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\begin{center}
{\large\bf\textsf{Generic Riemannian Submersions}}
\footnotetext{{\bf{2000 AMS Classification}}: 53 C 15, 53 C 40, 53 C 50}
\footnotetext{{\bf{Keywords}}: Riemannian submersions, generic Riemannian submersions, totally geodesic map.}
\end{center}


\begin{center}
Shahid Ali$^1$ and Tanveer Fatima$^2$\\
Department of Mathematics\\
Aligarh Muslim University\\
Aligarh 202002 (India)\\\vspace{.3cm}

\bf{E-mails}: $^1$shahid07ali@gmail.com, $^2$fatima.tanveer.maths@gmail.com \\
\end{center}

\begin{abstract}
B. Sahin [10] introduced the notion of semi-invariant Riemannian submersions as a generalization of anti-invariant Riemmanian submersions [9]. As a generalization to semi-invariant Riemannian submersions we introduce the notion of generic submersion from an almost Hermitian manifold
onto a Riemannian manifold and investigate the geometry of foliations which arise from the
definition of a generic Riemannian submersion and find necessary and sufficient condition for total manifold to be a
generic product manifold. We also find necessary and sufficient conditions for a generic submersion
to be totally geodesic.
\end{abstract}

\vspace{.2cm}
\parindent=0mm
%\begin{center}
{\bf{1. Introduction}}\\
%\end{center}

\parindent=8mm

Riemannian submersion between Riemannian manifolds were studied by O'Niell [8] and Gray [5].
Later on, such submersions have been studied widely in differential geometry. Riemannian submersions
between Riemannian manifolds equipped with an additional structure of almost complex type
was firstly studied by Watson [11]. Watson defined an almost Hermitian submersion between almost
Hermitian manifolds and he showed that the base manifold and each fiber have the same kind of structure
as the total space, in most cases. Almost Hermitian submersions have been extended to the
almost contact manifolds [3], locally conformal Kaehler manifolds [7] and quaternion Kaehler manifolds [6].\\

\parindent=8mm

Let $M$ be a complex $m$-dimensional almost Hermitian manifold with Hermitian metric $g$ and an almost
complex structure $J$ and $B$ be a complex $n$-dimensional almost Hermitian manifold with metric
$g_B$ and an almost complex structure $J'$. A Riemannian submersion $\pi:M\rightarrow B$
is called an almost Hermitian submersion if $\pi$ is an almost complex mapping, i.e., $\pi_*\circ J=J'\circ \pi_*.$
The main result of this notion is that the vertical and horizontal distributions are invariant under $J.$
On the other hand, Escobales [4] studied Riemannian submersions from complex projective
space onto a Riemannian manifold under the assumption that the fibres are connected, complex, totally
geodesic submanifolds. In fact this assumption also implies that the vertical distribution is invariant
with respect to the almost complex structure.\\

\parindent=8mm

The submersions mentioned above have one common property that the vertical and horizontal distributions
are invariant. Recently B. Sahin [9] introduced the notion of anti-invariant Riemannian submersions
which are Riemannian submersions from almost Hermitian manifolds such that the vertical
distributions (or, for that matter the fibers) are anti-invariant under the almost complex structure of the total manifold
and as a generalization of anti-invariant submersions and almost Hermitian submersions,
B. Sahin introduced the notion of semi-invariant Riemannian submersion when the base manifold is an almost
Hermitian manifold [10]. He has shown that such submersions are useful to investigate the geometry of the
total manifold of the submersion. In the present article, we introduce the notion of a submersion from an almost Hermitian manifold under the assumption that fibers are generic submanifold of the total space and call it as a generic Riemannian submersion, and one sees it as a generalization of semi-invariant Riemannian submersions. The paper is organized as follows; In section 2 we give some basic notions of almost Hermitian manifolds and a brief introduction of Riemannian submersion between Riemannian manifolds. In section 3, we give the definition of generic submersion and study the geometry of the distributions and their leaves. In the end of this section, we obtain decomposition theorems and find the conditions for generic submersion to be totally geodesic map.\\

\vspace{.2cm}
\parindent=0mm
%\begin{center}
{\bf{2. Preliminaries}}\\
%\end{center}

\parindent=8mm

Let $(\bar{M},g)$ be an almost Hermitian manifold. That is $\bar{M}$ admits
a tensor field $J$ of type (1,1) and a Riemannian metric $g$ such that
$$J^2=-I,~~g(X,Y)=g(JX,JY),~\forall~X,Y\in \Gamma (T\bar{M}).~~\eqno(2.1)$$
An almost Hermitian manifold $\bar{M}$ is called Kaehler manifold if
$$(\bar{\nabla}_XJ)Y=0,~\forall~X,Y\in \Gamma (T\bar{M})\eqno(2.2)$$
where $\bar{\nabla}$ is the Levi-Civita connection on $\bar{M}.$ \\

Let $(\bar M,g,J)$ be an almost Hermitian manifold and $M$ be real submanifold of $\bar M$, and let $${\cal D}_p=T_pM\cap J T_pM,~\forall ~p\in M$$ such that ${\cal D}_p$ is the maximal complex subspace of $\Gamma(T_pM)$. Then the submanifold $M$ is called generic submanifold if dim(${\cal D}_p$) is constant at each point $p\in M$ and ${\cal D}_p$ defines a differentiable distribution on $M$, called the holomorphic distribution.\\

We denote by $\cal D^\perp$ the orthogonal complementary distribution to $\cal D$ in $\Gamma(TM)$ and note that $J{\cal D}^\perp\cap{\cal D}^\perp=\{0\}.$ We call $\cal D^\perp$ purely real distribution on $M$.\\

For any $U\in \Gamma(TM$), we put $$JU=PU+FU, $$ where $PU$ and $FU$ are the tangential and normal parts of $JU$ respectively.\\

