meteorology.tex

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\begin{document}


%{\large

\begin{center}
{\bf Chen invariants for Riemannian submersions and their applications in meteorology}

\bigskip

Mehmet G\"{U}LBAHAR, \c{S}emsi EKEN MER\.{I}\c{C} and Erol KILI\c{C}
\end{center}

\bigskip

\begin{quote}
\noindent {\bf Abstract.}
 In this paper,  an optimal inequality
involving the delta curvature
 is exposed.
 An application of Riemannian submersions dealing meteorology
 is presented. Some characterizations about the vertical
motion and the horizontal divergence are obtained.

\end{quote}

{\rm \ }

\begin{quote}
\medskip

\noindent {\em AMS 2000 Mathematics Subject Classification\/}:
53C12, 53C40, 86A10.

\noindent {\em Keywords\/}: Curvature,  Riemannian submersion, Vertical motion, Horizontal Divergence.
\end{quote}

\section{Introduction\label{sect-intro}}

In 1993, B.-Y. Chen \cite{C:Arch} initially introduced a new invariant the so-called delta curvature $\delta$
 for an $n$-dimensional Riemannian manifold
$M$ by
\begin{equation}
\delta^{k}(p)=\tau (p)-\left( {\rm inf}\tau \left( \Pi _{k}\right)
\right) \left( p\right) ,  \label{eq-k-Chen-inv}
\end{equation}
where  $2\leq k\leq n-1$, $\tau (p)$ is the scalar curvature at $p\in M$ and
\[
\left( {\rm inf}\tau \left( \Pi _{k}\right) \right) \left( p\right) ={\rm inf%
}\{\tau (\Pi _{k})\,|\,\Pi _{k}\ {\rm is\ a}\ k\mbox{-}{\rm plane\ section}%
\subset T_{p}M\}.
\]
Furthermore, he gave a relation involving the delta curvature, the main
intrinsic and extrinsic
invariants of submanifolds
in a real space form (cf. Lemma 3.2 in \cite{C:Arch}). Then, this curvature drew attention of
many authors and the notion of discovering simple basic relationships
between intrinsic and extrinsic invariants of a submanifold becomes one
of the most fundamental problems in submanifold theory  (cf. \cite{Alegre}, \cite{Chen:95}, \cite{Chen:07}, \cite{Chen:11}, \cite{Chen:Dil:05},
\cite{Gulbahar:1}, \cite{Hong:05}, \cite{Poyraz}, \cite{Vilcu:2013}, etc.).




\medskip

Apart from isometric immersions and submanifolds theory, Riemannian
submersions have played a substantial role in differential geometry
since this frame of maps also makes possible to compare geometrical
properties between smooth manifolds. Besides the mathematical significance, Riemannian submersions
have important physical and engineering aspects. There exist very
nice applications of these mappings in the Kaluza-Klein theory
\cite{Fal:03,Kennedy,Wang}, in the statical machine learning process  \cite{Zhao}, in the medical
imaging cf. \cite{Memoli}, in the statical analysis  \cite{Bha}, in the robotic theory \cite{A,sahin}.



\medskip


Motivated by these facts, we firstly establish an optimal inequality involving
the delta curvature for Riemannian manifolds admitting a Riemannian submersion.
Then, we investigate this inequality for some special cases. Finally, we discuss this inequality
in meteorology and obtain some results dealing the vertical motion and horizontal divergence.




\section{Riemannian submersions \label{sect-prel}}

Let $(M,g)$ and $(B,\widetilde{g})$ be $m$ and $n$ dimensional Riemannian manifolds
 with Riemannian metrics $g$ and $\widetilde{g}$, respectively.
A smooth map $\pi:(M,g)\rightarrow (B,\widetilde{g})$
is called a \textit{Riemannian submersion}
if
\begin{enumerate}
\item[\textit{i)}] $\pi$ has maximal rank.

\item[\textit{ii)}] The differential $\pi_{*}$ preserves the lengths of horizontal vectors.
\end{enumerate}

Now, let $\pi:(M,g)\rightarrow (B,\widetilde{g})$ be a Riemannian submersion.
For any $b\in B$, $\pi^{-1}(b)$ is closed $r$-dimensional submanifold of $M$.
The submanifolds $\pi^{-1}(b)$ are called \textit{fibers}.
A vector field tangent to fibers is called \textit{vertical} and
a vector field orthogonal to fibers is called  \textit{horizontal}. If we
put
 \begin{eqnarray}
 \mathcal{V}_{p}=\textrm{kernel}(\pi_{*})
 \end{eqnarray}
at a point $p\in M$, then it can be obtained an
integrable distribution $\mathcal{V}$
corresponding to the foliation of $M$ determined by the fibres of $\pi$.
The distribution $\mathcal{V}_{p}$ is called \textit{vertical space} at $p\in M$.

\medskip

Let $\mathcal{H}$ be a complementary distribution of $\mathcal{V}$ determined by the
Riemannian metric $g$. For any $p\in M$, the distribution $\mathcal{H}$$_{p}=%
(\mathcal{V}$$_{p})^{\perp}$
is called \textit{horizontal space} on $M$ \cite{Koba:87}. Thus, we have the following orthogonal decomposition:
\begin{eqnarray}
TM={\mathcal{V}}\oplus {\mathcal{H}}. \label{Gauss}
\end{eqnarray}



A vector field $E$ on $M$ is called \textit{basic} if it is horizontal and $\pi-$ related
to a vector field $E_{*}$ on $B$ i.e., $\pi_{*}E_{p}=E_{*\pi(p)}$ for all $p\in M$.
Furthermore, it is known that if $E$ and $F$ are the basic vector fields respectively $\pi-$related to $E_{*}$ and $F_{*}$, one has
\begin{eqnarray}
g(E,F) & = & \widetilde{g}(E_{*},F_{*})\circ \pi.
\end{eqnarray}

Let $h$ and $v$ are  the projections of $\Gamma(TM)$ onto $\Gamma({\mathcal{H}})$
and  $\Gamma({\mathcal{V}})$, respectively.
The \textit{fundamental tensor fields} of $\pi$, denoted  by $A$ and $T$,
are defined respectively by
\begin{eqnarray}
A_{E}F=h\nabla _{hE}vF+v\nabla _{hE}hF,  \label{oneil:1} \\
T_{E}F=h\nabla _{vE}vF+v\nabla _{vE}hF  \label{oneil:2}
\end{eqnarray}
for any $E,F\in \Gamma(TM)$, where $\nabla$ is the Levi-Civita connection on $M$.

