.

# Characterization of warped product submanifolds of Lorentzian concircular
structure Manifolds

SHYAMAL KUMAR HUI∗, LAURIAN-IOAN PISCORAN and TANUMOY PAL

###### Abstract.

Recently Hui et al. ([10], [11]) studied contact CR-warped product
submanifolds and also warped product pseudo-slant submanifolds of a
$(LCS)_{n}$-manifold $\bar{M}$. In this paper we have studied the
characterization for both these classes of warped product submanifolds. It is
also shown that there do not exists any proper warped product bi-slant
submanifold of a $(LCS)_{n}$-manifold. Although we constructed an example of a
bi-slant submanifold of $(LCS)_{n}$-manifold.

###### Key words and phrases:

$(LCS)_{n}$-manifold, CR- submanifold, pseudo slant submanifold, bi-slant
submanifold, warped product submanifold
∗Corresponding author

###### 2010 Mathematics Subject Classification:

53C15, 53C25, 53C40.

##  1\. Introduction

As a generalization of Riemannian product manifold Bishop and O’Neill [6]
introduced the notion of warped product manifold and later it was studied in
([1], [2], [14], [17], [28]-[30]). The existence or non-existence of warped
product manifolds plays some important role in differential geometry as well
as physics.
As a generalization of LP-Sasakian manifold introduced independently by
Matsumoto [15] and also by Mihai and Rosca [16], Shaikh [19] introduced the
notion of Lorentzian concircular structure manifolds (briefly,
$(LCS)_{n}$-manifolds) with an example. Then Shaikh and Baishya ([21], [22])
investigated the applications of $(LCS)_{n}$-manifolds to the general theory
of relativity and cosmology. The $(LCS)_{n}$-manifolds are also studied in
([8], [20], [23]-[26]).
Due to important applications in applied mathematics and theoretical physics,
the geometry of submanifolds has become a subject of growing interest.
Analogous to almost Hermitian manifolds, the invariant and anti-invariant
subamnifolds are depend on the behaviour of almost contact metric structure
$\phi$. The study of the differential geometry of a contact CR-submanifold as
a generalization of invariant and anti-invariant subamnifold was introduced by
Bejancu [5]. In this connection it is mentioned that different class of
submanifolds of $(LCS)_{n}$-manifolds are studied in ([3], [4], [9], [10],
[12], [13], [27]). Recently Hui et al. ([10], [11]) studied contact CR-warped
product submanifolds and also warped product pseudo-slant submanifolds of a
$(LCS)_{n}$-manifold $\bar{M}$. In this paper we have studied the
characterization for both these classes of warped product submanifolds. An
example of bi-slant submanifold of $(LCS)_{n}$-manifold is constructed.
However, it is also shown that there do not exists any proper warped product
bi-slant submanifold of a $(LCS)_{n}$-manifold.

##  2\. Preliminaries

Let $\bar{M}$ be an $n$-dimensional Lorentzian manifold [18] admitting a unit
timelike concircular vector field $\xi$, called the characteristic vector
field of the manifold. Then we have

(2.1) $g(\xi,\xi)=-1.$

Since $\xi$ is a unit concircular vector field, it follows that there exists a
non-zero 1-form $\eta$ such that for

(2.2) $g(X,\xi)=\eta(X),$

the equation of the following form holds [31]

(2.3) $(\bar{\nabla}_{X}\eta)(Y)=\alpha\\{g(X,Y)+\eta(X)\eta(Y)\\},\ \ \
(\alpha\neq 0)$ (2.4) $\bar{\nabla}_{X}\xi=\alpha\\{X+\eta(X)\xi\\},\ \ \
\alpha\neq 0$

for all vector fields $X$, $Y$, where $\bar{\nabla}$ denotes the operator of
covariant differentiation with respect to the Lorentzian metric $g$ and
$\alpha$ is a non-zero scalar function satisfies

(2.5) ${\bar{\nabla}}_{X}\alpha=(X\alpha)=d\alpha(X)=\rho\eta(X),$

$\rho$ being a certain scalar function given by $\rho=-(\xi\alpha)$. If we put

(2.6) $\phi X=\frac{1}{\alpha}\bar{\nabla}_{X}\xi,$

then from (2.4) and (2.6) we have

(2.7) $\phi X=X+\eta(X)\xi,$ (2.8) $g(\phi X,Y)=g(X,\phi Y)$

from which it follows that $\phi$ is a symmetric (1,1) tensor and called the
structure tensor of the manifold. Thus the Lorentzian manifold $\bar{M}$
together with the unit timelike concircular vector field $\xi$, its associated
1-form $\eta$ and an (1,1) tensor field $\phi$ is said to be a Lorentzian
concircular structure manifold (briefly, $(LCS)_{n}$-manifold), [19].
Especially, if we take $\alpha=1$, then we can obtain the LP-Sasakian
structure of Matsumoto [15]. In a $(LCS)_{n}$-manifold $(n>2)$ $\bar{M}$, the
following relations hold [19]:

(2.9) $\eta(\xi)=-1,\ \ \phi\xi=0,\ \ \ \eta(\phi X)=0,\ \ \ g(\phi X,\phi
Y)=g(X,Y)+\eta(X)\eta(Y),$ (2.10) $\phi^{2}X=X+\eta(X)\xi,$ (2.11)
$(\bar{\nabla}_{X}\phi)Y=\alpha\\{g(X,Y)\xi+2\eta(X)\eta(Y)\xi+\eta(Y)X\\},$
(2.12) $(X\rho)=d\rho(X)=\beta\eta(X)$

for all $X,\ Y,\ Z\in\Gamma(T\bar{M})$ and $\beta=-(\xi\rho)$ is a scalar
function.
Let $M$ be a submanifold of $\bar{M}$ with induced metric $g$. Also let
$\nabla$ and $\nabla^{\perp}$ are the induced connections on the tangent
bundle $TM$ and the normal bundle $T^{\perp}M$ of $M$ respectively. Then the
Gauss and Weingarten formulae are given by