Let $\Gamma(T^\perp M)$ be the normal bundle of $M$ in $\Gamma(T\bar M)$. For any $N\in \Gamma(T^\perp M)$, we set $$JN=tN+fN,$$ where $tN$ and $fN$ are the tangential and normal parts of $JN$ respectively.\\

For generic submanifolds, we have\\
\noindent$(i)$ $P{\cal D}={\cal D},~~F{\cal D}=\{0\},$\\
\noindent$(ii)$ $P{\cal D}^\perp\subset{\cal D}^\perp,~~F{\cal D}^\perp\subset T^\perp M.$\\

Let $\mu$ be the differentiable vector subbundle of $T^\perp M$, then by [2] we have
$$T^\perp M=F{\cal D}^\perp\oplus\mu,~~t(T^\perp M)={\cal D}^\perp.~~\eqno(2.3)$$
{\noindent\bf {Note.}} Throughout this article we use the definition of generic submanifold which is given by B. Y. Chen [2].\\

For the theory of Riemannian submersion
we follow B. O'Niell [8];\\

Let $(M,g)$ and  $(B,g_B)$ be Riemannian manifolds, where dim$(M)=m$ and dim$(B)=n$ with $m>n.$
A Riemannian submersion $\pi:M\to B$ is a map from $M$ onto $B$ satisfying the following axioms;
\begin{enumerate}
  \item [$(S_1)$] $\pi$ has maximal rank.
  \item [$(S_2)$] The differential $\pi_*$ preserves the length of the  horizontal vectors.
\end{enumerate}

For each $q\in B,~\pi^{-1}(q)$ is an $(m-n)$ dimensional submanifold of $M$. The submanifolds
$\pi^{-1}(q),~q\in B$ are called fibers. Vector fields on $M$ which are in ($ker\pi_*$) are tangent to the fibres and are called vertical vector fields; vectors which are orthogonal to the vertical distribution (or orthogonal to
fibers) are said to be horizontal.\\


{\noindent\bf{Definition 2.1}} A smooth vector field $X$ on $M$ is called basic if\\
\noindent$(i)$ $X$ is horizontal and\\
$(ii)$ $X$ is $\pi$-related to a vector field $X_*$ on $B$ $, i.e.,$ $\pi_*X_p=X_{*\pi(p)}$ for all $p\in M.$\\

 We denote the projection morphisms on the distributions ($ker\pi_*$) and $(ker\pi_*)^\bot$
by ${\cal V}$ and $\mathcal{H}$ respectively. Then for any $U\in\Gamma(TM),$ we put$$U={\cal V}U+{\cal H}U.~~\eqno(2.4)$$

We recall the following lemma from O'Neill [8].\\

{\noindent\bf Lemma 2.1} Let $\pi:M\to B$ be a Riemannian submersion between Riemannian manifolds
and $X, ~Y$ be basic vector fields of $M.$ Then we have

\begin{enumerate}
  \item [$(a)$] $g(X,Y)=g_B(X_*,Y_*)o\pi,$
  \item [$(b)$] the horizontal part $\mathcal{H}{[X,Y]}$ of $[X,Y]$ is a basic vector field and corresponds
to $[X_*,Y_*],~i.e.,~\pi_*(\mathcal{H}[X,Y])=[X_*,Y_*].$
  \item [$(c)$] $[V,X]$ is vertical for any $V\in {\Gamma(ker\pi_*)}.$
  \item [$(d)$] $\mathcal{H}(\nabla_XY)$ is the basic vector field corresponding to $\nabla_{X_*}^BY_*,$ where
  $\nabla$ and $\nabla^B$ are the Levi-Civita connections of $g$ and $g_B$ respectively.
\end{enumerate}

The geometry of Riemannian submersions is characterized by O'Neill's configuration tensors $\mathcal{T}$ and $\mathcal{A}$ of Riemannian submersion $\pi:M\rightarrow B$
defined for arbitrary vector fields $E$ and $F$ on $M$ by
$$\mathcal{A}_EF=\mathcal{H}\nabla_{\mathcal{H}E}{\cal V} F+{\cal V}\nabla_{\mathcal{H}E}\mathcal{H}F,\eqno(2.5)$$
$$\mathcal{T}_EF=\mathcal{H}{\nabla_{{\cal V} E}{\cal V} F}+{\cal V}{\nabla_{{\cal V} E}}\mathcal{H}F.\eqno(2.6)$$

\noindent It is easy to see that a Riemannian submersion $\pi:M\to B$ has totally geodesic fibers if and only if
 $\mathcal{T}$ vanishes identically. For any $E\in \Gamma (TM),~{\cal T}_E$ and $\mathcal{A}_E$ are skew symmetric
 operators on $(\Gamma(TM),g)$ reversing the horizontal and the vertical distributions. It is also seen
 that $\mathcal{T}$ is vertical, $\mathcal{T}_E={\cal T}_{{\cal V} E}$ and $\mathcal{A}_E$ is horizontal, $\mathcal{A}_E
 =\mathcal{A}_{\mathcal{H}E}.$ We note that the tensor fields $\mathcal{T}$ and $\mathcal{A}$ satisfy
 $$\mathcal{T}_UW=\mathcal{T}_WU,~\forall~U,~W\in \Gamma (ker \pi_*)\eqno(2.7)$$
 $$\mathcal{A}_XY=-\mathcal{A}_YX=\frac{1}{2}{\cal V}[X,Y],~\forall~X,~Y\in \Gamma (ker\pi_*)^\bot.\eqno(2.8)$$