\medskip

Now, let us define the following mappings:
\begin{eqnarray*}
T^{\mathcal{H}}:\Gamma({\mathcal{V}} )\times \Gamma({\mathcal{V}} )&%
\rightarrow &\Gamma({\mathcal{H}} ),\\
(U,V)&\rightarrow &T^{\mathcal{H}}(U,V)=h\nabla _{U}V,  \label{op:1}
\end{eqnarray*}
\begin{eqnarray*}
T^{\mathcal{V}}:\Gamma({\mathcal{V}} )\times \Gamma({\mathcal{H}} )&%
\rightarrow &\Gamma({\mathcal{V}} ),\\
(U,X)&\rightarrow &T^{\mathcal{V}}(U,X)=v\nabla _{U}X,  \label{op:2}
\end{eqnarray*}
and
\begin{eqnarray*}
A^{\mathcal{H}}:\Gamma({\mathcal{H}} )\times \Gamma({\mathcal{V}} )&\rightarrow %
&\Gamma({\mathcal{H}} ), \\
(X,U)&\rightarrow &A^{\mathcal{H}}(X,U)=h\nabla _{X}U,  \label{op:3}
\end{eqnarray*}
\begin{eqnarray*}
A^{\mathcal{V}}:\Gamma({\mathcal{H}} )\times \Gamma({\mathcal{H}} )&%
\rightarrow &\Gamma({\mathcal{V}}), \\
(X,Y)&\rightarrow &A^{\mathcal{V}}(X,Y)=v\nabla _{X}Y,  \label{op:4}
\end{eqnarray*}
Then, it is clear from $(\ref{oneil:1})$ and $(\ref{oneil:2})$ that $T^{\mathcal{H}}$ is a symmetric operator on
$\Gamma({\mathcal{V}} )\times\Gamma({\mathcal{V}} )$ and $A^{\mathcal{V}}$ is an
anti-symmetric operator on $\Gamma({\mathcal{H}} )\times\Gamma({\mathcal{H}} )$.
If  $(\ref{oneil:1})$ and $(\ref{oneil:2})$ are taken into account in $(\ref{Gauss})$, we can write
\begin{eqnarray}
&&\nabla _{U}V=T^{\mathcal{H}}(U,V)+v\nabla _{U}V,  \label{Gauss:1}
\\
&&\nabla _{V}X=h\nabla _{V}X+T^{\mathcal{V}}(U,X),  \label{Gauss:2} \\
&&\nabla _{X}U=A^{\mathcal{H}}(X,U)+v\nabla _{X}U,  \label{Gauss:3} \\
&&\nabla _{X}Y=h\nabla _{X}Y+A^{\mathcal{V}}(X,Y)  \label{Gauss:4}
\end{eqnarray}
for any $U,V\in \Gamma({\mathcal{V}})$ and $X,Y\in \Gamma({\mathcal{H}})$.


\medskip




\medskip

Let $\{U_{1},\ldots,U_{r},X_{1},\ldots,X_{n}\}$ be an orthonormal basis on $T_{p}M$,
where \\
${\mathcal{V}}=\textrm{Span}\{U_{1},\ldots,U_{r}\}$ and ${\mathcal{H}}
=\textrm{Span}\{X_{1},\ldots,X_{n}\}$. The mean curvature vector field $\hbar(p)$ of any
fibre is defined by
\begin{eqnarray}
\mathcal{\hbar}(p)=\frac{1}{r}\sum_{j=1}^{r}T^{\mathcal{H}}(U_{j},U_{j}). \label{mean:curv}
\end{eqnarray}
Note that each fiber is a minimal submanifold of $M$ if and only if
$\hbar(p)=0$ for all $p\in M$.
 Furthermore, each fiber  is called \textit{totally geodesic}
if both $T^{\mathcal{H}}$ and $T^{\mathcal{V}}$ vanish identically and it is called
\textit{totally umbilical} if
\begin{eqnarray*}
T^{\mathcal{H}}(U,V)=g\left( U,V\right) \hbar
\end{eqnarray*}
for all $U,V\in \Gamma(\mathcal{V})$.

\medskip

Now we recall the following Theorem \cite{Fal:04}:

\begin{theorem} \label{Fal:book:1}
Let $\pi:(M,g)\rightarrow(B,\widetilde{g})$ be a Riemann submersion. Then
the horizontal space $\mathcal{H}$ is an integrable distribution
if and only if $A$ vanishes identically.
\end{theorem}

\begin{remark}
As a consequence of Theorem \ref{Fal:book:1}, we see that both $A^{\mathcal{H}}$ and $A^{\mathcal{V}}$ are
related to  integrability of $\mathcal{H}$,
they are identically zero if and only if $\mathcal{H}$ is integrable.
\end{remark}


