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\markboth{B.\,\c{S}ahin and M.\,A.\,Akyol}{Golden maps and constancy of certain maps}

\title{Golden maps between Golden Riemannian manifolds and constancy of certain maps}

\author[B.\,\c{S}ahin and M.\,Akif Akyol]{Bayram \c{S}ahin\affil{1}\comma\corrauth and Mehmet Akif Akyol\affil{2}}

\address{\affilnum{1}\ Department of Mathematics, Faculty of Arts and Sciences, \.{I}n\"{o}n\"{u} University, TR-44\,280 Malatya, Turkey\\
\affilnum{2}\ Department of Mathematics, Faculty of Arts and Sciences, Bing\"{o}l University, TR-12\,000 Bing\"{o}l, Turkey}

\emails{{\tt bayram.sahin@inonu.edu.tr}\,\,(B.\,\c{S}ahin), {\tt makyol@bingol.edu.tr} (M.\,A.\,Akyol)}

\begin{abstract}
We first introduce Golden maps between Golden Riemannian manifolds,
give an example and show that such map is harmonic. Then we
investigate the constancy of certain maps from Golden Riemannian
manifolds to various manifolds by imposing the holomorphic-like map
condition. Then we consider the reverse case and show that all such
maps are constant.
\end{abstract}

\keywords{Golden Riemannian manifold, almost contact metric manifold, almost product manifold, holomorphic map}

\ams{53C15, 53C25, 53D15}

\maketitle



%%%% Start %%%%%%
\section{Introduction}
Manifolds equipped with certain differential-geometric structures
possess rich geometric structures and such manifolds and maps
between them have been studied widely in differential geometry.
Indeed, almost complex manifolds, almost contact manifolds and
almost product manifolds and maps between such manifolds have been
studied extensively by many authors. Such manifolds are defined by a
$\left(1,1\right)  $-tensor field $\Phi$ such that the square of
$\Phi$ satisfies certain conditions, like $\Phi^{2}=-I,$
$\Phi^{2}=I$ or $\Phi^{2}=-I+\eta \otimes \xi,$
where $\eta$ and $\xi$ are 1-form and a vector field.

The number $\varphi=(1+\sqrt{5})/2=1,618...$ which is a
solution of the equation $x^{2}-x-1=0,$ represents the golden ratio.
Geometrically, Golden ratio implies that if a unit segment is divided
into two subsegments, then both the ratio of the entire segment to
the major subsegment and the ratio of the major subsegment to the
minor subsegment must equal $\varphi.$ The Golden ratio has been used in
many different areas, particularly, in arts and architecture. Being inspired
by the Golden ratio, the notion of Golden manifold $M$ was
defined in \cite{CH1} by a tensor field $\Phi$ on $M$ satisfying
$\Phi^{2}=\Phi+I$. The authors studied properties of Golden
manifolds and they showed that $\Phi$ is an automorphism of the
tangent bundle $TM$ and its eigenvalues are $\varphi$ and
$1-\varphi.$ They also defined Golden Riemannian manifolds and
investigated their submanifolds in \cite{CH2}. Moreover, the
integrability of Golden
structures has been investigated in \cite{GCS}.

To compare two manifolds or to study the geometry of a manifold $M$
by using another manifold $N$, it is useful to define a map $F$ from
$M$ to $N$ by imposing certain conditions on $F$. In this manner,
holomorphic maps between two complex manifolds have nice geometric
properties. For instance, any holomorphic map between Kaehler
manifolds is harmonic \cite{ES}. By adapting the notion of
holomophic maps, similar maps defined between almost contact
manifolds \cite{IP} and other manifolds endowed with a geometric
structure.

In this paper, we study a new map between Golden Riemannian
manifolds by imposing a holomorphic-like condition for the first time
as far as we know. We show that such map is a harmonic map, and then
we obtain certain conditions for such maps to be totally geodesic.
We also provide a simple elementary example. Moreover, we also check
the existence of such maps between Golden Riemannian manifolds and
another manifold equipped with a differentiable structure (almost
complex, almost contact, almost product, almost para-contact) and
surprisingly we find that there are no non-constant such maps.

