\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 165, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/165\hfil Harmonic-hyperbolic geometric flow]
{Harmonic-hyperbolic geometric flow}

\author[S. Azami \hfil EJDE-2017/165\hfilneg]
{Shahroud Azami}

\address{Shahroud Azami\newline
Department of Mathematics, Faculty of Sciences,
Imam Khomeini International University, 
Qazvin, Iran}
\email{azami@sci.ikiu.ac.ir}

\dedicatory{Communicated by Paul H. Rabinowitz}

\thanks{Submitted January 2, 2017. Published July 5, 2017.}
\subjclass[2010]{53C44, 58J45, 58J47}
\keywords{Hyperbolic geometric flow; quasilinear hyperbolic equation;
\hfill\break\indent strict hyperbolicity}

\begin{abstract}
 In this article we study a coupled system for hyperbolic geometric
 flow on a closed manifold $M$, with a harmonic flow map from $M$ 
 to some closed target manifold $N$.
 Then we show that this flow has a unique solution for a short-time.
 After that, we find evolution equations for Riemannian curvature tensor,
 Ricci curvature tensor, and scalar curvature of $M$ under  this flow.
 In the final section we give some examples of this flow on closed manifolds.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Let $(M^{m},g)$ and $(N^{n},\gamma)$ be smooth closed Riemannian manifolds.
Suppose that $N$ is isometrically embedded into Euclidean space
 $e_{N}:(N^{n},\gamma)\hookrightarrow\mathbb{R}^{d}$ for a sufficiently large $d$.
We identify maps $\varphi:M \to N$ with
 $e_{N}\circ \varphi:\hookrightarrow\mathbb{R}^{d}$.
Harmonic maps $\varphi: (M,g)\to(N,\gamma)$ are critical point of the
energy functional $E(\varphi)=\int_{M}|\nabla\varphi|^{2}d\mu$, where $d\mu$
is the volume form on $M$ with respect to the metric $g$ and
\begin{equation*}
|\nabla\varphi|^{2}:=\frac{1}{2}g^{ij}(\gamma_{\alpha\beta})_{\varphi}
\frac{\partial \varphi^{\alpha}}{\partial x^{i}}
\frac{\partial \varphi^{\beta}}{\partial x^{j}}.
\end{equation*}
Harmonic maps are generalizations of harmonic functions. For example the identity
and constant maps are harmonic maps, also, geodesics as the map $S^{1}\to M$
are harmonic maps. The first  major study of harmonic mapping between
Riemannian manifolds was made by Eells and Sampson \cite{ES}. They study
the harmonic map flow
\begin{equation}\label{ES}
\frac{\partial \varphi}{\partial t}=\tau_{g}\varphi,\quad
\varphi(0)=\varphi_{0}.
\end{equation}
where $\tau_{g} \varphi$ denotes the tension field of $\varphi$, and showed,
under suitable metric and curvature assumptions on the target manifold,
flow \eqref{ES} has unique solution.
The harmonic map flow is a nonlinear heat flow in geometric analysis.
Another, nonlinear heat flow and  wave flow in geometric analysis are
geometric flows. Geometric flows are important problem in differential geometry,
because by these flow we can find canonical metrics on Riemannian manifolds.
A geometric flow is an evolution of a geometric structure under a
differential equation with a functional on a manifold.

Let $M$ be an $n$-dimensional complete Riemannian  manifold  with the Riemannian
metric $g=(g_{ij})$. The Levi-Civita connection is given by the Christoffel symbols
\begin{equation}
\Gamma_{ij}^{k}=\frac{1}{2}g^{kl}\big\{\frac{\partial g_{jl}}{\partial x^{i}}
+\frac{\partial g_{il}}{\partial x^{j}}-\frac{\partial g_{ij}}{\partial x^{l}}\big\}
\end{equation}
and  Riemannian curvature tensor, Ricci curvature tensor, scalar curvature of
 $(M,g)$  as follows
\begin{gather*}
R_{ijl}^{k}=\frac{\partial \Gamma_{jl}^{k}}{\partial x^{i}}
 -\frac{\partial \Gamma_{il}^{k}}{\partial x^{j}}
 +\Gamma_{ip}^{k}\Gamma_{jl}^{p}-\Gamma_{jp}^{k}\Gamma_{il}^{p},\quad
 R_{ijkl}=g_{kp}R_{ijl}^{p},\\
R_{ik}=g^{jl}R_{ijkl},\quad R=g^{ij}R_{ij}.
\end{gather*}
The first important geometric flow is Ricci flow, defined as follows,
\begin{equation}\label{RF}
\frac{\partial }{\partial t}g=-2\operatorname{Ric},\quad g(0)=g_{0}
\end{equation}
where Ric denotes the Ricci curvature. The Ricci flow was  introduced by
 Hamilton in 1982 \cite{RH} and evolves a Riemannian metric by its Ricci curvature,
is a natural  analogue of the heat equation for metrics.
The existence solution of Ricci flow studied by  Hamilton (see \cite{RH}) and
 DeTurck (see \cite{DD}) on closed Riemannian manifolds.
Also evolution equation for geometric structures dependant to metric
investigated by some researcher (see \cite{BD}).

The second geometric flow is hyperbolic geometric flow which  is a system of
nonlinear evolution partial differential equations of second order,
it is very similar to wave equation flow metrics, defined as follows,
\begin{equation}\label{HGF}
\frac{\partial^{2} }{\partial t^{2}}g=-2\operatorname{Ric},\quad
g(0)=g_{0},\quad \frac{\partial g}{\partial t}(0)=k_{0}.
\end{equation}
where $k_{0}$ is a symmetric tensor on $M$ and this flow is similar to
Einstein equation
\begin{equation*}
\frac{\partial^{2} }{\partial t^{2}}g_{ij}
=-2R_{ij}-\frac{1}{2}g^{pq}\frac{\partial g_{ij}}{\partial t}
\frac{\partial g_{pq}}{\partial t}+g^{pq}\frac{\partial g_{ip}}{\partial t}
\frac{\partial g_{jq}}{\partial t}.
\end{equation*}
 The existences and uniqueness of \eqref{HGF} studied in \cite{DDK} on closed
Riemannian manifold.

