this generic notion of submanifolds, some of recent studies
can be found in [21]. Further the extended notion of pointwise bi-slant
submanifolds of Kaehler manifolds can be found in [15].

Bishop and O’Neill in $1960s$ introduced the concept of warped product
manifolds. These manifolds find their applications both in physics as well as
in mathematics. Since then the study of warped product submanifolds has been
investigated by many geometers (see, [3, 10, 11, 13]). In particular, Chen
started looking these warped products as submanifolds of different kinds of
manifolds (see, [8, 9]). In this connection, in Kaehlerian settings, he proved
besides CR- products the non-existence of warped products of the form
$N^{\perp}\times_{f}N^{T}$, where $N^{\perp},$ $N^{T}$ is a totally real and
holomorphic submanifold, respectively [34]. Now from the past two decades this
area of research is an active area of research among many of the geometry
groups. For the overall development of the subject we refer the reader to see
Chen’s book on it [14].

Now while importing the survey of warped products to slant cases, Sahin in
[30] proved the non-existence of semi-slant warped products in any Kaehler
manifold. Then in [32] he extended the study to pointwise semi-slant warped
products of Kaeherian manifolds. Sahin [33] and Atcken [1] investigated warped
product semi-slant submanifolds of locally product Riemannain manifolds. They
proved there is no warped product semi-slant submanifold of the form
$M_{T}\times_{f}M_{\theta}$ of a locally product Riemannian manifold $\bar{M}$
such that $M_{T}$ and $M_{\theta}$ are invariant and proper slant submanifolds
of $\bar{M}$, respectively. Moreover they provided non-trival examples and
proved a characterization theorem for warped product semi-slant submanifolds
of the form $M_{\theta}\times_{f}M_{T}$.

The main motivation for our paper is a recent study of Uddin, Alghamdi and
Solamy [35], in which they have studied the geometry of warped product
pointwise semi-slant submanifolds of locally product Riemannian manifolds. In
this paper, we try to generalize the notion to bi-slant warped products of
locally product Riemannain manifolds. We proved several results on pointwise
bi-slant submanifolds of locally product Riemannain manifolds, in addition, we
proved some characterization results for pointwise bi-slant submanifolds of
locally product Riemannain manifolds. Later, we also provide some non-trivial
examples of such submanifolds.

##  2\. Preliminaries

Let $\bar{M}$ be an m-dimensional differential manifold with a tensor field
$F$ of type (1,1) such that $F^{2}=I$ and $F\neq\pm I$. Then we say that
$\bar{M}$ is an almost product manifold with almost product structure $F$. If
an almost product manifold $\bar{M}$ has a Riemannian metric $g$ such that

$\displaystyle g(FX,FY)=g(X,Y),$ (2.1)

for any $X,Y\in\Gamma(T\bar{M})$, then $\bar{M}$ is called an almost product
Riemannian manifold[36], where $\Gamma(T\bar{M})$ denotes the set of all
vector fields of $\bar{M}$. Let $\bar{\nabla}$ denotes the Levi-Civita
connection on $\bar{M}$ with respect to the Riemannian metric $g$. if
$(\bar{\nabla}_{X}F)Y=0$, for any vector $X,Y\in\Gamma(T\bar{M})$, the
$\bar{M}$ is called a locally product Riemannian manifold [23].
Let $M$ be a Riemannian manifold isometrically immersed in $\bar{M}$ and we
denote by the symbol $g$ the Riemannian metric induced on $M$. Let
$\Gamma(TM)$ denote the Lie algebra of vector fields in $M$ and
$\Gamma(T^{\perp}M)$, the set of all vector fields normal to $M$. If $\nabla$
be the induced Levi-Civita connection on $M$, the Gauss and Weingarten
formulas are respectively given by

$\displaystyle\bar{\nabla}_{X}Y=\nabla_{X}Y+\sigma(X,Y),$ (2.2)

and

$\displaystyle\bar{\nabla}_{X}N=-A_{N}X+\nabla^{\perp}_{X}N,$ (2.3)

for any $X,Y\in\Gamma(TM)$ and $N\in\Gamma(T^{\perp}M)$, where
$\nabla^{\perp}$ is the normal connection on $T^{\perp}M$ and $A$ the shape
operator. The shape operator and the second fundamental form of $M$ are
related by

$\displaystyle g(A_{N}X,Y)=g(\sigma(X,Y),N),$ (2.4)

for any $X,Y\in\Gamma(TM)$ and $N\in\Gamma(T^{\perp}M)$, and $g$ denotes the
induced metric on $M$ as well as the metric on $\bar{M}$.
For a tangent vector field $X$ and a normal vector field $N$ of $M$, we can
write

$\displaystyle FX=TX+\omega X,\hskip 14.22636ptFN=BN+CN,$ (2.5)

where $TX$ and $\omega X$ (respectively, $BN\hskip
2.84544pt\textnormal{and}\hskip 2.84544ptCN$) are the tangential and normal
components of $FX$ (respectively, of $FN$).
Moreover, from (2.1) and (2.5), we have

$\displaystyle g(TX,Y)=g(X,TY),$ (2.6)

for any $X,Y\in\Gamma(TM)$.
We can now specify the following classes of submanifolds of locally product
Riemannian manifolds:
(1) A submanifold $M$ of a locally product Riemannian manifold $\bar{M}$ is
said to be slant (see [2, 7, 28]), if for each non-zero vector $X$ tangent to
$M$, the angle $\theta(X)$ between $FX$ and $T_{p}M$ is a constant, i.e., it
does not depend on the choice of $p\in M$ and $X\in T_{p}M$.
(2) A submanifold $M$ of a locally product Riemannian manifold $\bar{M}$ is
called semi-invariant submanifold (see [5, 27]) of $\tilde{M}$ if there exists
a differentiable distribution $D:p\rightarrow D_{p}\subset T_{p}M$ such that
$D$ is invariant with respect to $F$ and the complementary distribution
$D^{\perp}$ is anti-invariant with respect to $F$.
(3) A submanifold $M$ of a locally product Riemannian manifold $\bar{M}$ is
called semi-slant (see [23, 26]), if it is endowed with two orthogonal
distributions $D$ and $D^{\theta}$, where $D$ is invariant with respect to $F$
and $D^{\theta}$ is slant, i.e., $\theta(X)$ is the angle between $FX$ and
$D^{\theta}_{p}$ is constant for any $X\in D^{\theta}_{p}$ and $p\in M$.