(2.13) $\bar{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y)$

and

(2.14) $\bar{\nabla}_{X}V=-A_{V}X+\nabla^{\perp}_{X}V$

for all $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\perp}M)$, where $h$ and $A_{V}$
are second fundamental form and the shape operator (corresponding to the
normal vector field $V$) respectively for the immersion of $M$ into $\bar{M}$
and they are related by $g(h(X,Y),V)=g(A_{V}X,Y)$, for any $X,Y\in\Gamma(TM)$
and $V\in\Gamma(T^{\perp}M)$.
For any $X\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$, we can write

(2.15) $\displaystyle\text{(a)}\ \phi X=PX+QX,\ \ \text{(b)}\ \phi V=bV+cV$

where $\phi X,\ bV$ are the tangential components and $QX,\ cV$ are the normal
components.
A submanifold $M$ of a $(LCS)_{n}$-manifold $\bar{M}$ is said to be invariant
if $\phi(T_{p}M)\subseteq T_{p}M$, for every $p\in M$ and anti-invariant if
$\phi T_{p}M\subseteq T^{\bot}_{p}M$, for every $p\in M$.
A submanifold $M$ of a $(LCS)_{n}$-manifold $\bar{M}$ is said to be a CR-
submnaifold if there is a differential distribution
$\mathcal{D}:p\rightarrow\mathcal{D}_{p}\subseteq T_{p}M$ such that
$\mathcal{D}$ is an invariant distribution and the orthogonal complementary
distribution $\mathcal{D}^{\bot}$ is anti-invariant.
The normal space of a CR-submanifold $M$ is decomposed as
$T^{\bot}M=Q\mathcal{D^{\bot}}\oplus\nu$, where $\nu$ is the invariant normal
subbundle of $M$ with respect to $\phi$.
A submanifold $M$ of a $(LCS)_{n}$-manifold $\bar{M}$ is said to be slant if
for each non-zero vector $X\in T_{p}M$ the angle between $\phi X$ and $T_{p}M$
is a constant, i.e. it does not depend on the choice of $p\in M$.
A submanifold $M$ of a $(LCS)_{n}$-manifold $\bar{M}$ is said to be a pseudo-
slant submanifold if there exists a pair of orthogonal distributions
$\mathcal{D^{\bot}}$ and $\mathcal{D^{\theta}}$ such that
(i) TM admits the orthogonal direct decomposition
$TM=\mathcal{D^{\bot}}\oplus\mathcal{D^{\theta}}$,
(ii) The distribution $\mathcal{D^{\bot}}$ is anti-invariant,
(iii) The distribution $\mathcal{D^{\theta}}$ is slant with slant angle
$\theta\neq 0,\frac{\pi}{2}$.
From the definition it is clear that if $\theta=0$, then $M$ is a CR-
submanifold. We say that a pseudo-slant submanifold is proper if $\theta\neq
0,\frac{\pi}{2}$. The normal space of a pseudo-slant submanifold $M$ is
decomposed as
$T^{\bot}M=Q\mathcal{D^{\theta}}\oplus\phi\mathcal{D}^{\bot}\oplus\nu$.
On a slant submanifold $M$ of a $(LCS)_{n}$-manifold $\bar{M}$, we have [4]

(2.16) $\displaystyle P^{2}X$ $\displaystyle=$
$\displaystyle\cos^{2}\theta[X+\eta(X)\xi],$ (2.17) $\displaystyle Q^{2}X$
$\displaystyle=$ $\displaystyle\sin^{2}\theta[X+\eta(X)\xi],$

where $\theta$ is the slant angle of $M$ in $\bar{M}$.
From (2.16) and (2.17), we get

(2.18) $\displaystyle g(PX,PY)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta[g(X,Y)+\eta(X)\eta(Y)],$ (2.19) $\displaystyle
g(QX,QY)$ $\displaystyle=$
$\displaystyle\sin^{2}\theta[g(X,Y)+\eta(X)\eta(Y)],$

for any $X,\ Y\in\Gamma(TM)$.
Also for a slant submanifold from (2.15) and (2.16), we have

(2.20) $bQX=\sin^{2}\theta(X+\eta(X)\xi)\ \ \text{and}\ \ cQX=-QPX$

For a Riemannian manifold $\bar{M}$ of dimension $n$ and a smooth function $f$
on $\bar{M}$, $\nabla f$, the gradient of $f$ which is defined by

(2.21) $g(\nabla f,X)=X(f)(\text{or}=(Xf))$

for any $X\in\Gamma(TM)$.

######  Definition 2.1.

[6] Let $(N_{1},g_{1})$ and $(N_{2},g_{2})$ be two Riemannian manifolds with
Riemannian metric $g_{1}$ and $g_{2}$ respectively and $f$ be a positive
definite smooth function on $N_{1}$. The warped product of $N_{1}$ and $N_{2}$
is the Riemannian manifold $N_{1}\times_{f}N_{2}=(N_{1}\times N_{2},g)$, where

(2.22) $g=g_{1}+f^{2}g_{2}.$

A warped product manifold $N_{1}\times_{f}N_{2}$ is said to be trivial if the
warping function $f$ is constant.