Another important geometric flow is the harmonic-Ricci flow, defined as follows,
\begin{equation}\label{HRF}
\begin{gathered}
\frac{\partial }{\partial t}g=-2\operatorname{Ric}+2\alpha \nabla\varphi\otimes\nabla\varphi ,
\quad g(0)=g_{0},\\
\frac{\partial }{\partial t}\varphi=\tau_{g}\varphi,\quad \varphi(0)=\varphi_{0}.
\end{gathered}
\end{equation}
where  $\alpha$ is positive coupling constant, $\varphi$ is a map from $M$
to some closed target manifold $N$, and this flow studied in \cite{RM}.

Motivated by the above works, in this article we consider an $m$-dimensional,
closed smooth, Riemannian manifold $M$ whose metric $g=g(t)$ is evolving
according to the flow equation
\begin{equation}\label{e1}
\begin{gathered}
\frac{\partial^{2} }{\partial t^{2}}g=-2\operatorname{Ric}
+2\alpha \nabla\varphi\otimes\nabla\varphi ,\quad g(0)=g_{0},\quad
\frac{\partial g}{\partial t}(0)=k_{0}\\
\frac{\partial }{\partial t}\varphi=\tau_{g}\varphi,\quad
\varphi(0)=\varphi_{0}.
\end{gathered}
\end{equation}
where $k_{0}$ is a symmetric tensor on $M$,  $\operatorname{Ric}$ is
the Ricci tensor of the manifold, $\alpha$ is positive coupling constant,
$\varphi(t)$ a family of smooth maps from $M$ to $N$  and
$\tau_{g}\varphi$ denotes the tension field of the map $\varphi$
with respect to the evolving metric $g$.
Finally, $(\nabla\varphi\otimes\nabla \varphi)_{ij}
=\nabla_{i}\varphi^{\lambda}\nabla_{j}\varphi^{\lambda}$
is components of $\nabla_{i}\varphi^{\lambda}$.
This flow called harmonic-hyperbolic geometric flow and after this,
in short, we will display it with $(HG)_{\alpha}$ flow.

\section{Short-time existence and uniqueness for the  $(HG)_{\alpha}$ flow}

In this section we study  the existence and uniqueness of the
$(HG)_{\alpha}$ flow. We use a process similar to the one in the existence
and uniqueness of geometric flow, for the Ricci flow, hyperbolic geometric flow, and
harmonic-Ricci flow.

\begin{theorem} \label{thm2.1}
Let $(M, g_{0})$  and $(N,\gamma)$ be  compact Riemannian manifolds and
$ k_{0}$ be a symmetric tensor on $M$.   Then there exists a constant $T>0$
such that the initial value problem \eqref{e1} has a unique smooth solution
 metric $g$ and map $\varphi$ on  $M\times[0,T]$.
\end{theorem}