Let $R$, $\widetilde{R}$ and $\hat{R}$ are the curvature tensors on $M$,
 $B$ and  be the collection of all curvature
tensors on  fibers $\pi^{-1}(b)$ respectively,  and $\check{R}(X,Y)Z$
be the horizontal lift of $\widetilde{R}_{\pi(b)}(\pi_{*p}X_{b},\pi_{*p}Y_{b})Z_{b}$
at any point $b\in M$ satisfying
\begin{eqnarray*}
\pi_{*}(\check{R}(X,Y)Z) =  \widetilde {R}(\pi_{*}X,\pi_{*}Y)\pi_{*}Z.
\end{eqnarray*}
Then, there exist the following relations between these tensors:
\begin{eqnarray}
R\left( U,V,W,G\right)&=&\hat{R}\left( U,V,W,G\right) + g\left( (T^{%
\mathcal{H}}(U,G),T^{\mathcal{H}}(V,W)\right)  \nonumber \\
&&-g\left(T^{\mathcal{H}}(V,G),T^{\mathcal{H}}(U,W)\right),  \label{Cod:2} \\
R\left( X,Y,Z,H\right) &=& \check{R}(X,Y,Z,H)-2g\left(A^{\mathcal{V}%
}(X,Y),A^{\mathcal{V}}(Z,H)\right) \nonumber    \\
& & +g\left(A^{\mathcal{V}}(Y,Z),A^{\mathcal{V}}(X,H)\right) -g\left(A^{%
\mathcal{V}}(X,Z),A^{\mathcal{V}} (Y,H)\right), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \label{Cod:4} \\
R\left( X,V,Y,W\right) &=& g\left( \left( \nabla _{X}T \right) \left(
V,W\right) ,Y\right) +g\left( \left( \nabla _{V}A \right) \left( X,Y\right)
,W\right) \nonumber   \\
& & -g\left(T^{\mathcal{V}}(V,X),T^{\mathcal{V}}(W,Y)\right) +g\left(A^{%
\mathcal{H}}(X,V),A^{\mathcal{H}}(Y,W)\right),  \label{Cod:5}
\end{eqnarray}
for any $U,V,W,G\in \Gamma({\mathcal{V}})$ and $X,Y,Z,H\in\Gamma({\mathcal{H}})$.
Note that the above equalities are known as \textit{Gauss--Codazzi equations}
for a Riemannian submersion. With the help of Gauss--Codazzi equations,
we get the following  relations between
the sectional curvatures as follows:
\begin{eqnarray}
K\left( U,V \right)&=&\hat{K}\left( U,V \right)
-\left\Vert T^{%
\mathcal{H}}(U,V)\right\Vert ^{2}+g\left(T^{\mathcal{H}}(U,U),T^{\mathcal{H}%
}(V,V)\right), \label{sec:cur:1} \\
K(X,Y)&=& \check{K}(\check{X},\check{Y})+%
3\|A^{\mathcal{V}}(X,Y)\|^{2}, \\
 K\left( X,V \right)&=&-g\left( \left( \nabla _{X}T\right) \left(
V,V\right) ,X\right) +\left\Vert T^{\mathcal{V}}(V,X)\right\Vert
^{2}-\left\Vert A^{\mathcal{H}}(X,V)\right\Vert ^{2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \
\label{sec:cur:3}
\end{eqnarray}
where $K, \ \hat{K}$ and $\check{K}$ denote the
sectional curvatures in $M$, any fiber $\pi^{-1}(b)$ and  the horizontal
distribution $\mathcal{H}$, respectively.
The  scalar curvatures of the vertical and horizontal spaces
at a point $p \in M$ is given respectively by
\begin{eqnarray}
\hat{\tau }\left( p\right) =\sum\limits_{1\leq i<j\leq r} \hat{K}\left( U_{i},U_{j}\right)
\end{eqnarray}
and
\begin{eqnarray}
\check{\tau}\left( p\right) =\sum\limits_{1\leq i<j\leq n}\check{K}\left( X_{i},X_{j}\right).
\end{eqnarray}

Now, we recall the following definition of \cite{Besse}.
\begin{definition}
Let $\pi:(M,g)\rightarrow(B,\widetilde{g})$ be a Riemann submersion
and $X$ be a horizontal vector field on $\pi$. Then,
\textit{horizontal divergence} of $X$ is defined by
\begin{eqnarray}
div_{\mathcal{H}}(X)=\sum_{i=1}^{n}g(\nabla_{X_{i}}X,X_{i}).
\end{eqnarray}
\end{definition}



\begin{lemma} \label{Fal:book:2}\cite{Fal:04}
Let $\pi:(M,g)\rightarrow(B,\widetilde{g})$ be a Riemann submersion
and $\{U_{1},\ldots,U_{r}\}$ be any orthonormal basis of $\Gamma(\mathcal{V})$.
For any $E\in \Gamma(TM) $
and $X\in \Gamma({\mathcal{H}})$, we have
\begin{equation}
g\left( \nabla _{E} \mathcal{\hbar}, \textit{X}  \right) =\frac{1}{r}\sum_{j=1}^{r}
g\left( \left( \nabla _{E}T\right) \left( U_{j},U_{j}\right),X\right).\label{Bes:1}
\end{equation}
\end{lemma}

As a consequence of Lemma \ref{Fal:book:2}, we obtain that
 \begin{equation}
\textrm{div}_{\mathcal{H}}({\mathcal{\hbar}})=\frac{1}{r}\sum_{i=1}^{n}\sum_{j=1}^{r} g\left( \left( \nabla _{X_{i}}T%
\right) \left( U_{j},U_{j}\right) ,X_{i}\right).  \label{k:1a}
\end{equation}









\section{An optimal inequality for Riemannian submersions}


We begin this section with the following algebraic lemma:

\begin{lemma}
\label{lem:new} If $n>k\geq 2$ and $a_{1},\ldots ,a_{n},a$ are real
numbers such that
\begin{equation}
\left( \sum_{i=1}^{n}a_{i}\right) ^{2}=\left( n-k+1\right) \left(
\sum_{i=1}^{n}a_{i}^{2}+a\right) ,  \label{eq-new-a}
\end{equation}
then
\[
2\sum_{1\leq i<j\leq k}a_{i}a_{j}\geq a,
\]
with equality holding if and only if
\[
a_{1}+a_{2}+\cdots +a_{k}=a_{k+1}=\cdots =a_{n}.
\]
\end{lemma}

\begin{proof} By the Cauchy-Schwartz inequality, we have
\begin{equation}
\left( \!\sum_{i=1}^{n}a_{i}\!\right) ^{2}\leq
\!(n\!-\!k\!+\!1)((\!a_{1}\!+\!a_{2}\!+\!\cdots
\!+\!a_{k}\!)^{2}\!+\!a_{k+1}^{2}\!+\!\cdots \!+\!a_{n}^{2}).
\label{eq-new-b}
\end{equation}
From $(\ref{eq-new-a})$ and $(\ref{eq-new-b})$, we get
\[
\sum_{i=1}^{n}a_{i}^{2}+a\leq \left( a_{1}+a_{2}+\cdots +a_{k}\right)
^{2}+a_{k+1}^{2}+\cdots +a_{n}^{2}.
\]
The above equation is equivalent to
\[
2\sum_{1\leq i<j\leq k}a_{i}a_{j}\geq a.
\]
The equality holds if and only if $a_{1}+a_{2}+\cdots +a_{k}=a_{k+1}=\cdots
=a_{n}$.
\end{proof}