######  Proposition 2.1.

[18] Let $M=N_{1}\times_{f}N_{2}$ be a warped product manifold. Then

$\displaystyle\nabla_{U}X=\nabla_{X}U=(X\ln f)U,$

for any $X$, $Y\in\Gamma(TN_{1})$ and $U\in\Gamma(TN_{2})$.

######  Theorem 2.1 (Hiepko’s Theorem).

[7] Let $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ be two orthogonal
complementary distributions on a Riemannian manifold $M$. Suppose that
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are both involutive such that
$\mathcal{D}_{1}$ is a totally geodesic foliation and $\mathcal{D}_{2}$ is a
spherical foliation. Then $M$ is locally isometric to a non trivial warped
product $M_{1}\times_{f}M_{2}$, where $M_{1}$ and $M_{2}$ are integral
manifolds of $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$, respectively.

##  3\. characterization for contact CR-warped product submanifolds

In [11] it is shown that contact CR-warped product submanifolds of $\bar{M}$
of the form $N_{\bot}\times_{f}N_{T}$, where $N_{T}$ and $N_{\bot}$ are
invariant and anti-invariant submanifolds of $\bar{M}$ respectively, exists if
$\xi\in\Gamma(TN_{\bot})$ and does not exists if $\xi\in\Gamma(TN_{T})$. In
this section we find a characterization for a submanifolds $M$ of $\bar{M}$ to
be contact CR-warped product of the form $N_{\bot}\times_{f}N_{T}$ such that
$\xi\in\Gamma(TN_{\bot})$. First we prove the following Lemma:

######  Lemma 3.1.

Let $M=N_{\bot}\times_{f}N_{T}$ be a warped product submanifold of $\bar{M}$
such that $\xi\in\Gamma(TN_{\bot})$, then

(3.1) $\displaystyle g(h(X,Y),\phi Z)=-\alpha\eta(Z)g(X,Y)-(Z\ln f)g(\phi
X,Y)$

for $X,\ Y\in\Gamma(TN_{T})$ and $Z\in\Gamma(TN_{\bot})$.

###### Proof.

For $X,\ Y\in\Gamma(TN_{T})$ and $Z,\xi\in\Gamma(TN_{\bot})$ we have from
(2.11) and (2.13) that

$\displaystyle g(h(X,Y),\phi Z)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}Y,\phi Z)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}\phi Y,Z)-g((\bar{\nabla}_{X}\phi)Y,Z)$ $\displaystyle=$
$\displaystyle-g(\phi Y,\nabla_{X}Z)-\alpha g(X,Y)\eta(Z).$

By virtue of Proposition 2.1 from the above relation we get (3.1). ∎

Now interchanging $X$ by $\phi X$ and $Y$ by $\phi Y$, we get the following
respective relations

(3.2) $\displaystyle g(h(\phi X,Y),\phi Z)$ $\displaystyle=$
$\displaystyle-\alpha\eta(Z)g(\phi X,Y)-Z(\ln f)g(X,Y),$ (3.3) $\displaystyle
g(h(X,\phi Y),\phi Z)$ $\displaystyle=$ $\displaystyle-\alpha\eta(Z)g(\phi
X,Y)-Z(\ln f)g(X,Y),$ (3.4) $\displaystyle g(h(\phi X,\phi Y),\phi Z)$
$\displaystyle=$ $\displaystyle-\alpha\eta(Z)g(X,Y)-Z(\ln f)g(\phi X,Y).$

######  Corollary 3.1.

Let $M=N_{\bot}\times_{f}N_{T}$ be a warped product submanifold of $\bar{M}$
such that $\xi\in\Gamma(TN_{\bot})$. Then

$\displaystyle g(h(\phi X,Y),\phi Z)$ $\displaystyle=$ $\displaystyle
g(h(X,\phi Y),\phi Z)$ $\displaystyle\text{and}\ \ \ \ \ g(h(\phi X,\phi
Y),\phi Z)$ $\displaystyle=$ $\displaystyle g(h(X,Y),\phi Z)$

for $X,\ Y\in\Gamma(TN_{T})$ and $Z\in\Gamma(TN_{\bot})$.

Now we have the following characterization theorem:

######  Theorem 3.1.

Let $M$ be a contact CR-submanifold of a $(LCS)_{n}$-manifold $\bar{M}$ such
that $\xi$ is tangent to the anti-invariant distribution $\mathcal{D}^{\bot}$.
Then $M$ is locally a warped product submanifold if and only if

(3.5) $A_{\phi Z}X=-\alpha\eta(Z)X-(Z\mu)\phi X$

for any $X\in\Gamma(\mathcal{D})$ and $Z\in\Gamma(\mathcal{D}^{\bot})$ and
also for some smooth function $\mu$ on $M$ such that $(Y\mu)=0$ for any $Y\in
D$.

###### Proof.

If $M$ be a contact CR-warped product submanifold, then for any
$X\in\Gamma(TM_{T})$ and $Z,\ W\in\Gamma(TM_{\bot})$, we have

$g(A_{\phi Z}X,W)-g(h(X,W),\phi Z)=g(\bar{\nabla}_{W}X,\phi
Z)=g(\phi\bar{\nabla}_{W}X,Z).$

Using (2.11) in the above equation we get

(3.6) $g(A_{\phi Z}X,W)=g(\bar{\nabla}_{W}\phi X,Z).$

Then from (2.13) and Proposition 2.1 we have from (3.6) that $g(A_{\phi
Z}X,W)=0$ and therefore $A_{\phi Z}X$ has no component in $\Gamma(TN_{\bot})$.
Hence by virtue of Lemma 3.1, the relation (3.5) follows.

