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Abstract

In this paper, we prove that a map between Riemannian manifolds is an ${\textstyle f}$-harmonic morphism in a general sense if and only if it is horizontally weakly conformal, satisfying some conditions, and we investigate the properties of ${\textstyle f}$-harmonic morphism in a general sense.

Keywords

ff-harmonic maps; ff-harmonic morphism

2010 Mathematics Subject Classification

53A45; 53C20; 58E20

1. Introduction

${\textstyle f}$-harmonic morphisms are mappings between Riemannian manifolds which preserve Laplace’s equation  [4], [15] and [14]. They can be characterized as ${\textstyle f}$-harmonic maps which checks the property of horizontal weak conformality (called semiconformality).

In mathematical physics, ${\textstyle f}$-harmonic maps relate to the equations of the motion of a continuous system of spins [6] and the gradient Ricci-soliton structure  [16] and [1]. Recently the notion of ${\textstyle f}$-harmonic maps (resp bi-${\textstyle f}$-harmonic maps) was developed by N. Course  [7], M. Djaa and S. Ouakkas  [15], [8] and [4], and studied by many authors, including Y.J. Chiang [5], M. Rimoldi  [16], Y.L. Ou  [14], S. Feng [10], W.J. Lu  [13] and others.

The goal of this work is the characterization of ${\textstyle f}$-harmonic morphism (in a general sense) between Riemannian manifolds (Theorem 3.1), which generalizes the Fuglede–Ishihara characterization for harmonic morphisms [11] and [12], and we investigate the properties of${\textstyle f}$-harmonic morphism in a general sense. Also we construct some examples of ${\textstyle f}$-harmonic morphism (Example 3.2).

2. ${\textstyle f}$-harmonic maps2. f {\textstyle f} -harmonic maps

Consider a smooth map ${\textstyle \varphi :(M,g)\rightarrow (N,h)}$ between Riemannian manifolds and ${\textstyle f:(x,y,r)\in M\times N\times R\rightarrow f(x,y,r)\in (0,\infty )}$ a smooth positive function. For any compact domain ${\textstyle D}$ of ${\textstyle M}$ the ${\textstyle f}$-energy functional of ${\textstyle \varphi }$ is defined by

${\displaystyle E_{f}(\varphi ;D)=\int _{D}f(x,\varphi (x),e(\varphi )(x))\quad v_{g}{\mbox{,}}}$

(2.1)

where ${\textstyle v_{g}}$ is the volume element and

${\displaystyle e(\varphi )={\frac {1}{2}}\quad \sum _{i}h(d\varphi (e_{i}),d\varphi (e_{i})){\mbox{,}}}$

(2.2)

is the energy density of ${\textstyle \varphi }$, here ${\textstyle \lbrace e_{i}\rbrace }$ is an orthonormal frame on ${\textstyle (M,g)}$.

Definition 2.1 [4].

A map is called ${\textstyle f}$-harmonic if it is a critical point of the ${\textstyle f}$-energy functional over any compact subset ${\textstyle D}$ of ${\textstyle M}$.

2.1. The first variation of ${\textstyle f}$-energy functional2.1. The first variation of f {\textstyle f} -energy functional

Theorem 2.1 [4].

Let  ${\textstyle \varphi :(M,g)\rightarrow (N,h)}$be a smooth map and let  ${\textstyle \lbrace \varphi _{t}\rbrace _{t\in (-\epsilon ,\epsilon )}}$be a smooth variation of  ${\textstyle \varphi }$supported in  ${\textstyle D}$. If  ${\textstyle v={\frac {\partial \varphi _{t}}{\partial t}}\vert _{t=0}}$denote the variation vector field of  ${\textstyle \varphi }$, then

${\displaystyle {\frac {d}{dt}}E_{f}(\varphi _{t};D)\vert _{t=0}=-\int _{D}h(\tau _{f}(\varphi ),v)\quad v_{g}{\mbox{,}}}$

(2.3)

where

${\displaystyle \tau _{f}(\varphi )=({\frac {\partial f}{\partial r}})_{\varphi }\quad \tau (\varphi )+d\varphi (grad^{M}({\frac {\partial f}{\partial r}})_{\varphi })-(grad^{N}f)\circ \varphi {\mbox{,}}}$

here  ${\textstyle ({\frac {\partial f}{\partial r}})_{\varphi }(x)={\frac {\partial f}{\partial r}}(x,\varphi (x),e(\varphi )(x))}$and  ${\textstyle \tau (\varphi )=trace\nabla d\varphi }$.

Definition 2.2.

${\textstyle \tau _{f}(\varphi )}$ is called an ${\textstyle f}$-tension field of ${\textstyle \varphi }$.