######  Definition 2.1.

A submanifold $M$ of a locally product Riemannian manifolld $\bar{M}$ is
called pointwise slant [19], if at each point $p\in M$, the Wirtinger angle
$\theta(X)$ between $FX$ and $T_{p}M$ is independent of the choice of the non-
zero vector $X\in T_{p}M$. In this case, the Wirtinger angle gives rise to a
real valued function $\theta:TM-\\{0\\}\rightarrow\mathbb{R}$ which is called
Wirtinger function or slant function of the pointwise slant function.

We note that a pointwise slant submanifold of a locally product Riemannian
manifold is called slant, in the sense of [2, 28], if its Wirtinger function
$\theta$ is globally constant. We also note that every slant submanifold is a
pointwise slant slant submanifold.

From Chen’s result (Lemma 2.1) of [12], we can easily show that $M$ is a
pointwise slant submanifold of a locally product Riemannian manifold $\bar{M}$
if and only if

$\displaystyle T^{2}=(\cos^{2}\theta)I,$ (2.7)

for some real-valued function $\theta$ defined on $M$, where $I$ denotes the
identity transformation of the tangent bundle $TM$ of $M$. The following
relations are the consequences of (2.7) as

$\displaystyle g(TX,TY)=(\cos^{2}\theta)g(X,Y),\hskip 14.22636ptg(\omega
X,\omega Y)=(\sin^{2}\theta)g(X,Y).$ (2.8)

Also, for a pointwise bi-slant submanifold of locally product Riemannian
manifold, (2.5) and (2.7) yields

$\displaystyle B\omega X=(\sin^{2}\theta)X\hskip 14.22636ptC\omega X=-\omega
TX.$ (2.9)

##  3\. Pointwise bi-slant submanifolds

In this section, we define and study pointwise bi-slant submanifolds of a
locally product Riemannian manifold.

######  Definition 3.1.

Let $\bar{M}$ be a locally product Riemannian manifold and $M$ a real
submanifold of $\bar{M}$. The we say $M$ is a bi-slant submanifold if there
exists a pair of orthogonal distributions $D_{1}$ and $D_{2}$ of $M$, at a
point $p\in M$ such that
(a) $TM=D_{1}\oplus D_{2}$;
(b) $JD_{1}\perp D_{2}$ and $JD_{2}\perp D_{1}$;
(c) The distributions $D_{1},D_{2}$ are pointwise slant with slant functions
$\theta_{1},\theta_{2}$, respectively.

The pair $\\{\theta_{1},\theta_{2}\\}$ of slant functions is called the bi-
slant function. A pointwise bi-slant submanifold $M$ is called proper if its
bi-slant function satisfies $\theta_{1},\theta_{2}\neq 0,\frac{\pi}{2}$ and
both $\theta_{1},\theta_{2}$ are not constant on $M$.

Note that (2.5) and condition $(b)$ in the Definition 3.1 imply that

$\displaystyle T(D_{i})\subset D_{i},\ i=1,2.$ (3.1)

Given a pointwise bi-slant submanifold $M$ of locally product Riemannian
manifold $\bar{M}$, for any $X\in\Gamma(TM)$, we put

$\displaystyle X=P_{1}X+P_{2}X$ (3.2)

where $P_{i}$ is the projection from $\Gamma(TM)$ onto $D_{i}$. Clearly,
$P_{i}X$ is the components of $X$ in $D_{i},i=1,2$. In particular, if $X\in
D_{i}$, we have $X=P_{i}X$. If we put $T_{i}=P_{i}\circ T$, then we can find
(3.2) that

$\displaystyle FX=T_{1}X+T_{2}X+\omega X,$ (3.3)

for any $X\in\Gamma(TM)$.
From now onwards, we assume the ambient manifold $\bar{M}$ is locally product
Riemannian manifold and $M$ is pointwise bi-slant submanifold in $\bar{M}$.
Now we give the following useful lemma for later use.

######  Lemma 3.2.

Let $M$ be a pointwise bi-slant submanifold of a locally product Riemannian
manifold $\bar{M}$ with pointwise slant distributions $D_{1}$ and $D_{2}$ with
distinct slant functions $\theta_{1}$ and $\theta_{2}$, respectively. Then
(i) For $X,Y\in D_{1}$ and $Z\in D_{2}$, we have

$\displaystyle(\sin^{2}\theta_{2}-\sin^{2}\theta_{1})g(\nabla_{X}Y,Z)$
$\displaystyle=$ $\displaystyle g\big{\\{}(\sigma(X,Z),\omega
T_{1}Y)+(\sigma(X,T_{2}Z),\omega Y)\big{\\}}$
$\displaystyle+g\big{\\{}(\sigma(X,Y),\omega T_{2}Z)+(\sigma(X,T_{1}Y),\omega
Z)\big{\\}}.$

(ii) For $Z,W\in D_{2}$ and $X\in D_{1}$, we have

$\displaystyle(\sin^{2}\theta_{1}-\sin^{2}\theta_{2})g(\nabla_{Z}W,X)$
$\displaystyle=$ $\displaystyle g\big{\\{}(\sigma(X,Z),\omega
T_{2}W)+(\sigma(Z,T_{1}X),\omega W)\big{\\}}$
$\displaystyle+g\big{\\{}(\sigma(Z,W),\omega T_{1}X)+(\sigma(Z,T_{2}W),\omega
X)\big{\\}}.$

###### Proof.

For $X,Y\in D_{1}$ and $Z\in D_{2}$, we have

$\displaystyle
g(\nabla_{X}Y,Z)=g(\bar{\nabla}_{X}Y,Z)=g(F\bar{\nabla}_{X}Y,FZ).$

Using the locally product structure and (2.5), we have

$\displaystyle g(\nabla_{X}Y,Z)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}T_{1}Y,FZ)+g(\bar{\nabla}_{X}\omega
Y,T_{2}Z)+g(\bar{\nabla}_{X}\omega Y,\omega Z).$ $\displaystyle=$
$\displaystyle g(\bar{\nabla}_{X}T^{2}_{1}Y,Z)+g(\bar{\nabla}_{X}\omega
T_{1}Y,Z)-g(A_{\omega Y}X,T_{2}Z)$ $\displaystyle-g(\bar{\nabla}_{X}\omega
Z,\omega Y).$