Conversely, let $M$ be a contact CR-submanifold of $\bar{M}$ with the
invariant and anti-invariant distributions $D$ and $\mathcal{D}^{\bot}$ such
that the relation (3.5) holds. Then for any $X\in\Gamma(\mathcal{D})$ and $Z,\
W\in\Gamma(\mathcal{D}^{\bot})$, and using (2.13) we have

$\displaystyle g(\nabla_{Z}W,\phi X)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}\phi W,X)-g((\bar{\nabla}_{Z}\phi)W,X)$ $\displaystyle=$
$\displaystyle-g(\phi W,\bar{\nabla}_{Z}X)$ $\displaystyle=$
$\displaystyle-g(A_{\phi W}X,Z).$

Using (3.5) in above relation we get

$g(\nabla_{Z}W,\phi X)=0.$

Similarly, we get $g(\nabla_{W}Z,\phi X)=0.$ Thus we obtain

(3.7) $g(\nabla_{Z}W+\nabla_{W}Z,\phi X)=0,$

Which implies that $\nabla_{Z}W+\nabla_{W}Z\in\Gamma(\mathcal{D}^{\bot})$,
i.e., $\mathcal{D}^{\bot}$ is integrable and its leaves are totally geodesic
in $M$. Again for any $X,\ Y\in\Gamma(\mathcal{D})$ and
$Z\in\Gamma(\mathcal{D}^{\bot})$, we get

(3.8) $\displaystyle g(\nabla_{X}Y,Z)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}\phi Y,\phi Z)+\eta(Z)g(Y,\bar{\nabla}_{X}\xi).$

Using (2.6) in (3.8), we get

(3.9) $g(\nabla_{X}Y,Z)=g(h(X,\phi Y),\phi Z)+\alpha\eta(Z)g(Y,\phi X).\\\ $

Interchanging $X$ and $Y$ in (3.9), we get

(3.10) $g(\nabla_{Y}X,Z)=g(h(\phi X,Y),\phi Z)+\alpha\eta(Z)g(X,\phi Y).$

From (3.9) and (3.10), we have

(3.11) $g([X,Y],Z)=g(h(X,\phi Y),\phi Z)-g(h(\phi X,Y),\phi Z).$

Using (3.5) in (3.11), we get $g([X,Y],Z)=0$ and therefore $\mathcal{D}$ is
integrable on $M$.
Let us consider a leaf $N_{T}$ of $\mathcal{D}$ in $M$ and let $h^{T}$ be the
second fundamental form of $N_{T}$ in $M$, then we have

$\displaystyle g(h^{T}(X,Y),Z)$ $\displaystyle=$ $\displaystyle
g(\phi\bar{\nabla}_{Y}X,\phi Z)-\eta(Z)g(\bar{\nabla}_{Y}X,\xi)$
$\displaystyle=$ $\displaystyle-g(\phi X,\bar{\nabla}_{Y}\phi
Z)+\eta(Z)g(X,\bar{\nabla}_{Y}\xi).$

Using (2.6) and (2.14), (3) yields

(3.13) $g(h^{T}(X,Y),Z)=g(\phi X,A_{\phi Z}Y)+\alpha\eta(Z)g(X,\phi Y).$

From (3.5) and (3.13), we obtain

(3.14) $g(h^{T}(X,Y),Z)=-(Z\mu)g(X,Y).$

Using (2.21) in (3.14), we get

(3.15) $\displaystyle h^{T}(X,Y)$ $\displaystyle=$
$\displaystyle-(\nabla\mu)g(X,Y),$

where $\nabla\mu$ is the gradient of the function $\mu$ and therefore $N_{T}$
is totally umbilical in $M$ with mean curvature $-\nabla\mu$. Moreover, the
condition $(Y\mu)=0$, for any $Y\in\mathcal{D}$ implies that the leaves of
$\mathcal{D}$ are extrinsic spheres in $M$, i.e., the integral manifold
$N_{T}$ of $\mathcal{D}$ is umbilical and its mean curvature vector field is
non zero and parallel along $N_{T}$. Hence by Hiepko’s theorem $M$ is locally
a warped product $N_{\bot}\times_{f}N_{T}$ , where $N_{T}$ and $N_{\bot}$
denote the integral manifolds of the distributions $\mathcal{D}$ and
$\mathcal{D}^{\bot}$ respectively and $f$ is the warping function. Thus the
theorem is proved completely. ∎

##  4\. characterization for warped product pseudo slant submanifolds

Recently Hui et al. [10] studied warped product pseudo-slant submanifolds of
$(LCS)_{n}$-manifolds. In this section we obtain a characterization for a
submanifold $M$ of $\bar{M}$ to be a warped product pseudo-slant submanifold
of the form $N_{\theta}\times_{f}N_{\bot}$, where $N_{\theta}$ is a slant
submanifold tangent to $\xi$ and $N_{\bot}$ is an anti-invariant submanifolds
of $\bar{M}$.

