# Biwarped product submanifolds
of a Kähler manifold

Hakan Mete Taştan  Department of Mathematics
Faculty of Science
İstanbul University
Vezneciler, İstanbul, Turkey<EMAIL_ADDRESS>
###### Abstract.

We study biwarped product submanifolds which are special cases of multiply
warped product submanifolds in Kähler manifolds. We observe the non-existence
of such submanifolds under some circumstances. We show that there exists a
non-trivial biwarped product submanifold of a certain type by giving an
illustrate example. We also give a necessary and sufficient condition for such
submanifolds to be locally trivial. Moreover, we establish an inequality for
the squared norm of the second fundamental form in terms of the warping
functions for such submanifolds. The equality case is also discussed.

###### Key words and phrases:

warped product, multiply warped product, holomorphic distribution, totally
real submanifold, pointwise slant distribution, Kähler manifold.

###### 2010 Mathematics Subject Classification:

Primary 53B25, Secondary 53C55

##  1\. Introduction

Bishop and O’ Neill [4] introduced the concept of warped product of Riemannian
manifolds to construct a large class of complete manifolds of negative
curvature. This concept is also a generalization of the usual product of
Riemannian manifolds. Nölker [21] considered the notion of multiply warped
products as a generalization of the warped products. Since that time, multiply
warped products has been studied by many authors. For example, Ünal [38]
studied partially the geometry of the multiply warped products when the
metrics of such products are Lorentzian. Curvature properties of such products
were investigated by Dobarro and Ünal [14].

The concept of warped products or multiply warped products play very important
roles in physics as well as in differential geometry, especially in the theory
of relativity. Actually, the standard spacetime models such as Robertson-
Walker, Schwarschild, static and Kruscal are warped products. Also, the
simplest models of neighborhoods of stars and black holes are warped products
[23]. Moreover, many solutions to Einstein’s field equation can be expressed
in terms of warped products [2].

In differential geometry, especially in almost complex geometry, one of the
most intensively research areas is the theory of submanifolds. Actually, the
almost complex structure of an almost Hermitian manifold determines several
classes of submanifolds such as holomorphic(invariant), totally real(anti-
invariant) [40], CR- [3], generic [6], slant [7], semi-slant [24], hemi-
slant(pseudo-slant) [5, 29], pointwise slant [10, 15], bi-slant [5], skew CR-
and generic submanifolds [25]. Among them, the last one contains all other
classes.

The theory of warped product submanifolds has been becoming a popular research
area since Chen [8] studied the warped product CR-submanifolds in Kähler
manifolds. Actually, several classes of warped product submanifolds appeared
in the last fifteen years (see [26, 28, 29, 30, 31]). Also, warped product
submanifolds have been studying in different kinds of structures such as
nearly Kähler [33], para-Kähler [12], locally product Riemannian [32, 34],
cosymplectic [37], Sasakian [19], generalized Sasakian [27], trans-Sasakian
[17], $(\kappa,\mu)-$ [36], Kenmotsu [16, 20] and quaternion [18]. Most of the
studies related to the theory of warped product submanifolds can be found in
Chen’s coming book [9]. Recently, Chen and Dillen [11] studied multiply warped
product submanifolds in Kähler manifolds and they obtained very useful optimal
inequalities. We note that such submanifolds were also studied in Kenmotsu
manifolds [22].

In this paper, we consider and study biwarped product submanifolds in Kähler
manifolds. Here, a biwarped product means that a multiply warped product which
has only two fibers. We observe the non-existence of biwarped product
submanifolds under some circumstances. After giving an illustrate example, we
study such submanifolds in case of the base factor is holomorphic and one of
the two fibers is totally real and the other one is pointwise slant
submanifold. We also give characterization for this kind of submanifolds.
Moreover, we investigate the behavior of the second fundamental form of such
submanifolds and as a result, we give a necessary and sufficient condition for
such manifolds to be locally trivial. Furthermore, an inequality for the
squared norm of the second fundamental form in terms of the warping functions
for such submanifolds is obtained. The equality case is also considered.

##  2\. Preliminaries

In this section, we recall the fundamental definitions and notions needed
further study. Actually, in subsection 2.1, we will recall the definition of
the multiply warped product manifolds. In subsection 2.2, we will give the
basic background for submanifolds of Riemannian manifolds. The definition of a
Kähler manifold and the some classes of submanifolds of Kähler manifolds are
placed in subsection 2.3.

###  2.1. Multiply product manifolds

Let $(M_{i},g_{i})$ be Riemannian manifolds for any $i\in\\{0,1,...,k\\}$ and
let $f_{j}:M_{0}\rightarrow(0,\infty)$ be smooth functions for any
$j\in\\{1,2,...,k\\}.$ Then the _multiply product manifold_ [21]
$\bar{M}=M_{0}\times_{f_{1}}M_{1}\times...\times_{f_{k}}M_{k}$ is the product
manifold $\tilde{M}=M_{0}\times M_{1}\times...\times M_{k}$ endowed with the
metric

$g=\pi^{*}_{0}(g_{0})\oplus(f_{1}\circ\pi_{0})^{2}\pi^{*}_{1}(g_{1})\oplus...\oplus(f_{k}\circ\pi_{0})^{2}\pi^{*}_{k}(g_{k}).$

More precisely, for any vector fields $X$ and $Y$ of $\bar{M}$, we have

$g(X,Y)=g_{0}(\pi_{0*}X,\pi_{0*}Y)+\displaystyle\sum^{k}_{i=1}(f_{i}\circ\pi_{0})^{2}g_{i}(\pi_{i*}X,\pi_{i*}Y),$

where $\pi_{i}:\tilde{M}\rightarrow M_{i}$ is the canonical projection of
$\tilde{M}$ onto $M_{i}$, $\pi^{*}_{i}(g_{i})$ is the pullback of $g_{i}$ by
$\pi_{i}$ and the subscript $*$ denotes the derivative map of $\pi_{i}$ for
each $i.$ Each function $f_{j}$ is called a _warping function_ and each
manifold $(M_{j},g_{j})$, $j\in\\{1,2,...,k\\}$ is called a _fiber_ of the
multiply warped product $\bar{M}.$ The manifold $(M_{0},g_{0})$ is called a
_base manifold_ of $\bar{M}.$ As well known, the base manifold of $\bar{M}$ is
totally geodesic and the fibers of $\bar{M}$ are totally umbilic in $\bar{M}$.

Let $\bar{M}=M_{0}\times_{f_{1}}M_{1}\times...\times_{f_{k}}M_{k}$ be multiply
product manifold, if $k=1$, then we get a (singly) warped product [4]. We call
the multiply product manifolds as _biwarped product manifolds_ for $k=2.$ In
other words, a biwarped product manifold has the form
$M_{0}\times_{f_{1}}M_{1}\times_{f_{2}}M_{2}.$ We say that a biwarped product
manifold is _trivial_ , if the warping functions $f_{1}$ and $f_{2}$ are
constants. Note that biwarped product manifolds were also studied under the
name of _twice warped products_ [1].

Let $\bar{M}=M_{0}\times_{f_{1}}M_{1}\times_{f_{2}}M_{2}$ be a biwarped
product manifold with the Levi-Civita connection $\bar{\nabla}$ and
${}^{i}\nabla$ denote the Levi-Civita connection of $M_{i}$ for
$i\in\\{0,1,2\\}.$ By usual convenience, we denote the set of lifts of vector
fields on $M_{i}$ by $\mathcal{L}(M_{i})$ and use the same notation for a
vector field and for its lift. On the other hand, since the map $\pi_{0}$ is
an isometry and $\pi_{1}$ and $\pi_{2}$ are (positive) homotheties, they
preserve the Levi-Civita connections. Thus, there is no confusion using the
same notation for a connection on $M_{i}$ and for its pullback via $\pi_{i}.$
Then, the covariant derivative formulas for a biwarped product manifold are
given by the following.

######  Lemma 2.1.

([1]) Let $\bar{M}=M_{0}\times_{f_{1}}M_{1}\times_{f_{2}}M_{2}$ be a biwarped
product manifold. Then, we have

(2.1) $\displaystyle\bar{\nabla}_{U}V=\,^{0}\nabla_{U}V,$ (2.2)
$\displaystyle\bar{\nabla}_{V}X=\bar{\nabla}_{X}V=V(\ln f_{i})X,$ (2.3)
$\displaystyle\bar{\nabla}_{X}Z=\begin{cases}0&\mbox{if }i\neq j,\\\
{}^{i}\nabla_{X}Z-g(X,Z)\emph{grad}\,(\ln f_{i})&\mbox{if
}i=j,\end{cases}\qquad\qquad\qquad$

where $U,V\in\mathcal{L}(M_{0})$, $X\in\mathcal{L}(M_{i})$ and
$Z\in\mathcal{L}(M_{j})$.

We note that grad$(\ln f_{i})\in\mathcal{L}(M_{0})$ for $i=1,2$ [21].

