A Lower Bound for the Number of Central Configurations on $\H^2$
\begin{center
We study the indices of the geodesic central configurations on $\H^2$. We then show that central configurations are bounded away from the singularity set. With Morse's inequality, we obtain a lower bound for the number of central configurations on $\H^2$.
1702.05535.tex
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\title{A Lower Bound for the Number of Central Configurations  on $\H^2$ }
\begin{document}
\maketitle
\markboth{Shuqiang Zhu}{A Lower Bound for the Number of Central Configurations  on $\H^2$}
\author{\begin{center}
 {\textbf Shuqiang Zhu}\\
\bigskip
{\footnotesize 
Department of Mathematics and Statistics,
University of Victoria,
P.O.~Box 1700 STN CSC,
Victoria, BC, Canada, V8W 2Y2\\
\bigskip
 zhus@uvic.ca\\
}
\end{center}

 \begin{abstract}
 We study the indices of the geodesic central configurations on $\H^2$.  We then show that central configurations are bounded away from the singularity set. With  Morse's inequality, we obtain a lower bound for  the number of central configurations on $\H^2$. 
 \end{abstract}  
     \vspace{2mm}
   
      \textbf{Key Words:} celestial mechanics; curved $N$-body problem; central configurations; Morse theory.
           \vspace{8mm}

 \section{introduction}
 The Newtonian $N$-body problem is the study of the dynamics of $N$ particles moving according to Newton's laws of motion  in $\R^n$, where $n$ is always 2 or 3. After the discovery of Non-Euclidean geometry in 19th century, geometers considered the possibility of a three-dimensional sphere,  $\S^3$,  and a  three-dimensional hyperbolic sphere,  $\H^3$ universe. Thus the dynamics of $N$ particles in $\S^3$ and $\H^3$,  moving according to some attraction law, were considered. We  call this problem the curved $N$-body problem.  
 There have been  many publications in this field before the rise of general relativity.   This problem attracted attention later from the  point of view of quantum mechanics \cite{Sch40}  and the theory of integrable dynamical systems \cite{KH92,Shc06}.    
   Readers interested in its history  may read \cite{Alb13, Dia12-1, Shc06-1}.  On the topic of relative equilibria,  researchers studied  mainly the 2-dimensional ones \cite{Kil98} before Diacu's work.    Diacu wrote the equations of motion  in extrinsic coordinates in $\R^4$ for $\S^3$, and the Minkowski space $\R^{3,1}$ for $\H^3$. In this set up,  the matrix Lie group $SO(4)$ $(SO(3,1))$ serves as the symmetry group, which makes  the study the 3-dimensional relative equilibria easier. With this new approach, Diacu obtained many new results on relative equilibria  \cite{Dia12-1,Dia12-3, Dia13-1, DS14} and on other topics like singularity  \cite{Dia11}, homographic orbits  \cite{DP11}, rotopulsators \cite{DK13}, stability of orbits  \cite{DMPS13}, and the relationship between the Newtonian and the curved $N$-body problem \cite{Dia-A}. There are many following works like \cite{DSZ16, Zhu14} etc.  
   
 Based on Diacu's works, especially \cite{Dia12-1,Dia13-1},  the authors of \cite{DSZ-B}  proposed to  study central configurations.
   Roughly speaking, central configurations are special arrangements of the point particles such that the acceleration vector for each particle points toward a special geodesic, see  \cite[page 31]{DSZ-B}.  Like what happens in the Newtonian $N$-body problem \cite{LMS15}, 
    central configurations  are quite important in the study of the curved $N$-body problem.  For instance, each central configuration gives rise to a one-parameter family of relative equilibria, and central configurations are the bifurcation points in the topological classification of the curved $N$-body problem \cite{DSZ-B}. The interested readers may read \cite{DSZ-B, ZZ16, Zhu17-2} for more detail. 
 
 In this paper, we concentrate on central configurations in $\H^3$. Previous  studies show that their properties  are similar  to the properties of the central configurations in $\R^3$. For example, Moulton's theorem  \cite{Mou10} on geodesic central configurations in $\R^3$ can be extended to $\H^3$, while it can't  be extended to $\S^3$ \cite{DSZ-B}; In $\H^3$, only 2-dimensional central configurations give rise to relative equilibria \cite{ZZ16}, which 
 is the same as in $\R^3$ \cite{Win47}. 
 