######  Lemma 4.1.

Let $M$ be a proper pseudo-slant submanifold of a $(LCS)_{n}$-manifold
$\bar{M}$ with anti-invariant and proper slant dfistributions
$\mathcal{D}^{\bot}$ and $\mathcal{D}^{\theta}$, respectively such that
$\xi\in\Gamma(\mathcal{D}^{\theta})$. Then

(4.1) $g(\nabla_{X}Y,Z)=\sec^{2}\theta[g(h(X,PY),\phi Z)+g(h(X,Z),QPY)],$

for any $X,\ Y\in\Gamma(\mathcal{D}^{\theta})$ and $Z\in\Gamma(D^{\bot})$.

###### Proof.

For any $X,\ Y\in\Gamma(\mathcal{D}^{\theta})$ and
$Z\in\Gamma(\mathcal{D}^{\bot})$, we have

$\displaystyle g(\nabla_{X}Y,Z)$ $\displaystyle=$ $\displaystyle
g(\phi\bar{\nabla}_{X}Y,\phi Z)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}\phi Y,\phi Z)-g((\bar{\nabla}_{X}\phi)Y,Z)$
$\displaystyle=$ $\displaystyle g(\bar{\nabla}XPY,\phi
Z)+g(\bar{\nabla}_{X}QY,\phi Z)$ $\displaystyle=$ $\displaystyle
g(h(X,PY),\phi Z)+g(\bar{\nabla}X\phi QY,Z)-g((\bar{\nabla}_{X}\phi)QY,\phi
Z)$ $\displaystyle=$ $\displaystyle g(h(X,PY),\phi
Z)+g(\bar{\nabla}_{X}bQY,Z)+g(\bar{\nabla}_{X}cQY,Z).$

Using (2.20) in the above relation, we get

$\displaystyle g(\nabla_{X}Y,Z)$ $\displaystyle=$ $\displaystyle
g(h(X,PY),\phi Z)+\sin^{2}\theta
g(\bar{\nabla}_{X}Y,Z)-g(\bar{\nabla}_{X}QPY,Z).$

From which (4.1) follows. ∎

######  Corollary 4.1.

Let $M$ be a proper pseudo-slant submanifold of a $(LCS)_{n}$-manifold
$\bar{M}$ with anti-invariant and proper slant distributions
$\mathcal{D}^{\bot}$ and $\mathcal{D}^{\theta}$, respectively such that
$\xi\in\Gamma(\mathcal{D}^{\theta})$. Then the distribution
$\mathcal{D}^{\theta}$ defines a totally geodesic foliation if and only if

$g(h(X,PY),\phi Z)+g(h(X,Z),QPY)=0$

for every $X,\ Y\in\Gamma(\mathcal{D}^{\theta})$ and
$Z\in\Gamma(\mathcal{D}^{\bot})$.

######  Lemma 4.2.

Let $M=N_{\theta}\times_{f}N_{\bot}$ be a warped product submanifold of a
$(LCS)_{n}$-manifold $\bar{M}$, where $N_{\bot}$ and $N_{\theta}$ are ant-
invariant and proper slant submanifold of $\bar{M}$ such that
$\xi\in\Gamma(TN_{\theta})$. Then

(4.2) $\displaystyle g(h(X,Y),\phi Z)+g(h(X,Z),QY)$ $\displaystyle=$
$\displaystyle 0,$ (4.3) $\displaystyle g(h(Z,W),QX)+g(h(X,Z),QW)$
$\displaystyle=$ $\displaystyle(\phi X\ln f)g(Z,W),$ (4.4) $\displaystyle\ \ \
g(h(Z,W),QPX)+g(h(PX,Z),QW)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta[(X\ln f)+\alpha\eta(X)]g(Z,W).$

###### Proof.

For any $X,\ Y\in\Gamma(TN_{\theta})$ and $Z\in\Gamma(TN_{\bot})$, we have

$\displaystyle g(h(X,Y),\phi Z)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}Y,\phi Z)$ $\displaystyle=$ $\displaystyle
g(\phi\bar{\nabla}_{X}Y,Z)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}\phi Y,Z)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}PY,Z)+g(\bar{\nabla}_{X}QY,Z).$

Using Proposition 2.1 in the above relation, we get

(4.5) $g(h(X,Y),\phi Z)=(X\ln f)g(Z,PY)-g(h(X,Z),QY).$

Thus (4.2) follows from (4.5). Also, for any $X\in\Gamma(TN_{\theta})$ and
$Z,\ W\in\Gamma(TN_{\bot})$, we have

$\displaystyle g(h(Z,W),QX)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}W,\phi X)-g(\bar{\nabla}_{Z}W,PX)$ $\displaystyle=$
$\displaystyle g(\bar{\nabla}_{Z}\phi
W,X)-g((\bar{\nabla}_{Z}\phi)W,X)-g(\bar{\nabla}_{Z}W,PX)$ $\displaystyle=$
$\displaystyle-g(h(X,Z),\phi W)+g(W,\bar{\nabla}_{Z}PX).$

Using Proposition 2.1 in the above relation we get (4.3). Interchanging $X$ by
$PX$ in (4.3) we get (5.4) ∎

Now, we prove the following characterization theorem for warped product
pseudo-slant submanifolds.

######  Theorem 4.1.

Let $M$ be a proper pseudo-slant submanifold of a $(LCS)_{n}$-manifold
$\bar{M}$ with anti-invariant distribution $\mathcal{D}^{\bot}$ and proper
pseudo-slant distribution $\mathcal{D^{\theta}}$, respectively such that
$\xi\in\Gamma(\mathcal{D}^{\theta})$. Then $M$ is locally a mixed-geodesic
warped product submanifold of the form $N_{\theta}\times_{f}N_{\bot}$ if and
only if

(4.6) $A_{\phi Z}X=0\ \ \text{and}\ \
A_{QPX}Z=\cos^{2}\theta[(X\mu)+\alpha\eta(X)]Z,$

for any $X\in\Gamma(\mathcal{D^{\theta}}),\ Z\in\Gamma(\mathcal{D^{\bot}})$
and for some function $\mu$ on $M$ satisfying $(Z\mu)=0$, for any
$Z\in\Gamma(\mathcal{D}^{\bot})$.