Again using (2.5) and (2.7), we obtain

$\displaystyle g(\nabla_{X}Y,Z)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{1}g(\bar{\nabla}_{X}Y,Z)-\sin
2(\theta_{1})X(\theta_{1})g(Y,Z)-g(A_{\omega T_{1}Y}X,Z)$
$\displaystyle-g(A_{\omega Y}X,T_{2}Z)-g(\bar{\nabla}_{X}\omega
Z,FY)+g(\bar{\nabla}_{X}\omega Z,T_{1}Y).$

By using the orthogonality of two distributions and the symmetry of shape
operator, the above equation reduces to

$\displaystyle\sin^{2}\theta_{1}g(\nabla_{X}Y,Z)$ $\displaystyle=$
$\displaystyle-g(\sigma(X,Z),\omega T_{1}Y)-g(\sigma(X,T_{2}Z),\omega Y)$
$\displaystyle-g(\bar{\nabla}_{X}B\omega Z,Y)-g(\bar{\nabla}_{X}C\omega
Z,Y)-g(A_{\omega Z}X,T_{1}Y).$

Thus, from (2.9), we obtain

$\displaystyle\sin^{2}\theta_{1}g(\nabla_{X}Y,Z)$ $\displaystyle=$
$\displaystyle-g(\sigma(X,Z),\omega T_{1}Y)-g(\sigma(X,T_{2}Z),\omega Y)$
$\displaystyle-\sin^{2}\theta_{2}g(\bar{\nabla}_{X}Z,Y)-\sin
2(\theta_{2})X(\theta_{2})g(Y,Z)$ $\displaystyle+g(\bar{\nabla}_{X}\omega
T_{2}Z,Y)-g(A_{\omega Z}X,T_{1}Y).$

Using (2.3) and the orthogonality of vector fields, we have

$\displaystyle\sin^{2}\theta_{1}g(\nabla_{X}Y,Z)$ $\displaystyle=$
$\displaystyle-g(\sigma(X,Z),\omega T_{1}Y)-g(\sigma(X,T_{2}Z),\omega Y)$
$\displaystyle+\sin^{2}\theta_{2}g(\bar{\nabla}_{X}Y,Z)-g(A_{\omega
T_{2}Z}X,Y)$ $\displaystyle-g(A_{\omega Z}T_{1}Y,X).$

Now, part $(i)$ of the lemma follows from the above Equation by using (2.4).
In the similar fashion, we can prove part $(ii)$. ∎

The following corollary is the immediate consequence of the Lemma 1$(i)$

######  Corollary 3.3.

Let $M$ be a pointwise semi-slant submanifold of a locally product Riemannian
manifold $\bar{M}$. Then,

$\displaystyle\sin^{2}\theta g(\nabla_{X}Y,Z)=g(\sigma(X,Y),\omega
TZ)+g(\sigma(X,FY),\omega Z),$

for any $X,Y\in D_{1}$ and $Z\in D_{2}$.

###### Proof.

If we put $\theta_{1}=0$ and $\theta_{2}=\theta$, a slant function, then the
submanifold $M$ of locally product Riemannian manifold $\bar{M}$ becomes
pointwise semi-slant submanifold. In this case, the first two terms in the
right hand size of (3.2) vanish identically. Thus the relation (3.2) reduces
to

$\displaystyle\sin^{2}\theta g(\nabla_{X}Y,Z)=g(\sigma(X,Y),\omega
TZ)+g(\sigma(X,FY),\omega Z).$

The same result has been proved in [35]. ∎

##  4\. Warped product pointwise bi-slant submanifolds of locally product
Riemannian manifold

Let $(M_{1},g_{1})$ and $(M_{2},g_{2})$ be two Riemannian manifolds and $f>0$,
be a positive differentiable function on $M_{1}$. Consider the product
manifold $M_{1}\times M_{2}$ with its canonical projections $\pi:M_{1}\times
M_{2}\rightarrow M_{1}$ and $\rho:M_{1}\times M_{2}\rightarrow M_{2}$. The
warped product $M=M_{1}\times_{f}M_{2}$ is the product manifold $M_{1}\times
M_{2}$ equipped with the Riemannian metric $g$ such that

$\displaystyle
g(X,Y)=g_{1}(\pi_{\ast}(X),\pi_{\ast}(Y))+(f\circ\pi)^{2}g_{2}(\rho_{\ast}(X),\rho_{\ast}(Y))$

for any tangent vector $X,Y\in TM$, where $\ast$ is the symbol for the tangent
maps. It was proved in [4] that for any $X\in TM_{1}$ and $Z\in TM_{2}$, the
following holds

$\displaystyle\nabla_{X}Z=\nabla_{Z}X=(Xlnf)Z$ (4.1)

where $\nabla$ denotes the Levi-Civita connection of $g$ on $M$. A warped
product manifold $M=M_{1}\times_{f}M_{2}$ is said to be trivial if the warping
function $f$ is constant. If $M=M_{1}\times_{f}M_{2}$ is a warped product
manifold then $M_{1}$ is totally geodesic and $M_{2}$ is a totally umbilical
(see [4, 9]).

######  Lemma 4.1.

Let $M_{T}\times_{f}M_{\theta}$ be a warped product pointwise bi-slant
submanifold of a locally product Riemannian manifold $\bar{M}$ such that
$M_{T}$ and $M_{\theta}$ are pointwise slant submanifolds with slant functions
$\theta_{1}$ and $\theta_{2}$, respectively of $\bar{M}$. Then we have the
following

$\displaystyle g(\sigma(X,W),\omega T_{2}Z)+g(\sigma(X,T_{2}Z),\omega
W)=-(\sin 2\theta_{2})X(\theta_{2})g(Z,W)$ (4.2)

for any $X\in TM_{T}$ and $Z,W\in TM_{\theta}$.

