text:
The Moche of Ancient Peru: Media and Messages
When Aaron Dworkin's Sphinx Organization was founded in 1998, only about 1.16 percent of orchestras were black. Today that number is up to 2.5 percent. "We need to look at diversity as something that is critical to the evolution and survival of our field and our art form," said Dworkin. "We have a great deal of distance to go; we are not yet done by any means." Dworkin was at Harvard to receive the Luise Vosgerchian Teaching Award from the Office for the Arts.
Aaron Dworkin aims to spur diversity in U.S. orchestras
By Colleen Walsh Harvard Staff Writer
Date March 23, 2011 June 17, 2019
When people ask Aaron Dworkin why he cares so much about bringing diversity to classical music, he answers, "I am basically a black, white, Jewish, Irish Catholic, Jehovah's Witness who plays the violin. … I am the definition of diversity, and really had no choice but to do this work."
Dworkin is spreading African-American and Latino diversity as the founder and president of the Detroit-based Sphinx Organization, which focuses on youth development and diversity in performing and appreciating classical music.
Soundbytes: Catalyst Quartet at Harvard
The son of an unwed, white, Irish Catholic mother and an African-American Jehovah's Witness, he was given up for adoption by his parents two weeks after his birth to a white Jewish couple from New York, both professors in neural and behavioral science with a love of music.
Inspired by his adoptive mother, an amateur violinist, Dworkin took up the instrument at age 5. Three years later, while attending a concert by the violin virtuoso Isaac Stern at New York's Carnegie Hall, Dworkin said he was struck by the "sense of awe it built in me about music." The feeling has remained with him ever since.
But along with his love of the art form came an understanding of its lack of diversity, and a desire to make that change, leading to the Sphinx Organization.
Dworkin was at Harvard on March 11 to receive the University's Luise Vosgerchian Teaching Award, which honors a nationally recognized educator and is administered by the Office for the Arts at Harvard.
During a presentation in the Barker Center's Thompson Room, Dworkin discussed some grim numbers, signifying what he called a "stark underrepresentation" of blacks and Hispanics in the nation's classical music landscape.
Blacks and Latinos represent only 4 percent of members in the nation's 1,200 orchestras, according to a survey conducted by the League of American Orchestras. But the problem runs much deeper than just the diversity of the stage performers, said Dworkin. The survey also found similar statistics among top administrative positions. Only about 4 percent of music directors or orchestra conductors are black or Latino. The numbers are worse for executive director and artistic administrator positions. "Statistically," said Dworkin, "zero percent are black or Latino."
Even education and community relations directors, those charged with connecting an orchestra to its surrounding community, are rarely men or women of color. In addition, repertoires reflect no black or Latino composers, and audiences are largely composed of older, white members.
But with Dworkin's help, the tide is slowly shifting. His group includes the Sphinx Performance Academy, a full-scholarship, intensive chamber music and solo performance program designed for aspiring black and Latino string players; a professional development program that helps prepare young artists for a career in classical music; the annual Sphinx Competition, open to all junior high, high school, and college-age black and Latino string players in the nation; the Catalyst Quartet, composed of top laureates and alumni; and the Sphinx Symphony, an all-black and Latino orchestra made up of top professionals from around the country.
Dworkin said his group has played a significant role in doubling the number of black performers in the nation's orchestras. When his organization was founded in 1998, only about 1.16 percent of orchestras were black. Today that number is up to 2.5 percent. According to Dworkin, every new African-American member of an orchestra since 1998 has some tie to the Sphinx Organization.
"We need to look at diversity as something that is critical to the evolution and survival of our field and our art form," said Dworkin. "We have a great deal of distance to go; we are not yet done by any means."
Quoting civil rights leader Martin Luther King Jr., who met his wife while she was studying voice and violin at the New England Conservatory of Music in Boston, Dworkin said, "Change does not roll in on the wheels of inevitability but comes through continuous struggle. … Our lives begin to end the day we become silent about things that matter."
Fittingly, the Catalyst Quartet closed the discussion with a performance of Terry Riley's "Sunrise of the Planetary Dream Collector."


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text:
\section{Introduction}
Riemannian submersions between Riemannian manifolds were studied by O'Neill \cite{O} and Gray
\cite{Gray}, for recent developments on the geometry of Riemannian
submanifolds and Riemannian submersions, see:\cite{Chen} and \cite{FIP}, respectively. In \cite{watson}, the Riemannian submersions were considered between almost Hermitian manifolds by
Watson under the name of almost Hermitian submersions. In this case, the Riemannian submersion
is also an almost complex mapping and consequently the vertical and horizontal distribution are
invariant with respect to the almost complex structure of the total manifold of the submersion. The
study of anti-invariant Riemannian submersions from almost Hermitian manifolds were initiated by
\c{S}ahin \cite{s1}. In this case, the fibres are anti-invariant with respect to the almost complex structure
of the total manifold. Beside there are many notions related with anti-invariant
Riemannian submersion (see: \cite{Asa}, \cite{BEM}, \cite{CE}, \cite{GIP}, \cite{Gun1}, \cite{Gun2}, \cite{Gun3}, \cite{LPSS},
 \cite{Park1}, \cite{Park2}, \cite{Park3}, \cite{Park4}, \cite{Fatima-Shahid1}, \cite{s2}, \cite{s3}, \cite{Tastan1}).
In \cite{Chi4}, Chinea defined almost contact Riemannian submersions between
almost contact metric manifolds and examined the differential geometric properties of Riemannian
submersions between almost contact metric manifolds. More precisely, let $(M_1,\phi,\xi,\eta,g_1)$ and
$(M_2,\phi^{'},\xi^{'},\eta^{'},g_2)$ be almost contact manifolds with $dim M_1=2m+1$ and $dim M_2=2n+1$. A
Riemannian submersion $\pi:M_1\longrightarrow M_2$ is called the almost contact metric submersion if $\pi$ is an
almost contact mapping, i.e., $\phi^{'}\pi_*=\pi_*\phi$. An immediate consequence of the above definition is that the vertical and
horizontal distributions are $\phi$-invariant. Moreover, the characteristic vector field $\xi$ is horizontal.
We note that only $\phi$-holomorphic submersions have been considered on almost contact manifolds
\cite{Chi4}.

One the other hand, as a generalization of Riemannian submersion,
horizontally conformal submersions are defined as follows \cite{BW}: Suppose
that $(M,g_{M})$ and $(B,g_{B})$ are Riemannian manifolds and $\pi:M\longrightarrow B$ is a smooth submersion,
then $\pi$ is called a horizontally conformal submersion, if there is a positive function $\lambda$ such that
\begin{equation*}
\lambda^{2}g_{M}(X,Y)=g_{B}(\pi_{*}X,\pi_{*}Y)
\end{equation*}
for every $X,Y\in\Gamma((ker\pi_{*})^\perp).$ It is obvious that every
Riemannian submersion is a particular horizontally conformal submersion with
$\lambda=1$. We note that horizontally conformal submersions are special
horizontally conformal maps which were introduced independently by Fuglede
\cite{F} and Ishihara \cite{I}. We also note that a horizontally conformal
submersion $\pi:M\longrightarrow B$ is said to be horizontally homothetic if
the gradient of its dilation $\lambda$ is vertical, i.e.,
\begin{equation}
\mathcal{H}(grad\lambda)=0
\end{equation}
at $p\in M$, where $\mathcal{H}$ is the projection on the horizontal space
$(ker\pi_{*})^{\perp}$. For conformal submersion, see: \cite{BW}, \cite{G}, \cite{Ornea}.

As a generalization of holomorphic submersions, conformal holomorphic submersions were studied
by Gudmundsson and Wood \cite{GW}. They obtained necessary and sufficient conditions for con-
formal holomorphic submersions to be a harmonic morphism, see also \cite{Chi1}, \cite{Chi2} and \cite{Chi3} for the
harmonicity of conformal holomorphic submersions.

Recently, in \cite{As} we have introduced conformal anti-invariant submersions from almost Hermitian
manifolds onto Riemannian manifolds and investigated the geometry of such submersions. (See also:\cite{A})
We showed that the geometry of such submersions are different from anti-invariant Riemannian
submersions.  In this paper, we consider conformal anti-invariant $\xi^\perp-$submersions from an almost contact
metric manifold under the assumption that the fibers are anti-invariant with respect to the tensor field of
type $(1,1)$ of the almost contact manifold.

The paper is organized as follows. In the second section,
we gather main notions and formulas for other sections. In section 3, we introduce conformal
anti-invariant $\xi^\perp-$submersions from almost contact metric manifolds onto Riemannian manifolds,
investigates the geometry of leaves of the horizontal distribution and the vertical distribution and
find necessary and sufficient conditions for a conformal anti-invariant $\xi^\perp-$submersion to be totally
geodesic and harmonic, respectively. In section 4, we show that there are certain product structures
on the total space of a conformal anti-invariant $\xi^\perp-$submersion.

\section{Preliminaries}

In this section, we define almost contact metric manifolds, recall the notion of
(horizontally) conformal submersions between Riemannian manifolds and give a
brief review of basic facts of (horizontally) conformal submersions.

Let $(M,g_{M})$ be an almost contact metric manifold with structure tensors $(\phi,\xi,\eta,g_{M})$
where $\phi$ is a tensor field of type (1,1), $\xi$ is a vector field, $\eta$ is a 1-form and $g_{M}$ is the Riemannian metric on $M.$
Then these tensors satisfy \cite{B}
\begin{equation}
\phi\xi=0,\ \ \eta o\phi=0,\ \ \eta(\xi)=1 \label{e.q:2.1}
\end{equation}
\begin{equation}
\phi^{2}=-I+\eta\otimes\xi \ \ \ \text{and} \ \ g_{M}(\phi X,\phi Y)=g_{M}(X,Y)-\eta(X)\eta(Y),  \label{e.q:2.2}
\end{equation}
where $I$ denotes the identity endomorphism of $TM$ and $X,Y$ are any vector fields on $M$. Moreover, if $M$ is Sasakian \cite{SH}, then we have
\begin{equation}
(\nabla_{X}\phi)Y=-g_{M}(X,Y)\xi+\eta(Y)X \ \ \text{and} \ \ \nabla_{X}\xi=\phi X, \label{e.q:2.3}
\end{equation}
where $\nabla$ is the connection of Levi-Civita covariant differentiation.

Conformal submersions belong to a wide class of conformal maps that we are going to recall their
definition, but we will not study such maps in this paper.

\begin{definition}
\textit{(\cite{BW})} Let $\varphi:(M^{m},g)\longrightarrow (N^{n},h)$ be a
smooth map between Riemannian manifolds, and let $x\in M$. Then $\varphi$ is
called horizontally weakly conformal or semi conformal at $x$ if either

\begin{enumerate}
\item[(i)] $d\varphi_{x}=0$, or
\item[(ii)] $d\varphi_{x}$ maps horizontal space $\mathcal{H
_{x}=(ker(d\varphi_{x}))^\perp$ conformally onto $T_{\varphi_{*}}N$, i.e.,
d\varphi_{x}$ is surjective and there exists a number $\Lambda(x)\neq0$ such
that
\begin{equation}
h(d\varphi_{x}X,d\varphi_{x}Y)=\Lambda(x)g(X,Y)\mbox{ }(X,Y\in\mathcal{H
_{x}).
\end{equation}
\end{enumerate}
\end{definition}
A point $x$ is of type (i) in Definition if and only if it is a critical point of $\varphi$; we shall call
a point of type (ii) a \textit{regular point}. At a critical point,
d\varphi_{x}$ has rank $0$; at a regular point, $d\varphi_{x}$ has rank $n$
and $\varphi$ is submersion. The number $\Lambda(x)$ is called the \textit
square dilation} (of $\varphi$ at $x$); it is necessarily non-negative; its
square root $\lambda(x)=\sqrt{\Lambda(x)}$ is called the dilation (of
\varphi$ at $x$). The map $\varphi$ is called \textit{horizontally weakly
conformal} or \textit{semi conformal} (on $M$) if it is horizontally weakly
conformal at every point of $M$. It is clear that if $\varphi$ has no
critical points, then we call it a (\textit{horizontally}) conformal
submersion.

Next, we recall the following definition from \cite{G}. Let
$\pi:M\longrightarrow N$ be a submersion. A vector field $E$ on $M$ is said
to be projectable if there exists a vector field $\check{E}$ on $N$, such
that $d\pi(E_{x})=\check{E}_{\pi(x)}$ for all $x\in M$. In this case $E$ and
$\check{E}$ are called $\pi-$ related. A horizontal vector field $Y$ on
(M,g)$ is called basic, if it is projectable. It is well known fact, that is
$\check{Z}$ is a vector field on $N$, then there exists a unique basic
vector field $Z$ on $M$, such that $Z$ and $\check{Z}$ are $\pi-$ related.
The vector field $Z$ is called the horizontal lift of $\check{Z}$.