 On the other hand from (2.5) and (2.6) we have

 $$\nabla_VW={\cal T}_VW+\hat{\nabla}_VW~~\eqno(2.9)$$
 $$\nabla_VX=\mathcal{H}(\nabla_{ V} X)+\mathcal{T}_VX~~\eqno(2.10)$$
 $$\nabla_XV=\mathcal{A}_XV+{\cal V} \nabla_XV~~\eqno(2.11)$$
$$ \nabla_XY=\mathcal{H}(\nabla_XY)+\mathcal{A}_XY~~\eqno(2.12)$$

\noindent for any $X,~Y\in \Gamma (ker\pi_*)^\bot$ and $V,~W\in \Gamma (ker\pi_*),$ where $\hat{\nabla}_VW={\cal V} (\nabla_VW).$
If $X$ is basic, then $\mathcal{H}(\nabla_VX)=\mathcal{A}_XV.$\\

Next, we recall the following definitions;\\

{\noindent\bf Definition 2.1 [10]} A Riemannian submersion $\pi:(M,g,J)\to (B,g_B)$ is called a semi-invariant Riemannian submersion if there is a distribution  ${\cal D}_1\subset \Gamma(ker\pi_*)$ such that
$$(ker\pi_*)={\cal D}_1\oplus {\cal D}_2,~J{\cal D}_1={\cal D}_1,~J{\cal D}_2\subset (ker\pi_*)^\bot,$$ where ${\cal D}_2$ is the orthogonal complement of
${\cal D}_1$ in $\Gamma(ker\pi_*$).\\

{\noindent\bf Definition 2.2 [7]}  A Riemannian submersion $\pi:(M,g,J)\to (B,g_B)$ is called a slant submersion if there is a distribution ${\cal D}_1\subset \Gamma(ker\pi_*)$ such that
$$(ker\pi_*)={\cal D}_1\oplus {\cal D}_2,~J{\cal D}_1={\cal D}_1$$
and the angle $\theta=\theta(X)$ between $JX$ and the space $({\cal D}_2)_p$ is constant for any non zero $X\in({\cal D}_2)_p$ and $p\in M,$ where ${\cal D}_2$ is the orthogonal complement of ${\cal D}_1$ in $\Gamma(ker\pi_*).$ We call the angle $\theta$ a slant angle.\\

\noindent
Finally, we recall the notion of second fundamental form of a map between Riemannian manifolds.
Let $(M,g)$ and $(B,g_B)$ be Riemmanian manifolds and $\phi:M\to B$ be a smooth map between them. Then the differential $\phi_*$ of $\phi$ can be viewed as a section of the bundle Hom$((TM), \phi^{-1} (TB))\rightarrow M$, where $\phi^{-1}(TB)$ is the pullback bundle which has fibres $(\phi^{-1}(TB))_p=T_{\phi(p)}B,~p\in M. $ Hom $(TM, \phi^{-1} (TB))$ has a connection $\nabla$ induced from Levi-Civita connection $\nabla^M$ and the pull back connection. The second fundamental form $\phi$ is then given by
$$(\nabla \phi_*)(X,Y)=\nabla_X^\phi\phi_*(Y)-\phi_*(\nabla^M_XY),~~ \eqno (2.13)$$

\noindent for any $X,Y\in \Gamma (TM),$ where $\nabla^\phi$ is the pullback connection. It is known that the second fundamental form is symmetric. \\\\

\vspace{.2cm}
\parindent=0mm
%\begin{center}
{\bf{3. Generic Riemannian Submersions}}\\
%\end{center}

\parindent=8mm

In this section, we define generic Riemannian submersions from an almost Hermitian manifold onto a
Riemannian manifold which in fact generalizes both the semi-invariant and slant submersions, and investigate the integrability of the distributions and obtain
necessary and sufficient conditions for such submersions to be totally geodesic
map. We also obtain decomposition theorems for the total manifold of such submersions.\\

{\noindent\bf Definition 3.1} Let $M$ be a complex $m$-dimensional almost Hermitian
manifold with Hermitian metric $g$ and an almost complex structure $J$ and $B$ be a
Riemannian manifold with Riemannian metric $g_B.$ A Riemannian submersion $\pi:M\to B$ is
called a generic Riemannian submersion if there is a distribution ${\cal D}_1\subset \Gamma(ker\pi_*)$ such that
$$(ker\pi_*)={\cal D}_1\oplus {\cal D}_2,~J{\cal D}_1={\cal D}_1,$$
where ${\cal D}_2$ is the orthogonal complement of ${\cal D}_1$ in $\Gamma(ker\pi_*),$ and is purely real distribution on the fibres of the submersion $\pi.$\\

It is known that the distribution $(ker\pi_*)$ is integrable.
Hence above definition implies that the integral manifold (fiber) $\pi^{-1}(q),~q\in B$,
of $(ker\pi_*)$ is a generic submanifold of $M$. For generic submanifold we refer to [2].\\

For any $V\in \Gamma(ker \pi_*)$ we write
$$JV=\phi V+\omega V,~~\eqno(3.1)$$
where $\phi V\in \Gamma ({\cal D}_1)$ and $\omega V\in\Gamma (ker\pi_*)^\bot.$ We denote the complementary distribution to $\omega {\cal D}_2$ in $({\ker \pi_*})^\perp$ by $\mu$. Then we have $$(ker \pi_*)^\perp=\omega{\cal D}_2\oplus\mu,~~\eqno(3.2)$$ and that $\mu$ is invariant under $J$. Thus, for any $X\in\Gamma(ker \pi_*)^\perp$ we have
$$JX=BX+CX,~~\eqno(3.3)$$ \noindent where $BX\in \Gamma({\cal D}_2)$ and $CX\in \Gamma(\mu).$\\