###### Proof.

Let $M=N_{\theta}\times_{f}N_{\bot}$ be a mixed geodesic warped product
submanifold of a $(LCS)_{n}$-manifold $\bar{M}$ such that (4.6) holds. Then
for any $X,\ Y\in\Gamma(\mathcal{D}^{\theta})$ and
$Z\in\Gamma(\mathcal{D}^{\bot})$, from (4.2) and (5.4), we get (4.6).
Conversely, Let $M$ is a proper pseudo-slant submanifold of a
$(LCS)_{n}$-manifold $\bar{M}$ such that (4.6) holds.
Then for any for any $X,\ Y\in\Gamma(\mathcal{D}^{\theta})$ and for any
$Z\in\Gamma(\mathcal{D}^{\bot})$, from (4.1) and (4.6), we get
$g(\nabla_{X}Y,Z)=0$ and hence the leaves of $\mathcal{D}^{\theta}$ are
totally geodesic in $M$.
Also, for any $X\in\Gamma(\mathcal{D})$ and $Z,\
W\in\Gamma(\mathcal{D}^{\bot})$, we have

$\displaystyle g([Z,W],X)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}W,X)-g(\bar{\nabla}_{W}Z,X)$ $\displaystyle=$ $\displaystyle
g(\phi\bar{\nabla}_{Z}W,\phi X)-g(\phi\bar{\nabla}_{W}Z,\phi X)$
$\displaystyle=$ $\displaystyle g(\bar{\nabla}_{Z}\phi W,\phi
X)-g(\bar{\nabla}W\phi Z,\phi X)$ $\displaystyle=$ $\displaystyle-g(\phi
W,\bar{\nabla}_{Z}\phi X)+g(\phi Z,\bar{\nabla}_{W}\phi X)$ $\displaystyle=$
$\displaystyle-g(\phi W,\bar{\nabla}_{Z}PX)-g(\phi
W,\bar{\nabla}_{Z}QX)+g(\phi Z,\bar{\nabla}_{W}PX)+g(\phi
Z,\bar{\nabla}_{W}QX)$ $\displaystyle=$ $\displaystyle-g(\phi
W,h(Z,PX))-g(W,\bar{\nabla}_{Z}bQX)-g(W,\bar{\nabla}_{Z}cQX)$
$\displaystyle+g(\phi
Z,h(W,PX))+g(Z,\bar{\nabla}_{W}bQX)+g(Z,\bar{\nabla}_{W}cQX).$

Using (2.20) in the above relation, we get

$\displaystyle g([Z,W],X)$ $\displaystyle=$ $\displaystyle-g(A_{\phi
W}PX,Z)+g(A_{\phi Z}PX,W)+\sin^{2}\theta g([Z,W],X)$
$\displaystyle+g(A_{QPX}Z,W)-g(A_{QPX}W,Z).$

Using (4.6) in (4), we get

(4.8) $\cos^{2}\theta g([Z,W],X)=0.$

Since $\mathcal{D}^{\theta}$ is proper pseudo-slant so, $\theta\neq
0,\frac{\pi}{2}$. Therefore, $g([Z,W],X)=0$ and hence the anti-invariant
distribution $\mathcal{D}^{\bot}$ is integrable.
Now, let $h^{\bot}$ be the second fundamental form of a leaf $N_{\bot}$ of
$\mathcal{D}^{\bot}$ in $M$. Then for any $Z,\ W\in\Gamma(\mathcal{D}^{\bot})$
and $X\in\Gamma(\mathcal{D}^{\theta})$, we have

$\displaystyle g(h^{\bot}(Z,W),X)$ $\displaystyle=$ $\displaystyle
g(\phi\bar{\nabla}_{Z}W,\phi X)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}\phi W,\phi X)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}\phi W,PX)+g(\bar{\nabla}_{Z}\phi W,QX)$ $\displaystyle=$
$\displaystyle-g(A_{\phi W}Z,PX)-g(W,\bar{\nabla}_{Z}\phi QX)$
$\displaystyle=$ $\displaystyle-g(A_{\phi
W}PX,Z)-g(W,\bar{\nabla}_{Z}bQX)-g(W,\bar{\nabla}_{Z}cQX)$ $\displaystyle=$
$\displaystyle-g(A_{\phi W}PX,Z)+\sin^{2}\theta
g(\bar{\nabla}_{Z}W,X)-g(A_{QPX}W,Z).$

Therefore

(4.9) $\cos^{2}\theta g(h^{\bot}(Z,W),X)=-g(A_{\phi W}PX,Z)-g(A_{QPX}W,Z).$

Using (4.6) in (4.9), we get

(4.10) $\cos^{2}\theta
g(h^{\bot}(Z,W),X)=-\cos^{2}\theta[(X\mu)+\alpha\eta(X)]g(Z,W).$