\section{Preliminaries}
%\setcounter{equation}{0}
%\renewcommand{\theequation}{2.\arabic{equation}}

In this section, we give a brief information for almost complex
manifolds, almost contact metric manifolds, almost product
manifolds, almost para-contact metric manifolds, Golden Riemannian
manifolds. We note that throughout this paper all manifolds and
bundles, along with sections and connections, are assumed to be of
class $C^{\infty}$. A map is always a $C^{\infty}$ map between
manifolds.

\subsection{Almost complex manifolds}

Let $M^{\prime}$ be a $2n-$ dimensional real manifold. An almost
complex structure $J$ on $M^{\prime}$ is a tensor field
$J:TM^{\prime}\rightarrow
TM^{\prime}$ such that
\begin{equation}
J^{2}=-I, \label{2.1}
\end{equation}
where $I$ is the identity transformation. Then $\left(
M^{\prime},J\right)  $ is called an almost complex manifold \cite{YK}.

A smooth map $\varphi:$ $M_{1}^{\prime}\rightarrow M_{2}^{\prime}$
between almost complex manifolds $\left(
M_{1}^{\prime},J_{1}\right)  $ and $\left(
M_{2}^{\prime},J_{2}\right)  $ is called an almost complex (or
holomorphic) map if $d\varphi \left(J_{1}X\right)  =J_{2}d\varphi
\left(X\right)  $ for $X\in \Gamma \left(TM_{1}^{\prime}\right)
,$ where $J_{1}$ and $J_{2}$ are complex structures of
$M_{1}^{\prime}$ and $M_{2}^{\prime}$, respectively.

\subsection{Almost contact metric manifolds}

An $n-$dimensional differentiable manifold $M$ is said to have an
almost contact structure $(\phi,\xi,\eta)$ if it carries a tensor
field $\phi$ of type $(1,1)$, a vector field $\xi$ and $1-$ form
$\eta$ on $M$, respectively, such that
\begin{equation}
\phi^{2}=-I+\eta \otimes \xi,\phi \xi=0,\eta \circ \phi=0,\eta
\left(\xi \right)=1, \label{2.2}
\end{equation}
where $I$ is the identity transformation. The almost contact
structure is said to be normal if $N+2d\eta \otimes \xi=0,$ where
$N$ is the Nijenhuis tensor of $\phi.$ Suppose that a Riemannian
metric tensor $g$ is given in $M $ and
satisfies the condition
\begin{equation}
g\left(\phi X,\phi Y\right)  =g\left(X,Y\right)  -\eta \left(X\right) \eta \left(Y\right), \quad \eta \left(X\right)
=g\left(X,\xi \right). \label{2.3}
\end{equation}
Then $\left(M,\phi,\xi,\eta,g\right)  $ is called an almost
contact metric manifold \cite{B2}.

\subsection{Almost product manifolds}

Let $N$ be an $n-$dimensional manifold with a tensor of type $\left(1,1\right)  $ such that
\begin{equation}
F^{2}=I, \label{2.4}
\end{equation}
where $I$ is the identity transformation. Then we say that $N$ is an
almost
product manifold with almost product structure $F.$ We put
\begin{equation}
Q=\frac{1}{2}\left(I+F\right), \quad  Q^{^{\prime}}=\frac{1}{2}\left(
I-F\right). \label{2.5}
\end{equation}
Then we have
\begin{equation}
Q+Q^{^{\prime}}=I, \quad Q^{2}=Q, \quad Q^{^{\prime}2}=Q^{^{\prime}}, \quad QQ^{^{\prime}}=Q^{\prime}Q=0 \label{2.6}
\end{equation}
and
\begin{equation}
F=Q-Q^{\prime}. \label{2.7}
\end{equation}
If an almost product manifold $N$ admits a Riemannian metric $g$ such that
\[
g\left(FX,FY\right) =g\left(X,Y\right)
\]
for any vector fields $X$ and $Y$ on $N,$ then $N$ is called an
almost product Riemannian manifold \cite{YK}.