 In this paper, we extend another interesting result of the Newtonian  $N$-body problem to the curved $N$-body problem in $\H^3$. Recall that Smale \cite{Sma70-3} and  Palmore \cite{Pal73} have applied 
   Morse theory to obtain a lower bound for the number of central configurations of the Newtonian  $N$-body problem. Their idea is as follows.  First characterise central configurations as critical points of a certain function on a certain manifold. Then find the indices of some known critical points and the Poincar\'{e} polynomial of the manifold.  In the end, assuming the function is a Morse function,  Morse inequality, which relates the critical points and the topology of the manifold, gives rise to an estimation  for the number of the critical points. We apply the same idea to study  central configurations in $\H^3$. 
   
   The paper is organized as follows. In Section \ref{sec:cnbp}, we recall the basic setting of the curved $N$-body problem in $\H^3$ and some basic facts about central configurations. In Section \ref{sec:hes}, we show that the central configurations in $\H^3$ are critical points of a certain function  on $S_c$ and discuss 
    the computation of the Hessian. In Section \ref{sec:geo}, we study the 
     indices some known critical points, namely,  the $N!/2$ geodesic central configurations.
    We prove that the index of each of them is  $N-2$. 
   In Section \ref{sec:morse}, we show that there is a neighbourhood of the singularity set which contains no central configurations, which is essential for the application of Morse theory. Then with the Poincar\'{e} polynomial of $S_c$, we apply Morse inequality to obtain a lower bound for the number of central configurations on $\H^2$.  
  
  
  
 
   



 
\section{the curved $N$-body problem in $\H^3$ and central configurations  }\label{sec:cnbp}


Vectors are all column vectors, but written as row vectors in the  text. As done in \cite{Dia12-1,Dia13-1}, the equations will be written  in the Minkowski space $\R^{3,1}$.   For two vectors, $\q_1=(x_1,y_1,z_1,w_1)$ and $\q_2=(x_2,y_2,z_2,w_2)$, the inner products are given by
     \[ \q_1\cdot \q_2 = x_1x_2+y_1y_2+z_1z_2 - w_1w_2. \]
      We define the unit hyperbolic sphere $\H^3$ as

  \[ \H^3:=
                       \{(x,y,z,w)\in\mathbb R^{3,1}\ \! |\ \!
                       x^2+y^2+z^2-w^2=-1, \ w>0\}.\]
    Given the positive masses $m_1,\dots, m_N$, whose positions are described by  the
    configuration $\q=(\q_1,\dots,\q_N)\in(\H^3)^N$,
    $\q_i=(x_i,y_i,z_i,w_i),\ i=1,..., N$, we define the
    singularity set
     \begin{equation}
       \D=\cup_{1\le i<j\le N}\{\q\in (\H^3)^N\ \! ; \ \! \q_i=\q_j\}.
       \notag\end{equation}
 Let $d_{ij}$ be  the geodesic distance between the point masses $m_i$ and $m_j$, which is computed by
 \[ \cosh  d_{ij}(\q)= -\q_i \cdot \q_j.   \]
  The force function $U$ in $(\H^3)^N\setminus\D$ is
    $$
    U(\q):=\sum_{1\le i<j\le N}m_im_j\coth d_{ij}(\q).
    $$
   
    Define the kinetic energy as $T(\dot \q)= \frac{1}{2}\sum_{1\le i\le N}m_i \dot \q_i \cdot \dot \q_i$, $\dot \q=(\dot \q_1,...,\dot\q_N)$.  Then the curved $N$-body problem in $\H^3$ is given by the Lagrange system on $T((\H^3)^N\setminus\D)$, with
    \[  \ L(\q,\dot\q):=T(\dot\q)+U(\q).   \]
    
    

 Using variational methods, it is easy to  obtain the equations \cite{DSZ-B}:  
 \begin{equation*}
    \begin{cases}
    \ddot\q_i=\sum_{j=1,j\ne i}^N\frac{m_im_j [\q_j-\cosh d_{ij}\q_i]}{\sinh^3 d_{ij}}+ m_i(\dot{\q}_i\cdot \dot{\q}_i)\q_i\cr
    \q_i\cdot \q_i=-1, \ \ \ \ i=1,..., N.
    \end{cases}
    \end{equation*}
 The first part of the acceleration is from the gradient of the force function, $U(\q): (\H^3)^N \setminus \D  \to \R$,  and we will denote it by $\F_i$.  It is the sum of $\F_{ij}:=\frac{m_im_j [\q_j-\cosh d_{ij}\q_i]}{\sinh^3 d_{ij}}$ for $j\ne i$, see  \cite[page 17]{DSZ-B} for the derivation.