Thus, we get

$h^{\bot}(Z,W)=-[\overrightarrow{\nabla}^{\bot}\mu+\alpha\xi]g(Z,W),$

where $\overrightarrow{\nabla}^{\bot}\mu$ is gradient of the function $\mu$.
Therefore $N_{\bot}$ is totally umbilical in $M$ with the mean curvature
$H^{\bot}=-(\overrightarrow{\nabla}^{\bot}\mu+\alpha\xi)$.
Now, let $\mathcal{D}^{N}$ be the normal connection of $N_{\bot}$ in $M$. Then
for any $Y\in\Gamma(\mathcal{D}^{\theta})$ and
$Z\in\Gamma(\mathcal{D}^{\bot})$, we have

$g(\mathcal{D}_{Z}^{N}\overrightarrow{\nabla}^{\bot}\mu+\alpha\xi,Y)=g(\nabla_{Z}\overrightarrow{\nabla}^{\bot}\mu,Y)+\alpha
g(\nabla_{Z}\xi,Y).$

Also, from (2.6) and (2.13) we get $\nabla_{Z}\xi=0$.
Therefore,
$g(\mathcal{D}_{Z}^{N}\overrightarrow{\nabla}^{\bot}\mu+\alpha\xi,Y)=g(\nabla_{Z}\overrightarrow{\nabla}^{\bot}\mu,Y)=0$,
since $(Z\mu)=0$ for every $Z\in\Gamma(\mathcal{D}^{\bot})$ and hence the mean
curvature of $N_{\bot}$ is parallel.
Thus the leaves of the distribution $\mathcal{D}^{\bot}$ are totally umbilical
in $M$ with non-vanishing parallel mean curvature vector $H^{\bot}$, i.e.
$N_{\bot}$ is an extrinsic sphere in $M$. Therefore by Theorem 2.1, $M$ is a
warped product submanifold. ∎

##  5\. warped product bi-slant submanifolds of $(LCS)_{n}$-manifolds

######  Definition 5.1.

A submanifold $M$ of a $(LCS)_{n}$-manifold $\bar{M}$ is said to be a bi-slant
submanifold if there exists a pair of orthogonal distributions
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ of $M$ such that
$(\textit{i})TM=\mathcal{D}_{1}\oplus\mathcal{D}_{2}$
$(\textit{ii})\phi\mathcal{D}_{1}\bot\mathcal{D}_{2}\ \ \text{and}\ \
\phi\mathcal{D}_{2}\bot\mathcal{D}_{1}$
$(\textit{iii})\mathcal{D}_{1},\mathcal{D}_{2}$ are slant submanifolds with
slant angles $\theta_{1}$ and $\theta_{2}$, respectively.

If we assume $\theta_{1}=0$ and $\theta_{2}=\frac{\pi}{2}$, then $M$ is a CR-
submanifold and if $\theta_{1}=0$ and $\theta_{2}\neq 0,\frac{\pi}{2}$, then
$M$ is a semi-slant submanifold. Also, if $\theta_{1}=\frac{\pi}{2}$ and
$\theta_{2}\neq 0,\frac{\pi}{2}$, then $M$ is a pseudo-slant submanifold. A
bi-slant submanifold $M$ of a $(LCS)_{n}$-manifold $\bar{M}$ is said to be
proper if the slant distributions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are
of slant angles $\theta_{1},\ \theta_{2}\neq 0,\frac{\pi}{2}$.
For a proper bi-slant submanifold $M$ of a $(LCS)_{n}$-manifold, the normal
bundle of $M$ is decomposed as

$T^{\bot}M=Q\mathcal{D}_{1}\oplus Q\mathcal{D}_{2}\oplus\nu,$

where $\nu$ is the invariant normal subbundle of $M$.
Now we will construct a bi-slant submanifold of a $(LCS)_{n}$-manifold.

######  Example 5.1.

Consider the semi-Euclidean space ${\mathbb{R}}^{7}$ with the cartesian
coordinates $(x_{1},y_{1}\cdots,x_{3},y_{3},\,t)$ and paracontact structure

$\phi\left(\frac{\partial}{\partial x_{i}}\right)=\frac{\partial}{\partial
y_{i}},\,\,\,\,\phi\left(\frac{\partial}{\partial
y_{j}}\right)=\frac{\partial}{\partial
x_{j}},\,\,\phi\left(\frac{\partial}{\partial t}\right)=0,\,\,\,\,1\leq
i,j\leq 3.$

It is clear that ${\mathbb{R}}^{7}$ is a Lorentzian metric manifold manifold
with usual semi-Euclidean metric tensor. For any
$\theta_{1},\theta_{2}\in[0,\frac{\pi}{2}]$ let $M$ be a submanifold of
${\mathbb{R}}^{7}$ defined by

$\chi(u,v,w,s,\,t)=(w+u\cos\theta_{1},\,u\sin\theta_{1},\,s+v\cos\theta_{2},\,v\sin\theta_{2},\,0,\,0,\,t).$

Then the tangent space of $M$ is spanned by the following vectors

$\displaystyle Z_{1}$ $\displaystyle=$
$\displaystyle\cos\theta_{1}\frac{\partial}{\partial
x_{1}}+\sin\theta_{1}\frac{\partial}{\partial x_{2}},$ $\displaystyle Z_{2}$
$\displaystyle=$ $\displaystyle\cos\theta_{2}\frac{\partial}{\partial
y_{1}}+\sin\theta_{2}\frac{\partial}{\partial y_{2}},$ $\displaystyle Z_{3}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial x_{1}},$
$\displaystyle Z_{4}$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
y_{1}},$ $\displaystyle Z_{5}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial t}.$