\subsection{Almost para-contact metric manifolds}

An $n-$dimensional differentiable manifold $N^{\prime}$ is said to
have an almost para-contact structure $\left(
\phi^{\prime},\xi^{\prime},\eta ^{\prime}\right)  $ if it carries a
tensor field $\phi^{\prime}$ of type $\left(1,1\right),$ a
vector field $\xi^{\prime}$ and a $1-$form
$\eta^{\prime}$ on $N^{\prime}$, respectively, such that
\begin{equation}
\phi^{\prime2}=I-\eta^{\prime}\otimes \xi^{\prime},\text{
}\eta^{\prime}\left(
\xi^{\prime}\right)  =1,\text{ }\phi^{\prime}\xi^{\prime}=0, \quad
\eta^{\prime}\circ \phi^{\prime}=0 \label{2.8}
\end{equation}
where $I$ is the identity tranformation. Suppose that a Riemannian
metric tensor
$g^{\prime}$ is given in $N^{\prime}$ and satisfies the condition
\be
g^{\prime}\left(\phi^{\prime}X,\phi^{\prime}Y\right)
&=&g^{\prime}\left( X,Y\right)  -\eta^{\prime}\left(X\right)
\eta^{\prime}\left(Y\right)\label{2.9}
\\
g^{\prime}\left(X,\xi \right)&=&\eta^{\prime}\left(X\right),\quad
g^{\prime}\left(X,\phi^{\prime}Y\right)=-g^{\prime}\left(\phi^{\prime
}X,Y\right)  \label{2.10}
\ee
for any vector fields $X$ and $Y$ on $N^{\prime}$, then $\left(
N^{\prime
},\phi^{\prime},\xi^{\prime},\eta^{\prime},g^{\prime}\right)  $ is
called an almost para-contact metric manifold
\cite{Kaneyuki-Willams,Zamkovoy}.

\subsection{Golden Riemannian manifolds}

Let $\left(\bar{M},g\right)  $ be a Riemannian manifold. A golden
structure on $\left(\bar{M},g\right)  $ is an $\left(1,1\right)
$ tensor field $P$ which satisfies the equation
\begin{equation}
P^{2}=P+I, \label{2.11}
\end{equation}
where $I$ is the identity transformation. We say that the metric $g$
is $P$
compatible if
\begin{equation}
g\left(PX,Y\right)  =g\left(X,PY\right)  \label{2.12}
\end{equation}
for all $X,Y\in \Gamma \left(T\bar{M}\right).$ If we substitute
$PX$ into
$X$ in \eqref{2.12}, equation \eqref{2.12} may also written as
\[
g\left(PX,PY\right)  =g\left(P^{2}X,Y\right)  =g\left(\left(
P+I\right)  X,Y\right)  =g\left(PX,Y\right)  +g\left(X,Y\right).
\]
The Riemannian metric \eqref{2.12} is called $P-$compatible and $\left(
\bar {M},P,g\right)  $ is named a Golden Riemannian manifold
\cite{CH1}. It is known (\cite{CH1})  that a Golden structure $P$
is integrable if the Nijenhuis tensor $N_{P}$ vanishes. In
\cite{GCS}, the authors show that a Golden structure is integrable
if and only if $\nabla P=0$, where $\nabla$ is the Levi-Civita
connection of $g$.

\subsection{Harmonic maps}

Let $\left(M,g_{M}\right)  $ and $\left(N,g_{N}\right)  $ be
Riemannian manifolds and suppose that $P:M\rightarrow N$ is a
smooth mapping between them. Then the differential $dP$ of
$P$ can be viewed as a section of the bundle $Hom\left(
TM,P^{-1}TN\right)  \rightarrow M,$ where $P^{-1}TN$ is
the pullback bundle which has fibres $\left( P^{-1}TN\right)
_{p}=T_{P \left(p\right)  }N,$ $p\in M.$ $Hom\left(
TM,P^{-1}TN\right)  $ has a connection $\nabla$ induced from
the Levi-Civita connection $\nabla^{M}$ and the pullback connection.
Then the
second fundamental form of $P$ is given by
\begin{equation}
\nabla dP \left(X,Y\right)  =\nabla_{X}^{P}dP
\left(
Y\right)  -dP \left(\nabla_{X}^{M}Y\right)  \label{2.13}
\end{equation}
for $X,Y\in \Gamma \left(TM\right).$ It is known that the second
fundamental form is symmetric. A smooth map $P:\left(
M,g_{M}\right) \rightarrow \left(N,g_{N}\right)  $ is said to be
harmonic if trace $\nabla dP=0.$ The tension field of
$P$ is the section $\tau \left(
P \right)  $ of $\Gamma \left(P^{-1}TN\right)  $ defined by
\begin{equation}
\tau \left(P \right)  =\text{div}dP=\sum
\limits_{i=1}^{m}\nabla
dP \left(e_{i},e_{i}\right), \label{2.14}
\end{equation}
where $\left \{e_{1},...,e_{m}\right \}$ is a local orthonormal
frame on $M.$ Then it follows that $P$ is harmonic if and only
if $\tau \left( P \right)  =0.$ For more information, see
\cite{BW}.