Theorem 2.2 [4].

Let  ${\textstyle \varphi :(M,g)\rightarrow (N,h)}$be a smooth map between Riemannian manifolds and  ${\textstyle f:M\times N\times R\rightarrow (0,\infty )}$be a smooth function. Then  ${\textstyle \varphi }$is an  ${\textstyle f}$-harmonic map if and only if

${\displaystyle \tau _{f}(\varphi )=({\frac {\partial f}{\partial r}})_{\varphi }\quad \tau (\varphi )+d\varphi (grad^{M}({\frac {\partial f}{\partial r}})_{\varphi })-(grad^{N}f)\circ \varphi =0{\mbox{.}}}$

(2.4)

Remarks 2.1.

• If ${\textstyle f(x,y,r)=f_{1}(x)f_{2}(y)\quad r}$, then any smooth map ${\textstyle \varphi :(M^{m},g)\rightarrow (N^{n},h)}$${\textstyle (m\not =2)}$ is an ${\textstyle f}$-harmonic map if and only if ${\textstyle \varphi :(M^{m},f_{1}^{\frac {2}{m-2}}g)\rightarrow (N^{n},f_{2}h)}$ is a harmonic map.
• If ${\textstyle f(x,y,r)=f(x)\quad r}$, then any smooth map ${\textstyle \varphi :(M^{m},g)\rightarrow (N^{n},h)}$${\textstyle (m\not =2)}$ is an ${\textstyle f}$-harmonic map if and only if ${\textstyle \varphi :(M^{m},f^{\frac {2}{m-2}}g)\rightarrow (N^{n},h)}$ is a harmonic map (see  [9]).

Examples 2.1 [14].

Inhomogeneous Heisenberg spin system

• ${\textstyle \varphi :(R^{3},ds_{0})\rightarrow (N^{n},h)}$ is an ${\textstyle f}$-harmonic map if and only if

${\displaystyle \varphi :(R^{3},f_{1}^{2}ds_{0})\rightarrow (N^{n},h)}$

is a harmonic map with ${\textstyle f(x,y,r)=f_{1}(x)r}$.

• ${\textstyle \varphi :S^{3}\setminus \lbrace N\rbrace \equiv (R^{3},{\frac {4ds_{0}}{(1+\vert x\vert ^{2})^{2}}})\rightarrow (N^{n},h)}$ is a harmonic map if and only if

${\displaystyle \varphi :(R^{3},ds_{0})\rightarrow (N^{n},h)}$

is an ${\textstyle f}$-harmonic map with ${\textstyle f(x,y,r)={\frac {2}{(1+\vert x\vert ^{2})}}r}$.

• When ${\textstyle (N^{n},h)=S^{2}}$, we have ${\textstyle 1}$-${\textstyle 1}$ correspondence between the set of harmonic maps ${\textstyle S^{3}\rightarrow S^{2}}$ and the set of stationary solutions of the inhomogeneous Heisenberg spin system on ${\textstyle R^{3}}$.

${\displaystyle \varphi :H^{3}\equiv (D^{3},{\frac {4ds_{0}}{(1+\vert x\vert ^{2})^{2}}})\rightarrow (N^{n},h)}$

is a harmonic map if and only if

${\displaystyle \varphi :(D^{3},ds_{0})\rightarrow (N^{n},h)}$

is an ${\textstyle f}$-harmonic map with ${\textstyle f(x,y,r)={\frac {2}{(1+\vert x\vert ^{2})}}r}$.

3. ${\textstyle f}$-harmonic morphisms3. f {\textstyle f} -harmonic morphisms

Let ${\textstyle \varphi :(M^{m},g)\rightarrow (N^{n},h)}$ be a smooth map between Riemannian manifolds and ${\textstyle C_{\varphi }=\lbrace x\in M\mid d_{x}\varphi =0\rbrace }$ be the set of critical points of ${\textstyle \varphi }$. Then ${\textstyle \varphi }$ is called horizontally weakly conformal or semi-conformal if for each ${\textstyle x\in M\setminus C_{\varphi }}$ the restriction of ${\textstyle d_{x}\varphi }$ to ${\textstyle H_{x}}$ is surjective and conformal, where the horizontal space ${\textstyle H_{x}}$ is the orthogonal complement of ${\textstyle V_{x}=Ker\quad d_{x}\varphi }$. The horizontal conformality of ${\textstyle \varphi }$ implies that there exists a function ${\textstyle \lambda :M\setminus C_{\varphi }\rightarrow R_{+}}$ such that for all ${\textstyle x\in M\setminus C_{\varphi }}$ and ${\textstyle X,Y\in H_{x}}$