\begin{proof}
 Using the gauge fixing idea as in the Ricci flow (see \cite{RH1}) and
the push-forward of  a solution of \eqref{e1} we can find a system of
nonlinear strictly-hyperbolic partial differential equations of
second order and then the short-time existence and uniqueness result on
a compact manifold, show that  the existence and uniqueness for this
system and in finally  similar to the proof of existence and uniqueness
for Ricci flow (see \cite{RH1})  the pull-back of this solution complete
the proof of theorem. For this end, let $({g}(t),{\varphi}(t))_{t\in[0,T)}$
is a solution of the $(HG)_{\alpha}$ flow with initial data
$(g(0),\varphi(0))=(g_{0}, \varphi_{0})$,
$\frac{\partial g_{ij}}{\partial t}(0)=k_{ij}(0)$.
Let $\psi_{t}:(M, \hat{g}(t))\to (M,g_{0})$ be solution of  the harmonic map
heat flow $\frac{\partial}{\partial t}\psi=\tau_{g}\psi$, with $\psi(0)=id_{M}$. Let
\begin{equation}
\hat{g}_{ij}(t)=\psi_{*}{g_{ij}},\quad \hat{\varphi}(t)=\psi_{*}{\varphi}(t)
 \end{equation}
be the push-forward of  ${g_{ij}}$ and ${\varphi}$ respectively.
We now find the evolution for $(\hat{g}_{ij}(t),\hat{ \varphi}(t))$. Denote by
$y(x,t)=\psi_{t}(x)=(y^{1}(x,t),\dots ,y^{n}(x,t))$ in locally coordinates. Then
\begin{equation}
\hat{g}_{ij}(x,t)=\frac{\partial y^{\alpha}}{\partial x^{i}}
\frac{\partial y^{\beta}}{\partial x^{j}}g_{\alpha\beta}(y,t)
 \end{equation}
by direct computations,  we have
\begin{align*}
\frac{\partial^{2} \hat{g}_{ij}}{\partial t^{2}}(x,t)
&=\frac{\partial^{2} {g}_{\alpha\beta}}{\partial t^{2}}
\frac{\partial y^{\alpha}}{\partial x^{i}}\frac{\partial y^{\beta}}{\partial x^{j}}
+\frac{\partial^{2} {g}_{\alpha\beta}}{\partial y^{\gamma}\partial y^{\lambda}}
 \frac{\partial y^{\alpha}}{\partial x^{i}}
 \frac{\partial y^{\beta}}{\partial x^{j}}
 \frac{\partial y^{\gamma}}{\partial t}
 \frac{\partial y^{\lambda}}{\partial t}\\
&\quad +2\frac{\partial^{2} {g}_{\alpha\beta}}{\partial y^{\gamma}\partial t}
 \frac{\partial y^{\alpha}}{\partial x^{i}}
 \frac{\partial y^{\beta}}{\partial x^{j}}
 \frac{\partial y^{\gamma}}{\partial t}
+\frac{\partial}{\partial x^{i}}({g}_{\alpha\beta}
 \frac{\partial y^{\beta}}{\partial x^{j}}
 \frac{\partial^{2} y^{\alpha}}{\partial t^{2}})
+\frac{\partial}{\partial x^{j}}({g}_{\alpha\beta}
 \frac{\partial y^{\beta}}{\partial x^{i}}
 \frac{\partial^{2} y^{\alpha}}{\partial t^{2}})\\
&\quad +\Big[\frac{\partial {g}_{\alpha\beta}}{\partial y^{\gamma}}
 \frac{\partial y^{\alpha}}{\partial x^{i}}
 \frac{\partial y^{\beta}}{\partial x^{j}}
-\frac{\partial}{\partial x^{i}}({g}_{\beta\gamma}
 \frac{\partial y^{\beta}}{\partial x^{j}})
-\frac{\partial}{\partial x^{j}}({g}_{\beta\gamma}
 \frac{\partial y^{\beta}}{\partial x^{i}})\Big]
 \frac{\partial^{2} y^{\gamma}}{\partial t^{2}}\\
&\quad +2\frac{\partial}{\partial x^{i}}(\frac{\partial y^{\alpha}}{\partial t})
 \frac{\partial y^{\beta}}{\partial x^{j}}
 (\frac{\partial {g}_{\alpha\beta}}{\partial t}
+\frac{\partial  {g}_{\alpha\beta}}{\partial y^{\gamma}}
 \frac{\partial y^{\gamma}}{\partial t})\\
&\quad +2\frac{\partial y^{\alpha}}{\partial x^{i}}
 \frac{\partial}{\partial x^{j}}(\frac{\partial y^{\beta}}{\partial t})
 (\frac{\partial  {g}_{\alpha\beta}}{\partial t}
+\frac{\partial  {g}_{\alpha\beta}}{\partial y^{\gamma}}
 \frac{\partial y^{\gamma}}{\partial t})+2{g}_{\alpha\beta}
 \frac{\partial}{\partial x^{i}}(\frac{\partial y^{\alpha}}{\partial t})
 \frac{\partial}{\partial x^{j}}(\frac{\partial y^{\beta}}{\partial t}).
\end{align*}
For the normal coordinates $\{x^{i}\}$ around  a fixe point $p\in M$,
we have $\frac{\partial g_{ij}}{\partial x^{k}}(p)=0$  and
\begin{equation}
\frac{\partial  {g}_{\alpha\beta}}{\partial y^{\gamma}}
\frac{\partial y^{\alpha}}{\partial x^{i}}\frac{\partial y^{\beta}}{\partial x^{j}}
-\frac{\partial}{\partial x^{i}}({g}_{\beta\gamma}
 \frac{\partial y^{\beta}}{\partial x^{j}})
-\frac{\partial}{\partial x^{j}}({g}_{\beta\gamma}
 \frac{\partial y^{\beta}}{\partial x^{i}})=0,
\quad \forall i,j,\gamma =1,2,\dots ,n.
\end{equation}
Let $y(x,t)$ be a solution of the  equation
\begin{equation}
\begin{gathered}
\frac{\partial^{2} y^{\alpha}}{\partial t^{2}}
=\frac{\partial y^{\alpha}}{\partial x^{k}}
g^{il}(\hat{\Gamma}_{jl}^{k} -\mathring{\Gamma}^k_{jl} )\\
y^{\alpha}(x,0)=x^{\alpha},\quad
 \frac{\partial}{\partial t}y^{\alpha}(x,0)=y_{1}^{\alpha}(x)
\end{gathered}
\end{equation}
and define the vector field
\begin{equation}
V_{i}=g_{ik}g^{jl}(\hat{\Gamma}_{jl}^{k}-\mathring{\Gamma}_{jl}^k)
\end{equation}
where $\hat{\Gamma}_{jl}^{k}$ and $\mathring{\Gamma}_{jl}^k$ are the connection
coefficients corresponding to the metrics $\hat{g}_{ij}(x,t)$
and  $g_{ij}(x,0)$, respectively, $y_{1}^{\alpha}(x)\in C^{\infty}(M)$.
Since $\frac{\partial^{2} }{\partial t^{2}}g_{ij}=-2R_{ij}
+2\alpha \nabla_{i}\varphi\nabla_{j}\varphi $,
therefore the evolution equation for $\hat{g}_{ij}$ is
\begin{equation}
\frac{\partial^{2} }{\partial t^{2}}\hat{g}_{ij}
=-2\hat{R}_{ij}+2\alpha\nabla_{i}\hat{\varphi}\nabla_{j}\hat{\varphi}
+\hat{\nabla}_{i}V_{j}+\hat{\nabla}_{j}V_{i}+F(Dy,D_{t}D_{x}y),
\end{equation}
where
\begin{equation*}
Dy=\big(\frac{\partial y^{\alpha}}{\partial t},
\frac{\partial y^{\alpha}}{\partial x^{i}}\big),\quad
D_{t}D_{x}y =\big(\frac{\partial^{2} y^{\alpha}}{\partial x^{i}\partial t}\big),
\quad \alpha,\,i=1,2,\dots ,n.
\end{equation*}
The relation
\[
\hat{\Gamma}_{jl}^{k}=\frac{\partial y^{\alpha}}{\partial x^{j}}
\frac{\partial y^{\beta}}{\partial x^{i}}
\frac{\partial x^{k}}{\partial y^{\gamma}}\Gamma_{\alpha\beta}^{\gamma}
+\frac{\partial x^{k}}{\partial y^{\alpha}}
\frac{\partial^{2}y^{\alpha}}{\partial x^{j}\partial x^{i}}
\]
implies
\begin{equation}\label{e8}
\frac{\partial^{2} y^{\alpha}}{\partial t^{2}}
=g^{jl}\big(\frac{\partial^{2}y^{\alpha}}{\partial x^{j}\partial x^{i}}
-\mathring{\Gamma}_{jl}^k  \frac{\partial y^{\alpha}}{\partial x^{j}}
+\Gamma_{\alpha\beta}^{\gamma} \frac{\partial y^{\beta}}{\partial x^{j}}
\frac{\partial y^{\gamma}}{\partial x^{i}} \big)
\end{equation}
and
\begin{equation}\label{e9}
\frac{\partial^{2} }{\partial t^{2}}\hat{g}_{ij}
=\hat{g}^{kl}\frac{\partial^{2}\hat{g}_{ij}}{\partial x^{k}\partial x^{l}}
+2\alpha\nabla_{i}\hat{\varphi}\nabla_{j}\hat{\varphi}
+G(\hat{g},D_{x}\hat{g})+F(Dy,D_{t}D_{x}y),
\end{equation}
where $\hat{g}=(\hat{g}_{ij})$,
$D_{x}\hat{g}=(\frac{\partial \hat{g}_{ij}}{\partial x^{k}})$ for
$i,j,k=1,2,\dots ,n$. Hence, both \eqref{e8} and \eqref{e9} are
clearly strictly hyperbolic system.
On the other hand,
\begin{equation*}
\frac{\partial \hat{\varphi}}{\partial t}
=\psi_{*}(\frac{\partial {\varphi}}{\partial t})+L_{V} \hat{\varphi}
=\tau_{\hat{g}} \hat{\varphi}+\langle \nabla \hat{\varphi}, V\rangle
=\tau_{\hat{g}}\hat{\varphi}+d \hat{\varphi}(V).
\end{equation*}
Using normal coordinates on $(N,\gamma)$ results that
${}^{N}\Gamma_{\mu\nu}^{\lambda}=0$ at the base point and hence
$\tau_{\hat{g}}\hat{\varphi}=\Delta_{\hat{g}}\hat{\varphi}$
which implies that
\begin{equation} \label{e10}
\begin{aligned}
\frac{\partial \hat{\varphi}}{\partial t}
&=\Delta_{\hat{g}}\hat{\varphi}+d \hat{\varphi}(V)
=\hat{g}^{kl}(\partial_{k}\partial_{l}\hat{\varphi}^{\lambda}-\hat{\Gamma}_{kl}^{j}
 \nabla_{j}\hat{\varphi}^{\lambda})
+ \nabla_{j}\hat{\varphi}^{\lambda}\hat{g}^{kl}(\hat{\Gamma}_{kl}^{j}
 -\mathring{\Gamma}_{jl}^k  )\\
&=\hat{g}^{kl}(\partial_{k}\partial_{l}\hat{\varphi}^{\lambda}
 -\mathring{\Gamma}_{jl}^k  \nabla_{j}\hat{\varphi}^{\lambda})
\end{aligned}
\end{equation}
and it is strictly hyperbolic equation. Since the equations \eqref{e8}, \eqref{e9}
 and \eqref{e10} are strictly hyperbolic and the manifold $M$ is compact,
 it follows from the standard  theory of hyperbolic equations
(see \cite{SK}) that the system \eqref{e1} has a unique smooth solution for
a short time. So, the proof of the theorem is complete.
\end{proof}