\section{Golden maps between Golden manifolds}
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%\renewcommand{\theequation}{3.\arabic{equation}}

In this section, we give a new notion, namely a Golden map, and show
that such map is harmonic. We also investigate conditions for a
Golden map to be totally geodesic.

\begin{definition}
Let $\varphi$ be a smooth map from a Golden Riemannian manifold
$\left( M,P,g\right)  $ to a Golden Riemannian manifold $\left(
N,P^{\prime },g^{\prime}\right)  $. Then $\varphi$ is called a
Golden map if the following
condition is satisfied.
\begin{equation}
d\varphi\,P=P^{\prime}d\varphi\,.\label{eq:100}
\end{equation}
\end{definition}


We provide the following elementary example.

\begin{example}
Let $\varphi:%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{4}\rightarrow%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}$ be a map defined by%
\[
\varphi \left(x_{1},x_{2},x_{3},x_{4}\right)  =\left(\frac{x_{1}+x_{2}}
{4},\frac{x_{3}+x_{4}}{4}\right).
\]
Then, by direct calculations
\[
\ker d\varphi\,=span\left\{  X_{1}=\frac{\partial}{\partial x_{1}}
-\frac{\partial}{\partial x_{2}},\text{ }X_{2}=\frac{\partial}{\partial x_{3}
}-\frac{\partial}{\partial x_{4}}\right\}
\]
and
\[
\left(\ker d\varphi\,\right)  ^{\bot}=span\left \{
Z_{1}=\frac{\partial }{\partial x_{1}}+\frac{\partial}{\partial
x_{2}},\text{ }Z_{2}=\frac {\partial}{\partial
x_{3}}+\frac{\partial}{\partial x_{4}}\right \}.
\]
Then considering Golden structures on $
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{4}$ and $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}$ defined by
\[
\Phi \left(a_{1},a_{2},a_{3},a_{4}\right)  =\left(\phi
a_{1},\phi a_{2},\left(1-\phi \right)  a_{3},\left(1-\phi
\right)  a_{4}\right)
\]
and
\[
\Phi^{\prime}\left(a_{1},a_{2}\right)  =\left(\phi a_{1},\left(
1-\phi \right)  a_{2}\right),
\]
where $\phi$ and $1-\phi$ are eigenvalues of Golden structures
\cite{CH1}. It
is easy to see that $d\varphi\,\left(PZ_{1}\right)  =P^{\prime}
d\varphi\,\left(Z_{1}\right)  $ and $d\varphi\,\left(
PZ_{2}\right)  =P^{\prime}d\varphi\,\left(Z_{2}\right).$ Thus
$\varphi$ is a Golden map.
\end{example}
From now on, when we mention a Golden Riemannian manifold, we will
assume that its almost Golden structure is integrable.