${\displaystyle h(d_{x}\varphi (X),d_{x}\varphi (Y))=\lambda (x)^{2}g(X,Y){\mbox{.}}}$

(3.1)

${\textstyle \varphi }$ is horizontally weakly conformal at ${\textstyle x}$ with dilation ${\textstyle \lambda (x)}$ if and only if in any local coordinates ${\textstyle (y^{\alpha })}$ on a neighborhood of ${\textstyle \varphi (x)}$ we have

${\displaystyle g(grad^{M}\varphi ^{\alpha },grad^{M}\varphi ^{\beta })=\lambda ^{2}(h^{\alpha \beta }\circ \varphi )\quad (\alpha ,\beta =1,\ldots ,n){\mbox{.}}}$

(3.2)

Definition 3.1.

Let ${\textstyle f:M\times R\times R\rightarrow (0,\infty )}$, ${\textstyle (x,t,r)\mapsto f(x,t,r)}$ be a smooth function and ${\textstyle U}$ be an open subset of ${\textstyle M}$. A ${\textstyle C^{2}}$-function ${\textstyle u:U\rightarrow R}$ is called ${\textstyle f}$-harmonic if

${\displaystyle \Delta _{f}^{M}u\equiv ({\frac {\partial f}{\partial r}})_{u}\quad \Delta ^{M}u+du(grad^{M}({\frac {\partial f}{\partial r}})_{u})-({\frac {\partial f}{\partial t}})_{u}=0}$

(3.3)

where

${\displaystyle ({\frac {\partial f}{\partial r}})_{u}:U\rightarrow (0,+\infty )}$

Definition 3.2.

A map ${\textstyle \varphi :(M^{m},g)\rightarrow (N^{n},h)}$ between Riemannian manifolds is said to be an ${\textstyle f}$-harmonic morphism if for every open subset ${\textstyle V}$ of ${\textstyle N}$ with ${\textstyle \varphi ^{-1}(V)\not =0\nsupseteqq }$ and every harmonic function ${\textstyle v:V\rightarrow R}$, the composition ${\textstyle v\circ \varphi :\varphi ^{-1}(V)\rightarrow R}$ is ${\textstyle f}$-harmonic.

Example 3.1 [14].

Let ${\textstyle \varphi ,\psi ,\phi :R^{3}\rightarrow R^{2}}$ be defined as

${\displaystyle \varphi (x,y,z)=(x,y){\mbox{,}}}$

Then both ${\textstyle \varphi }$ and ${\textstyle \psi }$ are ${\textstyle f}$-harmonic with ${\textstyle f=r\quad e^{z}}$, ${\textstyle \varphi }$ is a horizontally conformal submersion whilst ${\textstyle \psi }$ is not. Also, ${\textstyle \phi }$ is an ${\textstyle f}$-harmonic map with ${\textstyle f=r\quad e^{(y-z)}}$, which is a submersion but not horizontally weakly conformal.

Theorem 3.1.

Let  ${\textstyle \varphi :(M^{m},g)\rightarrow (N^{n},h)}$be a smooth map between Riemannian manifolds and  ${\textstyle f:M\times R\times R\rightarrow (0,\infty )}$be a smooth function such that

${\displaystyle \lbrace {\begin{array}{cc}{\frac {\partial f}{\partial r}}(x,t,r)\not =0{\mbox{,}}&{\mbox{for all   }}(x,t,r)\in M\times R\times R{\mbox{;}}\\{\frac {\partial f}{\partial t}}(x,t,0)=0{\mbox{,}}&{\mbox{for all   }}(x,t)\in M\times R{\mbox{.}}\end{array}}}$

(3.4)

Then, the following are equivalent:

• ${\textstyle \varphi }$is an  ${\textstyle f}$-harmonic morphism;
• ${\textstyle \varphi }$is a horizontally weakly conformal satisfying

${\displaystyle ({\frac {\partial f}{\partial r}})_{\varphi ^{\alpha }}\quad \tau (\varphi )^{\alpha }+g(grad^{M}({\frac {\partial f}{\partial r}})_{\varphi ^{\alpha }},grad^{M}\varphi ^{\alpha })-({\frac {\partial f}{\partial t}})_{\varphi ^{\alpha }}=0{\mbox{,}}}$

(3.5)

for all  ${\textstyle \alpha =1,\ldots ,n}$and in any local coordinates  ${\textstyle (y^{\alpha })}$on  ${\textstyle N}$;

• There exists a smooth positive function  ${\textstyle \lambda }$on  ${\textstyle M}$such that

${\displaystyle \Delta _{f}^{M}(v\circ \varphi )=({\frac {\partial f}{\partial r}})_{v\circ \varphi }\quad \lambda ^{2}\quad (\Delta ^{N}v)\circ \varphi {\mbox{,}}}$

for every smooth function  ${\textstyle v}$defined on an open subset  ${\textstyle V}$of  ${\textstyle N}$.