\section{Evolution equations of curvature tensor along the $(HG)_{\alpha}$  flow}

Next, we consider the techniques and ideas used by Brendle \cite{BD}
for  evolution equation  along the Ricci flow, and  by  Dai and et al \cite{DDK}
for the evolution  equation along the hyperbolic geometric flow.
We find  the evolution formula  for Riemannian curvature tensor,
Ricci curvature tensor and scalar curvature of $(M,g)$ under the $(HG)_{\alpha}$
flow.

\begin{theorem} \label{thm3.1}
Under the $(HG)_{\alpha}$  flow, the Riemannian curvature tensor $R_{ijkl}$
of  $(M,g)$ satisfies the evolution equation
\begin{equation} \label{q1}
\begin{aligned}
&\frac{\partial^{2}}{\partial t^{2}}R_{ijkl} \\
&=\Delta R_{ijkl}+2(B_{ijkl}-B_{ijlk}-B_{iljk}+B_{ikjl})\\
&\quad -g^{pq}(R_{pjkl}R_{qi}+R_{ipkl}R_{qj}+R_{ijpl}R_{qk}+R_{ijkp}R_{ql})\\
&\quad +2g_{pq}\Big(\frac{\partial}{\partial t}\Gamma_{il}^{p}
 \frac{\partial}{\partial t}\Gamma_{jk}^{q}
 -\frac{\partial}{\partial t}\Gamma_{jl}^{p}
 \frac{\partial}{\partial t}\Gamma_{ik}^{q} \Big)\\
&\quad +\alpha \Big[\frac{\partial^{2}(\nabla_{k}\varphi\nabla_{j}\varphi)}
 {\partial x^{i}\partial x^{l}}
-\frac{\partial^{2}(\nabla_{j}\varphi\nabla_{l}\varphi)}
 {\partial x^{i}\partial x^{k}}  - \frac{\partial^{2}(\nabla_{k}\varphi\nabla_{i}\varphi)}{\partial x^{j}\partial x^{l}}
-\frac{\partial^{2}(\nabla_{i}\varphi\nabla_{l}\varphi) }
 {\partial x^{j}\partial x^{k}}
\Big]
\end{aligned}
\end{equation}
where $B_{ijkl}=g^{pr}g^{qs}R_{piqj}R_{rksl}$ and $\Delta$ is the Laplacian
with respect to the evolving metric $g$.
\end{theorem}