The fundamental tensors of a submersion were introduced in \cite{O}. They
play a similar role to that of the second fundamental form of an immersion.
More precisely, O'Neill's tensors $T$ and $A$ defined for vector fields $E,F$
on $M$ by
\begin{equation}  \label{A}
A_{E}F=\mathcal{V}\nabla_{\mathcal{H}E}\mathcal{H}F +\mathcal{H}\nabla_
\mathcal{H}E}\mathcal{V}F
\end{equation}
\begin{equation}  \label{T}
T_{E}F=\mathcal{H}\nabla_{\mathcal{V}E}\mathcal{V}F +\mathcal{V}\nabla_
\mathcal{V}E}\mathcal{H}F
\end{equation}
where $\mathcal{V}$ and $\mathcal{H}$ are the vertical and horizontal
projections (see \cite{FIP}). On the other hand, from (\ref{A}) and (\ref{T
), we have
\begin{equation}  \label{nvw}
\nabla_{V}W=T_{V}W+\hat{\nabla}_{V}W
\end{equation}
\begin{equation}  \label{nvx}
\nabla_{V}X=\mathcal{H}\nabla_{V}X+T_{V}X
\end{equation}
\begin{equation}  \label{nxv}
\nabla_{X}V=A_{X}V +\mathcal{V}\nabla_{X}V
\end{equation}
\begin{equation}  \label{nxy}
\nabla_{X}Y=\mathcal{H}\nabla_{X}Y+A_{X}Y
\end{equation}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V,W\in\Gamma(ker\pi_{*})$, where $\hat
\nabla}_{V}W=\mathcal{V}\nabla_{V}W$. If $X$ is basic, then $\mathcal{H
\nabla_{V}X=A_{X}V$. It is easily seen that for $x\in M$, $X\in \mathcal{H
_{x}$ and $\mathcal{V}_{x}$ the linear operators $T_{V}$,
A_{X}:T_{X}M\longrightarrow T_{X}M$ are skew-symmetric, that is
\begin{equation*}
-g(T_{V}E,F)=g(E,T_{V}F)\text{ and }-g(A_{X}E,F)=g(E,A_{X}F)
\end{equation*}
for all $E,F\in T_{x}M$. We also see that the restriction of $T$ to the
vertical distribution $T\mid_{V\times V}$ is exactly the second fundamental
form of the fibres of $\pi$. Since $T_{V}$ skew-symmetric we get: $\pi$ has
totally geodesic fibres if and only if $T\equiv0$. For the special case when
$\pi$ is horizontally conformal we have the following:

\begin{proposition}
\textit{(\cite{G})} Let $\pi:(M^{m},g)\longrightarrow (N^{n},h)$ be a
horizontally conformal submersion with dilation $\nabla$ and $X,Y$ be
horizontal vectors, then
\begin{equation}
A_{X}Y=\frac{1}{2}\{\mathcal{V}[X,Y]-\lambda^{2}g(X,Y)grad_{\mathcal{V}}
\frac{1}{\lambda^{2}})\}.
\end{equation}
\end{proposition}

We see that the skew-symmetric part of $A\mid_{(ker\pi_{*})^\perp\times(ker\pi_{*})^\perp}$
measures the obstruction integrability of the horizontal distribution
(ker\pi_{*})^\perp$.

Let $(M,g_{M})$ and $(N,g_{N})$ be Riemannian manifolds and suppose that
\pi:M \longrightarrow N$ is a smooth map between them. The
differential $\pi_{*}$ of $\pi$ can be viewed a section of the
bundle $Hom(TM,\pi^{-1}TN) \longrightarrow M$, where $\pi^{-1}TN$ is
the pullback bundle which has fibres $(\pi^{-1}TN)_{p}=T_{\pi(p)}N$,
$p\in M$. $Hom(TM,\pi^{-1}TN)$ has a connection $\nabla$ induced from
the Levi-Civita connection $\nabla^{M}$ and the pullback connection. Then
the second fundamental form of $\pi$ is given by
$$\nabla\pi_*:\Gamma(TM)\times\Gamma(TM)\longrightarrow \Gamma(TN)$$ defined by
\begin{equation}
(\nabla\pi_{*})(X,Y)=\nabla^{\pi}_{X}\pi_{*}(Y)-\pi_{*}(\nabla^{M}_{X}Y) \label{nfixy}
\end{equation}
for $X,Y\in\Gamma(TM)$, where $\nabla^{\pi}$ is the pullback connection.
It is known that the second fundamental form is symmetric.
\begin{lemma}\cite{Urakawa}
Let $(M,g_{M})$ and $(N,g_{N})$ be Riemannian manifolds and suppose that
\varphi:M \longrightarrow N$ is a smooth map between them. Then we have
\begin{equation}
\nabla^{\varphi}_{X}\varphi_{*}(Y)-\nabla^{\varphi}_{Y}\varphi_{*}(X)-\varphi_{*}([X,Y])=0
\end{equation}
for $X,Y\in\Gamma(TM)$.
\end{lemma}
A smooth map $\pi:(M,g_{M}) \longrightarrow (N,g_{N})$ is said to be harmonic if
trace(\nabla\pi_{*})=0$. On the other hand, the tension field of $\pi
$ is the section $\tau(\pi)$ of $\Gamma(\pi^{-1}TN)$ defined by
\begin{equation}
\tau(\pi)=div\pi_{*}=\sum_{i=1}^{m}(\nabla\pi_{*})(e_{i},e_{i}),
\end{equation}
where $\{e_{1},...,e_{m}\}$ is the orthonormal frame on $M$. Then it follows
that $\pi$ is harmonic if and only if $\tau(\pi)=0$ (for details,
see \cite{BW}). Finally, we recall the following lemma from \cite{BW}.

\begin{lemma}
\label{lem1} Suppose that $\pi:M \longrightarrow N$ is a horizontally conformal submersion. Then, for any
horizontal vector fields $X,Y$ and vertical fields $V,W$ we have
\begin{enumerate}
\item[(i)] $(\nabla \pi_*)(X,Y)=X(\ln\lambda)\pi_* Y+Y(\ln\lambda)\pi_* X-g(X,Y)\pi_*(grad\ln\lambda)$;
\item[(ii)] $(\nabla \pi_*)(V,W)=-\pi_* (T_{V}W)$;
\item[(iii)] $(\nabla \pi_*)(X,V)=-\pi_*(\nabla^{M}_{X}V)=-\pi_*(A_{X}V)$.
\end{enumerate}
\end{lemma}

\section{Conformal Anti-invariant $\xi^\perp-$submersions}

In this section, we define conformal anti-invariant $\xi^\perp-$submersions from an almost
contact metric manifold onto a Riemannian manifold and investigate the integrability
of distributions and obtain a necessary and sufficient condition for such
submersions to be totally geodesic map. We also investigate the harmonicity
of such submersions.

\begin{definition}
\label{def} Let $(M,\phi,\xi,\eta,g_M)$ be an almost contact metric manifold and
and $(N,g_{N})$ be a Riemannian manifold. We suppose that there exist a horizontally conformal submersion
$\pi:M \longrightarrow N$ such that $\xi$ is normal to $ker\pi_{*}$ and $ker\pi_{*}$ is anti-invariant with respect to $\phi$, i.e.,
$\phi(ker\pi_{*})\subset(ker\pi_{*})^{\perp}.$ Then we say that $\pi$ is a conformal anti-invariant $\xi^\perp-$submersion.
\end{definition}

Here, we assume that if $\pi:(M,\phi,\xi,\eta,g_M)\longrightarrow (N,g_{N})$ is a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold
$(M,\phi,\xi,\eta,g_M)$ to a Riemannian manifold $(N,g_{N})$. Then from Definition \ref{def},
we have $\phi(ker\pi_{*})^\perp\cap ker\pi_{*}\neq{0}.$ We denote the complementary orthogonal
distribution to $\phi(ker\pi_{*})$ in $(ker\pi_{*})^\perp$ by $\mu.$ Then we have
\begin{equation}
(ker\pi_{*})^\perp=\phi(ker\pi_{*})\oplus\mu. \label{e.q:3.1}
\end{equation}
We can easily to see that $\mu$ is an invariant distribution of $(ker\pi_{*})^\perp,$ with respect to $\phi.$ Hence $\mu$ contains $\xi.$ Thus, for
$X\in\Gamma((ker\pi_{*})^\perp)$, we have
\begin{equation}
\phi X=\mathcal{B}X+\mathcal{C}X, \label{e.q:3.2}
\end{equation}
where $\mathcal{B}X\in\Gamma(ker\pi_{*})$ and $\mathcal{C}X\in\Gamma(\mu).$ On the other hand, since $\pi_{*}((ker\pi_{*})^\perp)=TN$ and $\pi$ is a conformal submersion, using (\ref{e.q:3.2}) we derive $\frac{1}{\lambda^2}g_{N}(\pi_{*}\phi V,\pi_{*}\mathcal{C}X)=0$ for any $X\in\Gamma((ker\pi_{*})^\perp)$ and
$V\in\Gamma(ker\pi_{*}),$ which implies that
\begin{equation}
TN=\pi_{*}(\phi ker\pi_{*})\oplus \pi_{*}(\mu). \label{e.q:3.3}
\end{equation}
\begin{remark}
We note that every anti-invariant $\xi^\perp-$submersion from an almost contact manifold onto
a Riemannian manifold is a conformal anti-invariant $\xi^\perp-$submersion with $\lambda=I$, where $I$ denotes the identity function \cite{Lee}.
\end{remark}
\begin{lemma}
Let $\pi$ be a conformal anti-invariant $\xi^\perp$-submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$. Then we have
\begin{equation}
A_X\xi=-\mathcal{B}X,  \label{axxi}
\end{equation}
\begin{equation}
T_V\xi=0  \label{tvxi},
\end{equation}
\begin{equation}
g_{M}(\mathcal{C}Y,\phi V)=0, \label{cywv}
\end{equation}
\begin{equation}
g_{M}(\nabla_{X}\mathcal{C}Y,\phi V)=-g_{M}(\mathcal{C}Y,\phi A_{X}V)  \label{nxcyjv}
\end{equation}
for $X,Y,\xi\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$.
\end{lemma}
\begin{proof}
By virtue of (\ref{e.q:2.3}), (\ref{nxy}) and (\ref{e.q:3.2}) we have (\ref{axxi}). Using (\ref{e.q:2.3}) and (\ref{nvx}) we get (\ref{tvxi}).
By using (\ref{e.q:2.2}), for $Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$, we have
$$g_{M}(\mathcal{C}Y,\phi V)=g_{M}(\phi Y-\mathcal{B}Y,\phi V)=g_{M}(\phi Y,\phi V)=g_{M}(Y,V)+\eta(Y)\eta(V)=g_{M}(Y,V)=0,$$ since
$\mathcal{B}Y\in\Gamma(ker\pi_{*})$ and $\phi V,\xi\in\Gamma((ker\pi_{*})^\perp).$ Differentiating (\ref{cywv}) with respect to $X,$ we get
\begin{align*}
g_{M}(\nabla_{X}\mathcal{C}Y,\phi V)&=-g_{M}(\mathcal{C}Y,\nabla_{X}\phi V)\\
&=-g_{M}(\mathcal{C}Y,(\nabla_{X}\phi) V)-g_{M}(\mathcal{C}Y,\phi(\nabla_{X}V))\\
&=-g_{M}(\mathcal{C}Y,\phi(\nabla_{X}V))\\
&=-g_{M}(\mathcal{C}Y,\phi A_{X}V)-g_{M}(\mathcal{C}Y,\phi\mathcal{V}\nabla_{X}V)\\
&=-g_{M}(\mathcal{C}Y,\phi A_{X}V)
\end{align*}
due to $\phi\mathcal{V}\nabla_{X}V\in\Gamma(\phi ker\pi_{*}).$ Our assertion is complete.
\end{proof}
Since the distribution $ker\pi_{*}$ is integrable, we only study the integrability of the distribution $(ker\pi_{*})^\perp$ and
then we investigate the geometry of leaves of $ker\pi_{*}$ and $(ker\pi_{*})^\perp$.