Let $\pi:(M,g)\rightarrow (B,\widetilde{g})$ be a Riemannian submersion between Riemannian
manifolds $(M,g)$ and  $(B,\widetilde{g})$.
 Suppose $\{U_{1},\ldots,U_{r},X_{1},\ldots,X_{n}\}$ be an orthonormal basis on $T_{p}M$,
where ${\mathcal{V}}=\textrm{Span}\{U_{1},\ldots,U_{r}\}$ and ${\mathcal{H}}
=\textrm{Span}\{X_{1},\ldots,X_{n}\}$.  Then, we have
\begin{eqnarray}
&&\|T^{\mathcal{H}}\|^{2}=\sum_{i,j=1}^{r}g\left(T^{\mathcal{H}%
}(U_{i},U_{j}),T^{\mathcal{H}}(U_{i},U_{j})\right),  \label{N:1} \\
&&\|T^{\mathcal{V}}\|^{2}=\sum_{i=1}^{r}\sum_{j=1}^{n}g\left(T^{\mathcal{V}%
}(U_{i},X_{j}),T^{\mathcal{V}}(U_{i},X_{j})\right),  \label{N:2} \\
&&\|A^{\mathcal{H}}\|^{2}=\sum_{i=1}^{r}\sum_{j=1}^{n}g\left(A^{\mathcal{H}%
}(X_{j},U_{i}),A^{\mathcal{H}}(X_{j},U_{i})\right),  \label{N:3} \\
&&\|A^{\mathcal{V}}\|^{2}=\sum_{i,j=1}^{n}g\left(A^{\mathcal{V}%
}(X_{i},X_{j}),A^{\mathcal{V}}(X_{i},X_{j})\right).  \label{N:4}
\end{eqnarray}
Putting $(\ref{sec:cur:1})-(\ref{mean:curv})$, $(\ref{k:1a})$
and $(\ref{N:1})-(\ref{N:4})$ in
\begin{eqnarray*}
\tau(p)=\sum_{1\leq i<j\leq n}[K(U_{i},U_{j})+K(X_{i},U_{j})+K(X_{i},X_{j})],
\end{eqnarray*}
we obtain the following lemma:

\begin{lemma}
\label{lemma:2} Let $\left( M,g\right) $ and $(B,\widetilde{g})$ be
a Riemannian manifolds admitting a Riemannian submersion $\pi :(M,g)\rightarrow
(B,\widetilde{g})$. For any point $p\in M$, we have
\begin{eqnarray}
2\tau \left( p\right) & = & 2\hat{\tau }\left( p\right) +2\check{\tau}%
(p) +r^{2}\left\Vert \hbar(p)\right\Vert ^{2} -\left\Vert%
T^{\mathcal{H}}\right\Vert ^{2} +3\left\Vert A^{\mathcal{V}%
}\right\Vert ^{2}  \nonumber \\
& & - r \ \textrm{div}_{\mathcal{H}}{(\hbar(p))}+ \Vert T^{\mathcal{V}}\Vert^{2}%
-\|A^{\mathcal{H}}\|^{2}. \label{scalar:1}
\end{eqnarray}
\end{lemma}




 Now, we are going to give  an optimal inequality involving the $\delta-$curvature
for Riemannian manifolds admitting a Riemannian submersion.

\begin{theorem} \label{Th:opt}
Let $\pi:(M,g)\rightarrow (B,\widetilde{g})$ be a Riemannian submersion. Then,
for each point $p\in M$ and each $k$-plane section $L_{k}\subset \mathcal{V}_{p}$
$(r>k\geq2)$, we have
\begin{eqnarray}
\delta(k)&\leq& \hat{\tau}(p)-\hat{\tau}(L_{k})+\check{\tau}(p)+\frac{r^{2}(r-k)}{2(r-k+1)}\|\hbar\|^{2}
-\frac{r}{2}div_{\mathcal{H}}(\mathcal{\hbar}) \nonumber \\
&&+\frac{3}{2}\|A^{\mathcal{V}}\|^{2}+\frac{1}{2}\|T^{\mathcal{V}}\|^{2}. \label{delta:1}
\end{eqnarray}
The equality of $(\ref{delta:1})$ holds at $p\in M$ if and only if $A^{\mathcal H}$ vanishes identically and the shape operators
$S_{X_{1}},\ldots,S_{X_{n}}$ of \ $\mathcal{V}_{p}$ take forms as follows:
\begin{equation}
S_{X_{1}}=\left(
\begin{array}{cc}
\begin{array}{cccc}
T _{11}^{1} & 0 & \cdots & 0 \\
0 & T_{22}^{1} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & T_{kk}^{1}%
\end{array}
& {\LARGE 0} \\
{\LARGE 0} & \left( \displaystyle\sum_{i=1}^{k}T_{ii}^{1}\right)
I_{r-k}%
\end{array}
\right) ,  \label{shape-1}
\end{equation}
\begin{equation}
S_{X_{s}}=\left(
\begin{array}{cc}
\begin{array}{cccc}
T_{11}^{s} & T_{12}^{s} & \cdots & T_{1k}^{s} \\
T_{12}^{s} & T_{22}^{s} & \cdots & T_{2k}^{s} \\
\vdots & \vdots & \ddots & \vdots \\
T_{1k}^{s} & T_{2k}^{s} & \cdots & -\displaystyle%
\sum_{i=1}^{k-1}T_{ii}^{s}%
\end{array}
& 0 \\
0 & 0_{n-k}%
\end{array}
\right) ,\qquad s\in \left\{ 2,\ldots ,n\right\} .  \label{shape-2}
\end{equation}
\end{theorem}
\begin{proof}
Let $L_{k}$ be a $k$-plane section of $\mathcal{V}_{p}$. We choose an
orthonormal basis \\
$\{U_{1},\ldots,U_{r},X_{1},\ldots,X_{n}\}$  on $T_{p}M$
such that ${\mathcal{V}}=\textrm{Span}\{U_{1},\ldots,U_{r}\}$ and \\
${\mathcal{H}}
=\textrm{Span}\{X_{1},\ldots,X_{n}\}$. We write
\begin{eqnarray}
T_{ij}^{s}=g(T^{\mathcal{H}}(U_{i},U_{j}),X_{s})
\end{eqnarray}
for any $i,j\in\{1,\ldots,r\}$ and $s\in \{1,\ldots,n\}$. Suppose that the
mean curvature vector $\hbar(p)$ is in the direction of $X_{1}$
and $X_{1},...,X_{n}$ diagonalize the shape operator $S_{X_{1}}$.
 If we put
\begin{eqnarray}
\eta&=&2\tau(p)-2\hat{\tau}(p)-2\check{\tau}(p)-\frac{r^{2}(r-k)}{(r-k+1)}\|\hbar\|^{2}
+r\ div_{\mathcal{H}}(\mathcal{\hbar}(p))\nonumber \\
&& -3\left\Vert A^{\mathcal{V}%
}\right\Vert ^{2}- \Vert T^{\mathcal{V}}\Vert^{2}%
+\|A^{\mathcal{H}}\|^{2} \label{eta}
\end{eqnarray}
in $(\ref{scalar:1})$, it follows that
\begin{eqnarray}
r^{2}\|\hbar\|^{2}=(n-k+1)(\eta+\Vert T^{\mathcal{H}}\Vert^{2}). \label{delta:proof:1}
\end{eqnarray}
The equation $(\ref{delta:proof:1})$ is equivalent to
\begin{equation}
\left( \sum_{i=1}^{r}T_{ii}^{1}\right) ^{2}=\left(n-k+1\right)
\left( \eta+\sum_{i=1}^{r}\left(T_{ii}^{1}\right)
^{2}+\sum_{s=2}^{n}\sum_{i,j=1}^{r}\left(T_{ij}^{s}\right)
^{2} \right).  \label{delta:proof:2}
\end{equation}
 Applying Lemma~\ref{lem:new} to equation (\ref{delta:proof:2}), we get
 \begin{equation}
2\sum_{1\leq i<j\leq k}T_{ii}^{n+1}T_{jj}^{n+1}\geq \eta
+\sum_{s=2}^{n}\sum_{i,j=1}^{r}\left(T_{ij}^{s}\right) ^{2}.
\label{delta:proof:3}
\end{equation}
On the other hand, we have from $(\ref{scalar:1})$ that
\begin{equation}
\tau \left(L_{k}\right) =\hat{\tau }\left(L_{k}\right)
+\sum_{1\leq i<j\leq k}T_{ii}^{1}T_{jj}^{1}+\sum_{s=2}^{n}\sum_{1\leq i<j\leq k}\left( T
_{ii}^{s}T_{jj}^{s}-\left(T_{ij}^{s}\right) ^{2}\right) .
\label{delta:proof:4}
\end{equation}
From $(\ref{delta:proof:3})$ and $(\ref{delta:proof:4})$, we get
\begin{eqnarray}
\tau \left(L_{k}\right) &\geq &\hat{\tau }\left(L_{k}\right) +%
\frac{1}{2}\eta +\sum_{s=2}^{n}\sum_{j>k}\{(T
_{1j}^{s})^{2}+(T_{2j}^{s})^{2}+\cdots +(T_{kj}^{s})^{2}\}
\nonumber \\
&&\quad +\ \frac{1}{2}\sum_{s=2}^{n}(T_{11}^{s}+T_{22}^{s}+\cdots +T_{kk}^{s})^{2}+
\frac{1}{2}\sum_{s=2}^{n}%
\sum_{i,j>k}(T_{ij}^{s})^{2}.   \label{delta:proof:5}
\end{eqnarray}
In view of $(\ref{delta:proof:5})$, we see that
\begin{equation}
\tau \left( \Pi _{k}\right) \geq \widetilde{\tau }\left( \Pi _{k}\right) +%
\frac{1}{2}\eta .  \label{delta:proof:6}
\end{equation}
From $(\ref{delta:proof:1})$ and $(\ref{delta:proof:6})$, we obtain $(\ref{delta:1})$.