\begin{proof}
 The Christoffel symbol  of metric $g$ is
$\Gamma_{jl}^{h}=\frac{1}{2}g^{hm}\big(\frac{\partial g_{mj}}{\partial x^{l}}+\frac{\partial g_{ml}}{\partial x^{j}}-\frac{\partial g_{jl}}{\partial x^{m}}\big)
$, therefore by direct computations,
\begin{align*}
\frac{\partial^{2}}{\partial t^{2}}\Gamma_{jl}^{h}
&=\frac{1}{2}\frac{\partial^{2}g^{hm}}{\partial t^{2}}
 \Big(\frac{\partial g_{mj}}{\partial x^{l}}
 +\frac{\partial g_{ml}}{\partial x^{j}}
 -\frac{\partial g_{jl}}{\partial x^{m}}\Big)
 +\frac{\partial g^{hm}}{\partial t}
 \Big(\frac{\partial^{2}g_{mj}}{\partial x^{l}\partial t}
 +\frac{\partial^{2} g_{ml}}{\partial x^{j}\partial t}
 -\frac{\partial^{2} g_{jl}}{\partial x^{m}\partial t}\Big)\\
&\quad +\frac{1}{2}g^{hm}\Big(\frac{\partial }{\partial x^{l}}
 (\frac{\partial^{2}g_{mj}}{\partial t^{2}})
 +\frac{\partial }{\partial x^{j}}(\frac{\partial^{2}g_{ml}}{\partial t^{2}})
 -\frac{\partial }{\partial x^{m}}(\frac{\partial^{2}g_{jl}}{\partial t^{2}})\Big).
\end{align*}
On the other hand,
$R_{ijl}^{h}=\frac{\partial \Gamma_{jl}^{h}}{\partial x^{i}}
 -\frac{\partial \Gamma_{il}^{h}}{\partial x^{j}}
 +\Gamma_{ip}^{h}\Gamma_{jl}^{p}-\Gamma_{jp}^{h}\Gamma_{il}^{p}$ 
and the Riemannian curvature tensor of $(M,g)$ is
$R_{ijkl}=g_{hk}R_{ijl}^{h}$, thus with a double differentiation respect 
to $t$ we have
\begin{equation} \label{q3}
\begin{aligned}
&\frac{\partial^{2}}{\partial t^{2}}R_{ijkl} \\
&=g_{hk}\Big[ \frac{\partial }{\partial x^{i}}
 (\frac{\partial^{2}\Gamma_{jl}^{h}}{\partial t^{2}})
 -\frac{\partial }{\partial x^{j}}
 (\frac{\partial^{2}\Gamma_{il}^{h}}{\partial t^{2}})
 +\frac{\partial^{2}}{\partial t^{2}}(\Gamma_{ip}^{h}\Gamma_{jl}^{p}
 -\Gamma_{jp}^{h}\Gamma_{il}^{p})\Big]\\
&\quad +2\frac{\partial g_{hk}}{\partial t}
\Big[ \frac{\partial }{\partial x^{i}}
 (\frac{\partial\Gamma_{jl}^{h}}{\partial t})-
\frac{\partial }{\partial x^{j}}(\frac{\partial\Gamma_{il}^{h}}{\partial t})
+\frac{\partial}{\partial t}(\Gamma_{ip}^{h}\Gamma_{jl}^{p}
 -\Gamma_{jp}^{h}\Gamma_{il}^{p})\Big]
+R_{ijl}^{h}\frac{\partial^{2} g_{hk}}{\partial t^{2}}.
\end{aligned}
\end{equation}
We choose the normal coordinates  around a fixed point $p$ on $M$, then
$\frac{\partial g_{ij}}{\partial x^{k}}(p)=0$ and $\Gamma_{ij}^{k}(p)=0$.
 Since $\frac{\partial^{2} }{\partial t^{2}}g
=-2\operatorname{Ric}+2\alpha \nabla\varphi\otimes\nabla\varphi $,
then we can rewrite \eqref{q3} as follows:
\begin{equation} \label{q4}
\begin{aligned}
&\frac{\partial^{2}}{\partial t^{2}}R_{ijkl} \\ 
&=\frac{1}{2}\Big[\frac{\partial^{2}}{\partial x^{i}\partial x^{l}}
 (-2R_{kj}+2\alpha\nabla_{k}\varphi\nabla_{j}\varphi)
 -\frac{\partial^{2}}{\partial x^{i}\partial x^{k}}
  (-2R_{jl}+2\alpha\nabla_{j}\varphi\nabla_{l}\varphi)  \Big]\\
&\quad -\frac{1}{2}\Big[ \frac{\partial^{2}}{\partial x^{j}\partial x^{l}}
 (-2R_{ki}+2\alpha\nabla_{k}\varphi\nabla_{i}\varphi)
-\frac{\partial^{2}}{\partial x^{j}\partial x^{k}}
 (-2R_{il}+2\alpha\nabla_{i}\varphi\nabla_{l}\varphi)\Big]\\
&\quad -g^{pm}\frac{\partial^{2}g_{kp}}{\partial x^{i}\partial t}
\Big( \frac{\partial^{2}g_{mj}}{\partial x^{l}\partial t}
 +\frac{\partial^{2}g_{ml}}{\partial x^{j}\partial t}
 -\frac{\partial^{2}g_{jl}}{\partial x^{m}\partial t}  \Big)\\
&\quad  +g^{pm}\frac{\partial^{2}g_{kp}}{\partial x^{j}\partial t}
 \Big(\frac{\partial^{2}g_{mi}}{\partial x^{l}\partial t}
 +\frac{\partial^{2}g_{ml}}{\partial x^{i}\partial t}
 -\frac{\partial^{2}g_{il}}{\partial x^{m}\partial t}  \Big)\\
&\quad +2g_{hk}\Big(\frac{\partial}{\partial t}\Gamma_{ip}^{h}
 \frac{\partial}{\partial t}\Gamma_{jl}^{p}
 -\frac{\partial}{\partial t}\Gamma_{jp}^{h}
 \frac{\partial}{\partial t}\Gamma_{il}^{p} \Big).
\end{aligned}
\end{equation}
For the other side, we have
\begin{equation}\label{q5}
\frac{\partial^{2}}{\partial x^{i}\partial x^{l}} R_{jk}
=\nabla_{i}\nabla_{l}R_{jk}+R_{jp}\nabla_{i}\Gamma_{lk}^{p}
+R_{kp}\nabla_{i}\Gamma_{lj}^{p},
\end{equation}
and
\begin{equation} \label{q6}
\begin{aligned}
&-g^{pm}\frac{\partial^{2}g_{kp}}{\partial x^{i}\partial t}
\Big( \frac{\partial^{2}g_{mj}}{\partial x^{l}\partial t}
 +\frac{\partial^{2}g_{ml}}{\partial x^{j}\partial t}
 -\frac{\partial^{2}g_{jl}}{\partial x^{m}\partial t}  \Big)\\
&+g^{pm}\frac{\partial^{2}g_{kp}}{\partial x^{j}\partial t}
\Big( \frac{\partial^{2}g_{mi}}{\partial x^{l}\partial t}
 +\frac{\partial^{2}g_{ml}}{\partial x^{i}\partial t}
 -\frac{\partial^{2}g_{il}}{\partial x^{m}\partial t}  \Big)\\
&+2g_{hk}\Big(\frac{\partial}{\partial t}\Gamma_{ip}^{h}
 \frac{\partial}{\partial t}\Gamma_{jl}^{p}
 -\frac{\partial}{\partial t}\Gamma_{jp}^{h}
 \frac{\partial}{\partial t}\Gamma_{il}^{p} \Big)\\
&=2g_{pq}\Big(\frac{\partial}{\partial t}\Gamma_{il}^{p}
 \frac{\partial}{\partial t}\Gamma_{jk}^{q}
 -\frac{\partial}{\partial t}\Gamma_{jl}^{p}
 \frac{\partial}{\partial t}\Gamma_{ik}^{q} \Big).
\end{aligned}
\end{equation}
Plugging \eqref{q5} and \eqref{q6} in \eqref{q4} leads to
\begin{equation}
\begin{aligned}
&\frac{\partial^{2}}{\partial t^{2}}R_{ijkl}\\
&=-\nabla_{i}\nabla_{l}R_{jk}+
\nabla_{i}\nabla_{k}R_{jl}+\nabla_{j}\nabla_{l}R_{ki}-\nabla_{j}\nabla_{k}R_{il}\\
&\quad -g^{pq}(R_{ijql}R_{kp}+R_{ijkq}R_{kp})+2g_{pq}
 \Big(\frac{\partial}{\partial t}\Gamma_{il}^{p}
  \frac{\partial}{\partial t}\Gamma_{jk}^{q}
 -\frac{\partial}{\partial t}\Gamma_{jl}^{p}
 \frac{\partial}{\partial t}\Gamma_{ik}^{q} \Big)\\
&\quad +\alpha \Big[\frac{\partial^{2}(\nabla_{k}
 \varphi\nabla_{j}\varphi)}{\partial x^{i}\partial x^{l}}
-\frac{\partial^{2}(\nabla_{j}\varphi\nabla_{l}\varphi)}
 {\partial x^{i}\partial x^{k}}
 - \frac{\partial^{2}(\nabla_{k}\varphi\nabla_{i}\varphi)}
 {\partial x^{j}\partial x^{l}}
 -\frac{\partial^{2}(\nabla_{i}\varphi\nabla_{l}\varphi) }
 {\partial x^{j}\partial x^{k}} \Big]\\
&=\Delta R_{ijkl}+2(B_{ijkl}-B_{ijlk}-B_{iljk}+B_{ikjl})\\
&\quad -g^{pq}(R_{pjkl}R_{qi}+R_{ipkl}R_{qj}+R_{ijpl}R_{qk}+R_{ijkp}R_{ql})\\
&\quad +2g_{pq}\Big(\frac{\partial}{\partial t}\Gamma_{il}^{p}
 \frac{\partial}{\partial t}\Gamma_{jk}^{q}
 -\frac{\partial}{\partial t}\Gamma_{jl}^{p}
 \frac{\partial}{\partial t}\Gamma_{ik}^{q} \Big)\\
&\quad +\alpha \Big[\frac{\partial^{2}(\nabla_{k}\varphi\nabla_{j}\varphi)}
 {\partial x^{i}\partial x^{l}}
-\frac{\partial^{2}(\nabla_{j}\varphi\nabla_{l}\varphi)}
 {\partial x^{i}\partial x^{k}}
 - \frac{\partial^{2}(\nabla_{k}\varphi\nabla_{i}\varphi)}
 {\partial x^{j}\partial x^{l}}
-\frac{\partial^{2}(\nabla_{i}\varphi\nabla_{l}\varphi) }
 {\partial x^{j}\partial x^{k}}\Big]
\end{aligned}
\end{equation}
where $B_{ijkl}=g^{pr}g^{qs}R_{piqj}R_{rksl}$, so the proof is complete.
\end{proof}