###  2.2. Submanifolds of Riemannian manifolds

Let $M$ be an isometrically immersed submanifold in a Riemannian manifold
$(\bar{M},g)$. Let $\bar{\nabla}$ is the Levi-Civita connection of $\bar{M}$
with respect to the metric $g$ and let $\nabla$ and $\nabla^{\bot}$ be the
induced, and induced normal connection on $M$, respectively. Then, for all
$U,V\in TM$ and $\xi\in T^{\bot}M$, the Gauss and Weingarten formulas are
given respectively by

(2.4) $\begin{array}[]{c}\bar{\nabla}_{U}V=\nabla_{U}V+h(U,V)\end{array}$

and

(2.5)
$\begin{array}[]{c}\bar{\nabla}_{U}\xi=-A_{\xi}U+\nabla_{U}^{\bot}\xi\end{array},$

where $TM$ is the tangent bundle and $T^{\bot}M$ is the normal bundle of $M$
in $\bar{M}$. Additionally, $h$ is the _second fundamental form_ of $M$ and
$A_{\xi}$ is the Weingarten endomorphism associated with $\xi.$ The second
fundamental form $h$ and the _shape operator_ $A$ related by

(2.6) $\begin{array}[]{c}g(h(U,V),\xi)=g(A_{\xi}U,V)\end{array}.$

The _mean curvature vector field_ $H$ of $M$ is given by
$H=\frac{1}{m}(trace\,h),$ where $dim(M)=m.$ We say that the submanifold $M$
is _totally geodesic_ in $\bar{M}$ if $h=0$, and _minimal_ if $H=0.$ The
submanifold $M$ is called _totally umbilical_ if $h(U,V)=g(U,V)H$ for all
$U,V\in TM.$ If the manifold $M$ is totally umbilical and its mean curvature
vector field $H$ is _parallel_ , i.e. $g(\bar{\nabla}_{U}H,\xi)=0$ for all
$U\in TM$ and $\xi\in T^{\bot}M$, then the submanifold $M$ is said to be
_spherical_ or _extrinsic sphere_.

Let $\mathcal{D}^{1}$ and $\mathcal{D}^{2}$ be any two distributions on $M$.
Then we say that $M$ is $\mathcal{D}^{1}$-_geodesic_ , if $h(U,V)=0$ for all
$U,V\in\mathcal{D}^{1}$ and we say that
$(\mathcal{D}^{1},\mathcal{D}^{2})$-_mixed geodesic_ if $h(V,X)=0$ for
$V\in\mathcal{D}^{1}$ and $X\in\mathcal{D}^{2}.$ If for all
$V\in\mathcal{D}^{1}$ and $X\in\mathcal{D}^{2}$,
$\nabla_{X}V\in\mathcal{D}^{1}$, then $\mathcal{D}^{1}$ is called
$\mathcal{D}^{2}$-_parallel_. We say that $\mathcal{D}^{1}$ is _autoparallel_
if $\mathcal{D}^{1}$ is $\mathcal{D}^{1}$-parallel. If a distribution on $M$
is autoparallel, then by the Gauss formula it is totally geodesic.

###  2.3. Some classes of submanifolds of Kähler manifolds

Let $\bar{M}$ be an almost complex manifold with almost complex structure $J.$
If there is a Riemannian metric $g$ on $\bar{M}$ satisfying

(2.7) $\begin{array}[]{c}g(JX,JY)=g(X,Y)\end{array}$

for any $X,Y\in T\bar{M},$ then we say that $(\bar{M},J,g)$ is an almost
Hermitian manifold. Let $\bar{\nabla}$ be the Levi-Civita connection of the
almost Hermitian manifold $(\bar{M},J,g)$ with respect to $g.$ Then
$(\bar{M},J,g)$ is called a _Kähler manifold_ [40] if $J$ is parallel with
respect to $\bar{\nabla},$ i.e.,

(2.8) $\begin{array}[]{c}(\bar{\nabla}_{X}J)Y=0\end{array}$

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

Let $M$ be a Riemannian manifold isometrically immersed submanifold in a
Kähler manifold $(\bar{M},J,g)$. Then the submanifold $M$ is called a
_pointwise slant submanifold_ [10, 15] if for every point $p$ of $M,$ the
Wirtinger angle $\theta(V)$ between $JV$ and the tangent space $T_{p}M$ at $p$
is independent of the choice of the nonzero vector $V\in T_{p}M.$ In this
case, the angle $\theta$ can be viewed as a function on $M$ and it is called
the _slant function_ of $M$. We say that the pointwise slant submanifold $M$
is _proper_ neither $\cos\theta(p)=0$ nor $\sin\theta(p)=0$ at each point
$p\in M$. (This condition is different from Chen’s definition, see [10]).

Now, let $M$ be a submanifold of a Kähler manifold $(\bar{M},J,g)$. For any
$V\in TM$, we put

(2.9) $\begin{array}[]{c}JV=TV+FV\end{array}.$

Here $TV$ is the tangential part of $JV,$ and $FV$ is the normal part of $JV.$
Then $M$ is a pointwise slant submanifold of $\bar{M}$ if and only if, for any
$V\in TM$, we have

(2.10) $\begin{array}[]{c}T^{2}V=-\cos^{2}\\!\theta V\end{array}$

for some function $\theta$ defined on $M$ [10]. For a pointwise slant
submanifold of $\bar{M}$, using (2.9), (2.10) and the Kähler structure, it is
not difficult to prove the following two facts.

(2.11) $\begin{array}[]{c}g(TU,TV)=\cos^{2}\\!\theta g(U,V)\end{array},$
(2.12) $\begin{array}[]{c}g(FU,FV)=\sin^{2}\\!\theta g(U,V)\end{array}$

for $U,V\in TM$.

Let $M$ be a pointwise slant submanifold with slant function $\theta$ of a
Kähler manifold $(\bar{M},J,g).$ If the function $\theta$ is a constant, i.e.,
it is also independent of the choice of the point $p\in M$, then we say that
$M$ is a _slant submanifold_ [7].

If $\theta\equiv 0,$ then $M$ is called a _holomorphic_ or _complex
submanifold_ [40]. In that case, the tangent space $T_{p}M$ is invariant with
respect to the almost complex structure $J$ at each point $p\in M$, i.e.,
$J(T_{p}M)\subseteq T_{p}M.$

If $\theta\equiv\frac{\pi}{2},$ then $M$ is called a _totally real
submanifold_ of [40]. In which case, the tangent space $T_{p}M$ is anti-
invariant with respect to the almost complex structure $J$ for every point $p$
of $M$, i.e., $J(T_{p}M)\subseteq T^{\bot}_{p}M.$

##  3\. Generalized $J$-induced submanifolds of order 1

In this section, after renaming the generic submanifolds (in the sense of
Ronsse [25]), we will give some results concerning totally geodesicness and
integrability of the distributions which are involved in the definition of
such submanifolds.

The most general class of submanifolds determined by the almost complex
structure is the class of generic submanifols which was defined by Ronsse
[25]. There are two other classes of submanifolds with the same name. One of
these is the class of defined by Chen [6] and the other one is the class
defined by Yano and Kon [39]. Because of these facts, to avoid name confusion,
we call the generic submanifolds (in the sense of Ronsse [25]) as _generalized
structure induced_ or _generalized J-induced submanifolds_.

######  Definition 3.1.

([25]) Let $M$ be a submanifold of a Kähler manifold $(\bar{M},J,g).$ Then $M$
is said to be a _generalized structure induced_ or _generalized J-induced
submanifold_ if the tangent bundle $TM$ of $M$ has the form

$\begin{array}[]{c}TM=\mathcal{D}^{T}\oplus\mathcal{D}^{\bot}\oplus\mathcal{D}^{\theta_{1}}\oplus...\oplus\mathcal{D}^{\theta_{k}}\end{array},$

where $\mathcal{D}^{T}$ is a holomorphic, $\mathcal{D}^{\bot}$ is a totally
real and each of $\mathcal{D}^{\theta_{i}}$ is a pointwise slant distribution
on $M$ and $\theta_{i}$’s are distinct for $i=1,...,k.$ In addition, if each
of $\mathcal{D}^{\theta_{i}}$ is a slant distribution, then we say that $M$ is
a _skew CR-submanifold_ of $\bar{M}.$

In a special case, we have the following definition.

######  Definition 3.2.

A submanifold $M$ of a Kähler manifold $(\bar{M},J,g)$ is called a
_generalized structure induced submanifold of order 1_ or _generalized
$J$-induced submanifold of order 1_ if it is a generalized $J$-induced
submanifold with $k=1.$

In which case, we have

(3.1)
$\begin{array}[]{c}TM=\mathcal{D}^{T}\oplus\mathcal{D}^{\bot}\oplus\mathcal{D}^{\theta}\end{array}.$

Thus, the normal bundle $T^{\bot}M$ of $M$ is decomposed as

(3.2) $\begin{array}[]{c}T^{\bot}M=J(\mathcal{D}^{\bot})\oplus
F(\mathcal{D}^{\theta})\oplus\overline{\mathcal{D}}^{T}\end{array},$

where $\overline{\mathcal{D}}^{T}$ is the orthogonal complementary
distribution of $J(\mathcal{D}^{\bot})\oplus F(\mathcal{D}^{\theta})$ in
$T^{\bot}M$ and it is an invariant subbundle of $T^{\bot}M$ with respect to
$J.$

We say that a generalized $J$-induced submanifold of order 1 is _proper_ , if
$\mathcal{D}^{T}\neq\\{0\\}$, $\mathcal{D}^{\bot}\neq\\{0\\}$ and the slant
function $\theta$ belongs to open interval $(0,\frac{\pi}{2}).$

For the further study of generalized $J$-induced submanifolds of order 1 of a
Kähler manifold, we need the following lemma.

######  Lemma 3.3.

Let $M$ be a generalized $J$-induced submanifold of order 1 of a Kähler
manifold $(\bar{M},J,g)$. Then, we have

(3.3) $\begin{array}[]{c}g(\nabla_{U}V,Z)=\csc^{2}\\!\theta g(A_{FZ}JV-
A_{FTZ}V,U)\end{array},$ (3.4)
$\begin{array}[]{c}g(\nabla_{Z}W,V)=\csc^{2}\\!\theta
g(A_{FTW}V-A_{FW}JV,Z)\end{array},$

where $U,V\in\mathcal{D}^{T}$ and $Z,W\in\mathcal{D}^{\theta}$.