###### Proof.

For any $X\in TM_{T}$ and $Z,W\in TM_{\theta}$, we have

$\displaystyle g(\bar{\nabla}_{X}Z,W)=g(\nabla_{X}Z,W)=X(lnf)g(Z,W).$ (4.3)

On the other hand, we also have

$\displaystyle
g(\bar{\nabla}_{X}Z,W)=g(F\bar{\nabla}_{X}Z,FW)=g(\bar{\nabla}_{X}FZ,FW).$

Now for any $X\in TM_{T}$ and $Z,W\in TM_{\theta}$. Using (2.5), we obtain

$\displaystyle
g(\bar{\nabla}_{X}Z,W)=g(\bar{\nabla}_{X}T_{2}Z,T_{2}W)+g(\bar{\nabla}_{X}T_{2}Z,\omega
W)+g(\bar{\nabla}_{X}\omega Z,FW).$

Then from (2.1), (2.2), (4.1) and the locally product Riemannian structure, we
derive

$\displaystyle g(\bar{\nabla}_{X}Z,W)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{2}X(lnf)g(Z,W)+g(\sigma(X,T_{2}Z),\omega
W)+g(\bar{\nabla}_{X}F\omega Z,W)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{2}X(lnf)g(Z,W)+g(\sigma(X,T_{2}Z),\omega
W)+g(\bar{\nabla}_{X}B\omega Z,W)$ $\displaystyle+g(\bar{\nabla}_{X}C\omega
Z,W).$

Using (2.9), we find

$\displaystyle g(\bar{\nabla}_{X}Z,W)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{2}X(lnf)g(Z,W)+g(\sigma(X,T_{2}Z),\omega W)$
$\displaystyle+\sin^{2}\theta_{2}g(\bar{\nabla}_{X}Z,W)+\sin
2\theta_{2}X(\theta_{2})g(Z,W)$ $\displaystyle-g(\bar{\nabla}_{X}\omega
T_{2}Z,W).$

Thus the lemma follows from (4.3) and (4.1) by Using (2.3) and (4.1). ∎

######  Lemma 4.2.

Let $M_{T}\times_{f}M_{\theta}$ be a warped product pointwise bi-slant
submanifold of a locally product Riemannian manifold $\bar{M}$ such that
$M_{T}$ and $M_{\theta}$ are pointwise slant submanifolds with slant functions
$\theta_{1}$ and $\theta_{2}$, respectively of $\bar{M}$. Then we have the
following

$\displaystyle g(\sigma(X,Z),\omega W)+g(\sigma(X,W),\omega
Z)=-2(\tan\theta_{2})X(\theta_{2})g(T_{2}Z,W)$ (4.5)

for any $X\in TM_{T}$ and $Z,W\in TM_{\theta}$.

###### Proof.

The proof of this lemma follows by Interchanging $Z$ by $T_{2}Z$ for any $Z\in
TM_{2}$ in (4.2) and then by using (2.7). ∎

######  Lemma 4.3.

Let $M_{T}\times_{f}M_{\theta}$ be a warped product pointwise bi-slant
submanifold of a locally product Riemannian manifold $\bar{M}$ such that
$M_{T}$ and $M_{\theta}$ are pointwise slant submanifolds with slant functions
$\theta_{1}$ and $\theta_{2}$, respectively of $\bar{M}$. Then

$\displaystyle(i)\hskip 14.22636ptg(\sigma(X,Z),\omega W)=g(\sigma(X,W),\omega
Z),$ (4.6) $\displaystyle(ii)\hskip 14.22636ptg(\sigma(X,Z),\omega
Y)=-g(\sigma(X,Y),\omega Z),$ (4.7)

for any $X\in TM_{T}$ and $Z,W\in TM_{\theta}$.

###### Proof.

For any $X\in TM_{T}$ and $Z,W\in TM_{\theta}$, we have

$\displaystyle g(\sigma(X,Z),\omega W)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}X,\omega W)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}X,FW)-g(\bar{\nabla}_{Z}X,T_{2}W).$

Using (2.1),(2.5) and (4.1), we obtain

$\displaystyle g(\sigma(X,Z),\omega
W)=g(\bar{\nabla}_{Z}T_{1}X,W)+g(\bar{\nabla}_{Z}\omega
X,W)-X(lnf)g(Z,T_{2}W).$

On simplification and using (2.3), (2.4) and (4.1), we derive

$\displaystyle g(\sigma(X,Z),\omega W)=T_{1}X(lnf)g(Z,W)-g(\sigma(Z,W),\omega
X)-X(lnf)g(Z,T_{2}W).$

Then from polarization, we get

$\displaystyle g(\sigma(X,W),\omega Z)=T_{1}X(lnf)g(Z,W)-g(\sigma(Z,W),\omega
X)-X(lnf)g(T_{2}Z,W).$

Subtracting (4) from (4) and using (2.6), we obtain

$\displaystyle g(\sigma(X,Z),\omega W)-g(\sigma(X,W),\omega Z)=0.$

Hence, the proof follows from the above relation.
For part $(ii)$, the proof follows same as part $(i)$. ∎

######  Theorem 4.4.

Let $M=M_{T}\times_{f}M_{\theta}$ be a warped product pointwise bi-slant
submanifold of a locally product Riemannian manifold $\bar{M}$ such that
$M_{T}$ and $M_{\theta}$ are pointwise slant submanifolds with distinct slant
functions $\theta_{1}$ and $\theta_{2}$, respectively of $\bar{M}$. Then we
have

$\displaystyle g(A_{\omega T_{1}X}W+A_{\omega X}T_{2}W,Z)+g(A_{\omega
T_{2}W}X+A_{\omega W}T_{1}X,Z)$
$\displaystyle=(\sin^{2}\theta_{2}-\sin^{2}\theta_{1})X(lnf)g(Z,W).$ (4.10)

for any $X,Y\in TM_{T}$ and $Z,W\in TM_{\theta}$.