\begin{theorem}\label{teo1}
Let $\pi:(M,\phi,\xi,\eta,g_M)\longrightarrow (N,g_{N})$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$. Then the following assertions are equivalent to each other;
\begin{enumerate}
\item[(a)] $(ker\pi_{*})^\perp$ is integrable,
\item[(b)]$\begin{aligned}[t]
\frac{1}{\lambda^{2}}g_{N}(\nabla^{\pi}_{Y}\pi_{*}\mathcal{C}X-\nabla^{\pi}_{X}\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)
&=g_{M}(A_{X}\mathcal{B}Y-A_{Y}\mathcal{B}X-\mathcal{C}Y(\ln\lambda)X+\mathcal{C}X(\ln\lambda)Y\\
&-2g_M(\mathcal{C}X,Y)\ln\lambda-\eta(Y)X+\eta(X)Y,\phi V)
\end{aligned}$
\end{enumerate}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$.
\end{theorem}
\begin{proof}
From  (\ref{e.q:2.2}) and (\ref{e.q:2.3}), we obtain
\begin{equation}
g_{M}(\nabla_{X}Y,V)=g_{M}(\nabla_{X}\phi Y,\phi V)-\eta(Y)g_{M}(X,\phi V). \label{e.q:3.8}
\end{equation}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$. Then, from (\ref{e.q:3.2}) and (\ref{e.q:3.8}), we have
\begin{align*}
g_{M}([X,Y],V)&=g_{M}(\nabla_{X}\phi Y,\phi V)-g_{M}(\nabla_{Y}\phi X,\phi V)
-\eta(Y)g_{M}(X,\phi V)+\eta(X)g_{M}(Y,\phi V)\\
&=g_{M}(\nabla_{X}\mathcal{B}Y,\phi V)+g_{M}(\nabla_{X}\mathcal{C}Y,\phi V)
-g_{M}(\nabla_{Y}\mathcal{B}X,\phi V)-g_{M}(\nabla_{Y}\mathcal{C}X,\phi V)\\
&-\eta(Y)g_{M}(X,\phi V)+\eta(X)g_{M}(Y,\phi V).
\end{align*}
Using (\ref{nxv}) and if we take into account that $\pi$ is a conformal submersion, we obtain
\begin{align*}
g_{M}([X,Y],V)&=g_{M}(A_{X}\mathcal{B}Y-A_{Y}\mathcal{B}X,\phi V)+\frac{1}{\lambda^{2}}g_{N}(\pi_{*}(\nabla_{X}\mathcal{C}Y),\pi_{*}\phi V)\\
&-\frac{1}{\lambda^{2}}g_{N}(\pi_{*}(\nabla_{Y}\mathcal{C}X),\pi_{*}\phi V)-\eta(Y)g_{M}(X,\phi V)+\eta(X)g_{M}(Y,\phi V).
\end{align*}
Thus, from (\ref{nfixy}) and Lemma \ref{lem1} we derive
\begin{align*}
g_{M}([X,Y],V)&=g_{M}(A_{X}\mathcal{B}Y-A_{Y}\mathcal{B}X,\phi V)-g_{M}(\mathcal{H}gradln\lambda,X)g_{M}(\mathcal{C}Y,\phi V)\\
&-g_{M}(\mathcal{H}gradln\lambda,\mathcal{C}Y)g_{M}(X,\phi V)+g_{M}(X,\mathcal{C}Y)g_{M}(\mathcal{H}gradln\lambda,\phi V)\\
&+\frac{1}{\lambda^{2}}g_{N}(\nabla^{\pi}_X\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)+g_{M}(\mathcal{H}gradln\lambda,Y)g_{M}(\mathcal{C}X,\phi V)\\
&+g_{M}(\mathcal{H}gradln\lambda,\mathcal{C}X)g_{M}(Y,\phi V)-g_{M}(Y,\mathcal{C}X)g_{M}(\mathcal{H}gradln\lambda,\phi V)\\
&-\frac{1}{\lambda^{2}}g_{N}(\nabla^{\pi}_Y\pi_{*}\mathcal{C}X,\pi_{*}\phi V)-\eta(Y)g_{M}(X,\phi V)+\eta(X)g_{M}(Y,\phi V).
\end{align*}
Moreover, using (\ref{cywv}), we obtain
\begin{align*}
g_{M}([X,Y],V)&=g_{M}(A_{X}\mathcal{B}Y-A_{Y}\mathcal{B}X-\mathcal{C}Y(\ln\lambda)X+\mathcal{C}X(\ln\lambda)Y
-2g_M(\mathcal{C}X,Y)\ln\lambda\\
&-\eta(Y)X+\eta(X)Y,\phi V)-\frac{1}{\lambda^{2}}g_{N}(\nabla^{\pi}_Y\pi_{*}\mathcal{C}X-\nabla^{\pi}_X\pi_{*}\mathcal{C}Y,\pi_{*}\phi V).
\end{align*}
This show that $(a)\Leftrightarrow (b)$.
\end{proof}
From Theorem $\ref{teo1},$ we deduce the following which shows that a conformal anti-invariant $\xi^\perp-$submersion
with integrable $(ker\pi_{*})^{\perp}$ turns out to be a horizontally homothetic submersion.
\begin{theorem}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$. Then any two conditions below imply the third;
\begin{enumerate}
\item [(i)] $(ker\pi_{*})^\perp$ is integrable.
\item [(ii)] $\pi$ is horizontally homothetic submersion.
\item [(iii)]$\begin{aligned}[t]
g_{N}(\nabla^{\pi}_Y\pi_{*}\mathcal{C}X-\nabla^{\pi}_X\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)=\lambda^{2}g_{M}(A_{X}\mathcal{B}Y-A_{Y}\mathcal{B}X-\eta(Y)X+\eta(X)Y,\phi V)
\end{aligned}$
\end{enumerate}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$.
\end{theorem}
\begin{proof}
From Theorem \ref{teo1}, we have
\begin{align*}
g_{M}([X,Y],V)&=g_{M}(A_{X}\mathcal{B}Y-A_{Y}\mathcal{B}X-\mathcal{C}Y(\ln\lambda)X+\mathcal{C}X(\ln\lambda)Y
-2g_M(\mathcal{C}X,Y)\ln\lambda\\
&-\eta(Y)X+\eta(X)Y,\phi V)-\frac{1}{\lambda^{2}}g_{N}(\nabla^{\pi}_Y\pi_{*}\mathcal{C}X-\nabla^{\pi}_X\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)
\end{align*}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$. Now, if we have (i) and (iii), then we arrive at
\begin{align}
&-g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}Y)g_{M}(X,\phi V)+g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}X)g_{M}(Y,\phi V)\label{e.q:3.9} \\
&-2g_{M}(\mathcal{C}X,Y)g_{M}(\mathcal{H}grad\ln\lambda,\phi V)=0.\notag
\end{align}
Now, taking $Y=\phi V$ in (\ref{e.q:3.9}) for $V\in\Gamma(ker\pi_{*})$, using (\ref{e.q:2.2}) and (\ref{cywv}), we get
\begin{align*}
g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}X)g_{M}(\phi V, \phi V))&=g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}X)\{g_{M}(V,V)-\eta(V)\eta(V)\}\\
&=g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}X)g_{M}(V,V)=0.
\end{align*}
Hence $\lambda$ is a constant on $\Gamma(\mu)$. On the other hand, taking $Y=\mathcal{C}X$ in (\ref{e.q:3.9})
for $X\in\Gamma(\mu)$ and using (\ref{cywv}) we derive
\begin{align*}
&-g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}^{2}Y)g_{M}(X,\phi V)+g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}X)g_{M}(\mathcal{C}X,\phi V)\\
&-2g_{M}(\mathcal{C}X,\mathcal{C}X)g_{M}(\mathcal{H}grad\ln\lambda,\phi V)=0,
\end{align*}
thus, we arrive at $$2g_{M}(\mathcal{C}X,\mathcal{C}X)g_{M}(\mathcal{H}grad\ln\lambda,\phi V)=0.$$ From above equation, $\lambda$ is a constant on $\Gamma(\phi ker\pi_{*}).$ Similarly, one can obtain the other assertions.
\end{proof}
\begin{remark}
We assume that $(ker\pi_*)^\perp=\phi ker\pi_*\oplus\{\xi\}.$ Using (\ref{e.q:3.2}) one can prove that $\mathcal{C}X=0.$
\end{remark}
Hence we have the following corollary.
\begin{corollary}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$ with
$(ker\pi_{*})^{\perp}=\phi(ker\pi_{*})\oplus<\xi>$. Then the following assertions are equivalent to each other;
\begin{enumerate}
\item [(i)] $(ker\pi_{*})^\perp$ is integrable
\item [(ii)] $A_{X}\phi Y+\eta(X)Y=A_{Y}\phi X+\eta(Y)X$
\item [(iii)] $(\nabla \pi_{*})(X,\phi Y)+\eta(Y)\pi_{*}X=(\nabla \pi_{*})(Y,\phi X)+\eta(X)\pi_{*}Y$
\end{enumerate}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$.
\end{corollary}
For the geometry of leaves of the horizontal distribution, we have the following theorem.
\begin{theorem}\label{teo2}
Let $\pi:(M,\phi,\xi,\eta,g_M)\longrightarrow (N,g_{N})$ be a conformal anti-invariant $\xi^\perp$-submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$. Then the following assertions are equivalent to each other;
\begin{enumerate}
\item [(i)] $(ker\pi_{*})^\perp$ defines a totally geodesic foliation on $M$.
\item [(ii)]$\begin{aligned}[t]
-\frac{1}{\lambda^{2}}g_{N}(\nabla^{\pi}_{X}\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)=g_{M}(A_{X}BY-\mathcal{C}Y(\ln\lambda)X+g_M(X,\mathcal{C}Y)\ln\lambda-\eta(Y)X,\phi V)
\end{aligned}$
\end{enumerate}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$.
\end{theorem}
\begin{proof}
 By using (\ref{e.q:2.2}), (\ref{nxv}), (\ref{nxy}), (\ref{e.q:3.1}), (\ref{e.q:3.2}) and (\ref{e.q:3.8}), have
\begin{align*}
g_{M}(\nabla_{X}Y,V)&=g_{M}(A_{X}BY,\phi V)+g_{M}(\nabla_{X}\mathcal{C}Y,\phi V)-\eta(Y)g_{M}(X,\phi V)
\end{align*}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$. Since $\pi$ is a conformal submersion, using (\ref{nfixy}) and Lemma (\ref{lem1}) we arrive at
\begin{align*}
g_{M}(\nabla_{X}Y,V)&=g_{M}(A_{X}BY,\phi V)-\frac{1}{\lambda^2}g_{M}(\mathcal{H}grad\ln\lambda,X)g_{N}(\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)\\
&-\frac{1}{\lambda^2}g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}Y)g_{N}(\pi_{*}X,\pi_{*}\phi V)\\
&+\frac{1}{\lambda^2}g_{M}(X,\mathcal{C}Y)g_{N}(\pi_{*}(\mathcal{H}grad\ln\lambda),\pi_{*}\phi V)\\
&+\frac{1}{\lambda^2}g_{N}(\nabla^{\pi}_X\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)-\eta(Y)g_{M}(X,\phi V).
\end{align*}
Moreover, using Definiton \ref{def} and (\ref{cywv}) we obtain
\begin{align*}
g_{M}(\nabla_{X}Y,V)&=g_{M}(A_{X}BY-\mathcal{C}Y(\ln\lambda)X+g_M(X,\mathcal{C}Y)\ln\lambda-\eta(Y)X,\phi V)\\
&+\frac{1}{\lambda^2}g_{N}(\nabla_{\pi_{*}X}\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)
\end{align*}
which tells that $(i)\Leftrightarrow(ii)$.
\end{proof}
From Theorem \ref{teo2}, we also deduce the following characterization.
\begin{theorem}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$
 onto a Riemannian manifold $(N,g_{N})$. Then any two conditions below imply the third;
\begin{enumerate}
\item [(i)] $(ker\pi_{*})^\perp$ defines a totally geodesic foliation on $M$.
\item [(ii)] $\pi$ is a horizontally homothetic submersion.
\item [(iii)] $g_{N}(\nabla^{\pi}_X\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)=\lambda^{2}g_{M}(-A_{X}\mathcal{B}Y+\eta(Y)X,\phi V)$
\end{enumerate}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$.
\end{theorem}
\begin{proof}
For $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$, from Theorem \ref{teo2}, we have
\begin{align*}
g_{M}(\nabla_{X}Y,V)&=g_{M}(A_{X}BY-\mathcal{C}Y(\ln\lambda)X+g_M(X,\mathcal{C}Y)\ln\lambda-\eta(Y)X,\phi V)\\
&+\frac{1}{\lambda^2}g_{N}(\nabla^{\pi}_X\pi_{*}\mathcal{C}Y,\pi_{*}\phi V).
\end{align*}
Now, if we have (i) and (iii), then we obtain
\begin{equation}
-g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}Y)g_{M}(X,\phi V)+g_{M}(\mathcal{H}grad\ln\lambda,\phi V)g_{M}(X,\mathcal{C}Y)=0. \label{e.q:3.10}
\end{equation}
Now, taking $X=\mathcal{C}Y)$ in (\ref{e.q:3.10}) and using (\ref{cywv}), we get
$g_{M}(\mathcal{H}grad\ln\lambda,\phi V)g_{M}(X,\mathcal{C}Y)=0.$ Hence, $\lambda$ is a constant on $\Gamma(\phi ker\pi_{*})$. On the other hand, taking $X=\phi V$ in (\ref{e.q:3.10}) and using (\ref{cywv}) we derive
\begin{align*}
g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}Y)g_{M}(\phi V, \phi V))&=g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}Y)\{g_{M}(V,V)-\eta(V)\eta(V)\}\\
&=g_{M}(\mathcal{H}grad\ln\lambda,\mathcal{C}Y)g_{M}(V,V)=0.
\end{align*}
From above equation, $\lambda$ is a constant on $\Gamma(\mu)$. Similarly, one can obtain the other assertions.
\end{proof}
In particular, as an analogue of a conformal Lagrangian submersion in \cite{As}, we have the following corollary.
\begin{corollary}\label{cor1}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$ with
$(ker\pi_{*})^{\perp}=\phi(ker\pi_{*})\oplus<\xi>$. Then the following assertions are equivalent to each other;
\begin{enumerate}
\item [(i)] $(ker\pi_{*})^\perp$ defines a totally geodesic foliation on $M$.
\item [(ii)] $A_{X}\mathcal{B}Y=\eta(Y)X$
\item [(iii)]$(\nabla \pi_{*})(X,\phi V)=-\eta(Y)\pi_{*}X$
\end{enumerate}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$.
\end{corollary}