###### Proof.

Using (2.4), (2.7) and (2.8), we have

$\displaystyle g(\nabla_{U}V,Z)$
$\displaystyle=g(\bar{\nabla}_{U}J,JZ)=g(\bar{\nabla}_{U}JV,TZ)+g(\bar{\nabla}_{U}JV,FZ)$
$\displaystyle=-g(\bar{\nabla}_{U}V,T^{2}Z)-g(\bar{\nabla}_{U}V,FTZ)+g(\bar{\nabla}_{U}JV,FZ).$

for $U,V\in\mathcal{D}^{T}$ and $Z\in\mathcal{D}^{\theta}$. Now, using (2.10)
and (2.5), we obtain

$\begin{array}[]{c}\sin^{2}\\!\theta g(\nabla_{U}V,Z)=g(A_{FZ}JV-
A_{FTZ}V,U)\end{array}.$

This gives (3.3). The other assertion can be obtained in a similar way. ∎

######  Lemma 3.4.

Let $M$ be a generalized $J$-induced submanifold of order 1 of a Kähler
manifold $(\bar{M},J,g)$. Then, we have

(3.5) $\displaystyle g(\nabla_{U}V,X)=g(A_{JX}U,JV),$ (3.6) $\displaystyle
g(\nabla_{U}X,Z)=-\sec^{2}\\!\theta g(A_{JX}TZ-A_{FTZ}X,U),$ (3.7)
$\displaystyle g(\nabla_{X}Y,U)=-g(A_{JY}X,JU),$ (3.8) $\displaystyle
g(\nabla_{Z}W,Y)=\sec^{2}\\!\theta g(A_{JY}TW-A_{FTW}Y,Z),$ (3.9)
$\displaystyle g(\nabla_{Z}Y,U)=-g(A_{JY}JU,Z),$ (3.10) $\displaystyle
g(\nabla_{X}Y,Z)=-\sec^{2}\\!\theta\\{g(A_{JY}X,TZ)-g(A_{FTZ}X,Y)\\},$ (3.11)
$\displaystyle g(\nabla_{X}U,Z)=\csc^{2}\\!\theta\\{g(A_{FZ}JU-
A_{FTZ}U,X)\\},$

where $U,V\in\mathcal{D}^{T}$, $X,Y\in\mathcal{D}^{\bot}$ and
$Z,W\in\mathcal{D}^{\theta}$.

###### Proof.

The proofs of all equations are same as the proofs of equations of Lemma 1,
Lemma 2 and Lemma 3 of [30]. So, we omit them. ∎

As applications of Lemma (3.3) and Lemma (3.4), we have the following results.

######  Theorem 3.5.

Let $M$ be a generalized $J$-induced submanifold of order 1 of a Kähler
manifold $(\bar{M},J,g)$. Then the holomorphic distribution $\mathcal{D}^{T}$
is totally geodesic if and only if the following equations hold

(3.12) $\begin{array}[]{c}g(A_{JX}U,JV)=0\end{array},$ (3.13)
$\begin{array}[]{c}g(A_{FZ}U,JV)=g(A_{FTZ}U,V)\end{array}$

for $U,V\in\mathcal{D}^{T}$, $X\in\mathcal{D}^{\bot}$ and
$Z\in\mathcal{D}^{\theta}$.

###### Proof.

Let $M$ be a generalized $J$-induced submanifold of order 1 of a Kähler
manifold $(\bar{M},J,g)$. Then, the holomorphic distribution $\mathcal{D}^{T}$
is totally geodesic if and only if $g(\nabla_{U}V,X)=0$ and
$g(\nabla_{U}V,Z)=0$ for all $U,V\in\mathcal{D}^{T}$, $X\in\mathcal{D}^{\bot}$
and $Z\in\mathcal{D}^{\theta}$. Thus, both assertions follow from (3.5) and
(3.3), respectively. ∎

######  Theorem 3.6.

Let $M$ be a generalized $J$-induced submanifold of order 1 of a Kähler
manifold $(\bar{M},J,g)$. Then the pointwise slant distribution
$\mathcal{D}^{\theta}$ is integrable if and only if the following equations
hold

(3.14)
$\begin{array}[]{c}g(A_{FTZ}V-A_{FZ}JV,W)=g(A_{FTW}V-A_{FW}JV,Z)\end{array},$
(3.15) $\begin{array}[]{c}g(A_{JX}TW-A_{FTW}X,Z)=g(A_{JX}TZ-
A_{FTZ}X,W)\end{array}$

for $V\in\mathcal{D}^{T}$, $X\in\mathcal{D}^{\bot}$ and
$Z,W\in\mathcal{D}^{\theta}$.

###### Proof.

Let $M$ be a generalized $J$-induced submanifold of order 1 of a Kähler
manifold $(\bar{M},J,g)$. Then, the pointwise slant distribution
$\mathcal{D}^{\theta}$ is integrable if and only if $g([Z,W],V)=0$ and
$g([Z,W],X)=0$ for all $U\in\mathcal{D}^{T}$, $X\in\mathcal{D}^{\bot}$ and
$Z,W\in\mathcal{D}^{\theta}$. Thus, both assertions follow from (3.4) and
(3.8), respectively. ∎

######  Remark 3.7.

From Lemma 1 of [25] or Theorem 4.2 of [35], we know that the totally real
distribution $\mathcal{D}^{\bot}$ is always integrable.

##  4\. Biwarped product submanifolds of Kähler manifolds

In this section, we check that the existence of biwarped product submanifolds
in the form $M_{0}\times_{f_{1}}M_{1}\times_{f_{2}}M_{2}$ of a Kähler manifold
$(\bar{M},J,g)$, where $M_{0},M_{1}$ and $M_{2}$ are one of the submanifolds
given in subsection 2.3.

###  4.1. Existence problems

Chen proved that there do not exist (non-trivial) warped product submanifolds
in the form $M_{\bot}\times_{f}M_{T}$ in a Kähler manifold $\bar{M}$ such that
$M_{\bot}$ is a totally real and $M_{T}$ is a holomorphic submanifold of
$\bar{M}$ in Theorem 3.1 of [7], Thus, we obtain the following result.

######  Corollary 4.1.

There do not exist (non-trivial) biwarped product submanifolds of type
$M_{\bot}\times_{f}M_{T}\times_{\sigma}M_{\theta}$ of a Kähler manifold
$(\bar{M},J,g)$ such that $M_{\bot}$ is a totally real, $M_{T}$ is a
holomorphic and $M_{\theta}$ is a pointwise slant or slant submanifold of
$\bar{M}$.

In Theorem 3.1 of [29], Şahin showed that there is no (non-trivial) warped
product submanifolds of the form $M_{\phi}\times_{f}M_{T}$ in a Kähler
manifold $\bar{M}$ such that $M_{\phi}$ is a proper slant and $M_{T}$ is a
holomorphic submanifold of $\bar{M}$. In Theorem 4.1 of [30], he also proved
that there exist no (non-trivial) warped product submanifolds of the form
$M_{\theta}\times_{f}M_{T}$ in a Kähler manifold $\bar{M}$ such that
$M_{\theta}$ is a proper pointwise slant and $M_{T}$ is a holomorphic
submanifold of $\bar{M}.$ Hence, we conclude that:

######  Corollary 4.2.

Let $\bar{M}$ be a Kähler manifold. Then there exist no (non-trivial) biwarped
product submanifolds in the form
$M_{\theta}\times_{f}M_{T}\times_{\sigma}M_{\bot}$ of $\bar{M}$ such that
$M_{\theta}$ is a proper pointwise slant or proper slant, $M_{T}$ is a
holomorphic and $M_{\bot}$ is a totally real submanifold of $\bar{M}$.

On the other hand, in Theorem 3.2 of [29], Şahin proved that there exists no
(non-trivial) warped product submanifolds of the form
$M_{T}\times_{f}M_{\phi}$ in a Kähler manifold $\bar{M}$ such that $M_{T}$ is
a holomorphic and $M_{\phi}$ is a proper slant submanifold of $\bar{M}.$ Thus,
we get the following result.

######  Corollary 4.3.

There exist no (non-trivial) biwarped product submanifolds of type
$M_{T}\times_{f}M_{\phi}\times_{\sigma}M_{\bot}$ of a Kähler manifold
$(\bar{M},J,g)$ such that $M_{T}$ is a holomorphic, $M_{\phi}$ is a proper
slant and $M_{\bot}$ is a totally real submanifold of $\bar{M}$.

Now, we consider (non-trivial) biwarped product submanifolds of the form
$M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ in a Kähler manifold
$(\bar{M},J,g)$ such that $M_{T}$ is a holomorphic, $M_{\bot}$ is a totally
real and $M_{\phi}$ is a pointwise slant submanifold of $\bar{M}$. We first
present an example of such a submanifold.