\medskip

If the equality case of  $(\ref{delta:1})$ holds, then
we have $A^{\mathcal H}$ vanishes identically and
\begin{equation}
\left\{
\begin{array}{l}
T_{1j}^{1}=T_{2j}^{1}=T_{kj}^{1}=0,\qquad
j=k+1,\ldots ,r, \\
T_{ij}^{s}=0,\qquad i,j=k+1,\ldots ,r, \\
T_{11}^{r}+T_{22}^{r}+\cdots +T_{kk}^{r}=0%
\end{array}%
\right.  \label{eq-bi-6}
\end{equation}%
for $%
s=2,\ldots ,n$. Applying Lemma~\ref{lem:new}, we also have
\begin{equation}
T_{11}^{1}+T_{22}^{1}+\cdots +T_{kk}^{1}=T
_{ll}^{1},\qquad l=k+1,\ldots ,n.  \label{eq-bi-7}
\end{equation}%
Thus, with respect to a suitable orthonormal basis $\left\{ X_{1},\ldots
,X_{m}\right\} $ on $\mathcal{H}_{p}$, the shape operator of $\mathcal{V}_{p}$ becomes of the form given by $
(\ref{shape-1})$ and $(\ref{shape-2})$. The proof of the converse part is
straightforward.
\end{proof}

In particular case of $k=2$, we have the following:
\begin{corollary}
Let $\pi:(M,g)\rightarrow (B,\widetilde{g})$ be a Riemannian submersion. Then,
for each point $p\in M$ and each plane section $L\subset \mathcal{V}_{p}$,
 we have
\begin{eqnarray}
\delta(2)&\leq& \hat{\tau}(p)-\hat{K}(L)+\check{\tau}(p)+\frac{r^{2}(r-2)}{2(r-1)}\|\hbar\|^{2}
-\frac{r}{2}div_{\mathcal{H}}(\mathcal{\hbar}) \nonumber \\
&&+\frac{3}{2}\|A^{\mathcal{V}}\|^{2}+\frac{1}{2}\|T^{\mathcal{V}}\|^{2}. \label{delta:cor1}
\end{eqnarray}
The equality of $(\ref{delta:cor1})$ holds at $p\in M$ if and only if $A^{\mathcal H}$ vanishes identically and the shape operators
$S_{X_{1}},\ldots,S_{X_{n}}$ of \ $\mathcal{V}_{p}$ take forms
\begin{equation}
S_{X_{1}}=\left(
\begin{array}{ccc}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & \left( a+b\right) I_{r-2}%
\end{array}
\right), \label{shape:s:1}
\end{equation}
\begin{equation}
S_{X_{s}}=\left(
\begin{array}{ccc}
c_{s} & d_{s} & 0 \\
d_{s} & -c_{s} & 0 \\
0 & 0 & 0_{r-2}%
\end{array}
\right) ,\qquad s\in \left\{ 2,\ldots ,n\right\} .\label{shape:s:2}
\end{equation}
\end{corollary}

In particular case of $k=r-1$, we have the following

\begin{corollary}
\label{th-Ricci-H}
Let $\pi:(M,g)\rightarrow (B,\widetilde{g})$ be a Riemannian submersion.
 For each vertical unit vector $U$,  we have
\begin{eqnarray}
{\rm Ric}_{\mathcal{V}}\left( U\right) \leq \hat{{\rm Ric}}(U)+\check{\tau}(p)+\frac{r^{2}}{4}\|\hbar\|^{2}
-\frac{r}{2}div_{\mathcal{H}}(\mathcal{\hbar})+
\frac{3}{2}\|A^{\mathcal{V}}\|^{2}+\frac{1}{2}\|T^{\mathcal{V}}\|^{2}.  \ \ \ \ \ \  \label{eq-Ricci-H}
\end{eqnarray}
The equality case of {\em (\ref{eq-Ricci-H})}\ holds for all
unit vectors $U$ $\in \mathcal{V}_{p}$ if and only if
$A^{\mathcal H}$ vanishes identically and we have either
\begin{enumerate}
\item[(i)] if $r=2$, $\pi$ has totally umbilical fibers at $p\in M$,
\item[(i)] if $r\neq 2$, $\pi$ has totally geodesic fibers at $p\in M$.
\end{enumerate}

\end{corollary}
\begin{proof}
Let $L_{r-1}$  be a $(r-1)$-plane section of $\mathcal{V}_{p}$.
We get from Theorem \ref{Th:opt} that
\begin{eqnarray}
\delta(r-1)&\leq& \hat{\tau}(p)-\hat{\tau}(L_{r-1})+\check{\tau}(p)+\frac{r^{2}}{4}\|\hbar\|^{2}
-\frac{r}{2}div_{\mathcal{H}}(\mathcal{\hbar}) \nonumber \\
&&+\frac{3}{2}\|A^{\mathcal{V}}\|^{2}+\frac{1}{2}\|T^{\mathcal{V}}\|^{2}. \label{Ricci:proof:1}
\end{eqnarray}
Now, let $U$ be a unit vertical vector field such that $U=U_{r}$.
By a straightforward computation, we obtain $(\ref{eq-Ricci-H})$.