Then we have

$\displaystyle\phi Z_{1}$ $\displaystyle=$
$\displaystyle\cos\theta_{1}\frac{\partial}{\partial
y_{1}}+\sin\theta_{1}\frac{\partial}{\partial y_{2}},$ $\displaystyle\phi
Z_{2}$ $\displaystyle=$ $\displaystyle\cos\theta_{2}\frac{\partial}{\partial
x_{1}}+\sin\theta_{2}\frac{\partial}{\partial x_{2}},$ $\displaystyle\phi
Z_{3}$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial y_{1}},$
$\displaystyle\phi Z_{4}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial x_{1}},$ $\displaystyle\phi Z_{5}$
$\displaystyle=$ $\displaystyle 0.$

We take ${\mathcal{D}_{1}}={\rm{Span}}\\{Z_{1},\,Z_{4}\\}$ and
${\mathcal{D}_{2}}={\rm{Span}}\\{Z_{2},\ Z_{3}\\}$, then $g(Z_{1},\phi
Z_{4})=\cos\theta_{1}$ and $g(Z_{2},\phi Z_{3})=\cos\theta_{2}$. Thus the
distributions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are slant with slant
angles $\theta_{1}$ and $\theta_{2}$ respectively and hence $M$ is a bi-slant
submanifold.

######  Lemma 5.1.

Let $M$ be a proper bi-slant submanifold of a $(LCS)_{n}$-manifold $\bar{M}$
with the slant distributions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ such that
$\xi\in\Gamma(\mathcal{D}_{1})$. Then

$\displaystyle\cos^{2}\theta_{2}g(\nabla_{X_{1}}X_{2},Y_{2})$ $\displaystyle=$
$\displaystyle g(\nabla_{X_{1}}PX_{2},PY_{2})+g(h(X_{1},PX_{2}),QY_{2})$
$\displaystyle+g(h(X_{1},Y_{2}),QPX_{2}),$

for any $X_{1}\in\Gamma(\mathcal{D}_{1})$ and $X_{2},\
Y_{2}\in\Gamma(\mathcal{D}_{2})$, where $\theta_{1}$ and $\theta_{2}$ are the
slant angles of slant distributions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$
respectively.

###### Proof.

For any $X_{1}\in\Gamma(\mathcal{D}_{1})$ and $X_{2},\
Y_{2}\in\Gamma(\mathcal{D}_{2})$, we have

$\displaystyle g(\nabla_{X_{1}}X_{2},Y_{2})$ $\displaystyle=$ $\displaystyle
g(\phi\bar{\nabla}_{X_{1}}X_{2},\phi Y_{2})$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X_{1}}\phi X_{2},\phi Y_{2})$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X_{1}}PX_{2},PY_{2})+g(\bar{\nabla}_{X_{1}}PX_{2},QY_{2})+g(\bar{\nabla}-{X_{1}}QX_{2},\phi
Y_{2}).$ $\displaystyle=$ $\displaystyle
g(\nabla_{X_{1}}PX_{2},PY_{2})+g(h(X_{1},PX_{2}),QY_{2})$
$\displaystyle+g(\bar{\nabla}_{X_{1}}bQX_{2},Y_{2})+g(\bar{\nabla}_{X_{1}}cQX_{2},Y_{2}).$

Using (2.20) in the above relation we get (5.1). ∎

######  Theorem 5.1.

There does not exists a proper warped product bi-slant submanifold
$M=M_{1}\times_{f}M_{2}$ of $\bar{M}$ such that $\xi\in\Gamma(TM_{1})$.

###### Proof.

Let $M=M_{1}\times_{f}M_{2}$ be a proper warped product bi-slant submanifold
of $\bar{M}$. Then for $X_{1}\in\Gamma(TM_{1})$ and $X_{2},\
Y_{2}\in\Gamma(TM_{2})$, we have

$\displaystyle g(h(X_{1},PX_{2}),QY_{2})$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X_{1}}PX_{2},\phi Y_{2})+g(PX_{2}\bar{\nabla}_{X_{1}}PY_{2})$
$\displaystyle=$
$\displaystyle\cos^{2}\theta_{2}g(\bar{\nabla}_{X_{1}}X_{2},Y_{2})-g(h(X_{1},Y_{2}),QPX_{2})+g(PX_{2},\bar{\nabla}_{X_{1}}PY_{2}).$

Using Proposition 2.1 in the above relation, we get

(5.2)
$g(h(X_{1},PX_{2}),QY_{2})+g(h(X_{1},Y_{2}),QPX_{2})=2\cos^{2}\theta_{2}(X_{1}\ln
f)g(X_{2},Y_{2}).$

Again using Proposition 2.1 in (5.1), we get

(5.3) $g(h(X_{1},PX_{2}),QY_{2})+g(h(X_{1},Y_{2}),QPX_{2})=0.$

From (5.2) and (5.3), we get

(5.4) $\cos^{2}\theta_{2}(X_{1}\ln f)=0.$

Since $M$ is a proper warped product bi-slant submanifold so,
$\theta_{2}\neq\frac{\pi}{2}$. Therefore $(X_{1}\ln f)=0$ for every
$X_{1}\in\Gamma(TM_{1})$ and hence $M$ does not exists. ∎

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S. K. Hui and T.PaL
Department of Mathematics, The University of Burdwan, Golapbag, Burdwan –
713104, West Bengal, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

L.-I. Piscoran
North University Center of Baia Mare, Technical University of Cluj Napoca,
Department of Mathematics and Computer Science, Victoriei 76, 430122 Baia
Mare, Romania
E-mail<EMAIL_ADDRESS>