Proof.

To prove Theorem 3.1 we need the following lemma.

Lemma 3.1 [2].

Let  ${\textstyle y_{0}}$be a point in  ${\textstyle N^{n}}$and  ${\textstyle (y^{\gamma })}$be local coordinates centered at  ${\textstyle y_{0}}$. Then for any constants  ${\textstyle \lbrace c_{\gamma },c_{\alpha \beta }\rbrace _{\alpha ,\beta ,\gamma =1}^{n}}$with  ${\textstyle c_{\alpha \beta }=c_{\beta \alpha }}$and  ${\textstyle \sum _{\alpha =1}^{n}c_{\alpha \alpha }=0}$, there exists a neighborhood  ${\textstyle V}$of  ${\textstyle y_{0}}$in  ${\textstyle N}$and a harmonic function  ${\textstyle v:V\rightarrow R}$such that

${\displaystyle {\frac {\partial v}{\partial y^{\alpha }}}(y_{0})=c_{\alpha }{\mbox{,}}\quad {\frac {\partial ^{2}v}{\partial y^{\alpha }\partial y^{\beta }}}(y_{0})=c_{\alpha \beta }{\mbox{,}}}$

(3.6)

for all  ${\textstyle \alpha ,\beta ,\gamma =1,\ldots ,n}$.

Suppose that ${\textstyle \varphi :(M^{m},g)\rightarrow (N^{n},h)}$ is an ${\textstyle f}$-harmonic morphism. For ${\textstyle x_{0}\in M}$ we consider a system of local coordinates ${\textstyle (x^{i})}$ and ${\textstyle (y^{\alpha })}$ around ${\textstyle x_{0}}$ and ${\textstyle y_{0}=\varphi (x_{0})}$ respectively, where we assume that ${\textstyle (y^{\alpha })}$ are normal. By Lemma 3.1, for a sequence ${\textstyle (c_{\gamma },c_{\alpha \beta })_{\alpha ,\beta ,\gamma =1}^{n}}$ with ${\textstyle c_{\gamma }=0}$, ${\textstyle c_{\alpha \beta }=c_{\beta \alpha }}$ and ${\textstyle \sum _{\alpha }c_{\alpha \alpha }=0}$, we can choose a local harmonic function ${\textstyle v}$ such that

${\displaystyle {\frac {\partial v}{\partial y^{\alpha }}}(y_{0})=0{\mbox{,}}\quad {\frac {\partial ^{2}v}{\partial y^{\alpha }\partial y^{\beta }}}(y_{0})=c_{\alpha \beta }{\mbox{,}}}$

(3.7)

for all ${\textstyle \alpha ,\beta =1,\ldots ,n}$. By assumption ${\textstyle v\circ \varphi }$ is ${\textstyle f}$-harmonic in a neighborhood of ${\textstyle x_{0}}$, from Definition 3.1 we have

${\displaystyle {\begin{array}{l}\displaystyle 0=\Delta _{f}^{M}(v\circ \varphi )=({\frac {\partial f}{\partial r}})_{v\circ \varphi }\quad \Delta ^{M}(v\circ \varphi )+\\\displaystyle +d(v\circ \varphi )(grad^{M}({\frac {\partial f}{\partial r}})_{v\circ \varphi })-({\frac {\partial f}{\partial t}})_{v\circ \varphi }{\mbox{.}}\end{array}}}$

(3.8)

In particular at ${\textstyle x_{0}}$

${\displaystyle d(v\circ \varphi )(grad^{M}({\frac {\partial f}{\partial r}})_{v\circ \varphi })=0{\mbox{,}}}$

(3.9)

since ${\textstyle {\frac {\partial v}{\partial y^{\alpha }}}(y_{0})=0}$, ${\textstyle e(v\circ \varphi )=0}$ and ${\textstyle {\frac {\partial f}{\partial t}}(x,t,0)=0}$ for all ${\textstyle (x,t)\in M\times R}$.