######  Example 4.4.

Consider the Kähler manifold $\mathbb{R}^{14}$ with usual Kähler structure.
For $u,v\neq 0,1$ and $x,z,w\in(0,\frac{\pi}{2})$ we consider a submanifold
$M$ in $\mathbb{R}^{14}$ given by

$\displaystyle
y_{1}=u\cos\\!z,\,y_{2}=v\cos\\!z,\,y_{3}=u\cos\\!w,\,y_{4}=v\cos\\!w,$
$\displaystyle
y_{5}=u\sin\\!z,\,y_{6}=v\sin\\!z,\,y_{7}=u\sin\\!w,\,y_{8}=v\sin\\!w,$
$\displaystyle
y_{9}=z,\,y_{10}=w,\,y_{11}=u\cos\\!x,\,y_{12}=v\cos\\!x,\,y_{13}=u\sin\\!x,\,y_{14}=v\sin\\!x.$

Then, we see that the local frame of the tangent bundle $TM$ of $M$ is spanned
by

$\displaystyle U$ $\displaystyle=\cos\\!z\frac{\partial}{\partial
y_{1}}+\cos\\!w\frac{\partial}{\partial
y_{3}}+\sin\\!z\frac{\partial}{\partial
y_{5}}+\sin\\!w\frac{\partial}{\partial
y_{7}}+\cos\\!x\frac{\partial}{\partial
y_{11}}+\sin\\!x\frac{\partial}{\partial y_{13}},$ $\displaystyle V$
$\displaystyle=\cos\\!z\frac{\partial}{\partial
y_{2}}+\cos\\!w\frac{\partial}{\partial
y_{4}}+\sin\\!z\frac{\partial}{\partial
y_{6}}+\sin\\!w\frac{\partial}{\partial
y_{8}}+\cos\\!x\frac{\partial}{\partial
y_{12}}+\sin\\!x\frac{\partial}{\partial y_{14}},$ $\displaystyle X$
$\displaystyle=-u\sin\\!x\frac{\partial}{\partial
y_{11}}-v\sin\\!x\frac{\partial}{\partial
y_{12}}+u\cos\\!x\frac{\partial}{\partial
y_{13}}+v\cos\\!x\frac{\partial}{\partial y_{14}},$ $\displaystyle Z$
$\displaystyle=-u\sin\\!z\frac{\partial}{\partial
y_{1}}-v\sin\\!z\frac{\partial}{\partial
y_{2}}+u\cos\\!z\frac{\partial}{\partial
y_{5}}+v\cos\\!z\frac{\partial}{\partial y_{6}}+\frac{\partial}{\partial
y_{9}},$ $\displaystyle W$ $\displaystyle=-u\sin\\!w\frac{\partial}{\partial
y_{3}}-v\sin\\!w\frac{\partial}{\partial
y_{4}}+u\cos\\!w\frac{\partial}{\partial
y_{7}}+v\cos\\!w\frac{\partial}{\partial y_{8}}+\frac{\partial}{\partial
y_{10}},$

where $(y_{1},...,y_{14})$ are natural coordinates of $\mathbb{R}^{14}$. Then
$\mathcal{D}^{T}=span\\{U,V\\}$ is a holomorphic,
$\mathcal{D}^{\bot}=span\\{X\\}$ is a totally real and
$\mathcal{D}^{\theta}=span\\{Z,W\\}$ is a (proper) pointwise slant
distribution with the slant function
$\theta=\cos^{-1}(\frac{1}{1+u^{2}+v^{2}}).$ Thus, $M$ is a generalized
$J$-induced submanifold of order 1 of $\mathbb{R}^{14}.$ Also, one can easily
see the distribution $\mathcal{D}^{T}$ is totally geodesic and the
distributions $\mathcal{D}^{\bot}$ and $\mathcal{D}^{\theta}$ are integrable.
If we denote the integral submanifolds of $\mathcal{D}^{T},\mathcal{D}^{\bot}$
and $\mathcal{D}^{\theta}$ by $M_{T},M_{\bot}$ and $M_{\theta}$, respectively,
then the induced metric tensor of $M$ is

$\begin{array}[]{lll}ds^{2}&=&3(du^{2}+dv^{2})+(u^{2}+v^{2})dx^{2}+(1+u^{2}+v^{2})(dz^{2}+dw^{2})\\\
&=&g_{M_{T}}+(u^{2}+v^{2})g_{M_{\bot}}+(1+u^{2}+v^{2})g_{M_{\theta}}.\end{array}$

Thus, $M=M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ is a (non-trivial)
biwarped product generalized $J$-induced submanifold of order 1 of
$\mathbb{R}^{14}$ with warping functions $f=\sqrt{u^{2}+v^{2}}$ and
$\sigma=\sqrt{1+u^{2}+v^{2}}.$

##  5\. Biwarped product generalized $J$-induced submanifolds of order 1
in the form $M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ of a Kähler
manifold

In this section, we give a characterization for biwarped product generalized
$J$-induced submanifolds of order 1 in the form
$M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$, where $M_{T}$ is a
holomorphic, $M_{\bot}$ is a totally real and $M_{\theta}$ is a pointwise
slant submanifold of a Kähler manifold $(\bar{M},J,g)$. After that, we
investigate the behavior of the second fundamental form of such submanifolds
and as a result, we give a necessary and sufficient condition for such
manifolds to be locally trivial. Now, we give one of the main theorems of this
paper. We first recall the following fact given in [13] to prove our theorem.

######  Remark 5.1.

(Remark 2.1 [13]) Suppose that the tangent bundle of a Riemannian manifold $M$
splints into an orthogonal sum
$TM=\mathcal{D}_{0}\oplus\mathcal{D}_{1}\oplus...\oplus\mathcal{D}_{k}$ of
non-trivial distributions such that each $\mathcal{D}_{j}$ is spherical and
its complement in $TM$ is autoparallel for $j\in\\{1,2,...,k\\}.$ Then the
manifold $M$ is locally isometric to a multiply warped product
$M_{0}\times_{f_{1}}M_{1}\times...\times_{f_{k}}M_{k}.$

######  Theorem 5.2.

Let $M$ be a proper generalized $J$-induced submanifold of order 1 of a Kähler
manifold $(\bar{M},J,g).$ Then $M$ is a locally biwarped product submanifold
of the form $M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ if and only if

(5.1) $\displaystyle A_{JX}V=-JV(\lambda)X,$ (5.2) $\displaystyle
A_{FTZ}V-A_{FZ}JV=-\sin^{2}\\!\theta V(\mu)Z,$

for some functions $\lambda$ and $\mu$ satisfying $X(\lambda)=Z(\lambda)=0$
and $X(\mu)=Z(\mu)=0,$ and

(5.3) $\displaystyle g(A_{JY}TZ,X)=g(A_{FTZ}Y,X),$ (5.4) $\displaystyle
g(A_{JY}TW,Z)=g(A_{FTZ}Y,W).$

for $V\in\mathcal{D}^{T}$, $X,Y\in\mathcal{D}^{\bot}$ and
$Z,W\in\mathcal{D}^{\theta}$.

###### Proof.

Let $M$ be a biwarped product proper generalized $J$-induced submanifold of
order 1 of a Kähler manifold $(\bar{M},J,g)$ in the form
$M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$. Then for any
$V\in\mathcal{D}^{T}$, $X\in\mathcal{D}^{\bot}$ and $Z\in\mathcal{D}^{\theta}$
using (2.4)$\sim$(2.5) and (2.7)$\sim$(2.8), we have

$\displaystyle
g(A_{JX}V,U)=-g(\bar{\nabla}_{U}JX,V)=g(\bar{\nabla}_{U}X,JV)=g(\nabla_{U}X,JV).$

Here, we know $\nabla_{U}X=U(\ln f)X$ from (2.2). Thus, we obtain

(5.5) $\begin{array}[]{c}g(A_{JX}V,U)=U(\ln f)g(X,JV)=0\end{array},$

since $g(X,JV)=0.$ Similarly, we have

$\displaystyle
g(A_{JX}V,Z)=-g(\bar{\nabla}_{Z}JX,V)=g(\bar{\nabla}_{Z}X,JV)=g(\nabla_{Z}X,JV).$

Thus, we obtain

(5.6) $\begin{array}[]{c}g(A_{JX}V,Z)=0\end{array},$

since $\nabla_{Z}X=0$ from (2.3). Next, by a similar argument, for
$Y\in\mathcal{D}^{\bot}$, we have

$\displaystyle
g(A_{JX}V,Y)=-g(\bar{\nabla}_{Y}JX,V)=g(\bar{\nabla}_{Y}X,JV)=-g(\nabla_{Y}JV,X).$

Using (2.2), we obtain

(5.7) $\begin{array}[]{c}g(A_{JX}V,Y)=g(-JV(\ln f)X,Y)\end{array}.$

Moreover, we have $X(\ln f)=Z(\ln f)=0$, since $f$ depends on only points of
$M_{T}.$ So, we conclude that $\lambda=\ln f.$ Thus, ffrom (5.5)$\sim$(5.7),
it follows that (5.1). Now, we prove (5.2). Using (2.4)$\sim$(2.5) and
(2.7)$\sim$(2.10), we have

$\displaystyle g(A_{FTZ}V-A_{FZ}JV,U)=$
$\displaystyle-g(\bar{\nabla}_{V}FTZ,U)+g(\bar{\nabla}_{JV}FZ,U)$
$\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{V}JTZ,U)+g(\bar{\nabla}_{V}T^{2}Z,U)$
$\displaystyle+g(\bar{\nabla}_{JV}JZ,U)-g(\bar{\nabla}_{JV}TZ,U)$
$\displaystyle=$ $\displaystyle+g(\bar{\nabla}_{V}TZ,JU)-\cos^{2}\\!\theta
g(\bar{\nabla}_{V}Z,U)$
$\displaystyle-g(\bar{\nabla}_{JV}Z,JU)-g(\bar{\nabla}_{JV}TZ,U)$
$\displaystyle=$ $\displaystyle+g(\nabla_{V}TZ,JU)-\cos^{2}\\!\theta
g(\nabla_{V}Z,U)$
$\displaystyle-g(\nabla_{JV}Z,JU)-g(\nabla_{JV}TZ,U).\qquad\qquad$