\begin{lemma}
Let $\varphi$ be a Golden map from a Golden Riemannian manifold
$\left( M,P,g\right)  $ to a Golden Riemannian manifold $\left(
N,P^{\prime },g^{\prime}\right)  $ such that
$d\varphi\,P=P^{\prime}d\varphi\,$ is
satisfied. Then we have
\begin{equation}
\left(\nabla d\varphi\,\right)  \left(X,PY\right)  =\left(
\nabla d\varphi\,\right)  \left(PX,Y\right)  \label{3.1}
\end{equation}
for $X,Y\in \Gamma \left(TM\right).$
\end{lemma}

\begin{proof}
For $X,Y\in \Gamma \left(TM\right),$ from \eqref{2.13} and \eqref{2.11} we have
\[
\left(\nabla d\varphi\,\right)  \left(X,PY\right)
=\nabla^{\varphi}_{X}  d\varphi\,\left( P^{2}Y-Y\right)
-d\varphi\,\left(\nabla_{X}^{M}PY\right).
\]
Then using (\ref{eq:100}) we get
\[
\left(\nabla d\varphi\,\right)  \left(X,PY\right)
=\nabla^{\varphi}_{X}P^{\prime2}d\varphi\left(Y\right)
-\nabla^{\varphi}_{X}d\varphi \left(Y\right)  -d\varphi\,\left(
\nabla_{X}^{M}PY\right). \]
Since  $P'$ is integrable, from (\ref{2.11}) we obtain
\[
\left(\nabla d\varphi\,\right)  \left(X,PY\right)
=P^{\prime}\nabla^{\varphi}_{X}d\varphi\left(Y\right)
-d\varphi\,\left(\nabla_{X}^{M}PY\right).
\]
Integrable $P$ and (\ref{eq:100}) also imply
\[
\left(\nabla d\varphi\,\right)  \left(X,PY\right)
=P^{\prime}\left( \nabla^{\varphi}_{X}d\varphi\left(Y\right)
-d\varphi\,\left(\nabla_{X}^{M}Y\right) \right).
\]
Thus using (\ref{2.13}) we arrive at
\begin{equation}
 \left(\nabla d\varphi\,\right)  \left(X,PY\right)=P^{\prime}\left(\nabla d\varphi\,\right)  \left(X,Y\right).\label{reveq:1}
\end{equation}
Since $\nabla d\varphi\,$ is symmetric, we obtain (\ref{3.1}).
\end{proof}

The following theorem shows that a Golden map is a harmonic map.

\begin{theorem}
Every Golden map from a Golden Riemannian manifold $\left(
M,P,g\right)  $ to a Golden Riemannian manifold $\left(
N,P^{\prime},g^{\prime}\right)  $ is harmonic.
\end{theorem}

\begin{proof}
First, from \eqref{3.1} and (\ref{reveq:1}), we find
\begin{equation}
\left(\nabla d\varphi\,\right)  \left(PX,PY\right)
=P^{\prime2}\left(
\nabla d\varphi\,\right)  \left(X,Y\right). \label{3.3}
\end{equation}
Let $\left \{  e_{1},e_{2},...,e_{n}\right \}  $ be a basis of
$T_{p}M,$ $p\in M. $ Since $P$ is an isomorphism, from \cite{Nomizu}
we know that $\left \{ Pe_{1},Pe_{2},...,Pe_{n}\right \}  $ is also
a basis of $T_{p}M,$ $p\in M.$
Then from (\ref{2.11}) and (\ref{3.3}), we have
\bee
\sum_{i=1}^{n}\left(\nabla d\varphi\,\right)  \left(Pe_{i}
,Pe_{i}\right)   &=&\sum_{i=1}^{n}P^{\prime}\left(\nabla
d \varphi\right)  \left(e_{i},e_{i}\right)
+\sum_{i=1}^{n}\left(\nabla
d\varphi\,\right)  \left(e_{i},e_{i}\right)
\\
\tau \left(\varphi \right)   &=&P^{\prime}\tau \left(\varphi
\right) +\tau \left(\varphi \right),
\eee
which gives
\[
P^{\prime}\tau \left(\varphi \right)  =0.
\]
Since $P^{\prime}$ is an isomorphism on $N$, we get
\[
\tau \left(\varphi \right)  =0,
\]
which completes the proof.
\end{proof}

\begin{remark}
We note that  for any $C^2$ real valued function $f$ defined on an open subset of a Riemannian manifold $M$, the equation $\bigtriangleup f=0$ is called Laplace's equation and solutions are called harmonic functions on $U$. Let $ F:M \longrightarrow N$ be a smooth map between Riemannian manifolds. Then $F$ is called a harmonic morphism if, for every harmonic function $f:V \longrightarrow \mathbf{R}$ defined an open subset $V$ of $N$ with $F^{-1}(V)$ non-empty, the composition $f \circ F$ is harmonic on $F^{-1}(V)$. A smooth map $F:M \longrightarrow N$ between Riemannian manifolds is a harmonic morphism if and only if $F$ is both harmonic and horizontally weakly conformal \cite{Fuglede} and \cite{Ishihara}. In this respect, a Golden map is a good candidate for a harmonic morphism.
\end{remark}