From (3.1), (3.2) and (3.3) we have\\
{\noindent\bf{Lemma 3.1.}} For a generic submersion $\pi:M\rightarrow B$, we have\\
$(i)$~~$\phi {\cal D}_1={\cal D}_1,~\omega {\cal D}_1=0,~\phi {\cal D}_2\subset {\cal D}_2,~B(ker\pi_*)^\perp={\cal D}_2,$\\
$(ii)$~~$\phi^2+B\omega=-id,~C^2+\omega B=-id,$\\
$(iii)$~~$\omega\phi+C\omega=0,~BC+\phi B=0.$\\

We define the covariant derivative of $\phi$ and $\omega$ as follows;$$(\nabla_V\phi)W=\hat{\nabla}_V\phi W-\phi \hat{\nabla}_VW$$$$
(\nabla_V\omega)W={\cal H}(\nabla_V\omega W)-\omega\hat\nabla_V W.$$

\noindent Then by using (2.9), (2.10), (3.1) and (3.3), we get
$$(\nabla_V\phi)W=B{\cal T}_VW-{\cal T}_V\omega W,$$
$$(\nabla_V\omega)W=C{\cal T}_VW-{\cal T}_V\phi W,$$ \noindent for any $V,~W\in\Gamma (ker \pi_*).$\\


\noindent Next, we have the following lemma;\\

{\noindent\bf{Lemma 3.2.}} Let $\pi$ be generic Riemannian submersion from a Kaehler manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then
\begin{itemize}
  \item [$(i)$] ${\cal T}_V\phi W+{\cal A}_{\omega W}V=C{\cal T}_VW+\omega\hat\nabla_VW$\\
$ \mbox{and} ~\hat\nabla_V\phi W+{\cal T}_V\omega W=B{\cal T}_VW+\phi\hat\nabla_VW$
  \item  [$(ii)$] ${\cal A}_XBY+{\cal H}(\nabla_XCY)=C({\cal H}(\nabla_XY))+\omega{\cal A}_XY$\\
$\mbox{and}~{\cal V}(\nabla_XBY)+{\cal A}_XCY=B({\cal H}(\nabla_XY))+\phi{\cal A}_XY,$
  \item [$(iii)$] ${\cal A}_X\phi V+{\cal H}(\nabla_V\omega W)=C{\cal A}_XV+\omega({\cal V}(\nabla_XV))$\\
$\mbox{and}~{\cal V}(\nabla_X\phi V)+{\cal A}_X\omega V=B{\cal A}_XV+\phi({\cal V}(\nabla_XV)),$
\end{itemize}

\noindent for any $X,~Y\in \Gamma (ker\pi_*)^\perp$ and $V,~W\in \Gamma (ker \pi_*).$\\

{\noindent\bf{Proof (i).}} Since $M$ is a Kaehler manifold, then for any $V,~W\in \Gamma (ker \pi_*)$ using (2.2) and (3.1) we have
$$\begin{array}{lll} J\nabla_VW&=&\nabla_VJW\vspace{.3cm}\\
&=&\nabla_V\phi W+\nabla_V\omega W.\\
\end{array}$$
\noindent Further, on using (2.9) and (2.10) we get
$${\cal T}_V\phi W+\hat\nabla_V\phi W+{\cal A}_{\omega W}V+{\cal T}_V\omega W=J({\cal T}_VW+\hat\nabla_VW).$$
Since ${\cal T}_VW$ and $\hat\nabla_VW$ are the horizontal and vertical, therefore again using (3.1) and (3.3), we get$${\cal T}_V\phi W+\hat\nabla_V\phi W+{\cal A}_{\omega W}V+{\cal T}_V\omega W= B{\cal T}_VW+C{\cal T}_VW+\phi\hat\nabla_VW+\omega\hat\nabla_VW~~\eqno(3.4)$$
By comparing the vertical and horizontal parts in (3.4), we get the result.\\

\noindent Proof of $(ii)$ and $(iii)$ follows on the similar lines as in $(i)$.\\

{\noindent\bf{Lemma 3.3.}} Let $\pi:M\to B$ be a generic Riemannian submersion from a Kaehler manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then
$$g(J{\cal T}_UV,\xi)=g({\cal T}_UJV,\xi),$$ for any  $U\in\Gamma(ker\pi_*),$ $V\in\Gamma({\cal D}_1)$ and $\xi\in \Gamma(\mu).$\\

{\noindent\bf{Proof.}} Since $M$ is a Kaehler manifold, then for any $V\in\Gamma({\cal D}_1)$ and $U\in\Gamma(ker\pi_*)$ using (2.2) we have$$J\nabla_UV=\nabla_UJV.$$
\noindent On using (2.9) we get$$J({\cal T}_UV+\hat\nabla_UV)={\cal T}_UJV+\hat\nabla_UJV.$$
\noindent Taking inner product with $\xi\in\Gamma(\mu)$, we get
$$g(J{\cal T}_UV,\xi)+g(J\hat\nabla_UV,\xi)=g({\cal T}_UJV,\xi)+g(\hat\nabla_UJV,\xi).~~\eqno(3.5)$$
\noindent Since $\mu$ is invariant under $J$, then the result follows from (3.5).\\

Now, we investigate the integrability of the distributions ${\cal D}_1$ and $ {\cal D}_2.$ Since we have seen that the fibers
of generic submersions from Kaehler manifolds are generic Riemannian submanifolds and $\cal T$ is the
second fundamental form of the fibers, we have the following theorem;\\

{\noindent\bf Theorem 3.1.} Let $\pi$ be a generic Riemannian submersion from a Kaehler
manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then the distribution ${\cal D}_1$
is integrable if and only if$$g({\cal T}_VJW, \omega{U})=g({\cal T}_W JV,\omega{U}),~~\eqno(3.6)$$for any $V,~W\in \Gamma ({\cal D}_1)$ and ${U}\in \Gamma ({\cal D}_2).$\\