 \begin{definition}
 A configuration $\q\in (\H^3)^N\setminus \D$ is called a \emph{central configuration} if there is some constant $\lambda$ such that
 \begin{equation}\label{equ:CCE}
 \nabla_{\q_i}U(\q)=\lambda\nabla_{\q_i} I(\q),\ i=1,..., N,
 \end{equation}
 where $\nabla f $ is the gradient of a function  $f: (\H^3)^N\setminus \D \to \R$, and $I(\q)$ is the moment of inertia defined by $I(\q)= \sum _{i=1}^N m_i (x_i^2+y_i^2)$. We will refer to these conditions as the central configuration equations.  
\end{definition}

A central configuration $\q$ with all $\q_i$ lying on one geodesic is called a \textit{geodesic central configuration}.
% % % % % why define it in this way



The definition of  central configurations of the curved $N$-body problem is based on the work of Smale \cite{Sma70-2, Sma70-3}, see \cite[page 25]{DSZ-B}. A central configuration gives rise to a one-parameter family of relative equilibria, see  \cite[page 29]{DSZ-B}. They also influence the topology of the integral manifolds  \cite{Sma70-2}.  


Obviously,  the two functions $U$ and $I$ are both invariant under $SO(2)\times SO(1,1)$. Let 
$\chi=(\chi_1, \chi_2)\in SO(2)\times SO(1,1)$. The action  is defined by  
\[ \chi \q = (\chi \q_1, ..., \chi \q_N), \  \chi \q_i=(\chi_1 (x_i,y_i)^T, \chi_2 (z_i,w_i)^T  ).   \]
It is easy to see that if $\q$ is a central configuration, so is $\chi \q$. We call two such central configurations equivalent. In this paper, when we say a central configuration, we mean a class of central configurations as defined by the above equivalence relation.  


For our purpose, we will need the following three results. It is not hard to verify Theorem 1 and Corollary 1 by using the central configuration equations \eqref{equ:CCE}.

\begin{theorem}\cite[page 48]{DSZ-B}
 Let $\q=(\q_1,\dots,\q_N),\ \q_i=(x_i,y_i,z_i,w_i),\ i=1,...,N,$ be a central configuration in $\H^3$. Then we have the relationships
 \begin{equation}\label{equ:mc}
 \sum_{i=1}^Nm_ix_iz_i=\sum_{i=1}^Nm_ix_iw_i=\sum_{i=1}^Nm_iy_iz_i=\sum_{i=1}^Nm_iy_iw_i=0.
 \end{equation}
 \end{theorem}

 Define  $\H_{xyw}^2:=\{(x,y,z,w) \in \H^3: z=0\}$.

 \begin{theorem}[\cite{ZZ16}] \label{thm:h2cc}
     Each central configuration in $\H^3$ is equivalent to some central configuration on $\H^2_{xyw}$.
     \end{theorem}


     
Define  $\H_{xw}^1:=\{(x,y,z,w) \in \H^3: y=z=0\}$.
      
\begin{corollary}\cite[page 37]{DSZ-B} \label{cor:h1cc}
Each geodesic central configuration in $\H^3$ is equivalent to some central configuration on $\H^1_{xw}$.
\end{corollary}

 Thus to study central configurations in $\H^3$, it is enough to study them on $\H^2_{xyw}$ for non-geodesic ones, and on $\H^1_{xw}$ for geodesic ones. 
 From now on, unless specified otherwise,  we use $\H^2$ to indicate $\H^2_{xyw}\in \R^{2,1}$, and use $\H^1$ for $\H^1_{xw}\in \R^{1,1}$. We only study central configurations on $\H^2$.  Then two central configurations $\q, \tilde{\q}$ on $\H^2$ are
 equivalent if there is some element $\chi$ in $SO(2)$ such that $\chi \q=\tilde{\q}$. The action is defined by 
 $ \chi \q = (\chi \q_1, ..., \chi \q_N), \  \chi \q_i=(\chi (x_i,y_i)^T, w_i). $
 It is easy to find that the expression of $\nabla I$ is
 \[   \nabla_{\q_i} I(\q)=   2m_i \left( x_iw_i^2,
           y_iw_i^2,
             w_i(x_i^2+ y_i^2) \right), \]    
 see  \cite[page 26]{DSZ-B} for the derivation. Let $r_i=(x_i^2+y_i^2)^{1/2}\ge 0$. 
 