We now give a necessary and sufficient condition for a map $\varphi$ to be
totally geodesic. We recall that a map $\varphi$ is totally geodesic
if $\nabla d \varphi=0$. 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 $\varphi$ be Golden map from a Golden Riemannian manifold
$\left( M,P,g\right)  $ to a Golden Riemannian manifold $\left(
N,P^{\prime
},g^{\prime}\right).$ Then $\varphi$ is totally geodesic if and only if
\begin{equation}
\left(\nabla d\varphi\,\right)  \left(PX,PY\right)
=P^{\prime}\left(
\nabla d\varphi\,\right)  \left(X,Y\right),  \label{3.4}
\end{equation}
for $X,Y\in \Gamma \left(TM\right).$
\end{theorem}

\begin{proof}
It is obvious from \eqref{3.3}.
\end{proof}

\section{Constancy of some maps from Golden Riemannian manifolds}
%\setcounter{equation}{0}
%\renewcommand{\theequation}{4.\arabic{equation}}

In this section we investigate constancy of certain maps from Golden
Riemannian manifolds or to Golden manifolds by imposing
holomorphic-like conditions. We first check the situation for a map
between Golden Riemannian manifolds and almost complex manifolds.

\begin{theorem}
Let $\varphi$ be a smooth map from a Golden Riemannian manifold
$\left( \bar{M},P,g\right)  $ to an almost complex manifold $\left(
M^{\prime },J\right)  $ such that the condition $d\varphi
P=Jd\varphi$ is satisfied. Then $\varphi$ is a constant map.
\end{theorem}

\begin{proof}
Let $\left(\bar{M},P,g\right)  $ be a Golden Riemannian manifold
and $\left(M^{\prime},J\right)  $ an almost complex manifold.
Suppose that
$\varphi:\bar{M}\rightarrow M^{\prime}$ satisfies
\begin{equation}
d\varphi \left(PX\right)  =Jd\varphi \left(X\right),\quad X\in
\Gamma \left(T\bar{M}\right). \label{4.1}
\end{equation}
Then apply $J$ to the above equation and using \eqref{2.1} and \eqref{2.11}, we get
\begin{equation}
d\varphi \left(PX\right)  +d\varphi \left(X\right)  =-d\varphi
\left(
X\right), \quad X\in \Gamma \left(T\bar{M}\right).  \label{4.2}
\end{equation}
Applying $J$ to \eqref{4.2} again and using \eqref{4.1}, we have
\begin{equation}
d\varphi \left(PX\right)  +d\varphi \left(X\right)  +Jd\varphi
\left( X\right)  =-Jd\varphi \left(X\right), \quad X\in
\Gamma \left(T\bar
{M}\right). \label{4.3}
\end{equation}
Then \eqref{4.1} implies that
\begin{equation}
-3d\varphi \left(PX\right)  =d\varphi \left(X\right), \quad X\in
\Gamma \left(T\bar{M}\right). \label{4.4}
\end{equation}
From \eqref{4.2} and \eqref{4.4} we obtain
\[
d\varphi \left(X\right)=0,
\]
which shows that $\varphi$ is constant.
\end{proof}

In a similar way, we have the following result.

\begin{theorem}
Let $\varphi$ be a smooth map from an almost complex manifold
$\left( M^{\prime},J\right)  $ to a Golden Riemannian manifold
$\left(\bar {M},P,g\right)  $ such that the condition $d\varphi
J=Pd\varphi$ is satisfied. Then $\varphi$ is a constant map.
\end{theorem}

\begin{remark}Besides holomorphic maps, anti-holomorphic maps have also been studied by many authors
under the condition $J^{\prime}dF=-dFJ$, where $F$ is a
map between almost complex manifolds and $J,J^{\prime}$ are almost
complex structures, respectively. By following similar computations
above, one can find that  the notion of anti-holomorphic maps does
not work for Golden manifolds.
\end{remark}