{\noindent\bf Proof.} Since  $M$ is a Kaehler manifold, then for any $V,~W\in\Gamma({\cal D}_1)$, (2.2) and (2.9) gives
$$\begin{array}{lll} J[V,W]&=&J\nabla_VW-J\nabla_WV\vspace{.3cm}\\
&=&\nabla_VJW-\nabla_WJV\vspace{.3cm}\\
&=&{\cal T}_VJW-{\cal T}_WJV+\hat\nabla_VJW-\hat\nabla_WJV\vspace{.3cm}\\
\end{array}$$

\noindent Therefore, $${\cal T}_VJW-{\cal T}_WJV=J[V,W]+\hat\nabla_WJV-\hat\nabla_VJW~~\eqno (3.7)$$
Now if ${\cal D}_1$ is integrable then $J[V,W]\in{\Gamma(\cal D}_1)$ as $[V,W]\in\Gamma({\cal D}_1)$. Hence in (3.7) right hand side is vertical while the left hand side is horizontal.
 \noindent On comparing the horizontal and vertical part we get $${\cal T}_VJW={\cal T}_WJV,$$ for any $V,~W\in\Gamma({\cal D}_1)$. In particular, we get$$g({\cal T}_VJW,\omega U)=g({\cal T}_WJV,\omega U).$$

Conversely, suppose if (3.7) holds, i.e.,$$g({\cal T}_VJW-{\cal T}_WJV,\omega U)=0$$\noindent which shows that $${\cal T}_VJW-{\cal T}_WJV~\in \Gamma(\mu).$$

Now for any $\xi\in\Gamma( \mu)$, using the Lemma 3.3 and (2.7) we have$$g({\cal T}_VJW-{\cal T}_WJV,\xi)=g(J{\cal T}_VW-J{\cal T}_WV, \xi)=0,$$ which implies that ${\cal T}_VJW-{\cal T}_WJV=0,$ for any $V,~W\in{\cal D}_1$.\\

Thus from (3.7), we get$$J[V,W]=\hat\nabla_VJW-\hat\nabla_WJV.$$Since $\hat\nabla_VJW-\hat\nabla_WJV$ lies in $\Gamma(ker \pi_*)$, this implies that $[V,W]$ lies in $\Gamma({\cal D}_1)$ and hence ${\cal D}_1$ is integrable. \\

{\noindent\bf Theorem 3.2.} Let $\pi$ be a generic Riemannian submersion from a Kaehler
manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then the distribution ${\cal D}_2$
is integrable if and only if
$$\hat{\nabla}_W\phi V-\hat{\nabla}_V\phi \omega+{\cal T}_W\omega  V-{\cal T}_V\omega W\in \Gamma ({\cal D}_2),~~\eqno(3.8)$$
for any $V,~W\in \Gamma({\cal D}_2).$\\

{\noindent\bf Proof.} For any $V,~W\in \gamma({\cal D}_2),$ using (2.2), (2.9), (2.10), (3.1) and (3.3) we have
$$\begin{array}{lll}~~~~~~~~~~~\nabla_VW&=&-J\nabla_VJW\vspace{.3cm}\\
&=&-J(\nabla_V\phi W+\nabla_V\omega W)\vspace{.3cm}\\
&=&-J({\cal T}_V\phi W+\hat{\nabla}_V\phi W+\mathcal{A}_{\omega W}V+{\cal T}_V\omega W)\vspace{.3cm}\\
&=&-(B{\cal T}_V\phi W+C{\cal T}_V\phi W+\phi\hat{\nabla}_V\phi W+\omega\hat{\nabla}_V\phi W\vspace{.3cm}\\
&+&B\mathcal{A}_{\omega W}V+C\mathcal{A}_{\omega W}V+\phi {\cal T}_V\omega W+\omega {\cal T}_V \omega W).\vspace{.3cm}~~~~~~~~~~~~~~~~~~~~~~(3.9)\\
\end{array}$$
\noindent\text{Similarly, we get}
$$~~~~~~~~~~~~~~~~~\begin{array}{lll}\nabla_WV&=&-(B{\cal T}_W\phi V+C{\cal T}_W\phi V+\phi\hat{\nabla}_W \phi V+\omega\hat{\nabla}_W\phi V\vspace{.3cm}\\
&+&B\mathcal{A}_{\omega V}W+C\mathcal{A}_{\omega V}W+\phi {\cal T}_W\omega V+\omega {\cal T}_W \omega V).\vspace{.3cm}~~~~~~~~~~~~~~~~~~~~~~(3.10)\\
\end{array}$$
\text{From (3.9) and (3.10), we get}
$$\begin{array}{lll}[V,W]&=&B({\cal T}_W\phi V-{\cal T}_V\phi W+\mathcal{A}_{\omega V}W-A_{\omega W}V)\vspace{.3cm}\\
&+&C({\cal T}_W\phi V-{\cal T}_V\phi W+\mathcal{A}_{\omega V}W-A_{\omega W}V)\vspace{.3cm}\\
&+&\phi(\hat{\nabla}_W\phi V-\hat{\nabla}_V\phi W+{\cal T}_W\omega V-{\cal T}_V\omega W)\vspace{.3cm}\\
&+&\omega(\hat{\nabla}_W\phi V-\hat{\nabla}_V\phi W+{\cal T}_W\omega V-{\cal T}_V\omega W),\vspace{.3cm}~~~~~~~~~~~~~~~\\
\end{array}$$
for any $V,~W\in \Gamma({\cal D}_1)\subset \Gamma(ker\pi_*)$. As ($ker\pi_*$) is integrable therefore $[V,W]\in\Gamma(ker \pi_*),$ comparing the vertical part, we get
$$[V,W]=B({\cal T}_W\phi V-{\cal T}_V\phi W+\mathcal{A}_{\omega V}W-A_{\omega W}V)
+\phi(\hat{\nabla}_W\phi V-\hat{\nabla}_V\phi W+{\cal T}_W\omega V-{\cal T}_V\omega W).\eqno(3.11)$$From (3.11) it follows that the distribution ${\cal D}_2$ is integrable if and only if
$$\hat{\nabla}_W\phi V-\hat{\nabla}_V\phi W+{\cal T}_W\omega V-{\cal T}_V\omega W\in \Gamma ({\cal D}_2),$$
for any $V,~W\in \Gamma({\cal D}_2).$\\