###### Proof.

For any $X,Y\in TM_{T}$ and $Z,W\in TM_{\theta}$, we have

$\displaystyle g(\bar{\nabla}_{Z}X,W)=g(\nabla_{Z}X,W)=X(lnf)g(Z,W).$ (4.11)

On the other hand, for $X,Y\in TM_{T}$ and $Z,W\in TM_{\theta}$, we have

$\displaystyle
g(\bar{\nabla}_{Z}X,W)=g(F\bar{\nabla}_{Z}X,FW)=g(\bar{\nabla}_{Z}FX,FW).$

Therefore, by using (2.5), we get

$\displaystyle
g(\bar{\nabla}_{Z}X,W)=g(\bar{\nabla}_{Z}T_{1}X,FW)+g(\bar{\nabla}_{Z}\omega
X,T_{2}W)+g(\bar{\nabla}_{Z}\omega X,\omega W).$

Using (2.1), (2.3) and definition of locally product Riemannian manifold, we
obtain

$\displaystyle g(\bar{\nabla}_{Z}X,W)=g(\bar{\nabla}_{Z}FT_{1}X,W)-g(A_{\omega
X}Z,T_{2}W)-g(\bar{\nabla}_{Z}\omega W,\omega X).$

From (2.5) and symmetry of shape operator, we derive

$\displaystyle g(\bar{\nabla}_{Z}X,W)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}T_{1}^{2}X,W)+g(\bar{\nabla}_{Z}\omega T_{1}X,W)-g(A_{\omega
X}T_{2}W,Z)$ $\displaystyle-g(F\bar{\nabla}_{Z}\omega
W,X)+g(\bar{\nabla}_{Z}\omega W,T_{1}X)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{1}g(\bar{\nabla}_{Z}X,W)-\sin
2\theta_{1}Z(\theta_{1})g(X,W)-g(A_{\omega T_{1}X}Z,W)$
$\displaystyle-g(A_{\omega X}T_{2}W,Z)-g(\bar{\nabla}_{Z}F\omega
W,X)-g(A_{\omega W}Z,T_{1}X).$

Using (2.2), (2.5), (4.1), (4.11) and the orthogonality of vector fields and
symmetry of shape operator, we get

$\displaystyle\sin^{2}\theta_{1}X(lnf)g(Z,W)$ $\displaystyle=$
$\displaystyle-g(A_{\omega T_{1}X}W+A_{\omega
X}T_{2}W,Z)-g(\bar{\nabla}_{Z}B\omega W,X)$
$\displaystyle-g(\bar{\nabla}_{Z}C\omega W,X)-g(A_{\omega W}T_{1}X,Z).$

Using (2.9), we arrive at

$\displaystyle\sin^{2}\theta_{1}X(lnf)g(Z,W)$ $\displaystyle=$
$\displaystyle-g(A_{\omega T_{1}X}W+A_{\omega
X}T_{2}W,Z)-\sin^{2}\theta_{2}g(\bar{\nabla}_{Z}W,X)$ $\displaystyle-\sin
2\theta_{2}Z(\theta_{2})g(X,W)+g(\bar{\nabla}_{Z}\omega T_{2}W,X)$
$\displaystyle-g(A_{\omega W}T_{1}X,Z).$

Further, using orthogonality of vector fields and the relation (2.2), (2.3)
and (4.1), we obtain

$\displaystyle\sin^{2}\theta_{1}X(lnf)g(Z,W)$ $\displaystyle=$
$\displaystyle-g(A_{\omega T_{1}X}W+A_{\omega X}T_{2}W,Z)$
$\displaystyle+\sin^{2}\theta_{2}X(lnf)g(Z,W)-g(A_{\omega T_{2}W}Z,X)$
$\displaystyle-g(A_{\omega W}T_{1}X,Z).$

Again using the symmetry of shape operator, we obtain (4.4) from the above
relation. Hence the proof is complete. ∎

##  5\. Characterization for warped product pointwise bi-slant submanifolds of
locally product Riemannian manifolds.

In this section, we will prove the characterization for warped product
pointwise bi-slant submanifolds of locally product Riemannian manifolds. For
this, we need the following well known theorem of Hiepko’s.

######  Theorem 5.1.

[20] Let $D_{1}$ and $D_{2}$ be two orthogonal distribution on a Riemannian
manifold $M$. Suppose that $D_{1}$ and $D_{2}$ both are involutive such that
$D_{1}$ is totally geodesic folation and $D_{2}$ is a spherical foliation.
Then $M$ is locally isometric to a non-trial warped product
$M_{1}\times_{f}M_{2}$, where, $M_{1}$ and $M_{2}$ are integral manifolds of
$D_{1}$ and $D_{2}$, respectively.

The following result provides a characterization for warped product pointwise
bi-slant submanifolds of locally product Riemannian manifolds.

######  Theorem 5.2.

Let $M$ be a proper pointwise bi-slant submanifold of a locally product
Riemannian manifold $\bar{M}$ with pointwise slant distributions $D_{1}$ and
$D_{2}$. Then $M$ is locally a warped product pointwise bi-slant submanifold
of the form $M_{T}\times_{f}M_{\theta}$, where $M_{T}$ and $M_{\theta}$ are
pointwise slant submanifolds with distinct slant functions $\theta_{1}$ and
$\theta_{2}$, respectively of $\bar{M}$ if and only if the shape operator of
$M$ satisfies

$\displaystyle A_{\omega T_{1}X}Z+A_{\omega X}T_{2}Z+A_{\omega
T_{2}Z}X+A_{\omega Z}T_{1}X=(\sin^{2}\theta_{2}-\sin^{2}\theta_{1})X(\mu)Z$
(5.1)

for $X\in D_{1}$, $Z\in D_{2}$ and for a smooth function $\mu$ on $M$
satisfying $W(\mu)=0$ for any $W\in D_{2}$.

###### Proof.

Let $M=M_{T}\times_{f}M_{\theta}$ be a warped product pointwise bi-slant
submanifold of a locally product Riemannian manifold $\bar{M}$. Then from
Lemma 4.2$(ii)$, we have

$\displaystyle g(A_{\omega Y}Z+A_{\omega Z}Y,X)=0$ (5.2)

for any $X,Y\in TM_{1}$ and $Z\in TM_{2}$. Interchanging $Y$ by $T_{1}Y$ in
(5.2), we obtain

$\displaystyle g(A_{\omega T_{1}Y}Z+A_{\omega Z}T_{1}Y,X)=0.$ (5.3)

Again interchanging $Z$ by $T_{2}Z$ in (5.2), we obtain

$\displaystyle g(A_{\omega Y}T_{2}Z+A_{\omega T_{2}Z}Y,X)=0.$ (5.4)

Adding equations (5.3) and (5.4), we get

$\displaystyle g(A_{\omega T_{1}Y}Z+A_{\omega Z}T_{1}Y+A_{\omega
Y}T_{2}Z+A_{\omega T_{2}Z}Y,X)=0.$ (5.5)

Then (5.1) follows from (4.4) by using the above fact.