Here, if we use (2.2), we obtain

$\displaystyle g(A_{FTZ}V-A_{FZ}JV,U)=$ $\displaystyle
V(\ln\sigma)g(TZ,JU)-\cos^{2}\\!\theta JV(\ln\sigma)g(Z,U)$ $\displaystyle-
JV(\ln\sigma)g(Z,JU)-V(\ln\sigma)g(TZ,U).$

Hence, we get

(5.8) $\begin{array}[]{c}g(A_{FTZ}V-A_{FZ}JV,U)=0\end{array},$

since $g(TZ,JU)=g(Z,U)=g(Z,JU)=g(TZ,U)=0.$ Similarly, we have

$\displaystyle g(A_{FTZ}V-A_{FZ}JV,X)=$
$\displaystyle-g(\bar{\nabla}_{X}FTZ,V)+g(\bar{\nabla}_{X}FZ,JV)$
$\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{X}JTZ,V)+g(\bar{\nabla}_{X}T^{2}Z,V)$
$\displaystyle+g(\bar{\nabla}_{X}JZ,JV)-g(\bar{\nabla}_{X}TZ,JV)$
$\displaystyle=$ $\displaystyle+g(\bar{\nabla}_{X}TZ,JV)-\cos^{2}\\!\theta
g(\bar{\nabla}_{X}Z,V)$
$\displaystyle-g(\bar{\nabla}_{X}Z,V)-g(\bar{\nabla}_{X}TZ,JV)$
$\displaystyle=$ $\displaystyle+g(\nabla_{X}TZ,JV)-\cos^{2}\\!\theta
g(\nabla_{X}Z,V)$
$\displaystyle-g(\nabla_{X}Z,V)-g(\nabla_{X}TZ,JV).\qquad\qquad$

Here, we know $\nabla_{X}TZ=\nabla_{X}Z=0$ from (2.3). So, we get

(5.9) $\begin{array}[]{c}g(A_{FTZ}V-A_{FZ}JV,X)=0\end{array}.$

On the other hand, using (3.4) , we have

$\begin{array}[]{c}g(A_{FTZ}V-A_{FZ}JV,W)=-\sin^{2}\\!\theta
g(\nabla_{W}V,Z)\end{array}.$

Using (2.2), we get

(5.10) $\begin{array}[]{c}g(A_{FTZ}V-A_{FZ}JV,W)=g(-\sin^{2}\\!\theta
V(\ln\sigma)Z,W)\end{array}.$

Moreover, we have $X(\ln\sigma)=Z(\ln\sigma)=0$, since $\sigma$ depends on
only points of $M_{T}.$ So, we conclude that $\mu=\ln\sigma.$ Thus, (5.2)
follows form (5.8), (5.9) and (5.10). Next, we prove (5.3) and (5.4). Using
(3.10) and (2.3), we have

$\begin{array}[]{c}g(A_{JY}X,TZ)-g(A_{FTZ}X,Y)=-\cos^{2}\\!\theta
g(\nabla_{X}Z,Y)=0\end{array}.$

Thus, (5.3) follows. Similarly, using (5.5) and (2.3), we have

$\begin{array}[]{c}g(A_{JY}TW,Z)=g(A_{FTZ}Y,W)=-\cos^{2}\\!\theta
g(\nabla_{Z}Y,W)=0\end{array}.$

So, we get (5.4).

Conversely, assume that $M$ is a proper generalized $J$-induced submanifold of
order 1 of a Kähler manifold $(\bar{M},J,g)$ such that (5.1)$\sim$(5.4) hold.
Then we satisfy (3.12) and (3.13) by using (5.1) and (5.2), respectively.
Thus, by Theorem 3.5, the holomorphic distribution $\mathcal{D}^{T}$ is
totally geodesic and as a result it is integrable. By (5.3) and (5.4), we
easily satisfy (3.14) and (3.15). Thus, by Theorem 3.6, the pointwise slant
distribution $\mathcal{D}^{\theta}$ is integrable. Also, by Remark 3.7, the
totally real distribution $\mathcal{D}^{\bot}$ is always integrable. Let
$M_{T}$, $M_{\bot}$ and $M_{\theta}$ be the integral manifolds of
$\mathcal{D}^{T}$, $\mathcal{D}^{\bot}$ and $\mathcal{D}^{\theta}$,
respectively. If we denote the second fundamental form of $M_{\bot}$ in $M$ by
$h^{\bot}$, for $X,Y\in\mathcal{D}^{\bot}$ and $Z\in\mathcal{D}^{\theta}$,
using (2.4), (3.10) and (5.3), we have

(5.11) $\begin{array}[]{c}g(h^{\bot}(X,Y),Z)=g(\nabla_{X}Y,Z)=0\end{array}.$

For any $X,Y\in\mathcal{D}^{\bot}$ and $V\in\mathcal{D}^{T}$, using (2.4),
(3.6) and (5.1), we have

$\begin{array}[]{c}g(h^{\bot}(X,Y),V)=g(\nabla_{X}Y,V)=-g(A_{JY}X,JV)=-V(\lambda)g(X,Y)\end{array}.$

After some calculation, we obtain

(5.12)
$\begin{array}[]{c}g(h^{\bot}(X,Y),V)=g(-g(X,Y)\nabla\lambda,V)\end{array},$

where $\nabla\lambda$ is the gradient of $\lambda$. Thus, from (5.11) and
(5.12), we conclude that

$\begin{array}[]{c}h^{\bot}(X,Y)=-g(X,Y)\nabla\lambda\end{array}.$

This equation says that $M_{\bot}$ is totally umbilic in $M$ with the mean
curvature vector field $-\nabla\lambda.$ Now, we show that $-\nabla\lambda$ is
parallel. We have to satisfy $g(\nabla_{X}\nabla\lambda,E)=0$ for
$X\in\mathcal{D}^{\bot}$ and
$E\in(\mathcal{D}^{\bot})^{\bot}=\mathcal{D}^{T}\oplus\mathcal{D}^{\theta}.$
Here, we can put $E=V+Z,$ where $V\in\mathcal{D}^{T}$ and
$Z\in\mathcal{D}^{\theta}.$ By direct computations, we obtain

$\displaystyle g(\nabla_{X}\nabla\lambda,E)=$
$\displaystyle\\{Xg(\nabla\lambda,E)-g(\nabla\lambda,\nabla_{X}E)\\}$
$\displaystyle=$
$\displaystyle\\{X(E(\lambda))-[X,E]\lambda-g(\nabla\lambda,\nabla_{E}X)\\}$
$\displaystyle=$
$\displaystyle\\{[X,E]\lambda+E(X(\lambda))-[X,E]\lambda-g(\nabla\lambda,\nabla_{E}X)\\}$
$\displaystyle=$
$\displaystyle-g(\nabla\lambda,\nabla_{V}X)-g(\nabla\lambda,\nabla_{Z}X),$

since $X(\lambda)=0.$ Here, for any $U\in\mathcal{D}^{T}$, we have
$g(\nabla_{V}X,U)=-g(\nabla_{V}U,X)=0,$ since $M_{T}$ is totally geodesic in
$M$. Thus, $\nabla_{V}X\in\mathcal{D}^{\bot}$ or
$\nabla_{V}X\in\mathcal{D}^{\theta}.$ In either case, we have

(5.13) $\begin{array}[]{c}g(\nabla\lambda,\nabla_{V}X)=0\end{array}.$

On the other hand, from (3.9), we have $g(\nabla_{Z}X,U)=-g(A_{JX}JU,Z).$
Here, using (5.3), we obtain $g(\nabla_{Z}X,U)=0.$ That is;
$\nabla_{Z}X\in\mathcal{D}^{\bot}$ or $\nabla_{Z}X\in\mathcal{D}^{\theta}.$ In
either case, we get

(5.14) $\begin{array}[]{c}g(\nabla\lambda,\nabla_{Z}X)=0\end{array}.$

From (5.13) and (5.14), we find

$\begin{array}[]{c}g(\nabla_{X}\nabla\lambda,E)=0\end{array}.$

Thus, $M_{\bot}$ is spherical, since it is also totally umbilic. Consequently,
$\mathcal{D}^{\bot}$ is spherical.
Next, we show that $\mathcal{D}^{\theta}$ is spherical. Let $h^{\theta}$
denote the second fundamental form of $M_{\theta}$ in $M.$ Then for
$Z,W\in\mathcal{D}^{\theta}$ and $X\in\mathcal{D}^{\bot}$, using (2.4), (3.8)
and (5.3), we have

(5.15) $\begin{array}[]{c}g(h^{\theta}(Z,W),X)=g(\nabla_{Z}W,X)=0\end{array}.$

On the other hand, for any $V\in\mathcal{D}^{T}$, using (2.4) and (3.4), we
have

$\begin{array}[]{c}g(h^{\theta}(Z,W),V)=g(\nabla_{Z}W,V)=\csc^{2}\\!\theta
g(A_{FTW}V-A_{FW}JV,Z)\end{array}.$

Using (5.2), we obtain

$\begin{array}[]{c}g(h^{\theta}(Z,W),V)=-V(\mu)g(Z,W)\end{array}.$

After some calculation, we get

(5.16)
$\begin{array}[]{c}g(h^{\theta}(Z,W),V)=-g(g(Z,W)\nabla\mu,V)\end{array},$

where $\nabla\mu$ is the gradient of $\mu$. Thus, from (5.15) and (5.16), we
deduce that