In the sequel we are going to investigate the geometry of leaves of the distribution $ker\pi_{*}$.
\begin{theorem}\label{teo3}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$. Then the following assertions are equivalent to each other;
\begin{enumerate}
\item [(i)] $ker\pi_{*}$ defines a totally geodesic foliation on $M$.
\item [(ii)]$\begin{aligned}[t]
-\frac{1}{\lambda^2}g_{N}(\nabla^{\pi}_{\phi W}\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)&=g_{M}(\phi\mathcal{C}X(\ln\lambda)\phi V-T_{V}\mathcal{B}X, \phi V)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)
\end{aligned}$
\end{enumerate}
for $V,W\in\Gamma(ker\pi_{*})$ and $X\in\Gamma((ker\pi_{*})^\perp)$.
\end{theorem}
\begin{proof}
Since $g_{M}(W,\xi)=0,$ using (\ref{e.q:2.3}) we have $g_{M}(\nabla_{V}W,\xi)=-g_{M}(W,\nabla_{V}\xi)=-g_{M}(W,\phi V)=0$
for $V,W\in\Gamma(ker\pi_{*})$ and $\xi\in\Gamma((ker\pi_{*})^\perp).$  Thus we have
\begin{align*}
g_{M}(\nabla_{V}W,X)&=g_{M}(\phi\nabla_{V}W,\phi X)+\eta(\nabla_{V}W)\eta(X)\\
&=g_{M}(\phi\nabla_{V}\phi W,\phi X)\\
&=g_{M}(\nabla_{V}\phi W,\phi X)-g_{M}((\nabla_{V}\phi)W,\phi X).
\end{align*}
Using (\ref{e.q:2.3}), (\ref{nvw}) and (\ref{e.q:3.2}) we have
\begin{equation*}
g_{M}(\nabla_{V}W,X)=g_{M}(T_{V}\phi W,\mathcal{B}X)+g_{M}(\mathcal{H}\nabla_{V}\phi W,\mathcal{C}X).
\end{equation*}
Since $\nabla$ is torsion free and $[V,\phi W]\in\Gamma(ker\pi_{*})$ we obtain
\begin{equation*}
g_{M}(\nabla_{V}W,X)=g_{M}(T_{V}\phi W,\mathcal{B}X)+g_{M}(\nabla_{\phi W}V,\mathcal{C}X).
\end{equation*}
Using (\ref{e.q:2.3}) and (\ref{nxy}) we have
\begin{align*}
g_{M}(\nabla_{V}W,X)&=g_{M}(T_{V}\phi W,\mathcal{B}X)+g_{M}(\phi\nabla_{\phi W}V,\phi\mathcal{C}X)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)\\
&=g_{M}(T_{V}\phi W,\mathcal{B}X)+g_{M}(\nabla_{\phi W}\phi V,\phi\mathcal{C}X)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)
\end{align*}
here we have used that $\mu$ is invariant. Using (\ref{nfixy}) and Lemma \ref{lem1} (i) and if we take into account
that $\pi$ is a conformal submersion, we obtain
\begin{align*}
g_{M}(\nabla_{U}V,X)&=g_{M}(T_{V}\phi W,\mathcal{B}X)+\frac{1}{\lambda^2}g_{M}(\mathcal{H}gradln\lambda,\phi W)g_{N}(\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)\\
&-\frac{1}{\lambda^2}g_{M}(\mathcal{H}gradln\lambda,\phi V)g_{N}(\pi_{*}\phi W,\pi_{*}\phi\mathcal{C}X)\\
&+g_{M}(\phi W,\phi V)\frac{1}{\lambda^2}g_{N}(\pi_{*}(\mathcal{H}gradln\lambda),\pi_{*}\phi\mathcal{C}X)\\
&+\frac{1}{\lambda^2}g_{N}(\nabla_{\pi_{*}\phi W}\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X).
\end{align*}
Moreover, using Definition \ref{def} and (\ref{cywv}), we obtain
\begin{align*}
g_{M}(\nabla_{U}V,X)&=g_{M}(\phi\mathcal{C}X(\ln\lambda)\phi V-T_{V}\mathcal{B}X, \phi V)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)\\
&+\frac{1}{\lambda^2}g_{N}(\nabla^{\pi}_{\phi W}\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)
\end{align*}
which tells that $(i)\Leftrightarrow (ii)$.
\end{proof}
From Theorem \ref{teo3}, we deduce the following result.
\begin{theorem}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$. Then any two conditions below imply the third;
\begin{enumerate}
\item [(i)] $ker\pi_{*}$ defines a totally geodesic foliation on $M$.
\item [(ii)] $\lambda$ is a constant on $\Gamma(\mu)$.
\item [(iii)]$\begin{aligned}[t]
-\frac{1}{\lambda^2}g_{N}(\nabla^{\pi}_{\phi W}\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)&=g_{M}(T_{V}\phi W,\mathcal{B}X)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)
\end{aligned}$
\end{enumerate}
for $V,W\in\Gamma(ker\pi_{*})$ and $X\in\Gamma((ker\pi_{*})^\perp)$.
\end{theorem}
\begin{proof}
From Theorem (\ref{teo3}) we have
\begin{align*}
g_{M}(\nabla_{U}V,X)&=g_{M}(\phi\mathcal{C}X(\ln\lambda)\phi V-T_{V}\mathcal{B}X, \phi V)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)\\
&+\frac{1}{\lambda^2}g_{N}(\nabla^{\pi}_{\phi W}\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)
\end{align*}
for $U,V\in\Gamma(ker\pi_{*})$ and $X\in\Gamma((ker\pi_{*})^\perp).$ Now, if we have (i) and (iii), then we obtain
$$g_{M}(\phi W,\phi V)g_{M}(\mathcal{H}gradln\lambda,\phi\mathcal{C}X)=0.$$ From above equation, $\lambda$ is a constant on $\Gamma(\mu)$. Similarly, one can obtain the other assertions.
\end{proof}
As an analogue of a conformal Lagrangian submersion in \cite{As}, (\ref{e.q:3.3}) implies that $TN=\pi_{*}(\phi ker\pi_{*})$. Hence we have the following.
\begin{corollary}\label{cor2}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$ with
$(ker\pi_{*})^{\perp}=\phi(ker\pi_{*})\oplus<\xi>$. Then the following assertions are equivalent to each other;
\begin{enumerate}
\item [(i)] $ker\pi_{*}$ defines a totally geodesic foliation on $M$.
\item [(ii)] $T_{V}\phi W=0$
\end{enumerate}
for $V,W\in\Gamma(ker\pi_{*})$ and $X\in\Gamma((ker\pi_{*})^\perp)$.
\end{corollary}

Now we obtain necessary and sufficient condition for conformal anti-invariant $\xi^\perp-$submersion to be totally geodesic. We note that a differentiable map $\pi$ between two Riemannian manifolds is called totally geodesic if $\nabla \pi_{*}=0.$ A geometric interpretation of a totally geodesic map is that it maps every geodesic in the total manifold into a geodesic in the base manifold in proportion to arc lengths.
\begin{theorem}
Let $\pi:(M,\phi,\xi,\eta,g_{M})\longrightarrow (N,g_{N})$ be a conformal anti-invariant $\xi^\perp-$submersion, where $(M,\phi,\xi,\eta,g_{M})$ is a Sasakian manifold and $(N,g_{N})$ is a Riemannian manifold. Then $\pi$ is a totally geodesic map if
\begin{align}
-\nabla^{\pi}_{X}\pi_{*}Y_2&=\pi_{*}(\phi(A_{X}\phi Y_{1}+\mathcal{V}\nabla_{X}\mathcal{B}Y_{2} +A_{X}\mathcal{C}Y_{2})+\mathcal{C}(\mathcal{H}\nabla_{X}\phi Y_{1}+A_{X}\mathcal{B}Y_{2}+\mathcal{H}\nabla_{X}\mathcal{C}Y_{2}))\label{e.q:3.11}\\
&-\eta(Y_{2})\pi_{*}X-\{g_{M}(X,\phi Y_1)+g_M(X,\mathcal{C}Y_2)\}\pi_{*}\xi\notag
\end{align}
for any $X\in\Gamma((ker\pi_*)^\perp), Y=Y_{1}+Y_{2}\in\Gamma(TM)$, where $Y_{1}\in\Gamma(ker\pi_{*})$ and $Y_{2}\in\Gamma((ker\pi_{*})^\perp)$.
\end{theorem}
\begin{proof}
By virtue of (\ref{e.q:2.2}) and (\ref{nfixy}) we have
\begin{align*}
(\nabla \pi_{*})(X,Y)&=\nabla^{\pi}_{X}\pi_{*}Y+\pi_{*}(-\nabla_{X}Y)\\
&=\nabla^{\pi}_{X}\pi_{*}Y+\pi_{*}(\phi\nabla_{X}\phi Y-g(X,\phi Y)\xi-\eta(Y)X)
\end{align*}
for any $X\in\Gamma((ker\pi_*)^\perp), Y\in\Gamma(TM)$. Then from (\ref{nxv}), (\ref{nxy}) and (\ref{e.q:3.2}) we get
\begin{align*}
(\nabla \pi_{*})(X,Y)&=\nabla^\pi_{X}\pi_{*}Y_2+\pi_{*}(\phi A_{X}\phi Y_{1}+\mathcal{B}\mathcal{H}\nabla_{X}\phi Y_{1}
+\mathcal{C}\mathcal{H}\nabla_{X}\phi Y_{1}+\mathcal{B}A_{X}\mathcal{B}Y_{2}\\
&+\mathcal{C}A_{X}\mathcal{B}Y_{2}+\phi\mathcal{V}\nabla_{X}\mathcal{B}Y_{2}
+\phi A_{X}\mathcal{C}Y_{2}+\mathcal{B}\mathcal{H}\nabla_{X}\mathcal{C}Y_{2}+\mathcal{C}\mathcal{H}\nabla_{X}\mathcal{C}Y_{2})\\
&-\eta(Y_{2})\pi_{*}X-\{g_{M}(X,\phi Y_1)+g_M(X,\mathcal{C}Y_2)\}\pi_{*}\xi
\end{align*}
for any $Y=Y_{1}+Y_{2}\in\Gamma(TM)$, where $Y_{1}\in\Gamma(ker\pi_{*})$ and $Y_{2}\in\Gamma((ker\pi_{*})^\perp)$. Thus taking into account the vertical parts, we find
\begin{align*}
(\nabla \pi_{*})(X,Y)&=\nabla^\pi_{X}\pi_{*}Y+\pi_{*}(\phi(A_{X}\phi Y_{1}+\mathcal{V}\nabla_{X}\mathcal{B}Y_{2}+A_{X}\mathcal{C}Y_{2})\\
&+\mathcal{C}(\mathcal{H}\nabla_{X}\phi Y_{1}+A_{X}\mathcal{B}Y_{2}+\mathcal{H}\nabla_{X}\mathcal{C}Y_{2}))\\
&-\eta(Y_{2})\pi_{*}X-\{g_{M}(X,\phi Y_1)+g_M(X,\mathcal{C}Y_2)\}\pi_{*}\xi.
\end{align*}
Thus $(\nabla \pi_{*})(X,Y)=0$ if and only if the equation (\ref{e.q:3.11}) is satisfied.\\
\end{proof}