Conversely, if $M$ be a pointwise bi-slant subamnifold of a locally product
Riemannian manifold with pointwise slant distributions $D_{1}$ and $D_{2}$
such that (5.1) holds, then from Lemma 3.2$(i)$, we have

$\displaystyle(\sin^{2}\theta_{2}-\sin^{2}\theta_{1})g(\nabla_{X}Y,Z)$
$\displaystyle=$ $\displaystyle g(A_{\omega T_{1}Y}Z+A_{\omega Z}T_{1}Y$
$\displaystyle+A_{\omega Y}T_{2}Z+A_{\omega T_{2}Z}Y,X)$

for any $X,Y\in D_{1}$ and $Z\in D_{2}$. Using the above condition (5.1), we
have

$\displaystyle g(\nabla_{X}Y,Z)=X(\mu)g(X,Z)=0$

which indicates that the leaves of the distributions are totally geodesic in
$M$.
On the other hand, from Lemma 3.2$(ii)$, we have

$\displaystyle(\sin^{2}\theta_{1}-\sin^{2}\theta_{2})g(\nabla_{Z}W,X)$
$\displaystyle=$ $\displaystyle g(A_{\omega T_{2}W}X+A_{\omega W}T_{1}X$
$\displaystyle+A_{\omega T_{1}X}W+A_{\omega X}T_{2}W,Z).$

From the hypothesis of the theorem i.e., (5.1), we get

$\displaystyle g(\nabla_{Z}W,X)=-X(\mu)g(Z,W).$ (5.6)

By polarization, we arrive at

$\displaystyle g(\nabla_{W}Z,X)=-X(\mu)g(Z,W).$ (5.7)

On subtracting (5.7) from (5.6) and by the definition of Lie bracket, we
obtain $g([Z,W],X)=0$, which depicts that the distribution $D_{2}$ is
integrable. If we consider a leaf $M_{2}$ of $D_{2}$ and the second
fundamental form $\sigma_{2}$ of $M_{2}$ in $M$, then from (5.6), we have

$\displaystyle g(\sigma_{2}(Z,W),X)=g(\nabla_{Z}W,X)=-X(\mu)g(Z,W).$

Now, by the definition of the gradient we have
$\sigma_{2}(Z,W)=-\bar{\nabla}_{\mu}g(Z,W)$, such that $\bar{\nabla}_{\mu}$ is
the gradient of $\mu$. The above relations shows that the leaf of $M_{2}$ is
totally Umbilical in $M$ with the mean curvature vector
$H_{2}=-\bar{\nabla}_{\mu}$. Since $W(\mu)=0$ for any $W\in D_{2}$, which
clearly shows that the mean curvature is parallel. Thus, the spherical
condition is satisfied. Then by Hiepko’s Theorem $M$ is locally a warped
product pointwise bi-slant submanifold. Hence the proof is complete. ∎

The following immediate consequences of the above theorem are given below:

1\. In Theorem 5.2, if $\theta_{1}=0$ and $\theta_{2}=\theta$, a slant
function, then the submanifold $M$ of locally product Riemannian manifold
$\bar{M}$ becomes a pointwise semi-slant submanifold which has been studied in
[35]. In this case, the first two terms in the left hand side of (5.1) vanish
identically. Thus, the relation (5.1) is true for pointwise semi-slant warped
product and it reduces to

$\displaystyle A_{\omega TZ}X+A_{\omega Z}FX=(\sin^{2}\theta)X(\mu)Z$

for $X\in D_{1}$ and $Z\in D_{2}$, where $D_{1}$ and $D_{2}$ are complex and
proper pointwise slant distributions of $M$. The same has been proved in [35].

2\. In Theorem 5.2, if we consider $\theta_{1}=\theta$ a constant slant angle
and $\theta_{2}=\frac{\pi}{2}$, then it is a case of hemi-slant warped
products. In this case, the second and third term in the left hand side of
(5.1) vanish identically. Thus the relation (5.1) is true for hemi-slant
warped products and it reduces to

$\displaystyle A_{\omega TX}Z+A_{FZ}TX=(\cos^{2}\theta)X(\mu)Z$

for $X\in D_{\theta}$ and $Z\in D^{\perp}$, where $D_{\theta}$ and $D^{\perp}$
are proper slant and totally real distributions.

3\. In Theorem 5.2, if $\theta_{1}=0$ and $\theta_{2}=\frac{\pi}{2}$, then it
is a case of CR-warped product. In this case all the terms in the left hand
side of(5.1) vanish identically. Thus the relation (5.1) is true for CR-warped
products and it will be

$\displaystyle A_{FZ}FX=X(\mu)Z$

for $X\in D$ and $Z\in D^{\perp}$, where $D$ and $D^{\perp}$ are complex and
totally real distributions of $M$.

##  6\. Some examples on warped product pointwise bi-slant submanifolds of
locally product Riemannian manifold.

######  Example 1.

Let $\mathbb{R}^{4}$ be the Euclidean space with the cartesian coordinates
given by $(x_{1},x_{2},y_{1},y_{2})$ and the almost product structure

$\displaystyle F\bigg{(}\frac{\partial}{\partial
x_{i}}\bigg{)}=\frac{\partial}{\partial x_{i}},\hskip
14.22636ptF\bigg{(}\frac{\partial}{\partial
y_{j}}\bigg{)}=-\frac{\partial}{\partial y_{j}},1\leq i,j\leq 2.$

A submanifold $M$ of $\mathbb{R}^{4}$ defined by

$\displaystyle\chi(u,v,w)=(wu\cos v,wu\sin v,w\cos v,w\sin v).$

It is easy to see that the tangent bundle $TM$ of $M$ is spanned by the
following vectors

$\displaystyle v_{1}=w\cos v\frac{\partial}{\partial x_{1}}+w\sin
v\frac{\partial}{\partial x_{2}},$ $\displaystyle v_{2}=-wu\sin
v\frac{\partial}{\partial x_{1}}+wu\cos v\frac{\partial}{\partial x_{2}}-w\sin
v\frac{\partial}{\partial y_{1}}+w\cos v\frac{\partial}{\partial y_{2}},$
$\displaystyle v_{3}=u\cos v\frac{\partial}{\partial x_{1}}+u\sin
v\frac{\partial}{\partial x_{2}}+\cos v\frac{\partial}{\partial y_{1}}+\sin
v\frac{\partial}{\partial y_{2}}.$