\medskip

The equality of $(\ref{Ricci:proof:1})$ holds if and only if
the forms of shape operators $S_{X_{s}}$, $s=1,\ldots,n$, become
\begin{equation}
S_{X_{1}}=\left(
\begin{array}{ccccc}
T_{11}^{1} & 0 & \cdots & 0 & 0 \\
0 & T_{22}^{1} & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & T_{\left( r-1\right) \left( r-1\right) }^{1} & 0 \\
0 & 0 & \cdots & 0 & \left( \displaystyle\sum_{i=1}^{r-1}T
_{ii}^{1}\right)%
\end{array}
\right) ,  \label{eq-2}
\end{equation}
\begin{equation}
S_{X_{s}}=\left(
\begin{array}{ccccc}
T_{11}^{s} & T_{12}^{s} & \cdots & T_{1\left( r-1\right)
}^{r} & 0 \\
T_{12}^{s} & T_{22}^{s} & \cdots & T_{2\left(r-1\right)
}^{s} & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
T_{1\left( r-1\right) }^{s} & T_{2\left( r-1\right) }^{s} &
\cdots & -\displaystyle\sum_{i=1}^{r-2}T_{ii}^{s} & 0 \\
0 & 0 & \cdots & 0 & 0%
\end{array}
\right) ,\quad r\in \left\{2,\ldots,n\right\} .  \label{eq-3}
\end{equation}
From $(\ref{eq-2})$ and $(\ref{eq-3})$, we see that the equality in (\ref{eq-Ricci-H}) is valid
for a unit vertical vector field $U=U_{r}$ if and
only if
\begin{equation}
\left\{
\begin{array}{l}
T_{rr}^{s}=T_{11}^{s}+T_{22}^{s}+\cdots +T_{\left(
r-1\right) \left(r-1\right) }^{s} \\
T_{1r}^{s}=T_{2r}^{s}=\cdots =T_{\left( r-1\right)
r}^{s}=0.%
\end{array}
\right.  \label{eq-4}
\end{equation}
for $s\in \left\{1,\ldots, n\right\} $.

\medskip

Assuming the equality case of (\ref{eq-Ricci-H}) holds for
all unit vertical vector fields, in view of (\ref{eq-4}), for each $%
s\in \left\{1,\ldots,n\right\} $, we have
\begin{equation}
\left\{
\begin{array}{l}
2T_{ii}^{s}=T_{11}^{s}+T_{22}^{s}+\cdots +T_{rr}^{s},
\\
T_{ij}^{s}=0,\quad i\neq j\
\end{array}%
\right.  \label{eq-5}
\end{equation}%
for all $i\in \left\{ 1,...,r\right\} $ and $s\in \left\{1,\ldots
,n\right\} $. Thus, we have two cases, namely either $r=2$ or $r\neq 2$. In
the first case we see that $\pi$ has totally umbilical fibers, while in the second case $\pi$
has totally geodesic fibers. The proof of converse part is straightforward.
\end{proof}

\begin{remark} We note that  (\ref{eq-Ricci-H}) was also proved in \cite{semsi} (see Theorem 4.1 in \cite{semsi}).
In Theorem\ref{th-Ricci-H}, we gave a new proof for this inequality.
\end{remark}

\section{Some basic facts on Meteorology}

In this section, we shall present some basic facts related to notions the horizontal divergence,
the  conservation of mass and the continuity equation following \cite{Brune1,Brune2,Brune3,Hung,Ketchum,Ste,Vaughan}.

\medskip

This section is going to be handled on three subsections.

\subsection{Horizontal divergence and convergence}

\begin{definition} \cite{Wood}
Let $(M,g)$ be an $n$-dimensional Riemannian manifold
and $\{e_{1},\ldots,e_{n}\}$ be an orthonormal basis
of the tangent space $T_{p}M$ at $p \in M $. The divergence
 of a vector field $X$ in $T_{p}M$ is defined by
\begin{eqnarray}
\textrm{div}(X)=\textrm{trace}\left(\nabla X\right)\equiv
\sum_{i=1}^{n}g\left(\nabla_{e_{i}}X,e_{i}\right).
\end{eqnarray}
\end{definition}



\begin{theorem} \cite{Wood} \textit{[Divergence Theorem]}
Let $(M,g)$ be a Riemannian manifold. Suppose that $\mathfrak{D}$
to be a compact domain of $M$ with smooth boundary $\partial \mathfrak{D}$
and $N$ to be the unit normal vector to $\mathfrak{D}$. Then we have
\begin{eqnarray}
\int_{\mathfrak{D}}g(X,N)\mathrm{v}^{\partial \mathfrak{D}}=\iint_{\partial\mathfrak{D}}
\textrm{div}(X)\mathrm{v}^{M}
\end{eqnarray}
for any vector field $X \in \Gamma(\mathfrak{D})$.
Here, $\mathrm{v}^{M}$ is the measure of $(M,g)$.
\end{theorem}

In fluid kinematics, if a vector field $X$ is considered as
velocity of a fluid (a gas), then  sign of $\textrm{div}(X)$
describes the expansion or compression of  flow. Therefore, the total
expansion or compression of flow can be calculated by
the help of divergence theorem so divergence is a useful tool to measuring the net flow
of fluid diverging from a point or approaching a point. The first phenomenon
is called as \textit{ horizontal divergence} and the other is called as \textit{horizontal convergence}.

\begin{figure}[!htb]
\centering
\includegraphics[scale=.4]{1.eps}
\caption{Horizontal Divergence}
\label{fig:digraph}
\end{figure}

\begin{figure}[!htb]
\centering
\includegraphics[scale=.4]{2.eps}
\caption{Horizontal Convergence}
\label{fig:digraph}
\end{figure}

 \medskip

In meteorology, divergence and convergence are used to
measure the change in area of wind pressure or a fluid element. For example,
these notions occur in moving air masses. Air masses sometimes move away from
a line (or a point) or move towards a line.
If a horizontal divergence occurs, then air masses or a fluid element move
faster than those behind them and if a horizontal convergence
occurs, then they move slower than those behind them.