By (3.8),  and  and ${\textstyle {\frac {\partial f}{\partial r}}(x,t,r)\not =0}$, we have

${\displaystyle {\begin{array}{l}\displaystyle 0=\Delta ^{M}(v\circ \varphi )=dv(\tau (\varphi ))+trace_{g}\nabla dv(d\varphi \\\displaystyle d\varphi )=trace_{g}\nabla dv(d\varphi ,d\varphi ){\mbox{.}}\end{array}}}$

(3.11)

Since at ${\textstyle x_{0}}$

${\displaystyle \nabla dv=\sum _{\alpha ,\beta }{\frac {\partial ^{2}v}{\partial y^{\alpha }\partial y^{\beta }}}dy^{\alpha }\otimes dy^{\beta }=\sum _{\alpha ,\beta }c_{\alpha \beta }dy^{\alpha }\otimes dy^{\beta }{\mbox{,}}}$

(3.12)

from (3.7), (3.11) and (3.12), we obtain

${\displaystyle {\begin{array}{l}\displaystyle 0=\sum _{\alpha ,\beta }g(grad^{M}\varphi ^{\alpha },grad^{M}\varphi ^{\beta })c_{\alpha \beta }=\sum _{\alpha }g(grad^{M}\varphi ^{\alpha }\\\displaystyle grad^{M}\varphi ^{\alpha })c_{\alpha \alpha }+\sum _{\alpha \not =\beta }g(grad^{M}\varphi ^{\alpha },grad^{M}\varphi ^{\beta })c_{\alpha \beta }\end{array}}}$

(3.13)

by  and , we obtain

${\displaystyle {\begin{array}{l}\displaystyle 0=\sum _{\alpha }[g(grad^{M}\varphi ^{\alpha },grad^{M}\varphi ^{\alpha })-g(grad^{M}\varphi ^{1}\\\displaystyle grad^{M}\varphi ^{1})]c_{\alpha \alpha }+\sum _{\alpha \not =\beta }g(grad^{M}\varphi ^{\alpha },grad^{M}\varphi ^{\beta })c_{\alpha \beta }{\mbox{.}}\end{array}}}$

(3.15)

Let ${\textstyle \alpha _{0}\not =1}$ and let

${\displaystyle c_{\alpha \beta }=\lbrace {\begin{array}{cc}1{\mbox{,}}&{\mbox{if  }}\alpha =\beta =1{\mbox{;}}\\-1{\mbox{,}}&{\mbox{if  }}\alpha =\beta =\alpha _{0}{\mbox{;}}\\0{\mbox{,}}&{\mbox{if  }}\alpha =\beta \not =1,\alpha _{0}{\mbox{;}}\\0{\mbox{,}}&{\mbox{if  }}\alpha \not =\beta {\mbox{,}}\end{array}}}$

then by (3.15), we have

${\displaystyle g(grad^{M}\varphi ^{\alpha _{0}},grad^{M}\varphi ^{\alpha _{0}})=g(grad^{M}\varphi ^{1},grad^{M}\varphi ^{1}){\mbox{,}}}$

(3.16)

for all ${\textstyle \alpha =1,\ldots ,n}$. Let ${\textstyle \alpha _{0}\not =\beta _{0}}$ and

${\displaystyle c_{\alpha \beta }=\lbrace {\begin{array}{cc}1{\mbox{,}}&{\mbox{if  }}\alpha =\alpha _{0}{\mbox{  and  }}\beta =\beta _{0}{\mbox{;}}\\0{\mbox{,}}&{\mbox{if  }}\alpha \not =\alpha _{0}{\mbox{  or  }}\beta \not =\beta _{0}{\mbox{;}}\\0{\mbox{,}}&{\mbox{if  }}\alpha =\beta {\mbox{.}}\end{array}}}$

By (3.15), we have

${\displaystyle g(grad^{M}\varphi ^{\alpha _{0}},grad^{M}\varphi ^{\beta _{0}})=0{\mbox{,}}}$

(3.18)

and

${\displaystyle g(grad^{M}\varphi ^{\alpha },grad^{M}\varphi ^{\beta })=0{\mbox{,}}}$

(3.19)

for all ${\textstyle \alpha \not =\beta =1,\ldots ,n}$.  From  and (3.19) we deduce that ${\textstyle \varphi }$ is horizontally weakly conformal map such that

${\displaystyle g(grad^{M}\varphi ^{\alpha },grad^{M}\varphi ^{\beta })=\lambda ^{2}\quad \delta _{\alpha \beta }{\mbox{,}}}$

(3.20)

for all ${\textstyle \alpha ,\beta =1,\ldots ,n}$.