Then , clearly we obtain

$\displaystyle Fv_{1}=w\cos v\frac{\partial}{\partial x_{1}}+w\sin
v\frac{\partial}{\partial x_{2}},$ $\displaystyle Fv_{2}=-wu\sin
v\frac{\partial}{\partial x_{1}}+wu\cos v\frac{\partial}{\partial x_{2}}+w\sin
v\frac{\partial}{\partial y_{1}}-w\cos v\frac{\partial}{\partial y_{2}},$
$\displaystyle Fv_{3}=u\cos v\frac{\partial}{\partial x_{1}}+u\sin
v\frac{\partial}{\partial x_{2}}-\cos v\frac{\partial}{\partial y_{1}}-\sin
v\frac{\partial}{\partial y_{2}}.$

Then, we find that $D_{1}=span\\{v_{1},v_{3}\\}$ is a proper pointwise slant
distribution with slant angle
$\theta_{1}=\cos^{-1}\big{(}\frac{u}{\sqrt{1+u^{2}}}\big{)}$ and
$D_{2}=span\\{v_{2}\\}$ is again a proper pointwise slant distribution with
slant angle $\theta_{2}=\cos^{-1}\big{(}\frac{u^{2}-1}{u^{2}+1}\big{)}$. Thus,
$M$ is a proper pointwise bi-slant submanifold of $\mathbb{R}^{4}$.

It is easy to verify that both the distributions $D_{1}$ and $D_{2}$ are
integrable. If we denote the integrable manifolds of $D_{1}$ and $D_{2}$ by
$M_{T}$ and $M_{\theta}$, respectively. Then the metric tensor $g$ of product
manifold $M$ is given by

$\displaystyle g=g_{M_{T}}+w^{2}(1+u^{2})g_{M_{\theta}},$

where,

$\displaystyle g_{M_{T}}=w^{2}du^{2}+(1+u^{2})dw^{2}\hskip 14.22636ptand\hskip
14.22636ptg_{M_{\theta}}=dv^{2}.$

Hence, $M$ is a proper non-trival warped product pointwise bi-slant
submanifold of $\mathbb{R}^{4}$ with warping function
$f=\sqrt{w^{2}(1+u^{2})}$ and whose bi-slant angles $\theta_{1},\theta_{2}\neq
0,\frac{\pi}{2}$.

######  Example 2.

Let $\mathbb{R}^{6}=\mathbb{R}^{3}\times\mathbb{R}^{3}$ be a locally product
Riemannian manifold with cartesian coordinates
$(x_{1},x_{2},x_{3},y_{1},y_{3},y_{3})$. Consider a submanifold $M$ of
$\mathbb{R}^{6}$ defined by

$\displaystyle\chi(u,v,w)=(v\cos u,v\sin u,-v+w,w\cos u,w\sin u,v+w),$

with almost product structure $F$ defined by

$\displaystyle F\bigg{(}\frac{\partial}{\partial
x_{i}}\bigg{)}=-\bigg{(}\frac{\partial}{\partial x_{i}}\bigg{)},\hskip
14.22636ptF\bigg{(}\frac{\partial}{\partial
y_{j}}\bigg{)}=\bigg{(}\frac{\partial}{\partial y_{j}}\bigg{)},1\leq i,j\leq
3.$

It is easy to see that its tangent space $TM$ of $M$ is spanned by the
following vectors

$\displaystyle v_{1}=-v\sin u\frac{\partial}{\partial x_{1}}+v\cos
u\frac{\partial}{\partial x_{2}}-w\sin u\frac{\partial}{\partial y_{1}}+w\cos
u\frac{\partial}{\partial y_{2}},$ $\displaystyle v_{2}=\cos
u\frac{\partial}{\partial x_{1}}+\sin u\frac{\partial}{\partial
x_{2}}-\frac{\partial}{\partial x_{3}}+\frac{\partial}{\partial y_{3}},$
$\displaystyle v_{3}=\frac{\partial}{\partial x_{3}}+\cos
u\frac{\partial}{\partial y_{1}}+\sin u\frac{\partial}{\partial
y_{2}}+\frac{\partial}{\partial y_{3}}.$

Then, we have

$\displaystyle Fv_{1}=v\sin u\frac{\partial}{\partial x_{1}}-v\cos
u\frac{\partial}{\partial x_{2}}-w\sin u\frac{\partial}{\partial y_{1}}+w\cos
u\frac{\partial}{\partial y_{2}},$ $\displaystyle Fv_{2}=-\cos
u\frac{\partial}{\partial x_{1}}-\sin u\frac{\partial}{\partial
x_{2}}+\frac{\partial}{\partial x_{3}}+\frac{\partial}{\partial y_{3}},$
$\displaystyle Fv_{3}=-\frac{\partial}{\partial x_{3}}+\cos
u\frac{\partial}{\partial y_{1}}+\sin u\frac{\partial}{\partial
y_{2}}+\frac{\partial}{\partial y_{3}}.$

Let us put $D_{1}=span\\{v_{1}\\}$ is a proper slant distribution with slant
angle $\theta_{1}=\cos^{-1}\big{(}\frac{w^{2}-v^{2}}{w^{2}+v^{2}}\big{)}$ and
$D_{2}=span\\{v_{2},v_{3}\\}$ is again a proper slant distribution with slant
angle $\theta_{2}=\cos^{-1}\big{(}\frac{2}{3}\big{)}$. Hence the submanifold
$M$ defined by $\chi$ is a bi-slant submanifold.

It is easy to verify that both the distributions $D_{1}$ and $D_{2}$ are
integrable. If we denote the integrable manifolds of $D_{1}$ and $D_{2}$ by
$M_{T}$ and $M_{\theta}$, respectively. Then the metric tensor $g$ of product
manifold $M$ is given by

$\displaystyle g=g_{M_{\theta}}+(v^{2}+w^{2})g_{M_{T}}$

where

$\displaystyle g_{M_{\theta}}=3(dv^{2}+dw^{2})\hskip 14.22636ptand\hskip
14.22636ptg_{M_{T}}=du^{2}.$

Hence, $M$ is a proper non-trival warped product bi-slant submanifold of
$\mathbb{R}^{6}$ with warping function $f=\sqrt{v^{2}+w^{2}}$ and whose bi-
slant angles $\theta_{1},\theta_{2}\neq 0,\frac{\pi}{2}$.

Conflicts of Interest: The authors declare no conflict of interest.