We now present the following definition.
\begin{definition}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$ with
$(ker\pi_{*})^{\perp}=\phi(ker\pi_{*})\oplus<\xi>$. Then $\pi$ is called a $(\phi ker\pi_{*},\mu)$-totally geodesic map if
$$(\nabla \pi_{*})(\phi U,\xi)=0, for \ U\in\Gamma(ker\pi_{*})\ and \ \xi\in\Gamma((ker\pi_{*})^\perp).$$
\end{definition}
In the sequel we show that this notion has an important effect on the character of the conformal submersion.
\begin{theorem}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$ with
$(ker\pi_{*})^{\perp}=\phi(ker\pi_{*})\oplus<\xi>$. Then $\pi$ is a $(\phi ker\pi_{*},\mu)$-totally geodesic map if and only if $\pi$ is a horizontally homothetic map.
\end{theorem}
\begin{proof}
For $U\in\Gamma(ker\pi_{*})$ and $\xi\in\Gamma(\mu)$, from Lemma \ref{lem1}, we have
\begin{equation*}
(\nabla \pi_{*})(\phi U,\xi)=\phi U(\ln\lambda)\pi_{*}\xi+\xi(\ln\lambda)\pi_{*}\phi U-g_{M}(\phi U,\xi)\pi_{*}(grad\ln\lambda).
\end{equation*}
From above equation, if $\pi$ is a horizontally homothetic map then $(\nabla \pi_{*})(\phi U,\xi)=0.$ Conversely, if $(\nabla \pi_{*})(\phi U,\xi)=0,$ we obtain
\begin{equation}
\phi U(\ln\lambda)\pi_{*}\xi+\xi(\ln\lambda)\pi_{*}\phi U=0.\label{e.q:3.12}
\end{equation}
Taking inner product in (\ref{e.q:3.12}) with $\pi_{*}\phi U$ and if we take into account $\pi$ is a conformal submersion, we write
\begin{equation*}
g_{M}(grad\ln\lambda,\phi U)g_{N}(\pi_{*}\xi,\pi_{*}\phi U)+g_{M}(grad\ln\lambda,\xi)g_{N}(\pi_{*}\phi U,\pi_{*}\phi U)=0.
\end{equation*}
Above equation implies that $\lambda$ is a constant on $\Gamma(\mu).$ On the other hand, taking inner product in (\ref{e.q:3.12}) with $\pi_{*}\xi$, we have
\begin{equation*}
g_{M}(grad\ln\lambda,\phi U)g_{N}(\pi_{*}\xi,\pi_{*}\xi)+g_{M}(grad\ln\lambda,\xi)g_{N}(\pi_{*}\phi U,\pi_{*}\xi)=0.
\end{equation*}
From above equation, it follows that $\lambda$ is a constant on $\Gamma(\phi ker\pi_{*}).$ Thus $\lambda$ is a constant on $\Gamma((ker\pi_{*})^\perp).$ Hence proof is complete.
\end{proof}
Here we present another result on conformal anti-invariant $\xi^\perp-$submersion to be totally geodesic.
\begin{theorem}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ to a Riemannian manifold $(N,g_{N})$. $\pi$ is a totally geodesic map if and only if
\begin{enumerate}
\item [(a)] $T_{U}\phi V=0$ and $\mathcal{H}\nabla_{U}\phi V\in\Gamma(\phi ker\pi_{*})$,
\item [(b)] $\pi$ is a horizontally homotetic map,
\item [(c)] $A_{Z}\phi V=0$ and $\mathcal{H}\nabla_{Z}\phi V\in\Gamma(\phi ker\pi)$
\end{enumerate}
for $X,Y,Z\in\Gamma((ker\pi_{*})^\perp)$ and $U,V\in\Gamma(ker\pi_{*})$.
\end{theorem}
\begin{proof}
For any $U,V\in\Gamma(ker\pi_{*})$, using (\ref{e.q:2.3}) and (\ref{nfixy}) we have
\begin{align*}
(\nabla \pi_{*})(U,V)&=\nabla^{\pi}_{U}\pi_{*}V+\pi_{*}(-\nabla_{U}V)\\
&=\pi_{*}(\phi\nabla_{U}\phi V-g_{M}(U,\phi V)\xi-\eta(V)X)\\
&=\pi_{*}(\phi\nabla_{U}\phi V).
\end{align*}
Then from (\ref{nvw}) and (\ref{nvx}) we arrive at
\begin{equation*}
(\nabla \pi_{*})(U,V)=\pi_{*}(\phi T_{U}\phi V+\mathcal{C}\mathcal{H}\nabla_{U}\phi V).
\end{equation*}
From above equation, $(\nabla \pi_{*})(U,V)=0$ if and only if
\begin{equation}
\pi_{*}(\phi T_{U}\phi V+\mathcal{C}\mathcal{H}\nabla_{U}\phi V)=0. \label{e.q:3.13}
\end{equation}
Since $\phi$ is non-singular, $T_{U}\phi V=0$ and $\mathcal{H}\nabla_{U}\phi V\in\Gamma(\phi ker\pi_{*}).$ On the other hand, from Lemma \ref{lem1} we derive
\begin{equation*}
(\nabla \pi_{*})(X,Y)=X(\ln\lambda)\pi_{*}Y+Y(\ln\lambda)\pi_{*}X-g_{M}(X,Y)\pi_{*}(grad\ln\lambda)
\end{equation*}
for any $X,Y\in\Gamma(\mu).$ It is obvious that if $\pi$ is a horizontally homotetic map, it follows that $(\nabla \pi_{*})(X,Y)=0.$ Conversely, if $(\nabla \pi_{*})(X,Y)=0,$ taking $Y=\phi X$ in above equation, we get
\begin{equation*}
X(\ln\lambda)\pi_{*}\phi X+\phi X(\ln\lambda)\pi_{*}X=0.
\end{equation*}
Taking inner product in (\ref{e.q:3.13}) with $\pi_{*}\phi X,$ we obtain
\begin{equation}
g_{M}(grad\ln\lambda,X)\lambda^{2}g_{M}(\phi X,\phi X)+g_{M}(grad\ln\lambda,\phi X)\lambda^{2}g_{M}(X,\phi X)=0. \label{e.q:3.14}
\end{equation}
From (\ref{e.q:3.14}), $\lambda$ is a constant on $\Gamma(\mu).$ On the other hand, for $U,V\in\Gamma(ker\pi_{*})$, from Lemma \ref{lem1} we have
\begin{equation*}
(\nabla \pi_{*})(\phi U,\phi V)=\phi U(\ln\lambda)\pi_{*}\phi V+\phi V(\ln\lambda)\pi_{*}\phi U-g_{M}(\phi U,\phi V)\pi_{*}(grad\ln\lambda).
\end{equation*}
Again if $\pi$ is a horizontally homothetic map, then $(\nabla \pi_{*})(\phi U,\phi V)=0.$ Conversely, if $(\nabla \pi_{*})(\phi U,\phi V)=0$, putting $U$
instead of $V$ in above equation, we derive
\begin{equation}
2\phi U(\ln\lambda)\pi_{*}(\phi U)-g_{M}(\phi U,\phi U)\pi_{*}(grad\ln\lambda)=0. \label{e.q:3.15}
\end{equation}
Taking inner product in (\ref{e.q:3.15}) with $\pi_{*}\phi U$ and since $\pi$ is a conformal submersion, we have
\begin{equation*}
g_{M}(\phi U,\phi U)\lambda^{2}g_{M}(grad\ln\lambda,\phi U)=0.
\end{equation*}
From above equation, $\lambda$ is a constant on $\Gamma(\phi ker\pi_{*}).$ Thus $\lambda$ is a constant on $\Gamma((ker\pi_{*})^\perp).$ Now, for $Z\in\Gamma(\mu)$ and $V\in\Gamma(ker\pi_{*}),$ from (\ref{e.q:2.3}) and (\ref{nfixy}) we get
\begin{align*}
(\nabla \pi_{*})(Z,V)=\pi_{*}(\phi\nabla_{Z}\phi V).
\end{align*}
Using (\ref{nxv}) and (\ref{nxy}) we have
\begin{equation*}
(\nabla \pi_{*})(Z,V)=\pi_{*}(\phi A_{Z}\phi V+\mathcal{C}\mathcal{H}\nabla_{Z}\phi V).
\end{equation*}
Thus $(\nabla \pi_{*})(Z,V)=0$ if and only if
\begin{equation*}
\pi_{*}(\phi A_{Z}\phi V+\mathcal{C}\mathcal{H}\nabla_{Z}\phi V)=0.
\end{equation*}
Since $\phi$ is non-singular, $A_Z{\phi V}=0$ and $\mathcal{H}\nabla_Z{\phi V}\in\Gamma(\phi ker\pi_*).$ Thus proof is complete.
\end{proof}

Finally, in this section, We investigate the necessary and sufficient conditions for such submersions to be harmonic.
\begin{theorem}\label{teo5}
Let $\pi:(M^{2(m+n)+1},\phi,\xi,\eta,g_M)\longrightarrow (N^{m+2n+1},g_{N})$ be a conformal anti-invariant $\xi^\perp-$submersion, where $(M,\phi,\xi,\eta,g_{M})$ is a Sasakian manifold and $(N,g_{N})$ is a Riemannian manifold. Then the tension field $\tau$ of $\pi$ is
\begin{align}
\tau(\pi)&=-m\pi_{*}(\mu^{ker\pi_{*}})+(1-m-2n)\pi_{*}(grad\ln\lambda)\label{e.q:3.16}
\end{align}
where $\mu^{ker\pi_{*}}$ is the mean curvature vector field of the distribution of $ker\pi_{*}$.
\end{theorem}
\begin{proof}
Let $\{e_{1},...,e_{m},\phi e_{1},...,\phi e_{m},\xi,\mu_1,...,\mu_n,\phi \mu_1,...,\phi \mu_n\}$ be orthonormal basis of $\Gamma(TM)$ such that $\{e_{1},...,e_{m}\}$ be orthonormal basis of $\Gamma(ker\pi_{*})$, $\{\phi e_{1},...,\phi e_{m}\}$ be orthonormal basis of $\Gamma(\phi ker\pi_{*})$ and $\{\xi,\mu_1,...,\mu_n,\phi \mu_1,...,\phi \mu_n\}$ be orthonormal basis of $\Gamma(\mu)$. Then the trace of second fundamental form (restriction to $ker\pi_{*}\times ker\pi_{*}$) is given by
$$trace^{ker\pi_{*}}\nabla \pi_{*}=\sum_{i=1}^{m}(\nabla \pi_{*})(e_{i},e_{i}).$$ Then using (\ref{nfixy}) we obtain
\begin{equation}
trace^{ker\pi_{*}}\nabla \pi_{*}=-m\pi_{*}(\mu^{ker\pi_{*}}).\label{e.q:3.17}
\end{equation}
In a similar way, we have
\begin{equation*}
trace^{(ker\pi_{*})^{\perp}}\nabla \pi_{*}=\sum_{i=1}^{m}(\nabla \pi_{*})(\phi e_{i},\phi e_{i})+\sum_{i=1}^{2n}(\nabla \pi_{*})(\mu_{i},\mu_{i})+(\nabla \pi_{*})(\xi,\xi).
\end{equation*}
Using Lemma \ref{lem1} we arrive at
\begin{align*}
trace^{(ker\pi_{*})^{\perp}}\nabla \pi_{*}&=\sum_{i=1}^{m}2g_{M}(grad\ln\lambda,\phi e_{i})\pi_{*}\phi e_{i}-m\pi_{*}(grad\ln\lambda)\\
&+\sum_{i=1}^{2n}2g_{M}(grad\ln\lambda,\mu_{i})\pi_{*}\mu_{i}-2n\pi_{*}(grad\ln\lambda)\\
&+2\xi(\ln\lambda)\pi_{*}\xi-\pi_{*}(grad\ln\lambda).
\end{align*}
Since $\{\frac{1}{\lambda(p)}\pi_{*p}(\phi e_i),\frac{1}{\lambda(p)}\pi_{*p}(\mu_h),\frac{1}{\lambda(p)}\pi_{*p}\xi\}_{p\in M,\ 1\leq i\leq m,\, 1\leq h\leq n}$ is an orthonormal basis of $T_{\pi(p)}N$ and $\pi$ is a conformal anti-invariant
$\xi^\perp$-submersion, we derive
\begin{align}
trace^{(ker\pi_{*})^{\perp}}\nabla \pi_{*}&=\sum_{i=1}^{m}2g_{N}(\pi_*grad\ln\lambda,\frac{1}{\lambda}\pi_*\phi e_{i})\frac{1}{\lambda}\pi_{*}\phi e_{i}-m\pi_{*}(grad\ln\lambda)\notag\\
&+\sum_{i=1}^{2n}2g_{N}(\pi_*grad\ln\lambda,\frac{1}{\lambda}\pi_*\mu_{i})\frac{1}{\lambda}\pi_{*}\mu_{i}-2n\pi_{*}(grad\ln\lambda)\notag\\
&+2g_N(\pi_*grad\ln\lambda,\frac{1}{\lambda}\pi_{*}\xi)\frac{1}{\lambda}\pi_{*}\xi-\pi_{*}(grad\ln\lambda)\notag\\
&=(1-m-2n)\pi_*(grad\ln\lambda)\label{e.q:3.18}
\end{align}
Then proof follows from (\ref{e.q:3.17}) and (\ref{e.q:3.18}).\\
\end{proof}
From Theorem \ref{teo5} we deduce that:
\begin{theorem}
Let $\pi:(M^{2(m+n)+1},\phi,\xi,\eta,g_M)\longrightarrow (N^{m+2n+1},g_{N})$ be a conformal anti-invariant $\xi^\perp-$submersion, where $(M,\phi,\xi,\eta,g_{M})$ is a Sasakian manifold and $(N,g_{N})$ is a Riemannian manifold. Then any two conditions below imply the third:
\begin{enumerate}
\item [(i)] $\pi$ is harmonic
\item [(ii)] The fibres are minimal
\item [(iii)] $\pi$ is a horizontally homothetic map.
\end{enumerate}
\end{theorem}

We also have the following result.
\begin{corollary}
Let $\pi:(M^{2(m+n)+1},\phi,\xi,\eta,g_M)\longrightarrow (N^{m+2n+1},g_{N})$ be a conformal anti-invariant $\xi^\perp-$submersion, where $(M,\phi,\xi,\eta,g_{M})$ is a Sasakian manifold and $(N,g_{N})$ is a Riemannian manifold. $\pi$ is harmonic if and only if the fibres are minimal.
\end{corollary}

\section{Decomposition theorems}
In this section, we obtain decomposition theorems by using the existence of
conformal anti-invariant $\xi^\perp-$submersions. First, we recall the following results
from \cite{P}. Let $g$ be a Riemannian metric tensor on the manifold $B=M\times N$
and assume that the canonical foliations $D_{M}$ and $D_{N}$ intersect
perpendiculary everywhere. Then $g$ is the metric tensor of

(i) a twisted product $M\times_{f}N$ if and only if $D_{M}$ is a
totally geodesic foliation and $D_{N}$ is a totally umbilic foliation,

(ii) a warped product $M\times_{f}N$ if and only if $D_{M}$ is a
totally geodesic foliation and $D_{N}$ is a spheric foliation, i.e., it is
umbilic and its mean curvature vector field is parallel. We note that
in this case, from \cite{P} we have
\begin{equation}
\nabla_XU=X(\ln f)U \label{e.q:4.1}
\end{equation}
for $X\in\Gamma(TM)$ and $U\in\Gamma(TN),$ where $\nabla$ is the Levi-Civita connection on $M\times N.$

(iii) a usual product of Riemannian manifolds if and only if $D_{M}$ and
$D_{N}$ are totally geodesic foliations.