For every ${\textstyle C^{2}}$-function ${\textstyle v:V\rightarrow R}$ defined on an open subset ${\textstyle V}$ of ${\textstyle N}$, we have

${\displaystyle {\begin{array}{l}\displaystyle \Delta _{f}^{M}(v\circ \varphi )=({\frac {\partial f}{\partial r}})_{v\circ \varphi }\quad \Delta ^{M}(v\circ \varphi )+dv(d\varphi (grad^{M}({\frac {\partial f}{\partial r}})_{v\circ \varphi }))-\\\displaystyle -({\frac {\partial f}{\partial t}})_{v\circ \varphi }=({\frac {\partial f}{\partial r}})_{v\circ \varphi }dv(\tau (\varphi ))+({\frac {\partial f}{\partial r}})_{v\circ \varphi }trace_{g}\nabla dv(d\varphi \\\displaystyle d\varphi )+dv(d\varphi (grad^{M}({\frac {\partial f}{\partial r}})_{v\circ \varphi }))-({\frac {\partial f}{\partial t}})_{v\circ \varphi }{\mbox{.}}\end{array}}}$

(3.21)

Since, ${\textstyle \varphi }$ is a horizontally weakly conformal map, we obtain

${\displaystyle {\begin{array}{l}\displaystyle \Delta _{f}^{M}(v\circ \varphi )=({\frac {\partial f}{\partial r}})_{v\circ \varphi }\quad dv(\tau (\varphi ))+({\frac {\partial f}{\partial r}})_{v\circ \varphi }\quad \lambda ^{2}(\Delta ^{N}v)\circ \varphi +\\\displaystyle +dv(d\varphi (grad^{M}({\frac {\partial f}{\partial r}})_{v\circ \varphi }))-({\frac {\partial f}{\partial t}})_{v\circ \varphi }{\mbox{.}}\end{array}}}$

(3.22)

By special choice of harmonic function ${\textstyle v}$ we have

${\displaystyle ({\frac {\partial f}{\partial r}})_{v\circ \varphi }\quad dv(\tau (\varphi ))+dv(d\varphi (grad^{M}({\frac {\partial f}{\partial r}})_{v\circ \varphi }))-({\frac {\partial f}{\partial t}})_{v\circ \varphi }=0{\mbox{,}}}$

i.e., in any local coordinates ${\textstyle (y^{\alpha })}$ on ${\textstyle N}$, we have

${\displaystyle ({\frac {\partial f}{\partial r}})_{\varphi ^{\alpha }}\quad \tau (\varphi )^{\alpha }+g(grad^{M}({\frac {\partial f}{\partial r}})_{\varphi ^{\alpha }},grad^{M}\varphi ^{\alpha })-({\frac {\partial f}{\partial t}})_{\varphi ^{\alpha }}=0{\mbox{,}}}$

for all ${\textstyle \alpha =1,\ldots ,n}$. Thus, we obtain the implication ${\textstyle (1)\Longrightarrow (2)}$. Therefore, it follows from (3.22) that ${\textstyle (2)\Longrightarrow (3)}$. Finally, ${\textstyle (3)\Longrightarrow (1)}$ is clearly true.  □

Particular cases:

• If ${\textstyle f(x,t,r)=r}$ for all ${\textstyle (x,t,r)\in M\times R\times R}$, the condition (3.5) is equivalent to ${\textstyle \tau (\varphi )=0}$, i.e.  ${\textstyle \varphi }$ is harmonic. Then, a smooth map ${\textstyle \varphi :M\rightarrow N}$ between Riemannian manifolds is a harmonic morphism if and only if ${\textstyle \varphi :M\rightarrow N}$ is both harmonic and horizontally weakly conformal  [2].
• If ${\textstyle f(x,t,r)=f_{1}(x)r\quad }$ for all ${\textstyle (x,t,r)\in M\times R\times R}$, where ${\textstyle f_{1}\in C^{\infty }(M)}$ be a smooth positive function, the condition (3.5) is equivalent to ${\textstyle f_{1}\quad \tau (\varphi )+d\varphi (grad^{M}f_{1})=0}$, i.e.  ${\textstyle \varphi }$ is ${\textstyle f_{1}}$-harmonic. Then, a smooth map ${\textstyle \varphi :M\rightarrow N}$ between Riemannian manifolds is an ${\textstyle f_{1}}$-harmonic morphism if and only if ${\textstyle \varphi :M\rightarrow N}$ is both ${\textstyle f_{1}}$-harmonic and horizontally weakly conformal  [14].
• If ${\textstyle f(x,t,r)=f_{1}(x,t)r\quad }$ for all ${\textstyle (x,t,r)\in M\times R\times R}$, where ${\textstyle f_{1}\in C^{\infty }(M\times R)}$ be a smooth positive function, then ${\textstyle \varphi }$ is an ${\textstyle f}$-harmonic morphism if and only if ${\textstyle \varphi }$ is an ${\textstyle f_{1}}$-harmonic morphism  [3].
• If ${\textstyle f(x,t,r)=F(r)}$ for all ${\textstyle (x,t,r)\in M\times R\times R}$, where ${\textstyle F\in C^{\infty }(R)}$ be a smooth function such that ${\textstyle F^{'}>0}$, then, the following are equivalent:
• ${\textstyle \varphi }$ is an ${\textstyle f}$-harmonic morphism;
• ${\textstyle \varphi }$ is an ${\textstyle F}$-harmonic morphism;
• ${\textstyle \varphi }$ is horizontally weakly conformal satisfying