\begin{theorem} \label{thm3.2}
The evolution equation for Ricci curvature tensor under the $(HG)_{\alpha}$ 
flow is as follows:
\begin{equation} \label{q8}
\begin{aligned}
&\frac{\partial^{2}}{\partial t^{2}}R_{ij} \\
&=\Delta R_{ij}+2g^{pr}g^{qs}R_{piqj}R_{rs}-2g^{pq}R_{pi}R_{qj}\\
&\quad +2g^{kl}g_{pq}\Big(\frac{\partial}{\partial t}\Gamma_{il}^{p}
  \frac{\partial}{\partial t}\Gamma_{kj}^{q}
 -\frac{\partial}{\partial t}\Gamma_{kl}^{p}
 \frac{\partial}{\partial t}\Gamma_{ij}^{q} \Big)\\
&\quad +\alpha g^{kl}\Big[\frac{\partial^{2}(\nabla_{j}
 \varphi\nabla_{k}\varphi)}{\partial x^{i}\partial x^{l}}
-\frac{\partial^{2}(\nabla_{k}\varphi\nabla_{l}\varphi)}
 {\partial x^{i}\partial x^{j}}
- \frac{\partial^{2}(\nabla_{j}\varphi\nabla_{i}\varphi)}
 {\partial x^{k}\partial x^{l}} 
-\frac{\partial^{2}(\nabla_{i}\varphi\nabla_{l}\varphi) }
 {\partial x^{k}\partial x^{j}}\Big]\\
&\quad -2g^{kp}g^{lq}\frac{\partial g_{pq}}{\partial t}
 \frac{\partial R_{ikjl}}{\partial t}
 +2g^{kp}g^{rq}g^{sl}\frac{\partial g_{pq}}{\partial t}
 \frac{\partial g_{rs}}{\partial t}R_{ikjl} \\
&\quad -2\alpha g^{kp}g^{lq}\nabla_{p}\varphi\nabla_{q}\varphi R_{ikjl}.
\end{aligned}
\end{equation}
\end{theorem}