$\begin{array}[]{c}h^{\theta}(Z,W)=-g(Z,W)\nabla\mu\end{array}.$

It means that $M_{\theta}$ is totally umbilic in $M$ with the mean curvature
vector field $-\nabla\mu.$ What’s left is to show that $-\nabla\mu$ is
parallel. We have to satisfy $g(\nabla_{Z}\nabla\mu,E)=0$ for
$Z\in\mathcal{D}^{\theta}$ and
$E\in(\mathcal{D}^{\theta})^{\bot}=\mathcal{D}^{T}\oplus\mathcal{D}^{\bot}.$
Here, $E=V+X,$ for $V\in\mathcal{D}^{T}$ and $X\in\mathcal{D}^{\bot}.$ Upon
direct calculation, we obtain

$\displaystyle g(\nabla_{Z}\nabla\mu,E)=$
$\displaystyle\\{Zg(\nabla\mu,E)-g(\nabla\mu,\nabla_{Z}E)\\}$ $\displaystyle=$
$\displaystyle\\{Z(E(\mu))-[Z,E]\mu-g(\nabla\mu,\nabla_{E}Z)\\}$
$\displaystyle=$
$\displaystyle\\{[Z,E]\mu+E(Z(\mu))-[Z,E]\mu-g(\nabla\mu,\nabla_{E}Z)\\}$
$\displaystyle=$
$\displaystyle-g(\nabla\mu,\nabla_{V}Z)-g(\nabla\mu,\nabla_{X}Z),$

since $Z(\mu)=0.$ Here, for any $U\in\mathcal{D}^{T}$, using (5.1) and (3.11),
we have

$\begin{array}[]{c}g(\nabla_{X}Z,U)=-\csc^{2}\\!\theta\\{g(A_{FZ}JU-
A_{FTZ}U,X)\\}\end{array}.$

So, $\nabla_{X}Z\in\mathcal{D}^{\bot}$ or
$\nabla_{V}X\in\mathcal{D}^{\theta}.$ Hence

(5.17) $\begin{array}[]{c}g(\nabla\mu,\nabla_{X}Z)=0\end{array},$

since $\nabla\mu\in\mathcal{D}^{T}.$ On the other hand, since $M_{T}$ is
totally geodesic in $M$, we have $g(\nabla_{V}Z,U)=-g(\nabla_{V}U,Z)=0.$
Hence, $\nabla_{V}Z\in\mathcal{D}^{\bot}$ or
$\nabla_{V}Z\in\mathcal{D}^{\theta}.$ So, we get

(5.18) $\begin{array}[]{c}g(\nabla\mu,\nabla_{V}Z)=0\end{array}.$

By (5.17) and (5.18), we find

$\begin{array}[]{c}g(\nabla_{Z}\nabla\mu,E)=0\end{array}.$

Lastly, we prove that
$(\mathcal{D}^{\bot})^{\bot}=\mathcal{D}^{T}\oplus\mathcal{D}^{\theta}$ and
$(\mathcal{D}^{\theta})^{\bot}=\mathcal{D}^{T}\oplus\mathcal{D}^{\bot}$ are
autoparallel. In fact, $\mathcal{D}^{T}\oplus\mathcal{D}^{\theta}$ is
autoparallel if and only if all four types of covariant derivatives
$\nabla_{U}V,\nabla_{U}Z,\nabla_{Z}U,\nabla_{Z}W$ are again in
$\mathcal{D}^{T}\oplus\mathcal{D}^{\bot}$ for $U,V\in\mathcal{D}^{T}$ and
$X\in\mathcal{D}^{\bot}.$ This is equivalent to say that all four inner
products $g(\nabla_{U}V,X),g(\nabla_{U}Z,X),g(\nabla_{Z}U,X)$,
$g(\nabla_{Z}W,X)$ vanish, where $U,V\in\mathcal{D}^{T}$,
$Z,W\in\mathcal{D}^{\theta}$ and $X\in\mathcal{D}^{\bot}.$ Using (3.5), (3.5)
and (5.1), we get

$\begin{array}[]{c}g(\nabla_{U}V,X)=g(\nabla_{Z}U,X)=0\end{array}.$

By (3.6), (3.8) and (5.3), we get

$\begin{array}[]{c}g(\nabla_{U}Z,X)=g(\nabla_{Z}W,X)=0\end{array}.$

Thus, $\mathcal{D}^{T}\oplus\mathcal{D}^{\theta}$ is autoparallel. On the
other hand, $\mathcal{D}^{T}\oplus\mathcal{D}^{\bot}$ is autoparallel if and
only if all four inner products
$g(\nabla_{U}V,Z),g(\nabla_{U}X,Z),g(\nabla_{X}U,Z),g(\nabla_{X}Y,Z)$ vanish.
Firstly, we have already $g(\nabla_{U}X,Z)=0$ from above. Using (3.3), (3.11)
and (5.2), we get

$\begin{array}[]{c}g(\nabla_{U}V,Z)=g(\nabla_{X}U,Z)=0\end{array}.$

Using (3.10) and (5.3), we find

$\begin{array}[]{c}g(\nabla_{X}Y,Z)=0\end{array}.$

So, $\mathcal{D}^{T}\oplus\mathcal{D}^{\bot}$ is autoparallel. Thus, by Remark
5.1, $M$ is locally biwarped product submanifold of the form
$M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}.$ ∎

Let $M=M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ be a non-trivial
biwarped product generalized $J$-induced submanifold of order 1 of a Kähler
manifold $(\bar{M},J,g).$ Next, we investigate the behavior of the second
fundamental form $h$ of such submanifolds.

######  Lemma 5.3.

Let $M$ be a biwarped product generalized $J$-induced submanifold of order 1
in the form $M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ of a Kähler
manifold $(\bar{M},J,g)$. Then for the second fundamental form $h$ of $M$ in
$\bar{M}$, we have

(5.19) $\displaystyle g(h(U,V),JX)=0\,,\qquad\qquad$ (5.20) $\displaystyle
g(h(V,X),JY)=-JV(\ln f)g(X,Y)\,,$ (5.21) $\displaystyle
g(h(V,Z),JX)=0\,,\qquad\qquad\qquad\qquad$

where $U,V\in\mathcal{D}^{T}$, $X,Y\in\mathcal{D}^{\bot}$ and
$Z\in\mathcal{D}^{\theta}$.

###### Proof.

Using (2.4),(2.7) and (2.8), we have

$\begin{array}[]{c}g(h(U,V),JX)=g(\bar{\nabla}_{U}V,JX)=-g(\bar{\nabla}_{U}JX,V)=g(\bar{\nabla}_{U}X,JV)\end{array}$

for $U,V\in\mathcal{D}^{T}$ and $X\in\mathcal{D}^{\bot}$. Again, using (2.4),
we obtain $g(h(U,V),JX)=g(\nabla_{U}X,JV)$. Here, we know $\nabla_{U}X=U(\ln
f)X$ from (2.2). Thus, we obtain $g(h(U,V),JX)=U(\ln f)g(X,V)=0$, since
$g(X,V)=0.$ So, (5.19) follows. Now, let $V\in\mathcal{D}^{T}$ and
$X,Y\in\mathcal{D}^{\bot}$. Then using (2.4), (2.7) and (2.8), we have

$\begin{array}[]{c}g(h(V,X),JY)=g(\bar{\nabla}_{X}V,JY)=-g(\bar{\nabla}_{X}JV,Y)=-g(\nabla_{X}JV,Y)\end{array}.$

Here, we know $\nabla_{X}JV=JV(\ln f)X$ from (2.2). Thus, we get (5.20). The
last assertion can be obtained in a similar method. ∎

The previous lemma shows partially us the behavior of the second fundamental
form $h$ of the biwarped product generalized $J$-induced submanifolds of order
1 has the form $M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ in the
normal subbundle $J(\mathcal{D}^{T}).$

######  Lemma 5.4.

Let $M$ be a biwarped product generalized $J$-induced submanifold of order 1
in the form $M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ of a Kähler
manifold $(\bar{M},J,g)$. Then for the second fundamental form $h$ of $M$ in
$\bar{M}$, we have

(5.22) $\displaystyle g(h(U,V),FZ)=0\,,$ (5.23) $\displaystyle
g(h(V,X),FZ)=0\,,$ (5.24) $\displaystyle
g(h(V,Z),FW)=-JV(\ln\sigma)g(Z,W)-V(\ln\sigma)g(Z,TW)\,,\qquad\qquad$

where $U,V\in\mathcal{D}^{T}$, $X\in\mathcal{D}^{\bot}$ and
$Z,W\in\mathcal{D}^{\theta}$.