For the geometry of the leaves of the distributions ${\cal D}_1$ and ${\cal D}_2$ we have the following propositions;\\

{\noindent\bf Proposition 3.1.}  Let $\pi$ be a generic Riemannian submersion from a Kaehler
manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then the distribution ${\cal D}_1$
defines a totally geodesic foliation if and only if $$\hat{\nabla}_{V_1}\phi {W_1}\in\Gamma ({\cal D}_1)
~\text{and }~{\cal T}_{V_1}\phi {W_1}=0,$$ for any ${V_1},~{W_1}\in \Gamma ({\cal D}_1).$\\

{\noindent\bf Proof.} For ${V_1}, ~{W_1}\in \Gamma ({\cal D}_1),$ using (2.2), (2.9), (3.1) and (3.3) we have
\begin{align*}
~~~~~~~~~~~~~\nabla_{V_1}{W_1}&=-J\nabla_{V_1}J{W_1}\\
&=-J(\nabla_{V_1}\phi {W_1})\\
&=-J({\cal T}_{V_1}\phi {W_1}+\hat{\nabla}_{V_1}\phi {W_1})\\
&=-(B{\cal T}_{V_1}\phi {W_1}+C{\cal T}_{V_1}\phi {W_1}+\phi \hat{\nabla}_{V_1}\phi {W_1}+\omega\hat{\nabla}_{V_1}\phi {W_1})
\end{align*}

Hence $\nabla_{V_1}{W_1}\in\Gamma({\cal D}_1)$ if and only if $\hat{\nabla}_{V_1}\phi {W_1}\in\Gamma ({\cal D}_1)
~\text{and }~{\cal T}_{V_1}\phi {W_1}=0,$ which completes the proof.

{\noindent\bf Proposition 3.2.}  Let $\pi$ be a generic Riemannian submersion from a Kaehler
manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then the distribution ${\cal D}_2$ defines a totally geodesic foliation if and only if
$$T_{V_2}\phi W_2+\mathcal{A}_{\omega {W_2}}{V_2}\in \Gamma (\omega{\cal D}_2)~ \text{and}~\hat{\nabla}_{V_2}\phi {W_2}+{\cal T}_{V_2}\omega {W_2}=0$$
for any ${V_2},~{W_2}\in \Gamma ({\cal D}_2).$\\

{\noindent\bf Proof.} The proof follows from (3.9).\\

From Proposition 3.1 and Proposition 3.2, we have the following theorem;\\

{\noindent\bf Theorem 3.3.} Let $\pi$ be a generic Riemannian submersion from a Kaehler
manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then the fibers of $\pi$ are the locally Riemannian product of leaves of ${\cal D}_1$ and ${\cal D}_2$ if and only if $$\hat{\nabla}_{V_1}\phi {W_1}\in\Gamma ({\cal D}_1),~ T_{V_1}\phi {W_1}=0,~\mbox{ and}$$ $${\cal T}_{V_2}\phi {W_2}+\mathcal{A}_{\omega {W_2}}{V_2}\in
\Gamma( \omega{\cal D}_2),~\hat{\nabla}_{V_2}\phi {W_2}+{\cal T}_{V_2}\omega {W_2}=0,$$

\noindent for any ${V_1},~{W_1}\in \Gamma ({\cal D}_1)$
${V_2},~{W_2}\in \Gamma ({\cal D}_2).$\\

Now we discuss the geometry of the leaves of $(ker\pi_*)$ and $(ker\pi_*)^\bot.$\\

{\noindent\bf Proposition 3.3.} Let $\pi:M\to B$ be generic Riemannian submersion from a Kaehler
manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then the distribution
$(ker\pi_*)^\bot$ defines a totally geodesic foliation if and only if
$$\mathcal{A}_XBY+\mathcal{H}\nabla_XCY\in \Gamma (\mu)~~\text{and}~~{\cal V} \nabla_X BY+\mathcal{A}_XCY=0,$$
for any $X,~Y\in \Gamma(ker\pi_*)^\bot.$\\

{\noindent\bf Proof.} For any $X,Y\in \Gamma(ker\pi_*)^\bot,$ from (2.1) we have
$$\nabla_XY=-J\nabla_XJY$$
Then by using (3.3), (2.11) and (2.12), we get
\begin{align*}
~~~~~~~~\nabla_XY&=-J(\nabla_XBY+\nabla_XCY)\\
&=-J((\mathcal{A}_XBY+{\cal V}\nabla_XBY)+(\mathcal{H}\nabla_XCY+\mathcal{A}_XCY))\\
&=-(B\mathcal{A}_XBY+C\mathcal{A}_XBY+\phi({\cal V} \nabla_XBY)+\omega({\cal V}\nabla_XBY)\\
&~~~~+B(\mathcal{H}\nabla_XCY)+C(\mathcal{H}\nabla_XCY)
+\phi(\mathcal{A}_XCY)+\omega(\mathcal{A}_XCY))~~~~~(3.12)
\end{align*}
From (3.12) it follows that $(ker\pi_*)^\bot$ defines a totally geodesic foliation
if and only if
$$B(\mathcal{A}_XBY+\mathcal{H}\nabla_XCY)+\phi({\cal V} \nabla_XBY+\mathcal{A}_XCY)=0.$$
Which then yields
\begin{align*}
B(\mathcal{A}_XBY+\mathcal{H}\nabla_XCY)&=0\\
\phi({\cal V} \nabla_XBY+\mathcal{A}_XCY)&=0.
\end{align*}
Hence the result.\\