\begin{proposition}\label{prop:uncon_critical}
A central configuration $ \bar \q$ on $\H^2$ is a critical point of the function 
$U(\q)-\lambda I(\q): (\H^2)^N\setminus\Delta\to \R,$ 
where $\lambda$ is some constant depending on the configuration $\bar \q$.
\end{proposition} 
 
The value of $\lambda$ can be obtained in the following way.  Let $M$ be the matrix ${\rm diag}(m_1, m_1, m_1, ...,m_N, m_N, m_N )$. Introduce a metric in   $(\R^{2,1})^N$: 
$$\langle \q, \q \rangle=\sum_{i=1}^N m_i \q_i \cdot \q_i= \q\cdot M\q. $$ 
 
\begin{proposition}\label{prop:lam}
Let $\q$ be a central configuration on $\H^2$, then the value of $\lambda$ in the central configuration  equation is $ \frac{\langle M^{-1} \nabla U,  M^{-1} \nabla I  \rangle}{\langle M^{-1} \nabla I,  M^{-1} \nabla I  \rangle}$ and it is negative.  
\end{proposition} 

\begin{proof}
Using the explicit formula of $\nabla_{\q_i}I$, we see that $\langle M^{-1} \nabla I,  M^{-1} \nabla I  \rangle=0$ if and only if $\q_1=...\q_N=(0,0,1)$. Thus for a central configuration $\q$ that satisfies  the equation  $\nabla_{\q_i} U=\lambda \nabla_{\q_i} I$, we have $\langle M^{-1} \nabla I,  M^{-1} \nabla I  \rangle \ne 0$. Since 
$ \langle M^{-1} \nabla U,  M^{-1} \nabla I  \rangle = \lambda \langle M^{-1} \nabla I,  M^{-1} \nabla I  \rangle,$  we obtain $\lambda = \frac{\langle M^{-1} \nabla U,  M^{-1} \nabla I  \rangle}{\langle M^{-1} \nabla I,  M^{-1} \nabla I  \rangle}. $
 

Direct computation leads to
 \[ \langle M^{-1} \nabla I,  M^{-1} \nabla I  \rangle= \sum_{i=1}^N m_i (x_i^2 w_i^4 +y_i^2 w_i^4- (x_i^2+y_i^2)^2 w_i^2 )= \sum_{i=1}^N m_i  (x_i^2+y_i^2) w_i^2,\]
 \begin{equation*}
 \begin{split}
\langle M^{-1} \nabla U,  M^{-1} \nabla I  \rangle&= \sum_{i=1}^N m_i \left(  \sum_{j\ne i}^N m_j \frac{\q_j -\cosh d_{ij} \q_i}{\sinh^3 d_{ij} } \cdot  ( x_i w_i^2, y_i w_i^2, (x_i^2+y_i^2) w_i) \right) \\
        &= \sum_{ 1\le i<j \le N} m_im_j     \frac{  (x_ix_j +y_iy_j )(w_i^2+w_j^2)-w_iw_j(x_i^2 +y_i^2 +x_j^2 +y_j^2)}{\sinh^3 d_{ij} }\\
        &=\sum_{ 1\le i<j \le N} m_im_j  \frac{  (w_iw_j-\cosh d_{ij} )(w_i^2+w_j^2)-w_iw_j(w_i^2 +w_j^2-2)}{\sinh^3 d_{ij} }\\   
        &=\sum_{ 1\le i<j \le N} m_im_j  \frac{  -\cosh d_{ij}(w_i^2+w_j^2) + (w_i^2+w_j^2) -(w_i^2+w_j^2) +2w_iw_j}{\sinh^3 d_{ij} }\\ 
        &=\sum_{ 1\le i<j \le N} m_im_j  \frac{  (w_i^2+w_j^2)(1-\cosh d_{ij}) - (w_i-w_j)^2 }{\sinh^3 d_{ij} }<0.\\ 
 \end{split}
  \end{equation*}
Here we used the identities  $ \cosh d_{ij}=w_iw_j-(x_ix_j +y_iy_j ) $ and $x_i^2 +y_i^2-w_i^2=-1$. 
 This remark completes the proof.
\end{proof}