The following result shows that a smooth map satisfying  a
compatible condition between Golden Riemannian manifolds and almost
contact metric manifolds is also constant.

\begin{theorem}
Let $\varphi$ be a smooth map from a Golden Riemannian manifold
$\left( \bar{M},P,g\right)  $ to an almost contact metric manifold
$\left(M,\phi ,\xi,\eta,g\right)  $ such that the condition
$d\varphi P=\phi d\varphi$ is satisfied. Then $\varphi$ is a constant
map.
\end{theorem}

\begin{proof}
Let $\left(\bar{M},P,g\right)  $ be a Golden Riemannian manifold
and $\left(M,\phi,\xi,\eta,g\right)  $ an almost contact metric
manifold.
Suppose that $\varphi:\bar{M}$ $\rightarrow M$ satisfies
\begin{equation}
d\varphi \left(PX\right)  =\phi d\varphi \left(X\right),\quad
X\in
\Gamma \left(T\bar{M}\right). \label{4.5}
\end{equation}
Then apply $\phi$ to the above equation and using \eqref{2.2} and \eqref{2.11}, we get
\begin{equation}
d\varphi \left(PX\right)  =-2d\varphi \left(X\right)  +\eta
\left( d\varphi \left(X\right)  \right)  \xi, \quad X\in \Gamma
\left(T\bar
{M}\right). \label{4.6}
\end{equation}
Then applying $\phi$ to \eqref{4.6} again and using \eqref{4.5} and \eqref{2.2}, we have
\begin{equation}
-3d\varphi \left(PX\right)  =d\varphi \left(X\right). \label{4.7}
\end{equation}
From \eqref{4.6} and \eqref{4.7}, we obtain
\begin{equation}
-5d\varphi \left(PX\right)  =\eta \left(d\varphi \left(X\right)  \right)
\xi, \quad X\in \Gamma \left(T\bar{M}\right). \label{4.8}
\end{equation}
Again applying $\phi$ to \eqref{4.8}, we get
\begin{equation}
\phi d\varphi \left(X\right)  =0, \quad X\in \Gamma \left(
T\bar{M}\right).
\label{4.9}
\end{equation}
Using \eqref{4.9} in \eqref{4.8} we conclude that
\begin{equation}
\eta \left(d\varphi \left(X\right)  \right)  =0, \quad X\in
\Gamma \left(
T\bar{M}\right). \label{4.10}
\end{equation}
Then applying $\varphi$ to \eqref{4.10} we get
\[
-d\varphi \left(X\right)  +\eta \left(d\varphi \left(X\right)\right) \xi=0
\]
which gives
\[
d\varphi \left(X\right)  =0.
\]
This completes the proof.
\end{proof}

In a similar way, we have the following result.

\begin{theorem}
Let $\varphi$ be a smooth map from an almost contact metric manifold
$\left( M,\phi,\xi,\eta,g\right)  $ to a Golden Riemannian manifold
$\left(\bar {M},P,g\right)  $ such that the condition $d\varphi
\phi=Pd\varphi$ is satisfied. Then $\varphi$ is a constant map.
\end{theorem}
 We now check a similar situation for a map between
Golden Riemannian manifolds and almost product manifolds.

\begin{theorem}
Let $\varphi$ be a smooth map from a Golden Riemannian manifold
$\left( \bar{M},P,g\right)  $ to an almost product manifold $\left(
N,F\right)  $ such that the condition $d\varphi P=Fd\varphi$ is
satisfied. Then $\varphi$ is a constant map.
\end{theorem}