For the distribution $(ker\pi_*)$, we have\\

{\noindent\bf Proposition 3.4.} Let $\pi$ be a generic Riemannian submersion from a Kaehler
manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B).$ Then the distribution $(ker\pi_*)$
defines a totally geodesic foliation if and only if
$$T_V\phi W+\mathcal{A}_{\omega W}V\in \Gamma (\omega {\cal D}_2)~~\text{and}~~
\hat{\nabla}_V\phi W+{\cal T}_V\omega W\in \Gamma ({\cal D}_1),$$
for any $V,W\in \Gamma (ker\pi_*).$\\

{\noindent\bf Proof.} For any $V,~W\in \Gamma (ker\pi_*)$ using (2.2), (2.9), (2.10) and (3.1) we get
\begin{align*}
\nabla_VW&=-J\nabla_VJW\\
&=-J(\nabla_V\phi\omega+\nabla_V\omega W)\\
&=-J({\cal T}_V\phi W+\hat{\nabla}_V\phi W+\mathcal{A}_{\omega W}V+{\cal T}_V\omega W)\\
&=-(B{\cal T}_V\phi W+C{\cal T}_V\phi W+\phi\hat{\nabla}_V\phi W+\omega\hat{\nabla}_V\phi W\\
&~~~+B\mathcal{A}_{\omega W}V+C\mathcal{A}_{\omega W}V+\phi{\cal T}_V\omega W+\omega{\cal T}_V\omega W)
\end{align*}
\begin{align*}
~~~~~~~~~~~~~~~~~~~\nabla_VW&=-B({\cal T}_V\phi W+\mathcal{A}_{\omega W}V)-\phi(\hat{\nabla}_V\phi W+{\cal V} \nabla_V\omega W)\\
&~~~-C({\cal T}_V\phi W+\mathcal{A}_{\omega W}V)-\omega(\hat{\nabla}_V\phi W+{\cal V} \nabla_V\omega W)~~~~~~~~~(3.13)
\end{align*}

\noindent From above equation, it follows  that $(ker\pi_*)$ defines a totally geodesic foliation if and only if
$$C({\cal T}_V\phi W+\mathcal{A}_{\omega W}V)+\omega(\hat{\nabla}_V\phi W+{\cal V} \nabla_V\omega W)=0.$$
which implies
$${\cal T}_V\phi W+\mathcal{A}_{\omega W}V\in \Gamma (\omega {\cal D}_2)~\text{and}~
\hat{\nabla}_V\phi W+{\cal V} \nabla_V\omega W\in \Gamma ({\cal D}_1).$$

From Theorem 3.3 and Proposition 3.3, we have the following decomposition for total space;\\

{\noindent\bf Theorem 3.4.}  Let $\pi$ be a generic Riemannian submersion from a Kaehler
manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then the total space
$M$ is a generic product manifold of the leaves of ${\cal D}_1,~{\cal D}_2$ and $(ker\pi_*)^\bot,~i.e.,$
$M=M_{{\cal D}_1}\times M_{{\cal D}_2}\times M_{(ker\pi_*)^\bot}$, if and only if
$$\hat{\nabla}_{V_1}\phi {W_1}\in\Gamma ({\cal D}_1),~ T_{V_1}\phi {W_1}=0,$$  $${\cal T}_{V_2}\phi {W_2}+\mathcal{A}_{\omega {W_2}}{V_2}\in
\Gamma( \omega{\cal D}_2),~\hat{\nabla}_{V_2}\phi {W_2}+{\cal T}_{V_2}\omega {W_2}=0 ~\mbox{and}~$$
$$\mathcal{A}_XBY+\mathcal{H}\nabla_XCY\in \Gamma (\mu),~{\cal V} \nabla_X BY+\mathcal{A}_XCY=0,$$
for any ${V_1},~{W_1}\in \Gamma ({\cal D}_1)$, ${V_2},~{W_2}\in \Gamma ({\cal D}_2),$
and $X,~Y\in \Gamma(ker\pi_*)^\bot,$  where $M_{{\cal D}_1},~M_{{\cal D}_2}$ and $M_{(ker\pi_*)^\bot}$
are leaves of the distributions ${\cal D}_1,~{\cal D}_2$ and $(ker\pi_*)^\bot$ respectively.\\

From Proposition 3.3 and Proposition 3.4, we have \\

{\noindent\bf Theorem 3.5.}  Let $\pi$ be a generic Riemannian submersion from a Kaehler
manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then $M$ is generic product manifold
if and only if
$$\mathcal{A}_XBY+\mathcal{H}\nabla_XCY\in \Gamma (\mu),~{\cal V} \nabla_X BY+\mathcal{A}_XCY=0~\mbox{and}$$
$${\cal T}_V\phi W+\mathcal{A}_{\omega W}V\in \Gamma (\omega {\cal D}_2),~
\hat{\nabla}_V\phi W+{\cal V} \nabla_V\omega W\in \Gamma ({\cal D}_1),$$
for any $X,~Y\in \Gamma(ker\pi_*)^\bot$ and $V,~W\in \Gamma (ker\pi_*).$\\

Now we obtain necessary and sufficient condition for generic Riemannian
submersion to be totally geodesic. We recall that a differential map $\pi$
between Riemannian manifolds $(M,g)$ and $(B,_B)$ is called totally geodesic map if