Our first decomposition theorem for a conformal anti-invariant $\xi^\perp-$submersion comes
from Theorem \ref{teo2} and Theorem \ref{teo3} in terms of the second
fundamental forms of such submersions.

\begin{theorem}
Let $\pi:(M,\phi,\xi,\eta,g_{M})\longrightarrow (N,g_{N})$ is a conformal anti-invariant $\xi^\perp-$submersion, where $(M,\phi,\xi,\eta,g_{M})$ is a Sasakian manifold and $(N,g_{N})$ is a Riemannian manifold. Then $M$ is a locally product manifold if
\begin{align*}
-\frac{1}{\lambda^{2}}g_{N}(\nabla^{\pi}_{X}\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)=g_{M}(A_{X}BY-\mathcal{C}Y(\ln\lambda)X+g_M(X,\mathcal{C}Y)\ln\lambda-\eta(Y)X,\phi V)
\end{align*}
and
\begin{align*}
-\frac{1}{\lambda^2}g_{N}(\nabla^{\pi}_{\phi W}\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)&=g_{M}(\phi\mathcal{C}X(\ln\lambda)\phi V-T_{V}\mathcal{B}X, \phi V)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)
\end{align*}
for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $U,V\in\Gamma(ker\pi_{*})$, where $M_{(\ker \pi_{\ast})^{\perp}}$ and $M_{(\ker \pi_{\ast})}$ are integral
manifolds of the distributions $(\ker \pi_{\ast})^{\perp}$ and $(\ker \pi_{\ast})$.
Conversely, if $M$ is a locally product manifold of the form
$M_{(\ker \pi_{\ast})^{\perp}}\times M_{(\ker \pi_{\ast})}$ then we have
\begin{align*}
\frac{1}{\lambda^{2}}g_{N}(\nabla^{\pi}_{X}\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)=g_{M}(\mathcal{C}Y(\ln\lambda)X-g_M(X,\mathcal{C}Y)\ln\lambda+\eta(Y)X,\phi V)
\end{align*}
and
\begin{align*}
-\frac{1}{\lambda^2}g_{N}(\nabla^{\pi}_{\phi W}\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)&=g_{M}(\phi\mathcal{C}X(\ln\lambda)\phi V,\phi V)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X).
\end{align*}
\end{theorem}
From Corollary \ref{cor1} and Corollary \ref{cor2}, we have the following theorem.
\begin{theorem}
Let $\pi$ be a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ onto a Riemannian manifold $(N,g_{N})$ with
$(ker\pi_{*})^{\perp}=\phi(ker\pi_{*})\oplus<\xi>$. Then $M$ is a locally product manifold if $A_{X}\mathcal{B}Y=\eta(Y)X$ and $T_{V}\phi W=0$ for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V,W\in\Gamma(ker\pi_{*})$.
\end{theorem}

Next we obtain a decomposition theorem which is related to the
notion of twisted product mani-fold. But we first recall the adjoint
map of a map. Let $\pi:(M_1,g_1)\rightarrow(M_2,g_2)$ be a map between
Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2).$ Then the adjoint
map $^{*}\pi_*$ of $\pi_*$ is characterized by
$g_1(x,^{*}\pi_{*p_1}y)=g_2(\pi_{*p_1}x,y)$ for $x\in T_{p_1}M_1,\, y\in
T_{\pi(p_1)}M_2$ and $p_1\in M_1.$ Considering $\pi^{h}_*$ at each
$p_1\in M_1$ as a linear transformation
$$\pi^{h}_{*p_1}:((\ker \pi_*)^\perp(p_1),g_{1_{p_1((\ker \pi_*)^\perp(p_1))}})\rightarrow (range \pi_*(p_2),g_{2_{p_2((range \pi_*)(p_2))}}),$$
we will denote the adjoint of $\pi^{h}_{*}$ by $^{*}\pi^{h}_{*p_1}.$ Let $^{*}\pi_{*p_1}$ be adjoint of
$\pi_{*p_1}:(T_{p_1}M_1,g_{1_{p_1}})\rightarrow (T_{p_2}M_2,g_{2_{p_2}}).$ Then the linear transformation
$$(^*\pi_{*p_1})^h:range \pi_*(p_2)\rightarrow (\ker \pi_*)^\perp(p_1)$$ defined by $(^{*}\pi_{*p_1})^{h}y= ^{*}\pi_{*p_1}y,$ where
$y\in\Gamma(range\pi_{*p_1}), p_2=\pi(p_1),$ is an isomorphism and $(\pi^{h}_{*p_1})^{-1}=(^{*}\pi_{*p_1})^h= ^{*}\pi^{h}_{*p_1}.$
\begin{theorem}
Let $\pi:(M,\phi,\xi,\eta,g_{M})\longrightarrow (N,g_{N})$ is a conformal anti-invariant $\xi^\perp-$submersion, where $(M,\phi,\xi,\eta,g_{M})$ is a Sasakian manifold and $(N,g_{N})$ is a Riemannian manifold. Then $M$ is a twisted product manifold of the from $M_{(ker\pi_{*})}\times_{\lambda} M_{(ker\pi_{*})^\perp}$ if and only if
\begin{align}
-\frac{1}{\lambda^2}g_{N}(\nabla^{\pi}_{\phi W}\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)&=g_{M}(\phi\mathcal{C}X(\ln\lambda)\phi V-T_{V}\mathcal{B}X, \phi V)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)\label{e.q:4.2}
\end{align}
and
\begin{align}
g_{M}(X,Y)H&=-\mathcal{B}A_{X}\mathcal{B}Y+\mathcal{C}Y(\ln\lambda)\mathcal{B}X-\mathcal{B}\mathcal{H}(grad\ln\lambda)g_{M}(X,\mathcal{C}Y)\notag\\
&-\phi ^{*}\pi_*(\nabla^{\pi}_X\pi_{*}CY)+\eta(Y)\mathcal{B}X \label{e.q:4.3}
\end{align}
for $X,Y\in\Gamma((\ker \pi_{*})^{\perp})$ and $V,W\in\Gamma(\ker \pi_{*}),$
where $M_{(\ker \pi_{*})^{\perp}}$ and $M_{(\ker \pi_{*})}$ are integral
manifolds of the distributions $(\ker \pi_{*})^{\perp}$ and $(\ker \pi_{*})$
and $H$ is the mean curvature vector field of $M_{(\ker \pi_{*})^{\perp}}.$
\end{theorem}
\begin{proof}
Since $g_{M}(W,\xi)=0,$ using (\ref{e.q:2.3}) we have $g_{M}(\nabla_{V}W,\xi)=-g_{M}(W,\nabla_{V}\xi)=-g_{M}(W,\phi V)=0$
for $V,W\in\Gamma(ker\pi_{*})$ and $\xi\in\Gamma((ker\pi_{*})^\perp).$  Thus we have
\begin{align*}
g_{M}(\nabla_{V}W,X)&=g_{M}(\phi\nabla_{V}W,\phi X)+\eta(\nabla_{V}W)\eta(X)\\
&=g_{M}(\phi\nabla_{V}\phi W,\phi X)\\
&=g_{M}(\nabla_{V}\phi W,\phi X)-g_{M}((\nabla_{V}\phi)W,\phi X).
\end{align*}
Using (\ref{e.q:2.3}), (\ref{nvw}) and (\ref{e.q:3.2}) we have
\begin{equation*}
g_{M}(\nabla_{V}W,X)=g_{M}(T_{V}\phi W,\mathcal{B}X)+g_{M}(\mathcal{H}\nabla_{V}\phi W,\mathcal{C}X).
\end{equation*}
Since $\nabla$ is torsion free and $[V,\phi W]\in\Gamma(ker\pi_{*})$ we obtain
\begin{equation*}
g_{M}(\nabla_{V}W,X)=g_{M}(T_{V}\phi W,\mathcal{B}X)+g_{M}(\nabla_{\phi W}V,\mathcal{C}X).
\end{equation*}
Using (\ref{e.q:2.3}) and (\ref{nxy}) we have
\begin{align*}
g_{M}(\nabla_{V}W,X)&=g_{M}(T_{V}\phi W,\mathcal{B}X)+g_{M}(\phi\nabla_{\phi W}V,\phi\mathcal{C}X)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)\\
&=g_{M}(T_{V}\phi W,\mathcal{B}X)+g_{M}(\nabla_{\phi W}\phi V,\phi\mathcal{C}X)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)
\end{align*}
here we have used that $\mu$ is invariant. Using (\ref{nfixy}) and Lemma \ref{lem1} (i) and if we take into account
that $\pi$ is a conformal submersion, we obtain
\begin{align*}
g_{M}(\nabla_{U}V,X)&=g_{M}(T_{V}\phi W,\mathcal{B}X)+\frac{1}{\lambda^2}g_{M}(\mathcal{H}gradln\lambda,\phi W)g_{N}(\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)\\
&-\frac{1}{\lambda^2}g_{M}(\mathcal{H}gradln\lambda,\phi V)g_{N}(\pi_{*}\phi W,\pi_{*}\phi\mathcal{C}X)\\
&+g_{M}(\phi W,\phi V)\frac{1}{\lambda^2}g_{N}(\pi_{*}(\mathcal{H}gradln\lambda),\pi_{*}\phi\mathcal{C}X)\\
&+\frac{1}{\lambda^2}g_{N}(\nabla_{\pi_{*}\phi W}\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X).
\end{align*}
Moreover, using Definition \ref{def} and (\ref{cywv}), we obtain
\begin{align*}
g_{M}(\nabla_{U}V,X)&=g_{M}(\phi\mathcal{C}X(\ln\lambda)\phi V-T_{V}\mathcal{B}X, \phi V)+\eta(\nabla_{\phi W}V)\eta(\mathcal{C}X)\\
&+\frac{1}{\lambda^2}g_{N}(\nabla^{\pi}_{\phi W}\pi_{*}\phi V,\pi_{*}\phi\mathcal{C}X).
\end{align*}
Thus it follows that $M_{(ker\pi_{*})}$ is totally geodesic if and only if the equation (\ref{e.q:4.2}) is satisfied. On the other hand, for $X,Y\in\Gamma((ker\pi_{*})^\perp)$ and $V\in\Gamma(ker\pi_{*})$, by using (\ref{e.q:2.2}), (\ref{nxv}), (\ref{nxy}), (\ref{e.q:3.1}), (\ref{e.q:3.2}) and (\ref{e.q:3.8}), have
\begin{align*}
g_{M}(\nabla_{X}Y,V)&=g_{M}(A_{X}BY,\phi V)+g_{M}(\nabla_{X}\mathcal{C}Y,\phi V)-\eta(Y)g_{M}(X,\phi V).
\end{align*}
Since $\pi$ is a conformal submersion, using (\ref{nfixy}) and Lemma (\ref{lem1}) we arrive at
\begin{align*}
g_{M}(\nabla_{X}Y,V)&=g_{M}(A_{X}BY,\phi V)-\frac{1}{\lambda^2}g_{M}(\mathcal{H}gradln\lambda,X)g_{N}(\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)\\
&-\frac{1}{\lambda^2}g_{M}(\mathcal{H}gradln\lambda,\mathcal{C}Y)g_{N}(\pi_{*}X,\pi_{*}\phi V)\\
&+\frac{1}{\lambda^2}g_{M}(X,\mathcal{C}Y)g_{N}(\pi_{*}(\mathcal{H}gradln\lambda,\mathcal{C}Y),\pi_{*}\phi V)\\
&+\frac{1}{\lambda^2}g_{N}(\nabla_{\pi_{*}X}\pi_{*}\mathcal{C}Y,\pi_{*}\phi V)-\eta(Y)g_{M}(X,\phi V).
\end{align*}
Moreover, using Definiton \ref{def} and (\ref{cywv}) we obtain
\begin{align*}
g_{M}(\nabla_{X}Y,V)&=g_{M}(A_{X}BY-\mathcal{C}Y(\ln\lambda)X+g_M(X,\mathcal{C}Y)\ln\lambda-\eta(Y)X,\phi V)\\
&+\frac{1}{\lambda^2}g_{N}(\nabla_{\pi_{*}X}\pi_{*}\mathcal{C}Y,\pi_{*}\phi V).
\end{align*}
From above equation, $M_{(ker\pi_{*})^\perp}$ is totally umbilical if and only if the equation (\ref{e.q:4.3}) is satisfied.
\end{proof}
However, in the sequel, we show that the notion of conformal anti-invariant $\xi^\perp-$submersion puts some restrictions on the total space for locally warped product manifold.
\begin{theorem}
Let $\pi$ is a conformal anti-invariant $\xi^\perp-$submersion from a Sasakian manifold $(M,\phi,\xi,\eta,g_{M})$ to a Riemannian manifold $(N,g_{N})$. If $M$ is a locally warped product manifold of the from $M_{(ker\pi_{*})^\perp}\times_{\lambda} M_{(ker\pi_{*})}$, then either $\pi$ is a horizontally homothetic submersion or the fibres are one dimensional.
\end{theorem}
\begin{proof}
For $X\in\Gamma((ker\pi_{*})^\perp)$ and $V,W\in\Gamma(ker\pi_{*})$, $g_{M}(W,\xi)=0$ implies that from (\ref{e.q:2.3}), $g_{M}(\nabla_{V}W,\xi)=-g_{M}(W,\nabla_{V}\xi)=-g_{M}(W,\phi V)=0.$ Thus we have
\begin{align*}
g_{M}(\nabla_{V}W,X)&=g_{M}(\phi\nabla_{V}W,\phi X)+\eta(\nabla_{V}W)\eta(X)\\
&=g_{M}(\phi\nabla_{V}\phi W,\phi X)\\
&=g_{M}(\nabla_{V}\phi W,\phi X)-g_{M}((\nabla_{V}\phi)W,\phi X).
\end{align*}
Using (\ref{e.q:2.3}) and (\ref{e.q:4.1}), we get
\begin{align*}
g_{M}(\nabla_{V}W,X)&=g_{M}(\nabla_{V}\phi W,\phi X)-g_{M}(-g_{M}(V,W)\xi-\eta(W)V,\phi X)\\
-X(\ln\lambda)g_{M}(V,W)&=\phi W(\ln\lambda)g_{M}(V,\phi X).
\end{align*}
For $X\in\Gamma(\mu),$ we derive
\begin{equation*}
-X(\ln\lambda)g_{M}(V,W)=0.
\end{equation*}
From above equation, we conclude that $\lambda$ is a constant on $\Gamma(\mu).$ For $X=\phi V\in\Gamma(\phi ker\pi_{*})$ we obtain
\begin{equation}
\phi V(\ln\lambda)g_{M}(V,W)=\phi W(\ln\lambda)g_{M}(V,V).\label{e.q:4.4}
\end{equation}
Interchanging the roles of $W$ and $V$ in (\ref{e.q:4.4}) we arrive at
\begin{equation}
\phi V(\ln\lambda)g_{M}(V,W)=\phi V(\ln\lambda)g_{M}(W,W).\label{e.q:4.5}
\end{equation}
From (\ref{e.q:4.4}) and (\ref{e.q:4.5}) we get
\begin{equation}
\phi V(\ln\lambda)=\phi V(\ln\lambda)\frac{g_{M}(V,W)^{2}}{\parallel V \parallel^{2}\parallel W \parallel^{2} }.\label{e.q:4.6}
\end{equation}
From (\ref{e.q:4.6}), either $\lambda$ is a constant on $\Gamma(\phi ker\pi_{*})$ or $\Gamma(\phi ker\pi_{*})$ is 1-dimensional. Thus proof is complete.
\end{proof}