${\displaystyle F^{'}(e(\varphi ^{\alpha }))\quad \tau (\varphi )^{\alpha }+g(grad^{M}F^{'}(e(\varphi ^{\alpha })),grad^{M}\varphi ^{\alpha })=0{\mbox{,}}}$

for all ${\textstyle \alpha =1,\ldots ,n}$ and in any local coordinates ${\textstyle (y^{\alpha })}$ on ${\textstyle N}$;

• There exists a smooth positive function ${\textstyle \lambda }$ on ${\textstyle M}$ such that

${\displaystyle \Delta _{f}^{M}(v\circ \varphi )=F^{'}(e(v\circ \varphi ))\quad \lambda ^{2}\quad (\Delta ^{N}v)\circ \varphi {\mbox{,}}}$

for every smooth function ${\textstyle v}$ defined on an open subset ${\textstyle V}$ of ${\textstyle N}$.

Proposition 3.1.

Let  ${\textstyle \varphi :M\rightarrow N}$be an  ${\textstyle f}$-harmonic morphism between Riemannian manifolds with dilation  ${\textstyle \lambda _{1}}$,  ${\textstyle \psi :N\rightarrow P}$be a harmonic morphism between Riemannian manifolds with dilation  ${\textstyle \lambda _{2}}$and  ${\textstyle f}$be a smooth positive function in  ${\textstyle M\times R\times R}$satisfying   (3.4). Then, the composition  ${\textstyle \psi \circ \varphi :M\rightarrow P}$is an  ${\textstyle f}$-harmonic morphism with dilation  ${\textstyle \lambda _{1}(\lambda _{2}\circ \varphi )}$.

Proof.

This follows from

${\displaystyle \Delta _{f}^{M}(v\circ \varphi )=({\frac {\partial f}{\partial r}})_{v\circ \varphi }\quad \lambda _{1}^{2}\quad (\Delta ^{N}v)\circ \varphi {\mbox{,}}}$

for every smooth function ${\textstyle v}$ defined on an open subset ${\textstyle V}$ of ${\textstyle N}$ and

${\displaystyle \Delta ^{N}(u\circ \psi )=\lambda _{2}^{2}\quad (\Delta ^{P}u)\circ \psi {\mbox{,}}}$

for every smooth function ${\textstyle u}$ defined on an open subset ${\textstyle U}$ of ${\textstyle P}$. So that

${\displaystyle {\begin{array}{l}\displaystyle \Delta _{f}^{M}(u\circ \psi \circ \varphi )=({\frac {\partial f}{\partial r}})_{u\circ \psi \circ \varphi }\quad \lambda _{1}^{2}\quad (\Delta ^{N}(u\circ \psi ))\circ \varphi =\\\displaystyle =({\frac {\partial f}{\partial r}})_{u\circ \psi \circ \varphi }\lambda _{1}^{2}(\lambda _{2}\circ \varphi )^{2}(\Delta ^{P}u)\circ \psi \circ \varphi {\mbox{.}}\end{array}}}$

 □

Proposition 3.2.

Let  ${\textstyle (M,g)}$be a Riemannian manifold and  ${\textstyle f:M\times R\times R\rightarrow (0,+\infty )}$be a smooth positive function satisfying   (3.4). A smooth map

${\displaystyle \varphi :(M,g)\rightarrow (R^{n},\left\langle ,\right\rangle _{R^{n}}){\mbox{,}}\quad x\mapsto (\varphi ^{1}(x),\ldots ,\varphi ^{n}(x))}$

is an  ${\textstyle f}$-harmonic morphism if and only if its components  ${\textstyle \varphi ^{\alpha }}$are  ${\textstyle f}$-harmonic functions whose gradients are orthogonal and of the same norm at each point.

Proof.