\begin{proof}
 We have
\begin{align*}
\frac{\partial^{2}}{\partial t^{2}}R_{ij}
&= \frac{\partial^{2}}{\partial t^{2}}(g^{kl}R_{ikjl})\\
&= g^{kl}\frac{\partial^{2}}{\partial t^{2}}R_{ikjl}
 +2\frac{\partial g^{kl}}{\partial t}\frac{\partial R_{ikjl}}{\partial t}
 +R_{ikjl}\frac{\partial^{2}g^{kl}}{\partial t^{2}}.
\end{align*}
Since $\frac{\partial g^{kl}}{\partial t}
=-g^{kp}g^{lq}\frac{\partial g_{pq}}{\partial t}$ and
$\frac{\partial^{2}g^{kl}}{\partial t^{2}}
=-g^{kp}g^{lq}\frac{\partial^{2}g_{pq}}{\partial t^{2}}+2g^{kp}g^{rq}g^{sl}
\frac{\partial g_{pq}}{\partial t}\frac{\partial g_{rs}}{\partial t}$,  we have
\begin{equation} \label{q10}
\begin{aligned}
\frac{\partial^{2}}{\partial t^{2}}R_{ij}
&= g^{kl}\frac{\partial^{2}}{\partial t^{2}}R_{ikjl}
-2g^{kp}g^{lq}\frac{\partial g_{pq}}{\partial t}
 \frac{\partial R_{ikjl}}{\partial t}
-g^{kp}g^{lq}\frac{\partial^{2}g_{pq}}{\partial t^{2}}R_{ikjl}\\
&\quad +2g^{kp}g^{rq}g^{sl}\frac{\partial g_{pq}}{\partial t}
 \frac{\partial g_{rs}}{\partial t}R_{ikjl}
\end{aligned}
\end{equation}
by replacing \eqref{q1} and
$\frac{\partial^{2} }{\partial t^{2}}g_{ij}=-2R_{ij}
+2\alpha \nabla_{i}\varphi\nabla_{j}\varphi $  in \eqref{q10} the
 proof is complete.
\end{proof}

From $R=g^{ij}R_{ij}$ and  using \eqref{q8} we have the following result.

\begin{corollary} \label{coro3.3}
Under the $(HG)_{\alpha}$ flow, the evolution equation of the  scalar curvature  
satisfies
\begin{align*}
&\frac{\partial^{2}}{\partial t^{2}}R \\
&=\Delta R+2|\operatorname{Ric}|^{2}+2g^{ij}g^{kl}g_{pq}
 \Big(\frac{\partial}{\partial t}\Gamma_{il}^{p}
 \frac{\partial}{\partial t}\Gamma_{kj}^{q}
 -\frac{\partial}{\partial t}\Gamma_{kl}^{p}
 \frac{\partial}{\partial t}\Gamma_{ij}^{q} \Big)\\
&\quad +\alpha g^{ij} g^{kl}\Big[\frac{\partial^{2}(\nabla_{j}
 \varphi\nabla_{k}\varphi)}{\partial x^{i}\partial x^{l}}
-\frac{\partial^{2}(\nabla_{k}\varphi\nabla_{l}\varphi)}
{\partial x^{i}\partial x^{j}}  
- \frac{\partial^{2}(\nabla_{j}\varphi\nabla_{i}\varphi)}
{\partial x^{k}\partial x^{l}}
-\frac{\partial^{2}(\nabla_{i}\varphi\nabla_{l}\varphi) }
{\partial x^{k}\partial x^{j}}\Big] \\
&\quad -2g^{ij}g^{kp}g^{lq}\frac{\partial g_{pq}}
 {\partial t}\frac{\partial R_{ikjl}}{\partial t}
+4g^{kp}g^{rq}g^{sl}\frac{\partial g_{pq}}{\partial t}
 \frac{\partial g_{rs}}{\partial t}R_{kl}\\
&\quad -4\alpha g^{ij}g^{kp}g^{lq}\nabla_{p}\varphi\nabla_{q}\varphi R_{ikjl}
 -2g^{ip}g^{jq}\frac{\partial g_{pq}}{\partial t}\frac{\partial R_{ij}}{\partial t}.
\end{align*}
\end{corollary}