redpajama_set_name: RedPajamaArXiv


HSN Community / Forums / HSN TV / What's the latest you've waited to start your holiday shopping?
What's the latest you've waited to start your holiday shopping?
What's the latest you've waited to start your holiday shopping? Share your shopping dilemmas and join the conversation with Shivan Sarna during the 24 Hour Christmas In July Event July 13.
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One year we ordered everything on line and were left giftless, only to have to run out last minute, and the neighbor was ambushed to babysit.
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I've started in May for the last two Christmas holidays.??
I always started shopping in July and my husband didn't get it. I think he thought I was just having ealy Christmas fun. (he has only shopped for me..I take care of all the other shopping) After 10 years of my early Christmas fun, he asked if wouldn't it be better in December and enjoy all the seasonal fun. He volenteered to help. You know where this is going. We went out first week in December ready to do the shopping and he was out of patience, energy and common sence. After the trunk was loaded (4 hours later) we went home and unpacked. We took a hour break and I was ready to go back out and finish. He was gassed!!
That was 30 years ago and I start in shopping in July and have tons of time to enjoy my friends and family. I host cookie parties and enjoy the food, music and all the joys of Christmas.
Last year was the worst for me. Health issues and absolutely no ideas for my family and friends was holding me back. Finally, on December 10th I fired up my computer, bugged people for ideas and hit my keys for 2 days straight. I love internet shopping, no crowds and you know your item is there! I started already this year, but have to admit that last year was kind of fun.
One Christmas holiday we decided to go off to a tropical dive trip and omit gift giving. All our fellow divers decided on Christmas Eve to have a gift exchange. We had 3-4 hours to come up with gifts on a rustic island without shopping. We all scrambled to create, wrap and exchange gifts at sunset on the beach. Most were pretty seashells, travel or dive items, postcards…it was really fun and festive until one of the shell gifts woke up. Yes one of the guests had collected a shell with a live hermit crab inside. I will never forget the poor guy who got bit and his loud yell!
HI SHIVAN!!! I USUALLY SHOP FOR CHRISTMAS STARTING IN JANUARY!!!! I HAVE TO START EARLY IN ORDER TO GET THROUGH MY LIST!!! ONE YEAR, DUE TO SEVERAL SURGERIES, (FIVE), I WASN'T ABLE TO SHOP UNTIL MID DECEMBER!!! IT WAS A TERRIBLE EXPERIENCE AND I NEVER WANT TO REPEAT IT AGAIN!! YOU'RE LIMITED TO WHATEVER IS LEFT TO CHOOSE FROM. NOWADAYS, I SHOP ONLINE AND THAT'S A GREAT RELIEF!!!
I start Christmas shopping early and usually finish before Thanksgiving. Of course, I still purchase a few items, such as stocking stuffers, after Thanksgiving because I enjoy the hustle and bustle of Christmas.
If my Christmas shopping isn't finished by Nov 1 I am late. I hate the crowds so I always shop early!!!?? ??
waited, what waited? with the excellent sales at hsn, my christmas shopping is finished. christmas in july, and all of that. did i mention that i have a very small family? thank you hsn!!!


Mary Anne Rufino — Riverdale-on-Hudson Funeral Home, Inc.
Mary Anne Rufino (née Foley) died on January 22, 2019 at Schervier Nursing Care Center. Loving mother to Lou (Marijean). Predeceased by husband Lou and son Gregory. Born and raised in the Riverdale area and lived there until relocating with her husband and children to Rockland County. Through her son Gregory, Mary Anne was strongly involved with children of special needs and an advocate for their rights. She will be missed by her loving cousins, sister-in-law, nieces, nephews and friends.
The wake will be held on Friday, January 25th 2019 from 2-4 & 7-9 pm at Riverdale-on-Hudson Funeral Home, 6110 Riverdale Avenue, Bronx NY. Mass of Christian Burial Satrurday, January 26th 2019 at 9:45 am at St. Margaret of Cortona Church, 6000 Riverdale Avenue, Bronx NY. Interment Gate of Heaven Cemetery.


The 2014 International Champions Cup (or ICC) was a friendly association football tournament played in the United States and Canada. It began on July 24, 2014 when Olympiacos defeated Milan 3–0 at BMO Field in Toronto, and ended on August 4, 2014 at Sun Life Stadium in Miami Gardens, Florida. This tournament followed the 2013 edition and was staged throughout the United States, with one match held in Canada (Toronto). The participating teams were Liverpool, Manchester City and Manchester United of England; Milan, Internazionale and Roma of Italy; Olympiacos of Greece; and defending champions Real Madrid of Spain. Manchester United won the tournament, having beaten Liverpool 3–1 in the final.
"Manchester United F.C. v Real Madrid C.F."
"Real Madrid C.F. v A.S. Roma"
"Real Madrid C.F. v Inter Milan"


A writer, creative consultant, and at this point, an arts administrator and curator; Scott Holzman, the Creative Director of Chase Public is personally interested in seeing what can happen when people are put into a conversation with one another that they didn't really expect to have.
If you weren't familiar: Chase Public is a collaborative space for art and assembly. Their hosted events can really be anything, but it's often poetry readings. Everyone that has been involved with Chase Public has been a writer and that's where their gravitational center is. They also host concerts, film screenings, workshops, games. . . anything that makes sense to do in the space, they are interested in. Very rarely do they say no to something; more often than not we will just say, "let's look and see how we can make that idea work".
Chase Public's mission is to cultivate empathy through creative practice. They use the perspective and talent of artists and creative people, and the methodologies that they use to put their creativity out, and create a center for them to gather and share ideas and communicate.
Five-Dots: When was Chase Public established?
Scott Holzman: I was not involved with Chase Public in the beginning. The earliest artifact of Chase Public, that I've found, was a newsletter that they put out in April of 2010. Its' been around for a little while.
F-D: Who were the original founders?
SH: The founders were our current Board President, Mike Fleisch and two folks that aren't really around much any more: Dan Todd and Brad Salyers.
F-D:. . . And how long have you been involved?
SH: I think I hopped on in the fall of 2013.
F-D: How long have you been in the same position?
SH: When I came to Chase Public, it didn't exist in the same way that it does now. The first couple of years, Chase was only creating about four events a year, mostly poetry readings. That was something that I was interested in and gradually started showing up and then became more and more involved and present in what was going on. There weren't really any "positions" at Chase until about 2015. It was just a space that a few people utilized. For the last couple of years I've been, kind of, the person behind the curtain.
SH: Currently our board is: Mike Fleisch, Elese Daniel, and Nathan Swartzendruber and also, now, we have Marcus Donaldson helping out with communications.
F-D: How does the structure of the building inform the workplace practices and philosophies of Chase, did you have a plan when you moved in as to how you wanted to utilize the space?
SH: I'll preface this by stating that a lot of the questions I will probably answer in a comparative way with us having just recently moved into this location. I think one of the things that drew us to this space was how accessible it was. The opportunity to move into a ground level space was a huge plus for us. Being on the ground level does, in some way, impact the way things feel in here. Before when we were in the second floor space, it felt like this cool little clubhouse that you wouldn't know about if you were just walking past. With this space we have these big windows and doors and everyone that drives by, especially at night, with the space all lit up. It's nice, and perhaps because of our visibility we have formed a more public focus. I think things feel a lot more open now. Our old space had this really gross carpet and drop ceilings that weren't great; so to come into this space with these wonderful eighteen foot ceilings and nice polished floors— it just feels a lot more open and available.
To answer that second part of the question: I'd say forty percent, "yes" and sixty percent, "no". Certainly our programming is informed by our experience in Northside. One of the things I'm most excited about, now that we're settled into the space, is to see what kind of ideas people come to us with. I think so much of our identity comes from sharing. We aren't telling people what kind of events they should be enjoying. We're helping people who have ideas with the process of bringing those ideas out into the public. I'm excited to see who comes in with some idea and completely flips my brain out. I'm certain that it will happen, and I look forward to it.
F-D :How many events have you had at this location?
SH: About six or seven.
F-D: How has the reception been since the move?
SH: It's been pretty good—better than we thought it would be, to be honest. Even being as excited and optimistic as you can be about moving into a new space, there is always that nagging question: "What if this is a huge mistake?". But people have been really receptive and complimentary to the new thing. The crowds have been really good. It's been a really nice welcome for us.
Beyond that, I was afraid that we would lose some of the intimacy of the old space—some of the magic that was in that little, old room. It was a little nerve wracking but as it turns out, the magic came from the people that participated and came here. I'm obviously very happy that that still exists.
SH: Our mission is to cultivate empathy through creative practice. We use the perspective and talent of artists and creative people, and the methodologies that they use to put that out, and create a center for them to gather and share ideas and communicate.
Another big thing that's important to us is to think of art, curation, creativity, and attendance . . . any of those things, as a gift. Everything is always going to be free, here. It's this nice circle of us providing a space for free, artists providing the gift of their expression, and audiences providing the gift of their attention, focus, and feedback. Being able to do all of that in non market ways feels really good, healthy, and genuine.
F-D: Not to deter from your non capitalist ethos, but within all of your programming being free, how are you funding yourself?
SH: Generosity is what makes our funding possible.
F-D: That's a great answer.
F-D: Has the location of the space influenced your work selection or selection process in any way?
SH: Perhaps in a more macro sense than the question implies; being in the Midwest has affected us. This is a place for things that don't really have homes otherwise because there aren't ten thousand poets in the region. Being a place for these people to come together is really important to us. We feel there is a need for that.
F-D: What other poetry collectives or non delegated performance venues are you aware of in the region?
SH: I haven't yet come across or been told about another space that has chosen to operate in the same way that we do. A lot of poetry groups exist within academic circles and galleries, coffee house open mic scenarios, and DIY venues. Those are great and they're obviously essential to the culture. We feel as if having less structured opportunities that might not fit those formats—for example, having a flexible schedule for artists and poets that might be driving through town on a whim—works well for us. That flexibility is nice because we can be responsive to opportunities as they emerge. I don't think that I've ever been scheduled out more than six weeks out. We're just constantly responding to the people who want to use the space and let them do with it what they want. That doesn't answer your question at all. There are poetry venues within most major cities, but for whatever reason, Cincinnati didn't have one.
F-D: What sort of responsibility to do you feel toward performance artists in the area?
SH: I would love, absolutely love, to support and cultivate performance art in the city. We just haven't hit a point of sustainable endeavors into performance art. It comes in waves: we'll get a handful of artists interested and then it will fall down again. I'd love it if I regularly had people wanting to perform works in here.
SH: Does anyone ever answer this question like, "Yep! I've got an answer. Let me get my agenda!" [laughs]. It varies from day to day. I spend a lot of time in front of a screen. I spend a lot of time scheduling performances, getting coffee with people and talking about their ideas. I spend a lot of time moving chairs— that's a big part of my day. If there is a performance I have to make sure the space is ready for that. I spend a lot of time making lists. I'm very often, five of the seven nights of the week, going to some event around town. The moment that I stop knowing what is going on within the creative culture of the city is the moment that I'm not very good at my job.
SH: Just referencing our furniture we have a mid-century vibe going on here. I think it's nice that we have a lot of hand-me-down, crappy, versions of really nice stuff rather than having really nice versions of crappy stuff. I think it's comfortable in that way. Everything in here is second hand. It's comfortable in that nothing is too nice. I think that that applies to the building as well. It's so nice that that drainage pipe is exposed, these outlets are off kilter, not one thing in this building is level. It would feel wrong if it was correct.
F-D: Do you think there is an aesthetic or ethos in terms of programming?
SH: It's hard to say because I don't think that we curate things in a very traditional way. I'm not giving people month long runs of shows in which I hand select things based on a specific criteria. It may be someone who is traveling from Cleveland to Nashville on January 3rd and they wanna do a reading. I'll just ask to see some of their stuff, look at it, and try to find someone locally who matches or aligns with that artist or writer. I guess the overarching theme would be people who are willing to put themselves out here. It's flexible.
F-D: Do you think that there is a general philosophy or mindset that is applicable to the program and how you select performers?
SH: There is a series that we run: The Response Project. About once a month we round up a group or artists, writers, and thinkers from different backgrounds and then ask them to create ten minute responses to the same piece of art. The first piece we selected, Wanderer Above a Sea Of Fog by Caspar David Friedrich, we had eight participants: artists, poets, musicians, photographers, and professors, each come up with ten minutes in response to that painting. Immediately we knew that it was a good idea. I think that I just finished creating the line up for The Response on the 28th of May. There is a commonality or preference in our perspective that has emerged through The Response Project. That is, the intense power that develops when you put things together that don't necessarily belong in the same place and time. That commonality is present when you have a violinist followed by an indigenous rights activist followed by a performance poet followed by a sociologist create quick slices of material that's generated for the same purpose. It's different from highlighting a poet and artist next to each other with examples of their best, most polished work. Having in quick succession, two very different takes that were created in a month, and seeing them not totally finished, next to things that aren't totally finished, is nice. Nate May, a composer, once stated that (and I believe he was referencing The Response Project), Chase Public is "a place where everyone cares and no body minds". I thought that was a really nice compliment.
SH: Right up the block is a gallery, Wave Pool. Its been nice to be in such close proximity to them and we able to work on things together. I think we have very overlapping philosophies and very little overlapping programs. So, I don't perceive there to be any toes being stepped on. We have a pretty nice relationship with the Mercantile Library and the Contemporary Art Center. Sometimes I feel as if we are the little sibling to those places. They have a much more broad and national focus while we are doing similar things in a more local way.
F-D: Does the history of the community or social issues affect reading or performance selection?
SH: We're sensitive to the role that art organizations and non-profits have in neighborhood development and gentrification. We're working hard to minimize our impact in that regard, and be good, responsible members of an artistic community as well as the new geographic and cultural community we find ourselves in. We're trying to be good neighbors. Now that the weather is a little nicer, we're getting to know a lot more of our neighbors and we're getting a sense of the sort of programming that they would like to see. We prioritize those ideas on our calendar.
F-D: What are some of those interests?
SH: One thing we're exploring and taking some steps toward is specific and intentional programming for children and high school aged kids. In our old neighborhood, there were multiple amazing youth art nonprofits that did an amazing job, so we were never really focused on that. Now we're hearing that folks in the area see that as something they'd like to see more of, so we're exploring what we can do to work that into our offerings.
F-D: What influences outside of contemporary art or literature influence the programming?
SH: I think I'd say emerging design theory helps us think about the way that we are designing things. We are never trying to be restrictive, we try to be flexible and meet the needs as the needs reveal themselves. The gift theory is also really steering for us. For some ways, playfulness, is important. We aren't trying to be some austere stereotype of artist or poet. It doesn't have to be that serious. It can be important without being serious.
F-D: What does having a physical space to organize within your community mean to you and how do you make it a useful, productive space for everyone?
SH: I think when we decided that it was the right idea to move here, I had the realization that this space almost provides us a set of priorities. Could we do the work that we do without having our own space? Yes. We could move things around and schedule events in flux spaces. What having a space provides is a consistent presence; building that level of comfort with people in that they know what to expect and it removes a layer of anxiety in not knowing what your getting into. It lets us be steady. It's comfortable and that's important.
F-D: Do you see your space as relating to any current movement or direction in writing or culture? Which other organizations might your work be in conversation with?
SH: We are most certainly not alone in terms of having a community focus. The nice thing about that is it puts us in conversation with everyone. The point is to be in conversation, so I would say, that with the focus on those things that the conversation is wide open.
F-D: Are there any organizations that you feel a kinship to but, perhaps, haven't collaborated with?
SH: There is a creative re-use center in Over-The-Rhine called Indigo Hippo, PAR-Projects in Northside. . . it's kind of hard to think of a group that we haven't worked with that we feel related to because we have done some sort of collaboration with almost everyone at this point. It's basically more about if we just can't do it for some logistical reason. It's so hard for me to not think locally. I can think of plenty of organizations in Louisville or Indianapolis that we haven't worked with, that perhaps we'd like to, but I just can't get my head out of Cincinnati.
F-D: Why do you think that is?
SH: I think we're just so steeped here; it's probably actually a negative that we are so hyper focused. It would probably be nice to work with groups outside of here.
F-D: Somewhere in there. . .
SH: Yea, but I'm probably not going to think of something on the spot. That's, like, a poisonous question to ask a poet.
F-D: We ask because, typically, the people that state that they don't have one, typically explain their way into it. Those answers are always interesting because they're kind of stumbled upon.
SH: I mean. . . it's kind of like you're asking for a catch phrase. . .and it would have to be something around accessibility: in terms of creating a culture that is kind, supportive, accepting, and being literally and financially accessible. It needs to be a place for anyone. It would be somewhere in there.
SH: I run an art space called Chase Public.
SH: I have not received a regular pay check in four years.
F-D: How's that treating ya?
SH: Everything that's coming to mind is just intentionally cheesy. Like, don't say everything that comes into your mind.
SH: We are going to be having our first visual show. We moved into this big space that kind of begs to have something on its walls.
F-D: Is it going to be thematic? What kind of show is it?
SH: It'll go up in, probably, November. We have six illustrators and six writers who are meeting here weekly for three months to play Dungeons & Dragons together. At the end of the twelve week campaign, we will each have two months to create new work based on that experience which we will show, then put it all out as a book. We're excited about this as an opportunity to explore the relationship between play and creative practice, as well as the way that groups of strangers will come to know and understand one another by interacting through assumed characters and personalities. We have a mix of new and experienced players, and for the most part people don't really know each other. The players: Scot Holzman, Lindsay Nehls, John Sebastian, Geoff Woolf, Anissa Pulcheon, Christina Wald, Marcus Donaldson, Gabrielle Lanza, Aiesha Little, and Jay Kalagayan. Our Dungeon Masters are Alexa Justice and Jeb Brack.
F-D: That sounds amazing, what other events or shows are coming up for you?
SH: We have several concerts and readings this month, as well as a discussion with Jim Bowsher; but seeing as how there are several events each week throughout the month of June, it's probably easiest for your audience to simply visit our events page.
In addition: Cassandra was able to visit during a Poetry Reading including but not limited to Mark Mendoza and Cathy Wagner, as are pictured.