      
 \section{The gradient flow and the Hessian}\label{sec:hes}
 

In this section, we characterize central configurations as the restpoints of a certain gradient flow, i.e., the critical points of a certain function,  then we discuss the computation of the Hessian of these critical points. 
We  denote  by  $S_c$ the set $ \{ \q\in(\H^2)^N\setminus\Delta\ \! | \ \! I(\q)=c\}$.



\begin{proposition}
 $I^{-1}(c)$ is homeomorphic to a $(2N-1)$-dimensional sphere for each positive value of $c$.
\end{proposition}

\begin{proof}
Consider the homemorphism  $\pi: \R^2 \to \H^2$, $\pi (x,y)= (x,y, \sqrt{x^2+y^2+1})$. The map induces a homemorphism from $(\R^2)^N$ to $(\H^2)^N$, which we still denote by $\pi$. Thus the function $I: (\H^2)^N \to \R $ induces a function $\bar I: (\R^2)^N \to \R $ by $\bar I (\bar \q)= I (\pi \bar \q)$, where $\bar\q$ is a point in $(\R^2)^N$.  It is easy to see that $\bar I^{-1}(c) $ is homeomorphic to a $(2N-1)$-dimensional sphere for each positive value of $c$ and  $\bar I^{-1}(c) $ is homeomorphic to $ I^{-1}(c) $. This remark completes the proof. 
\end{proof}



Note that in the central configuration  equation, the value $\lambda$ can be also interpreted as Lagrange multiplier. More precisely, Consider the restricted function: 
 $$ U : \ S_c \to\R.$$  

\begin{proposition}\label{prop:grad}
 The vectorfield 
$$X=M^{-1} \nabla U - M^{-1}\frac{\langle M^{-1} \nabla U,  M^{-1} \nabla I  \rangle}{\langle M^{-1} \nabla I,  M^{-1} \nabla I  \rangle} \nabla I $$ 
is the gradient of $U|_{S_c}$, the restriction of $U(\q)$ on the set $S_c$, with respect to the metric $\langle \cdot, \cdot \rangle$. Moreover, the restpoints of this vectorfield are exactly the  central configurations  in $S_c$.
\end{proposition}

\begin{proof}
  Since $\langle  X, M^{-1} \nabla I\rangle=0$,  the vector field is  tangential to $S_c$.   For any $\v\in T_\q S_c$, we have $\langle\v, M^{-1} \nabla I\ \rangle=0$, thus
\[  \langle  X, \v \rangle=  \langle  M^{-1} \nabla U, \v \rangle = dU \v, \]  
where $dU $ is the differential of $U$. Thus $X$ is the gradient flow of $U$ on $S_c$.  The other statements are self-clear, a remark that completes the proof.
\end{proof}



The critical points of $U|_{S_c}$ are not isolated. Let $\q$ be a central configuration and $\phi$ an element of  $SO(2)$.  Then  $\phi \q$ is also a central configuration. Thus it follows that the critical points of  $U|_{S_c}$ are not isolated, but rather occur as manifolds of critical points.  This fact suggests that we can further look at the central configurations as critical points of $U$ subject to a quotient manifold. Note that both $U$, $I$,  and $S_c$ are invariant under the  $SO(2)$ action.   We thus have the following property. 

 \begin{proposition}
   There is a one-to-one correspondence between the classes of central configurations on $\H^2$ and the critical points of  $U$ on the quotient set $S_c/SO(2)$.
 \end{proposition} 

It is interesting to classify central configurations by their Morse index.  
Recall that if $x$ is a critical point of a smooth function $f$ on a manifold $\mathcal M$, there is  a {\it Hessian} quadratic form on the tangent space $T_x\mathcal M$ that is given in local coordinates by the symmetric matrix of the second derivatives:
\[       H(x) (\v) = \v^T D^2V(x)\v. \]
The {\it Morse index} $\ind(x)$ is the maximum dimension of a subspace of $T_x\mc M$ on which $H(x)$ is negative-definite. The {\it nullity} is the dimension of
\[   \ker H(x)=\{  \v: H(x) (\v, \u)=0 \ {\rm for \ all\  } \u \in  T_x\mc M \}, \] 
where $H(x) (\v, \u)= \v^T D^2V(x)\u$ is the symmetric bilinear form associated to $H(x)$.