\section{Examples}

In this section, we give some examples of $(HG)_{\alpha}$ flows.

\begin{example}\label{examp4.1} \rm
Let $(M, g(0))$ be a round two-sphere of constant Gauss curvature $1$. 
Consider, the $(HG)_{\alpha}$ flow, assuming that $(N, \gamma)=(M, g(0))$ 
and $\varphi(0)$ is the identity map, with  
$g(t)=c(t)g(0)$, $ c(0)=1$, $c'(0)=0$ and the fact the 
$\varphi(t)=\varphi(0)$ is harmonic map for all $g(t)$. 
The $(HG)_{\alpha}$ flow on $(M,g(0))$ reduces to
\begin{equation}
\frac{\partial^{2}c(t)}{\partial t^{2}}=-2+2\alpha
\end{equation}
and it has solution $c(t)=(-1+\alpha)t^{2}+1$ where
for $\alpha<1$, $c(t)$ goes to zero in finite time i.e. $(M, g(t))$ 
shrinks to a point, while the scalar curvature $R$  and the energy 
density $|\nabla \varphi|^{2}$ both go to infinity. For $\alpha=1$, 
the solution is stationary. For $\alpha>1$, $c(t)$ increasing.
\end{example}

\begin{example} \label{examp4.2} \rm
Let $(M^{4},g(t))=(S^{2}\times L, c(t)g_{S^{2}}\oplus d(t)g_{L})$ 
where $(S^{2},g_{S^{2}})$ is a round sphere with Gauss curvature $1$ and 
$(L, G_{L})$ is a surface  with constant Gauss curvature $-1$. 
 Consider, the $(HG)_{\alpha}$ flow, assuming that $(N, \gamma)=(M, g(0))$ 
and $\varphi(0)$ is the identity map. Then $\varphi(t)=\varphi(0)$ and 
$(HG)_{\alpha}$ flow results that
\begin{equation}
\begin{gathered}
\frac{\partial ^{2}}{\partial t^{2}}c(t)=-2+2\alpha,\quad
 c(0)=1,\quad c'(0)=0,\\
\frac{\partial ^{2}}{\partial t^{2}}d(t)=2+2\alpha,\quad 
d(0)=1,\quad d'(0)=0.
\end{gathered}
\end{equation}
If $0<\alpha<1$, then $\frac{\partial ^{2}}{\partial t^{2}}c(t)<0$ implies 
that $c(t)$ is decreasing and $\frac{\partial ^{2}}{\partial t^{2}}d(t)>0$ 
results that $d(t)$  is  increasing. If $\alpha=1$, then  $c(t)$ is 
stationary and $d(t)=2t^{2}+1$.
\end{example}

\begin{example} \label{examp4.3} \rm
Let $(M, g(0))$ be a arbitrary closed Riemannian manifold, $(N,\gamma)=(M, g(0))$ 
and $\varphi(0)$ is the identity map.
If the initial metric $g_{ij}(x,0)$ is Ricci flat, i.e. $R_{ij}(x,0)=0$, 
then $g_{ij}(x,t)=(\alpha t^{2}+t+1)g_{ij}(x,0)$ is obviously a solution 
to the evolution equation $(HG)_{\alpha}$ flow with 
$\frac{\partial g}{\partial t}(x,0)=g(x,0)$, therefore any Ricci 
flat metric is a stationary solution of the $(HG)_{\alpha}$ flow \eqref{e1}.
\end{example}

\begin{example} \label{examp4.4} \rm
A Riemannian metric $g_{ij}$ is called Einstein if $R_{ij}=\lambda g_{ij}$ 
for some constant $\lambda$. A smooth manifold $M$ with  an Einstein 
metric is called Einstein manifold.
Let $(M, g(0))$ be a  closed Riemannian manifold, the initial metric $g(0)$ 
 is Einstein that is for some constant $\lambda$ it holds
\begin{equation} 
R_{ij}(0)=\lambda g_{ij}(0) 
\end{equation}
and $(N,\gamma)=(M, g(0))$ and $\varphi(0)$ is the identity map.
The evolving metric under the $(HG)_{\alpha}$ flow will be steady state, 
or will expand homothetically for all time, or shrink in a finite time. 
Since, the initial metric is Einstein for some constant $\lambda$, 
let   $g_{ij}(t,x)=\rho(t) g_{ij}(0)$. By the definition of the
 Ricci tensor, we obtain
\begin{equation} 
R_{ij}(t)=R_{ij}(0)=\lambda g_{ij}(0). 
\end{equation}
In the present situation,  equation \eqref{e1} becomes
\begin{equation}
\frac{\partial^{2}(\rho(t)g_{ij}(0))}{\partial t^{2}}
=-2\lambda g_{ij}(0)+2\alpha g_{ij}(0),
 \end{equation}
this gives an ODE of second order
\begin{equation}
\frac{d^{2}\rho(t)}{\partial t^{2}}
=-2\lambda+2\alpha,\quad \rho(0)=1,\quad \rho'(0)=\nu,
 \end{equation}
if $\alpha$ is constant, then the solution of the initial value problem  is given by
\begin{equation}
\rho(t)=(\alpha-\lambda)t^{2}+\nu t+1.
 \end{equation}
Therefore the solution of  the $(HG)_{\alpha}$ flow remains Einstein.
\end{example}


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\end{document}