$$(\nabla \pi_*)(X,Y)=0,~\text{for all}~X,~Y\in \Gamma (TM).$$

{\noindent\bf Theorem 3.6.}  Let $\pi$ be a generic Riemannian submersion from a Kaehler
manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$. Then $\pi$ is a totally
geodesic map if and only if
\begin{align*}
\hat{\nabla}_V\phi W+{\cal T}_V\omega W&\in \Gamma {(\cal D}_1),\\
{\cal T}_V\phi W+\mathcal{A}_{\omega W}V&\in \Gamma(\omega{\cal D}_2),\\
\hat{\nabla}_VBX+{\cal T}_VCX&\in \Gamma ({\cal D}_1)~\text{and}\\
{\cal T}_VBX+\mathcal{A}_{CX}V&\in \Gamma({\omega{\cal D}_2}),
\end{align*}
for any $X\in \Gamma(ker\pi_*)^\bot$ and $V, ~W \in \Gamma(ker\pi_*).$\\

{\noindent\bf Proof.} Since $\pi$ is a Riemannian submersion, we have
$$(\nabla \pi_*)(X,Y)=0,~\text{for all}~ X,~Y\in\Gamma (ker\pi_*)^\bot.~~\eqno(3.14)$$

For any $V,~W\in \Gamma(ker\pi_*)$ using (2.2), (2.9), (2.10), (2.13) and (3.1), we get
\begin{align*}
(\nabla \pi_*)(V,W)&=-\pi_*(\nabla_VW)\\
&=-\pi_*(-J\nabla_VJW)\\
&=\pi_*(J\nabla_V(\phi W+\omega W))\\
&=\pi_*(J({\cal T}_V\phi W+\hat{\nabla}_V\phi W)+J(\mathcal{A}_{\omega W}V+{\cal T}_V\omega W))\\
&=\pi_*((B{\cal T}_V\phi W+C{\cal T}_V\phi W)+(\phi\hat{\nabla}_V\phi W+\omega\hat{\nabla}_V\phi W)\\
&+(B\mathcal{A}_{\omega W}V+C\mathcal{A}_{\omega W}V)+(\phi {\cal T}_V\omega W+\omega {\cal T}_V\omega W)).
\end{align*}

Thus $(\nabla\pi_*)(V,W)=0$ if and only if
$$\omega(\hat{\nabla}_V\phi W+{\cal T}_V\omega W)+C({\cal T}_V\phi W+\mathcal{A}_{\omega W}V)=0~~\eqno(3.15)$$

On the other hand using (2.2), (2.9),
(2.10) and (3.3) for any $X\in \Gamma(ker\pi_*)^\bot$ and $V\in \Gamma(ker\pi_*)$, we get
\begin{align*}
(\nabla \pi_*)(V,X)&=-\pi_*(\nabla_VX)\\
&=-\pi_*(-J\nabla_VJX)\\
&=\pi_*(J(\nabla_VBX+\nabla_VCX))\\
&=\pi_*((B{\cal T}_VBX+C{\cal T}_VBX)+(\phi\hat{\nabla}_VBX+\omega\hat{\nabla}_VBX)\\
&~~~+(B\mathcal{A}_{CX}V+C\mathcal{A}_{CX}V)+(\phi {\cal T}_VCX+\omega {\cal T}_VCX))
\end{align*}

Thus $(\nabla\pi_*)(V,X)=0$ if and only if
$$\omega(\hat{\nabla}_VBX+{\cal T}_VCX)+C({\cal T}_VBX+\mathcal{A}_{CX}V)=0.~~\eqno(3.16)$$
The result then follows from (3.14) and (3.15) and (3.16).\\

Now we recall that a Riemannian submersion from a Riemannian manifolds $(M,g)$ onto a Riemannian manifold $(B,g_B)$ is called a Riemannian submersion with totally umbilical fibres if$${\cal T}_VW=g(V,W)H,~~\eqno(3.17)$$for $V,~W\in\Gamma(ker \pi_*),$ where $H$ is mean curvature vector field of the fibres [10].\\

We then have\\

{\noindent\bf{Proposition 3.5.}} Let $\pi$ be a generic Riemannian subersion with totally umbilical fibres from a Kaehler manifold $(M,g,J)$ onto a Riemannian manifold $(B,g_B)$, then $H\in\Gamma (\omega{\cal D}_2)$.\\

{\noindent\bf{Proof.}} From (2.2), we have
$$~\nabla_VJW=J\nabla_VW,$$for~any~$V,~W\in \Gamma({\cal D}_1).$ Now using (3.1) and (3.3) we obtain$$\begin{array}{lll}{\cal T}_VJW+\hat\nabla_VJW&=&J({\cal T}_VW+\hat\nabla_VW) \vspace{.3cm}\\
&=&B{\cal T}_VW+C{\cal T}_VW+\phi\hat\nabla_VW+\omega\hat\nabla_VW.\vspace{.3cm}\\
\end{array}\eqno(3.18)$$Taking inner product in (3.18) with $X~\in \Gamma(\mu)$ and then using (3.17) we get
$$g({\cal T}_VJW,X)=g(C{\cal T}_VW,X)~~~~~~~~$$$$g(V,JW)g(H,X)=g(J{\cal T}_VW,X),~~~~~~~~~~~~~~$$$$g(V,JW)g(H,X)=g(J{\cal T}_VW,JX),~~~~~~~~~~~~$$$$g(V,JW)g(H,X)=-g(V,W)g(H,JX)~~~~~~\eqno(3.19)$$
Interchanging $V$ and $W$ in (3.19), we get$$g(W,JV)g(H,X)=-g(V,W)g(H,JX)~~~~~~~\eqno(3.20)$$
Combining (3.19) and (3.20) we get $g(H,JX)=0$ which shows that $H~\in \Gamma(\omega{\cal D}_2).$\\\\












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