\subsection{Conservation of mass}

Let $\mathrm{E}^{3}$ be $3$-dimensional Euclidean space with the usual coordinates
$(x,y,z)$. Consider an object of mass $m$ which is given by
\begin{eqnarray}
m=\rho \mathrm{v},
\end{eqnarray}
where $\rho$ is density  and $\mathrm{v}$ is volume of the object. Then we have
\begin{eqnarray}
\frac{dm}{dt}=\frac{d}{dt}(\rho \mathrm{v})=0,
\end{eqnarray}
which implies that
\begin{eqnarray}
\frac{1}{\rho}\frac{d\rho}{dt}+\frac{1}{\mathrm{v}}\frac{d\mathrm{v}}{dt}=0. \label{meter:1}
\end{eqnarray}
Putting $\mathrm{v}=\Delta x \Delta y \Delta z$, we get
\begin{eqnarray}
\frac{1}{\mathrm{v}}\frac{d\mathrm{v}}{dt}=\Delta x\frac{d\left(\Delta y \Delta z\right)}{dt}
+\Delta y\frac{d\left(\Delta x \Delta z\right)}{dt}
+\Delta z\frac{d\left(\Delta x \Delta y\right)}{dt}.
\end{eqnarray}
By a straightforward computation, we see that
\begin{eqnarray}
\frac{1}{\mathrm{v}}\frac{d\mathrm{v}}{dt}&=&\frac{du^{1}}{dx}
+\frac{du^{2}}{dy}+\frac{du^{3}}{dz} =\textrm{div}(U), \label{meter:2}
\end{eqnarray}
where $U=(u^{1},u^{2},u^{3})$ is a vector field on $E^{3}$. In view
of  $(\ref{meter:1})$ and $(\ref{meter:2})$, we have
\begin{eqnarray}
\frac{1}{\rho}\frac{d\rho}{dt}+\textrm{div}(U)=0. \label{meter:3}
\end{eqnarray}

As a consequence of $(\ref{meter:3})$, we obtain the followings:

\begin{enumerate}
\item[i)] $\textrm{div}(U)<0$ if and only if the density increases.

\item[ii)] $\textrm{div}(U)>0$ if and only if the density decreases.

\item[iii)] $\textrm{div}(U)=0$ if and only if the density doesn't change with time.
\end{enumerate}
\subsection{Continuity equation}
 Another important notion in meteorology is the continuity equation.
 The principle simple states that any matter can either be created or destroyed and
implies for the atmosphere that its mass may be redistributed but can never
be disappeared. Therefore,
this equation gives us that
 \begin{eqnarray}
 \textrm{div}(U)=0 \label{meter:4}
 \end{eqnarray}
for any vector field $U=(u^{1},u^{2},u^{3})$ on $\mathrm{E}^{3}$. It can be written from $(\ref{meter:4})$ that
\begin{eqnarray}
\textrm{div}_{H}(U)+\frac{\partial u^{3}}{\partial z}=0,\label{1}
\end{eqnarray}
where $\textrm{div}_{H}(v_{H})$ is the horizontal divergence of $U$ defined by
\begin{eqnarray}
\textrm{div}_{H}(v_{H})=\frac{\partial u^{1}}{\partial x}+\frac{\partial u^{2}}{\partial y}.
\end{eqnarray}
The equation given $(\ref{1})$ is also known as the \textit{continuity equation}
in literature.  Integrating  $(\ref{1})$,
 we have
 \begin{eqnarray}
 \omega (p_{1},p_{0})\equiv
 u^{3}(p_{1})-u^{3}(p_{0})=-\int_{p_{0}}^{p_{1}}(\frac{\partial u^{1}}{\partial x}+\frac{\partial u^{2}}{\partial y})
 dz,
 \end{eqnarray}
where $p_{1}$ and $p_{0}$ is some pressure levels on the atmosphere. If we assume that
$p_{0}$ is surface pressure then
$u^{3}(p_{0})=0$ and thus we get
\begin{eqnarray}
  \omega (p_{1})=-\int_{p_{0}}^{p_{1}}(\frac{\partial u^{1}}{\partial x}+\frac{\partial u^{2}}{\partial y})
 dz,
 \end{eqnarray}
 This formula tells us that $w$ at a given pressure level is proportional to the integral of the
horizontal divergence. Here, $\omega(p_{1})$ is called the \textit{vertical motion} at $p_{1}$.

\medskip
\begin{figure}[H]
\centering
\includegraphics[scale=.3]{r.eps}
\caption{Rising and descending motions}
\label{fig:digraph}
\end{figure}

If $\omega(p)<0$ at every point $p$ then this statement is called \textit{rising motion},  $\omega(p)>0$ at every point $p$ then
this statement is called \textit{descending motion}, (in this case, divergence is called convergence) in meteorology.
There is no divergence and it is clear that there is a local maximum or minimum of $w$
if $w(p)=0$. Therefore, one can consider that convergence and divergence roughly determine where air will be sinking or rising.
Rising air can be associated with clouds and precipitation; sinking air can be associated with clear, calm conditions and good weather.
Thus, convergence and divergence make the difference between a grey day and a sunny day.
The importance of convergence and divergence is related to pressure changes at the surface and height changes
of the constant-pressure levels.

\medskip

As a consequence of the above facts, we can state the followings:



\begin{figure}[H]
\centering
\includegraphics[scale=.5]{met.eps}
\caption{Minimum or maximum values of vertical motion}
\label{fig:digraph}
\end{figure}

\begin{equation*}
{\small
\begin{tabular}{|l|l|l|l|}
\hline
&  & &  \\
\textbf{Case}& \textbf{Vertical Motion} & \textbf{Horizontal Divergence(Convergence)} & \textbf{Point(s)}\\
&  & & \\ \hline
&  & & \\
\textrm{Rising Motion} & \textrm{Maximum Value}& \textrm{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\textrm{Minimum Value} & Warmest (ideal) \\
 &  & & \\
\hline
&  & & \\
\textrm{Rising Motion} & \textrm{Minimum Value}& \textrm{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\textrm{Maximum Value} & Coolest \\
 &  & & \\
\hline
&  & & \\
\textrm{Descending Motion} & \textrm{Maximum Value}& \textrm{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\textrm{Minimum Value} & Warmest \\
 &  & & \\
\hline
&  & & \\
\textrm{Descending Motion} & \textrm{Minumum Value}& \textrm{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\textrm{Maximum Value} & Coolest (ideal) \\
 &  & & \\
\hline
\end{tabular}%
\ }
\end{equation*}

\section{Main conclusions}

In this section, we shall present a solution way with the help of differential geometry tools for the following natural problem:

\bigskip

 ''Which conditions should provide to the horizontal divergence or the convergence receives  to the maximum value or minimum value?''