\begin{proof}
Let $\left(\bar{M},P,g\right)  $ be a Golden Riemannian manifold
and $\left(N,F\right)  $ an almost product manifold. Suppose that
$\varphi
:\bar{M}\rightarrow N$ satisfies
\begin{equation}
d\varphi \left(PX\right)  =Fd\varphi \left(X\right), \quad X\in
\Gamma \left(T\bar{M}\right). \label{4.11}
\end{equation}
Then apply $F$ to the above equation and using \eqref{2.4} and \eqref{2.11}, we get
\begin{equation}
d\varphi \left(PX\right)  =0, \quad X\in \Gamma \left(
T\bar{M}\right).
\label{4.12}
\end{equation}
Applying $F$ to \eqref{4.12} again and using \eqref{4.11}, we have
\begin{equation}
d\varphi \left(PX\right)  =-d\varphi(X), \quad X\in \Gamma \left(
T\bar
{M}\right). \label{4.13}
\end{equation}
From \eqref{4.13} and \eqref{4.12} we obtain
\[
d\varphi \left(X\right)=0,
\]
which shows that $\varphi$ is constant.
\end{proof}

In a similar way, we have the following result.

\begin{theorem}
Let $\varphi$ be a smooth map from an almost product manifold
$\left( N,F\right)  $ to a Golden Riemannian manifold $\left(
\bar{M},P,g\right)  $ such that the condition $d\varphi F=Pd\varphi$
is satisfied. Then $\varphi$ is a constant map.
\end{theorem}
Finally, we check the same problem for almost
para-contact metric manifolds.

\begin{theorem}
Let $\varphi$ be a smooth map from a Golden Riemannian manifold
$\left( \bar{M},P,g\right)  $ to an almost para-contact metric
manifold $\left(
N^{\prime},\phi^{\prime},\xi^{\prime},\eta^{\prime},g^{\prime}\right)
$ such that the condition $d\varphi P=\phi^{\prime}d\varphi$ is
satisfied. Then $\varphi$ is a constant map.
\end{theorem}

\begin{proof}
Let $\left(\bar{M},P,g\right)  $ be a Golden Riemannian manifold
and $\left(
N^{\prime},\phi^{\prime},\xi^{\prime},\eta^{\prime},g^{\prime
}\right)  $ an almost para-contact metric manifold. Suppose that
$\varphi
:\bar{M}$ $\rightarrow N^{\prime}$ satisfies
\begin{equation}
d\varphi \left(PX\right)  =\phi^{\prime}d\varphi \left(X\right),
\quad  X\in \Gamma \left(T\bar{M}\right). \label{4.14}
\end{equation}
Then apply $\phi^{\prime}$ to the above equation and using \eqref{2.8} and \eqref{2.11}, we
get
\begin{equation}
d\varphi \left(PX\right)  +d\varphi \left(X\right)  =d\varphi
\left( X\right)  -\eta^{\prime}\left(d\varphi \left(X\right)
\right)  \xi
^{\prime}, \quad X\in \Gamma \left(T\bar{M}\right). \label{4.15}
\end{equation}
Then applying $\phi^{\prime}$ to \eqref{4.15} again and using \eqref{4.14} and \eqref{2.8}, we
have
\begin{equation}
d\varphi \left(PX\right)  =-d\varphi \left(X\right). \label{4.16}
\end{equation}
From \eqref{4.15} and \eqref{4.16}, we obtain
\begin{equation}
d\varphi \left(X\right)  =\eta^{\prime}\left(d\varphi \left(
X\right) \right)  \xi^{\prime}, \quad X\in \Gamma \left(
T\bar{M}\right).
\label{4.17}
\end{equation}
Again applying $\phi^{\prime}$ to \eqref{4.17}, we get
\be
\phi^{\prime}d\varphi \left(X\right)&=&0, \quad X\in \Gamma \left(T\bar{M}\right). \label{4.18}
\ee
From \eqref{4.18} we have $d\varphi \left(X\right)=0$, which shows that $\varphi$ is constant.
\end{proof}

In a similar way, we have the following result.

\begin{theorem}
Let $\varphi$ be a smooth map from an almost para-contact metric
manifold $\left(
N^{\prime},\phi^{\prime},\xi^{\prime},\eta^{\prime},g^{\prime
}\right)  $ to a Golden Riemannian manifold $\left(
\bar{M},P,g\right)  $ such that the condition $d\varphi
\phi^{\prime}=Pd\varphi$ is satisfied. Then $\varphi$ is a constant
map.
\end{theorem}


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