###### Proof.

Using (2.4),(2.7) and (2.8), we have

$\begin{array}[]{c}g(h(U,V),FZ)=g(\bar{\nabla}_{U}V,FZ)=g(\bar{\nabla}_{U}V,JZ)-g(\bar{\nabla}_{U}V,TZ)\end{array}$

for $U,V\in\mathcal{D}^{T}$ and $Z\in\mathcal{D}^{\theta}$. After some
calculation, we obtain

$\begin{array}[]{c}g(h(U,V),FZ)=g(\nabla_{U}Z,JV)+g(\nabla_{U}TZ,V)\end{array}.$

Here, we know $\nabla_{U}Z=U(\ln\sigma)Z$ and $\nabla_{U}Z=U(\ln\sigma)TZ$
from (2.2). Thus, we get

$\begin{array}[]{c}g(h(U,V),FZ)=U(\ln\sigma)g(Z,JV)+U(\ln\sigma)g(TZ,V)=0\end{array},$

since $g(Z,JV)=g(TZ,V)=0.$ So, (5.22) follows. Now, we prove (5.24). Let
$V\in\mathcal{D}^{T}$ and $Z,W\in\mathcal{D}^{\theta}$. Then using (2.4),(2.7)
and (2.8), we have

$\begin{array}[]{c}g(h(V,Z),FW)=-g(\nabla_{Z}JV,W)-g(\nabla_{Z}V,TW)\end{array}.$

Again, by (2.2), we easily get (5.24). Similarly, we can obtain (5.23). ∎

The last lemma shows partially us the behavior of the second fundamental form
$h$ of the biwarped product generalized $J$-induced submanifolds of order 1 of
type $M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ in the normal
subbundle $F(\mathcal{D}^{\theta}).$

######  Remark 5.5.

Equation (5.19) of Lemma 5.3 and equation (5.22) of Lemma 5.4 also hold for
skew CR-warped product submanifolds of Kähler manifolds, see Lemma 4 of [30].
Moreover, equations (5.22) and (5.24) of Lemma 5.4 are also valid for warped
product pointwise semi-slant submanifolds in Kähler manifolds, see Lemma 5.3
of [31].

By (5.19) and (5.22), we immediately get the following result.

######  Corollary 5.6.

Let $M$ be a biwarped product generalized $J$-induced submanifold of order 1
in the form $M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ of a Kähler
manifold $(\bar{M},J,g)$ such that the invariant normal subbundle
$\overline{\mathcal{D}}^{T}=\\{0\\}.$ Then $M$ is $\mathcal{D}^{T}$-geodesic.

Lastly, we give a necessary and sufficient condition for such submanifolds to
be locally trivial.

######  Theorem 5.7.

Let $M$ be a biwarped product proper generalized $J$-induced submanifold in
the form $M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ of a Kähler
manifold $(\bar{M},J,g)$ such that the invariant normal subbundle
$\overline{\mathcal{D}}^{T}=\\{0\\}.$ Then $M$ is locally trivial if and only
if $M$ is both $(\mathcal{D}^{T},\mathcal{D}^{\bot})$ and
$(\mathcal{D}^{T},\mathcal{D}^{\theta})$-mixed geodesic.

###### Proof.

Let $M$ be a biwarped product proper generalized $J$-induced submanifold of
order 1 in the form $M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ of a
Kähler manifold $(\bar{M},J,g)$ such that the invariant normal subbundle
$\overline{\mathcal{D}}^{T}=\\{0\\}.$ If $M$ is locally trivial, then the
warping functions $f$ and $\sigma$ are constants. By (5.20), we have
$g(h(V,X),JY)=0$ for $V\in\mathcal{D}^{T}$ and $X,Y\in\mathcal{D}^{\bot}$,
since $JV(\ln f)=0.$ Taking into account the equation (3.2) and (5.23), we get
$h(V,X)=0$. It means that $M$ is $(\mathcal{D}^{T},\mathcal{D}^{\bot})$-mixed
geodesic.
On the other hand, for any $V\in\mathcal{D}^{T}$ and
$Z,W\in\mathcal{D}^{\theta},$ we have $g(h(V,Z),FW)=0$ from (5.24), since
$JV(\ln\sigma)=0$ and $V(\ln\sigma)=0.$ Taking into account the equation (3.2)
and (5.21), we obtain $h(V,Z)=0$. Which says us $M$ is
$(\mathcal{D}^{T},\mathcal{D}^{\theta})$-mixed geodesic.

Conversely, let $M$ be both $(\mathcal{D}^{T},\mathcal{D}^{\bot})$ and
$(\mathcal{D}^{T},\mathcal{D}^{\theta})$-mixed geodesic. Then, for any
$V\in\mathcal{D}^{T}$, from (5.20) we conclude that $JV(\ln f)=0$, since $M$
is $(\mathcal{D}^{T},\mathcal{D}^{\bot})$-mixed geodesic. Hence, it follows
that $f$ is a constant. Since $M$ is also
$(\mathcal{D}^{T},\mathcal{D}^{\theta})$-mixed geodesic, for
$V\in\mathcal{D}^{T}$ and $Z,W\in\mathcal{D}^{\theta},$ we have

(5.25)
$\begin{array}[]{c}JV(\ln\sigma)g(Z,W)+V(\ln\sigma)g(Z,TW)=0\end{array}$

from (5.24). If we put $V=JV$ in (5.25), we obtain

$\begin{array}[]{c}-V(\ln\sigma)g(Z,W)+JV(\ln\sigma)g(Z,TW)=0\end{array}.$

If we take $W=TW$ in the last equation and use (2.10), the last equation
becomes

(5.26) $\begin{array}[]{c}-V(\ln\sigma)g(Z,TW)-\cos^{2}\\!\theta
JV(\ln\sigma)g(Z,W)=0\end{array}.$

From (5.25) and (5.26), we get

(5.27) $\begin{array}[]{c}\sin^{2}\\!\theta JV(\ln\sigma)g(Z,W)=0\end{array}.$

Since $M$ is proper, $\sin\\!\theta\neq 0.$ So, we deduce that
$JV(\ln\sigma)=0$ from (5.27). Hence, it follows that $\sigma$ is a constant.
Thus, $M$ must be locally trivial, since we found the warping functions $f$
and $\sigma$ as constants. ∎

##  6\. An inequality for non-trivial biwarped product generalized
$J$-induced submanifolds of order 1 in the form
$M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$

In this section, by using the results given the preceding section, we shall
establish an inequality for the squared norm of the second fundamental form in
terms of the warping functions for biwarped product generalized $J$-induced
submanifolds of order 1 in the form
$M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$, where $M_{T}$ is a
holomorphic, $M_{\bot}$ is a totally real and $M_{\theta}$ is a pointwise
slant submanifold of a Kähler manifold$(\bar{M},J,g)$.

Let $M$ be a $(k+n+m)$-dimensional biwarped product generalized $J$-induced
submanifold of order 1 of type
$M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ of a Kähler manifold
$\bar{M}$ and let
$\\{e_{1},...,e_{k},\tilde{e}_{1},...,\tilde{e}_{n},\bar{e}_{1},...,\bar{e}_{m},e_{1}^{*},...,e_{k}^{*},J\tilde{e}_{1},...,J\tilde{e}_{n},\hat{e_{1}},...,\hat{e_{l}}\\}$
be a canonical orthonormal basis of $\bar{M}$ such that
$\\{e_{1},...,e_{k}\\}$ is an orthonormal basis of $\mathcal{D}^{T}$,
$\\{\tilde{e}_{1},...,\tilde{e}_{n}\\}$ is an orthonormal basis of
$\mathcal{D}^{\bot}$, $\\{\bar{e}_{1},...,\bar{e}_{k}\\}$ is an orthonormal
basis of $\mathcal{D}^{\theta}$, $\\{J\tilde{e}_{1},...,J\tilde{e}_{n}\\}$ is
an orthonormal basis of $J\mathcal{D}^{\bot}$, $\\{e_{1}^{*},...,e_{m}^{*}\\}$
is an orthonormal basis of $F\mathcal{D}^{\theta}$ and
$\\{\hat{e_{1}},...,\hat{e_{l}}\\}$ is an orthonormal basis of
$\overline{\mathcal{D}}^{T}.$ Here, $k=dim(\mathcal{D}^{T})$,
$n=dim(\mathcal{D}^{\bot})$, $m=dim(\mathcal{D}^{\theta})$ and
$l=dim(\overline{\mathcal{D}}^{T}).$

######  Remark 6.1.

In view of (2.7), we can observe that $\\{Je_{1},...,Je_{k}\\}$ is also an
orthonormal basis of $\mathcal{D}^{T}$. On the other hand, with the help of
(2.11) and (2.12), we can see that $\\{\sec\\!\theta
T\bar{e}_{1},...,\sec\\!\theta T\bar{e}_{m}\\}$ is also an orthonormal basis
of $\mathcal{D}^{\theta}$ and $\\{\csc\\!\theta F\bar{e}_{1},...,\csc\\!\theta
F\bar{e}_{m}\\}$ is also an orthonormal basis of $F\mathcal{D}^{\theta}$,
where $\theta$ is the slant function of $\mathcal{D}^{\theta}$.

######  Theorem 6.2.

Let $M$ be a biwarped product proper generalized $J$-induced submanifold of
order 1 in the form $M_{T}\times_{f}M_{\bot}\times_{\sigma}M_{\theta}$ of a
Kähler manifold $(\bar{M},J,g)$. Then the squared norm of the second
fundamental form $h$ of $M$ satisfies

(6.1) $\displaystyle\|h\|^{2}\geq 2\\{n\|\nabla(\ln
f)\|^{2}+m(\csc^{2}\\!\theta+\cot^{2}\\!\theta)\|\nabla(\ln\sigma)\|^{2}\\}$

where $n=dim(M_{\bot})$ and $m=dim(M_{\theta}).$ The equality case of (6.1)
holds identically if and only if the following assertions are true.
a) $M_{T}$ is a totally geodesic submanifold in $\bar{M}.$
b) $M_{\bot}$ and $M_{\theta}$ are totally umbilic submanifolds in $\bar{M}$
with their mean curvature vector fields $-\nabla(\ln f)$ and
$-\nabla(\ln\sigma)$, respectively.
c) $M$ is minimal in $\bar{M}.$
d) $M$ is $(\mathcal{D}^{\bot},\mathcal{D}^{\theta})$-mixed geodesic.