\medskip



 To obtain minimum or maximum values of the vertical motion (or horizontal divergence)
 it can be considered a Riemannian submersion on $\mathrm{E}^{3}$ to $\mathrm{E}^{2}$.
 Moreover, we can regard to different Riemannian submersions
such as  a Riemannian submersion on a three dimensional Riemannian manifold to two dimensional Riemannian manifold
as
\begin{eqnarray}
\pi:M^{3}\rightarrow N^{2}.
\end{eqnarray}
It can also be considered globally in high dimensional Riemannian manifolds with taking a Riemannian
submersion on $m$-dimensional Riemannian manifold to $n$-dimensional Riemannian manifold.


\medskip

Taking into account of the continuity equation and  $(\ref{delta:1})$, $(\ref{delta:cor1})$
and $(\ref{eq-Ricci-H})$ inequalities, we get some result dealing minimum or maximum values
of vertical motion for a manifold admitting a Riemannain submersion.



\medskip

As a consequence of
$(\ref{delta:1})$, we obtain the following:

\begin{corollary} \label{mm:1}
Let $\pi:\mathrm{E}^{n+r}\rightarrow \mathrm{E}^{n}$ be a Riemannian submersion.
 Then we have
 \begin{eqnarray}
\frac{r}{2} \omega (p)&\geq& \delta(k)-\frac{r^{2}(r-k)}{2(r-k+1)}\|\hbar\|^{2}
 -\frac{3}{2}\|A^{\mathcal{V}}\|^{2}-\frac{1}{2}\|T^{\mathcal{V}}\|^{2}. \label{delta:c1}
\end{eqnarray}
  The vertical motion at a point $p$ takes the minimum value
 if and only if $A^{\mathcal{H}}$ vanishes identically and the matrixes of shape operators
  of the vertical
space of $M$ take the form as $(\ref{shape-1})$ and $(\ref{shape-2})$.
\end{corollary}

As a consequence of
$(\ref{delta:cor1})$, we obtain the followings:
\begin{corollary} \label{mm:2}
Let $\pi:\mathrm{E}^{n+r}\rightarrow \mathrm{E}^{n}$ be a Riemannian submersion
with integrable horizontal distribution.
 Then we have
 \begin{eqnarray}
\frac{r}{2} \omega (p)&\geq& \delta(2)-\frac{r^{2}(r-2)}{2(r-1)}\|\hbar\|^{2}
 -\frac{1}{2}\|T^{\mathcal{V}}\|^{2}. \label{delta:c2}
\end{eqnarray}
  The vertical motion takes the minimum value
 if and only if  the matrixes of shape operators
 $S_{x_{1}},\ldots,S_{x_{n}}$ of the vertical
space of $M$ take the form as $(\ref{shape:s:1})$ and $(\ref{shape:s:2})$.
\end{corollary}

\begin{corollary} \label{mm:2b}
Let $\pi:\mathrm{E}^{n+r}\rightarrow \mathrm{E}^{n}$ be a Riemannian submersion
with totally geodesic leaves and integrable horizontal distribution.
 Then we have
 \begin{eqnarray}
\frac{r}{2} \omega (p)&=& \delta(2). \label{delta:c22}
\end{eqnarray}
\end{corollary}

From $(\ref{eq-Ricci-H})$, we get the followings:

\begin{corollary} \label{mm:3}
Let $\pi:\mathrm{E}^{n+r}\rightarrow \mathrm{E}^{n}$ be a Riemannian submersion.
For each vertical unit vector $U$,  we have
 \begin{eqnarray}
\frac{r}{2} \omega (p)  \geq {\rm Ric}_{\mathcal{V}}\left( U\right)-\frac{r^{2}}{4}\|\hbar\|^{2}
-\frac{3}{2}\|A^{\mathcal{V}}\|^{2}-\frac{1}{2}\|T^{\mathcal{V}}\|^{2}.  \ \ \ \ \ \  \label{eq-Ricci-cH}
\end{eqnarray}
 The equality case of {\em (\ref{eq-Ricci-cH})} holds for all
unit vectors $U$ $\in \mathcal{V}_{p}$ if and only if
$A^{\mathcal H}$ vanishes identically and we have either
\begin{enumerate}
\item[(i)] if $r=2$, $\pi$ has totally umbilical fibers at $p\in M$,
\item[(ii)] if $r\neq 2$, $\pi$ has totally geodesic fibers at $p\in M$.
\end{enumerate}
\end{corollary}
\begin{corollary} \label{mm:3}
Let $\pi:\mathrm{E}^{n+r}\rightarrow \mathrm{E}^{n}$ be a Riemannian submersion with
totally geodesic fibers.
For each vertical unit vector $U$,  we have
 \begin{eqnarray}
\frac{r}{2} \omega (p)  = {\rm Ric}_{\mathcal{V}}\left( U\right)
-\frac{3}{2}\|A^{\mathcal{V}}\|^{2}.  \ \ \ \ \ \  \label{eq-Ricci-cH}
\end{eqnarray}
\end{corollary}
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H. Zhao, A. R. Kelly, J. Zhou, J. Lu and Y. Y. Yang,
Graph attribute embedding via Riemannian submersion learning,
 Computer Vision and Image Understanding, 115 (2011),
962-975.

\end{thebibliography}

Mehmet G\"{U}LBAHAR \newline
Address: Department of Mathematics, Faculty of Science and Art, Siirt
University, \newline
Siirt, TURKEY. \newline
email:mehmetgulbahar85@gmail.com \newline



\c{S}emsi EKEN MER\.{I}\c{C} \newline
Address: Department of Mathematics,
Faculty of Science and Art, Mersin University,\newline
Mersin, TURKEY.\newline
e-mail: semsieken@hotmail.com \newline

Erol KILI\c{C} \newline
Address: Department of Mathematics,
Faculty of Science and Art, \.{I}n\"{o}n\"{u} University, \newline
Malatya, TURKEY.\newline
e-mail: erol.kilic@inonu.edu.tr \newline
\end{document}

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