## References

  * [1] M. Atceken,  Warped product semi-slant submanifolds in locally Riemannian product manifolds, Bull. Aust. Math. Soc., 77(2), (2008), 177-186.
  * [2] M. Atçeken,  Slant submanifolds of a Riemannian product manifold, Acta Math. Sci. Ser B Engl. Ed., 30(1), (2010), 215-224.
  * [3] F. R. Al-Solamy, V. A. Khan, S. Uddin,  Geometry of warped product semi-slant submanifolds of nearly Kaehler manifolds,  Results in Mathematics., 71(3), (2017), 783-799.
  * [4] R. L. Bishop, B. O’Neill,  Manifolds of negative curvature,  Trans. Am. Math. Soc., 145, (1969), 1-49.
  * [5] A. Bejancu,  Semi-invariant submanifolds of locally product Riemannian manifolds,  An. Univ. Timisoara. Ser Stiint. Math., 22, (1984), 3-11.
  * [6] J. L. Cabrerizo, A. Carriazo, L.M. Fernandez, M. Fernandez,  Semi-Slant submanifolds of a Sasakian manifold,  Geom. Dedicata., 78(2), (1999), 183-199.
  * [7] B. -Y. Chen,  Slant immersions,  Bull. Aust, Math. Soc., 41(1), (1990), 135-147.
  * [8] B. Y. Chen,  Geometry of warped product CR-submanifolds in Kaehler manifolds,  Monatsh. Math., 133, (2001), 177-195.
  * [9] B. Y. Chen,  Geometry of warped product CR-submanifolds in Kaehler manifolds II,  Monatsh. Math., 134, (2001), 103-119.
  * [10] B. Y. Chen,  Another general inequality for CR-warped products in complex space forms,  Hokkaido Math. J., 32(2), (2003), 415-444.
  * [11] B. Y. Chen,  CR-warped products in complex projective spaces with compact holomorphic factor,  Monatshefte für Mathematik., 141(3), (2004), 177-186.
  * [12] B. Y. Chen, O. J. Garay,  Pointwise slant submanifolds in almost Hermitian manifolds,  Turk. J. Math., 36, (2012), 630-640.
  * [13] B. Y. Chen,  Geometry of warped product submanifolds: A survey,  J. Adv. Math. Stud., 6(2), (2013), 1-8.
  * [14] B. -Y. Chen,  Differential geometry of warped product manifolds and submanifolds.  World Scientific., (2017).
  * [15] B. Y. Chen, S. Uddin,  Warped product pointwise bi-slant submanifolds of Kaehler manifolds,  arXiv preprint arXiv:1711.07117 (2017).
  * [16] B. -Y. Chen, M. H. Shahid, F. R. Al-Solamy,  Complex Geometry of Slant Submanifolds,  Springer, Berlin (2021).
  * [17] B. -Y. Chen, M. H. Shahid, F. R. Al-Solamy,  Contact Geometry of Slant Submanifolds,  Springer, Berlin (2021).
  * [18] F. Etayo,  On quasi-slant submanifolds of an almost Hermitian manifold,  Publ. Math. Debrecen., 53(1-2), (1998), 217-223.
  * [19] M. Gülbahar, E. Kilic, and S. S. ÇELİK  Special proper pointwise slant surfaces of a locally product Riemannian manifold, Turk. J. Math., 39(6), (2015), 884-899.
  * [20] S. Hiepko,  Eine inner kennzeichungder verzerrten produkte.  Math. Ann., 241, (1979), 209-215.
  * [21] Y. Kentaro, M. Kon,  Generic submanifolds.  Annali di Matematica Pura ed Applicata., 123(1), (1980), 59-92.
  * [22] V. A. Khan, M. A. Khan,  Semi-slant submanifolds of a nearly Kaehler manifold.  Turk. J. Math., 31(4), (2007), 341-353.
  * [23] H. Li, X. Liu,  ”Semi-slant submanifolds of a locally product manifold,” Georgian Math. J., 12, (2005), 273-282.
  * [24] M. A. Lone, M. H. Shahid, S. D. Sharma,  Hemi-slant submanifolds of nearly Kaehler manifolds.  Gulf Journal of Mathematics., 4(3), (2016).
  * [25] M. F. Naghi, M. S. Stankovic, F. Alghamdi,  Chen’s improved inequality for pointwise hemi-slant warped products in Kaehler manifolds.  Filomat., 34(3), (2020), 807-814.
  * [26] N. Papaghuic,  Semi-slant submanifolds of a Kahlerian manifold,  An. Stiint. Univ. Al. I. Cuza Iasi Mat., 40(1) (1994), 55-61.
  * [27] B. Sahin, M. Atceken,  Semi-invariant submanifolds of Riemannian product manifold, Balkan J. Geom. Appl., 8(1), (2003), 91-100.
  * [28] B. Sahin,  Slant submanifolds of an almost product Riemannian manifold,  J. Korean Math. Soc., 43(4), (2006), 717-732.
  * [29] B. Sahin,  Warped product submanifolds of Kaehler manifolds with a slant factor,  Ann. Polon. Math., 95(3), (2009), 207-226.
  * [30] B. Sahin,  Non-existence of warped product semi-slant submanifolds of Kaehler manifolds. Geometriae Dedicata., 117(1), (2006), 195-202.
  * [31] B. Sahin,  Warped product semi-slant submanifolds of a locally product Riemannian manifold,  Studia Sci. Math. Hungar., 46(2), (2009), 169-184.
  * [32] B. Sahin,  Warped product pointwise semi-slant submanifolds of Kaehler manifolds,  Port. Math., 70, (2013), 252-268.
  * [33] B. Sahin,  Warped product semi-slant submanifolds of a locally product Riemannian manifold,  Studia. Sci. Math. Hungar., 46(2), (2009), 169-184.
  * [34] S. Uddin, B. Y. Chen, F. R. Al-Solamy,  Warped product bi-slant immersions in Kaehler manifolds, Mediterr. J. Math., 14(2), (2017), 14-95.
  * [35] S. Uddin, F. Alghamdi, F. R. Al-Solamy,  Geometry of warped product pointwise semi-slant submanifolds of locally product Riemannian manifolds,  Journal of Geometry and Physics., 152, (2020), 103658.
  * [36] K. Yano, M. Kon,  Structures on manifolds,  Series in Pure mathematics, World scientific Publishing Co., Singapore, 1984.