Let us notice that the condition (3.5) of Theorem 3.1 becomes

${\displaystyle ({\frac {\partial f}{\partial r}})_{\varphi ^{\alpha }}\quad \Delta ^{M}\varphi ^{\alpha }+g(grad^{M}({\frac {\partial f}{\partial r}})_{\varphi ^{\alpha }},grad^{M}\varphi ^{\alpha })-({\frac {\partial f}{\partial t}})_{\varphi ^{\alpha }}=0{\mbox{,}}}$

for all ${\textstyle \alpha =1,\ldots ,n}$, i.e. the functions ${\textstyle \varphi ^{\alpha }}$ are ${\textstyle f}$-harmonic.  □

Using Proposition 3.2, we can construct many non-trivial examples on ${\textstyle R^{n}}$.

Example 3.2.

Let ${\textstyle M=R_{+}^{_{\ast }}\times R\times R}$, then the map

${\displaystyle \varphi :(M,\left\langle ,\right\rangle _{R^{3}})\rightarrow (R^{2},\left\langle ,\right\rangle _{R^{2}}){\mbox{,}}\quad (x,y,z)\mapsto ({\sqrt {x^{2}+y^{2}}},z){\mbox{,}}}$

is an ${\textstyle f}$-harmonic morphism with

${\displaystyle f(x,y,z,t,r)={\frac {F({\frac {y}{x}})\quad e^{-{\frac {1}{2}}(x^{2}+y^{2}+z^{2})+(t^{2}+1)\quad r}}{x\quad {\sqrt {(x^{2}+y^{2}+1)(z^{2}+1)}}}}{\mbox{,}}}$

where ${\textstyle F}$ is a smooth positive function. Indeed, we have

${\displaystyle \varphi ^{1}(x,y,z)={\sqrt {x^{2}+y^{2}}}{\mbox{,}}\quad \varphi ^{2}(x,y,z)=z{\mbox{,}}}$

Let ${\textstyle f(x,y,z,t,r)=h(x,y,z)\quad e^{(t^{2}+1)\quad r}}$ for all ${\textstyle (x,y,z,t,r)\in M\times R\times R}$, where ${\textstyle h}$ is a smooth positive function in ${\textstyle M}$, we get

${\displaystyle \lbrace {\begin{array}{cc}{\frac {\partial f}{\partial r}}(x,y,z,t,r)=h(x,y,z)\quad (t^{2}+1)\quad e^{(t^{2}+1)\quad r}\not =0{\mbox{,}}&\\{\frac {\partial f}{\partial t}}(x,y,z,t,0)=2\quad h(x,y,z)\quad t\quad r\quad e^{(t^{2}+1)\quad r}\vert _{r=0}=0{\mbox{,}}&\end{array}}}$

for all ${\textstyle (x,y,z,t,r)\in M\times R\times R}$,

${\displaystyle {\begin{array}{l}\displaystyle ({\frac {\partial f}{\partial r}})_{\varphi ^{1}}=h(x,y,z)\quad (x^{2}+y^{2}+1)\quad e^{\frac {x^{2}+y^{2}+1}{2}}{\mbox{,}}\quad ({\frac {\partial f}{\partial r}})_{\varphi ^{2}}=h(x\\\displaystyle y,z)(z^{2}+1)e^{\frac {z^{2}+1}{2}}{\mbox{,}}\end{array}}}$

According to Proposition 3.2, the map ${\textstyle \varphi }$ is ${\textstyle f}$-harmonic if and only if

${\displaystyle \lbrace {\begin{array}{cc}3\quad h\quad x^{2}+3\quad h\quad y^{2}+h+{\frac {\partial h}{\partial x}}\quad x^{3}+{\frac {\partial h}{\partial x}}\quad x\quad y^{2}+{\frac {\partial h}{\partial x}}\quad x+h\quad x^{4}&\\+\quad 2\quad h\quad x^{2}\quad y^{2}+{\frac {\partial h}{\partial y}}\quad x^{2}\quad y+{\frac {\partial h}{\partial y}}\quad y^{3}+{\frac {\partial h}{\partial y}}\quad y+h\quad y^{4}=0{\mbox{,}}&\\{\frac {\partial h}{\partial z}}\quad z^{2}+{\frac {\partial h}{\partial z}}+2\quad h\quad z+h\quad z^{3}=0{\mbox{.}}&\end{array}}}$

(3.23)

Let ${\textstyle F\in C^{\infty }(R)}$ be a smooth positive function, then the function of type

${\displaystyle h(x,y,z)={\frac {F({\frac {y}{x}})\quad e^{-{\frac {1}{2}}(x^{2}+y^{2}+z^{2})}}{x\quad {\sqrt {(x^{2}+y^{2}+1)(z^{2}+1)}}}}}$

satisfies the system of differential equations  (3.23).

Acknowledgments

The authors would like to thank the referees for their important and useful remarks and suggestions.

This note was supported by G.M.F.A.M.I Relizane Laboratory and Saida Laboratory of Geometry analysis and Applications.

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