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Wings tossed in a combination of medium, BBQ, and garlic sauces and then grilled.
Diced grilled chicken, applewood smoked bacon, mixed greens, tomatoes, bleu cheese crumbles, cucumbers, avocado, and egg tossed in bleu cheese dressing. Topped with potato sticks.
Melted aged Cheddar, pickles, lettuce, tomato, sliced onion, and burger sauce. Served with American fries.
Succulent dry rubbed baby back ribs, slow roasted in smokers with BBQ sauce, coleslaw, and American fries.
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American fries and tartar sauce.
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Mashed potatoes and vegetable topped with herb butter and lemon.
Topped with melted Cheddar cheese and Parmesan panko crumbs.


Is Laura Carmichael Lesbian ?
An English actress who is popular for starring as Lady Edith Crawley in the popular ITV and PBS series Downton Abbey. She is daughter to Andy Carmichael, a software consultant and Sarah, a radiographer. She completed her graduation from Bristol Old Vic Theater School in 2007. She has also made guest appearance in series like The Heart of Thomas Hardy and End of Our Street. Tinker Tailor Soldier Spy, Madame Bovary, Burn, Burn, Burn etc are some of the movies she has starred on. In 2013 she starred as the Arabella in The Fitzrovia Radio Hou which was shown in The Underglobe Theater.
No currently estimated rumor and appropriate details are there in media about her personal affairs and dating. She is not married yet. It seems like she is kind of person who wants privacy regarding her private matter. Since she has not married yet there is no question of getting divorce. She is an avid user of twitter and she has got more than 38 thousands of followers on her twitter account.


text:
Alumnus Nguyen Thi Mai Anh joins the developer circle in New Zealand
Tuesday, 24/08/2021, 15:08 (GMT+7)
Mai Anh's winning of a full scholarship from the renowned University of Waikato and landing a job as an analyst developer in New Zealand were part of her plan during her high school years at Vinschool.
Nguyen Thi Mai Anh, an alumnus of Vinschool Times City High School, was known years ago for winning a 100% scholarship worth NZ 85,000 (about VND 1.35 billion) from the University of Waikato (top 1.1% of best universities in the world according to QS rankings). Shortly after finishing her Bachelor's degree with Honours, Mai Anh landed a full-time job as an analyst developer at FNZ, a reputable tech firm in New Zealand.
As a Vinser, she had a strong preference for programming design and took initial steps towards pursuing it as a career by actively participating in multiple after-school clubs:
Head of Design – Vietnam National Model United Nations (VNMUN) 2016
Head of Design & Event Planning – LUMOS (Vinschool Student Council's event)
Chief Editor & Designer of Vinsers' Yearbook 2014 – 2017
Designer of the Monthly Vinschool Magazine
Representative of Waikato International Students' Association
"I kept telling myself to take an active part in after-school clubs to find out exactly what I was passionate about and what I was really good at. My winning in Computer Skills for the Office competition when I was in grade 11, getting 3rd prize in the Internet-based Olympiad of English (IOE), and experiences with actual projects all helped me figure out my passion, hence my decision to pursue Interface Design & Computer Science as my college major," shared Mai Anh.
Despite the challenges of living far from home, Mai Anh tried to adapt to her new life and arrange her schedule as best as she could. "I acquired effective time management skills when I was a Vinser and gradually developed good time management habits. I always make a detailed plan, put first things first, and take actions accordingly." Her plan worked well indeed since she was an excellent student of the University of Waikato during her 3 years there. At the end of her second year Mai Anh completed her internship at FNZ, an asset management service provider in Wellington, which is now her current company. That helped Mai Anh secure her dream job and realize her long-sought goal later on.
Knowing herself is probably a prerequisite for this girl's success. She knows what she really wants to learn about and turns it into what she wants to pursue as a career. "Choosing the right major increases your passion about your future career," said Mai Anh. There is no luck here; her efforts and the individualized orientation she received during her school years were important contributors to Mai Anh's journey to strive for excellence and make her dream come true.
"To pursue our passion and dream, we need to stay determined and believe in ourselves. Try hard, build your knowledge and skills, and success will find you one day!" said Mai Anh.
To meet students' diverse needs for career orientation and professional development and to create a special advantage for them as they enter college, Vinschool has, over the past few years, implemented the credit-based education model for high schoolers. According to Mai Anh, this allows Vinsers to focus on the subjects of their choice, making it possible for them to unleash their potential and discover their dream jobs.
"I know what I really want to do, and I always picture in my mind who I will become. That gives me the determination I need to make it a reality."
We wish Mai Anh further success and realization of her goals!
Practice public speaking in English with Helen O'Grady Academy
Future Innovators 2023: GATE Center's Recruitment of Talented Students

redpajama_set_name: RedPajamaCommonCrawl


text:
Section Branding
Legendary Rock Musicians And A Violinist Join Forces For 'A Night of Georgia Music'
September 27, 2019 4:15 PM |Updated: August 13, 2020 11:28 PM
Emilia Brock , and
Jesse Nighswonger
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From right, Rolling Stones touring keyboardist Chuck Leavell, violinist Robert McDuffie, former R.E.M. bassist Mike Mills, drummer Patrick Ferguson and guitarist William Tonks rehearse "A Night Of Georgia Music".
Two legendary rock musicians and an innovative classical violinist join On Second Thought to talk about how they are mashing up their respective disciplines. Along with original compositions, these musicians use songs from Georgia's most beloved musicians and bands and adapt them for an orchestra.
On Second Thought host Virginia Prescott speaks with Mike Mills, Chuck Leavell and Robert McDuffie.
"A Night of Georgia Music" continues a collaboration between Mike Mills, bassist songwriter and co-founder of R.E.M.; Chuck Leavell, former member of The Allman Brothers Band, now the music director and keyboardist for The Rolling Stones; and Robert McDuffie, violinist and founder of the Robert McDuffie Center for Strings at Mercer University.
They will be performing at the Atlanta Symphony Orchestra on Sept. 29 and have upcoming shows in Birmingham, Savannah and Augusta the following week, but first they joined On Second Thought.
Twitter: @OSTTalk
Facebook: OnSecondThought
Email: OnSecondThought@gpb.org
Former R.E.M. bassist Mike Mills.
Robert McDuffie
Tags: CHECK LEAVELL A night of Georgia Music GPB Loves Music Robert McDuffie mike mills
Virginia Prescott is the Gracie Award-winning host of On Second Thought for Georgia Public Broadcasting.
Emilia Brock
SENIOR PRODUCER, POLITICAL REWIND
Emilia Brock is senior producer for GPB's program, Political Rewind. Before that, she was supervising producer for GPB's On Second Thought.
Engineer and Audio Editor
Jesse Nighswonger works with On Second Thought and Political Rewind as an engineer and audio editor.
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On Second Thought For Friday, Sept. 27, 2019
"A Night of Georgia Music" continues a collaboration between Mike Mills, bassist songwriter and co-founder of R.E.M.; Chuck Leavell, former member of...
Hear Sibelius' Rousing 'Finlandia' — Symbolizing The Finnish Spirit — At Atlanta Symphony Orchestra
The Atlanta Symphony Orchestra is celebrating its 75th anniversary with special events throughout the season. Next week, Thomas Søndergård returns to...
Meet Peaches: A Prominent Figure In Atlanta's Entertainment Scene
Dee Dee Hibbler-Murray says her current job as entertainment consultant for the DeKalb County government is an "old music lady's dream come true." Every...
Virginia Prescott and
La'Raven Taylor
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The Superpro AC2 is a digitally based environmental controller featuring a short cycle timer for CO2 enrichment.
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