###### Proof.

By the decomposition (3.1), the squared norm of the second fundamental form
$h$ can be written as

$\displaystyle\|h\|^{2}=$
$\displaystyle\,\|h(\mathcal{D}^{T},\mathcal{D}^{T})\|^{2}+\|h(\mathcal{D}^{\bot},\mathcal{D}^{\bot})\|^{2}+\|h(\mathcal{D}^{\theta},\mathcal{D}^{\theta})\|^{2}$
$\displaystyle+2\bigg{\\{}\|h(\mathcal{D}^{T},\mathcal{D}^{\bot})\|^{2}+\|h(\mathcal{D}^{T},\mathcal{D}^{\theta})\|^{2}+\|h(\mathcal{D}^{\bot},\mathcal{D}^{\theta})\|^{2}\bigg{\\}}.$

In view of decomposition (3.2) and by (5.19)$\sim$(5.24), which can be
explicitly written as follows:

$\displaystyle\|h\|^{2}=$
$\displaystyle\displaystyle\sum^{n}_{a,b,c=1}\\!\\!\\!g(h(\tilde{e}_{a},\tilde{e}_{b}),J\tilde{e}_{c})^{2}+\displaystyle\sum^{n}_{a,b=1}\displaystyle\sum^{m}_{r=1}g(h(\tilde{e}_{a},\tilde{e}_{b}),e_{r}^{*})^{2}$
$\displaystyle+\displaystyle\sum^{m}_{r,s=1}\displaystyle\sum^{n}_{a=1}g(h(\bar{e}_{r},\bar{e}_{s}),J\tilde{e}_{a})^{2}+\displaystyle\sum^{m}_{r,s,q=1}g(h(\bar{e}_{r},\bar{e}_{s}),e_{q}^{*})^{2}$
(6.2)
$\displaystyle+2\displaystyle\sum^{k}_{i=1}\displaystyle\sum^{n}_{a,b=1}g(h(e_{i},\tilde{e}_{a}),J\tilde{e}_{b})^{2}+2\displaystyle\sum^{k}_{i=1}\displaystyle\sum^{m}_{r,s=1}g(h(e_{i},\bar{e}_{r}),e_{s}^{*})^{2}$
$\displaystyle+2\\!\displaystyle\sum^{n}_{a,b,c=1}g(h(\tilde{e}_{a},\tilde{e}_{b}),J\tilde{e}_{c})^{2}+2\\!\displaystyle\sum^{n}_{a,b=1}\displaystyle\sum^{m}_{r=1}g(h(\tilde{e}_{a},\tilde{e}_{b}),e_{r}^{*})^{2}$
$\displaystyle+\displaystyle\sum^{k+n+m}_{A,B=1}\displaystyle\sum^{l}_{t=1}g(h(e_{A},e_{B}),\hat{e}_{t})^{2}.$

Where the set $\\{e_{A}\\}_{1\leq A\leq(k+n+m)}$ is an orthonormal basis of
$M.$ Hence, we get

(6.3) $\begin{array}[]{c}\|h\|^{2}\geq
2\bigg{\\{}\displaystyle\sum^{k}_{i=1}\displaystyle\sum^{n}_{a,b=1}g(h(e_{i},\tilde{e}_{a}),J\tilde{e}_{b})^{2}+\displaystyle\sum^{k}_{i=1}\displaystyle\sum^{m}_{r,s=1}g(h(e_{i},\bar{e}_{r}),e_{s}^{*})^{2}\bigg{\\}}\end{array}.$

Using (5.20) and Remark (6.1), we arrive

$\begin{array}[]{c}\|h\|^{2}\geq
2\bigg{\\{}\displaystyle\sum^{k}_{i=1}\displaystyle\sum^{n}_{a,b=1}(Je_{i}(\ln
f)g(\tilde{e}_{a},\tilde{e}_{b}))^{2}+\displaystyle\sum^{k}_{i=1}\displaystyle\sum^{m}_{r,s=1}g(h(e_{i},\bar{e}_{r}),\csc\\!\theta
F\bar{e}_{s}))^{2}\bigg{\\}}\end{array}.$

from the inequality (6.3). Using (5.24) and after some calculation we find

(6.4) $\begin{array}[]{c}\|h\|^{2}\geq
2n\|\nabla(\ln\\!f)\|^{2}+2\displaystyle\sum^{k}_{i=1}\displaystyle\sum^{m}_{r,s=1}(\csc^{2}\\!\theta)\bigg{\\{}(Je_{i}(\ln\sigma)g(\bar{e}_{r},\bar{e}_{s}))^{2}\\\
\qquad\qquad\qquad+2Je_{i}(\ln\sigma)g(\bar{e}_{r},\bar{e}_{s})e_{i}(\ln\sigma)g(\bar{e}_{r},T\bar{e}_{s})+(e_{i}(\ln\sigma)g(\bar{e}_{r},T\bar{e}_{s}))^{2}\bigg{\\}}.\end{array}$

from the last inequality. Here,

$\displaystyle\displaystyle\sum^{k}_{i=1}\displaystyle\sum^{m}_{r,s=1}\bigg{\\{}(Je_{i}(\ln\sigma)g(\bar{e}_{r},\bar{e}_{s}))^{2}Je_{i}(\ln\sigma)g(\bar{e}_{r},\bar{e}_{s})e_{i}(\ln\sigma)g(\bar{e}_{r},T\bar{e}_{s})$
$\displaystyle=\displaystyle\sum^{k}_{i=1}\displaystyle\sum^{m}_{r,s=1}g(\nabla\ln\sigma,Je_{i})g(\nabla\ln\sigma,e_{i})g(\bar{e}_{r},\bar{e}_{s})g(\bar{e}_{r},T\bar{e}_{s})$
$\displaystyle=-\displaystyle\sum^{m}_{r,s=1}\bigg{\\{}\displaystyle\sum^{k}_{i=1}g(g(\nabla(\ln\sigma),e_{i})e_{i},J\nabla(\ln\sigma))\bigg{\\}}g(\bar{e}_{r},\bar{e}_{s})g(\bar{e}_{r},T\bar{e}_{s})$
$\displaystyle=-g(\nabla(\ln\sigma),J\nabla(\ln\sigma))\displaystyle\sum^{m}_{r,s=1}g(\bar{e}_{r},\bar{e}_{s})g(\bar{e}_{r},T\bar{e}_{s})=0.$

Thus, by Remark (6.1), the equation (2.11) and the last yield, we deduce the
inequality (6.1) from the inequality (6.4).

Next, from (6.2) we see that the equality case of (6.1) holds identically if
and only if the following conditions hold.

(6.5) $\begin{array}[]{c}h(\mathcal{D}^{T},\mathcal{D}^{T})=\\{0\\},\qquad
h(\mathcal{D}^{\bot},\mathcal{D}^{\bot})=\\{0\\},\qquad
h(\mathcal{D}^{\theta},\mathcal{D}^{\theta})=\\{0\\}\end{array}$

and

(6.6)
$\begin{array}[]{c}h(\mathcal{D}^{\bot},\mathcal{D}^{\theta})=\\{0\\}.\end{array}$

Since $M_{T}$ is totally geodesic in $M$, from the first condition in (6.5) it
follows that $M_{T}$ is also totally geodesic in $\bar{M}.$ So, assertion a)
follows. Now, let $h^{\bot}$ denote the second fundamental of $M_{\bot}$ in
$M$. We know that
$h^{\bot}(\mathcal{D}^{\bot},\mathcal{D}^{\bot})\subseteq\mathcal{D}^{T}$ from
[21]. Then for $V\in\mathcal{D}^{T}$ and $X,Y\in\mathcal{D}^{\bot},$ we have
$g(h^{\bot}(X,Y),V)=g(\nabla_{X}Y,V).$ Here, we know
$\nabla_{X}Y=^{\bot}\\!\nabla_{X}Y-g(X,Y)\nabla(\ln f)$ from (2.3), where
${}^{\bot}\nabla$ is an induced connection on $M_{\bot}.$ Hence, we obtain

$\begin{array}[]{c}g(h^{\bot}(X,Y),V)=-V(\ln f)g(X,Y)=-g(g(X,Y)\nabla(\ln
f),V)\end{array}.$

It follows that

(6.7) $\begin{array}[]{c}h^{\bot}(X,Y)=-g(X,Y)\nabla(\ln f)\end{array}$

from the last equation. Thus, combining the second condition in (6.5) and
(6.7), we can deduce that $M_{\bot}$ is a totally umbilic submanifold in
$\bar{M}$ with its mean curvature vector field $-\nabla(\ln f).$ By a similar
argument, we can find $M_{\theta}$ as a totally umbilic submanifold in
$\bar{M}$ with its mean curvature vector field $-\nabla(\ln\sigma).$ So,
assertion b) is obtained. Assertions c) and d) immediately follow from (6.5)
and (6.6), respectively. ∎

######  Remark 6.3.

In case $\mathcal{D}^{\theta}=\\{0\\}$, Theorem 6.2 coincides with Theorem 5.1
of [8]. In other words, Theorem 6.2 is a generalization of Theorem 4.2 of [8].
Moreover, Theorem 6.2 coincides with Theorem 5.2 of [31] if
$\mathcal{D}^{\bot}=\\{0\\}.$ Thus, Theorem 6.2 is also a generalization of
Theorem 5.2 of [31].

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