 We are interested in the function $U|_{S_c}$ given by restricting the potential to the manifold $S_c$, a $2N-1$-dimensional sphere. Instead of using the local coordinates of $S_c$, it is more convenient to use the coordinates of $\H^2$. Then the Hessian is given by  a $2N\times 2N$ matrix, also called $H(x)$, whose restriction to $T_\q S_c$ gives the correct values.
 
 
\begin{lemma}
 Let $\mc M$ be a smooth manifold,  $\mc N$ be a submanifold of $\mc M$, and $f_1$ be a smooth function on  $\mc N$. Assume that  $f$ is  a smooth function on  $\mc M$ and $f|_{\mc N}= f_1 +c$, where $c$ is some constant, and that $x \in \mc N$ is a critical point of $f$ and $f_1$. Denote the Hessian of $f$ and the Hessian of $f_1$ by $H(x)$ and $H_1(x)$ respectively. Then  $H(x)|_{T_{x} \mc N}= H_1(x)$.  
 \end{lemma}
 
\begin{proof}
  Let $(x_1, \cdots, x_n)$ be a local coordinate system of $\mc N$ near $x$. Extend this system to a  local coordinate system of $\mc M$ near $x$,  $(x_1, \cdots, x_n, y_1, \cdots, y_k)$. Since $x$ is the critical point of $f$ and $f_1$, we get
   \begin{equation*} 
   \begin{split}
 H(x)&=\frac{\partial ^2 f}{\partial x^i\partial x^j} dx^i\otimes dx^j + \frac{\partial ^2 f}{\partial x^i\partial y^j} dx^i\otimes dy^j +\frac{\partial ^2 f}{\partial y^i\partial y^j} dy^i\otimes dy^j,\\
 H_1(x)&=\frac{\partial ^2 f_1}{\partial x^i\partial x^j} dx^i\otimes dx^j= \frac{\partial ^2 f}{\partial x^i\partial x^j} dx^i\otimes dx^j,
   \end{split}
   \end{equation*}
  where we used  Einstein's  summation. Now let $\v=a^i\frac{\partial}{\partial x^i} \in T_x\mc N$, we obtain $H(x)(\v)=H_1(x)(\v)$ since $\frac{\partial ^2 f}{\partial x^i\partial y^j} dx^i\otimes dy^j(\v)= \frac{\partial ^2 f}{\partial y^i\partial y^j} dy^i\otimes dy^j(\v)=0$.  This remark completes the proof.
  \end{proof} 
 
 

\begin{lemma}\label{lemma:hes}
Let $\q$ be a  central configuration on $\H^2$, and $I(\q)=c$. Let $\lambda$ be the value of  $\frac{\langle M^{-1} \nabla U,  M^{-1} \nabla I  \rangle}{\langle M^{-1} \nabla I,  M^{-1} \nabla I  \rangle}$ at $\q$. Then the Hessian of $U|_{S_c}$ at the critical point $\q$ is given by $H(\q)(\v)= \v^T H(\q) \v$,  where $H(\q)$ is the 
$2N\times 2N$ matrix 
 \[ H(\q)=D^2U-\lambda D^2 I, \]
 and $D^2U$ and $D^2I$ are the second derivatives matrices of $U$ and $I$ in some coordinates of  $(\H^2)^N$. 
\end{lemma}

\begin{proof}
 Consider the manifold $(\H^2)^N\setminus \D$, the submanifold $S_c$, the smooth function $U-\lambda I: (\H^2)^N\setminus \D \to \R$, and the smooth function $U: S_c \to \R$.  By Proposition \ref{prop:uncon_critical} and Proposition \ref{prop:grad}, $\q$ is a critical point of the two functions. 
Restricting the first function to $S_c$, we have $U-\lambda I= U-\lambda c$. Thus by the above lemma, we see that on $T_\q S_c$, the two Hessians are the same, which is $H(\q)=D^2U-\lambda D^2 I$. This remark completes the proof.
\end{proof}


As noticed above, the critical points of $U|_{S_c}$ are not isolated, which implies that the  central configuration  are always degenerate as critical points. The following result describe the minimal degeneracy. 

\begin{proposition}\label{prop:null_occ}
Let $\q\in S_c$ be a central configuration. Then the nullity of $\q$ as a critical point of $U|_{S_c}$ satisfies 
\begin{equation}\label{equ:null_occ}
{\rm null} (\q)\ge 1.
\end{equation}  
 \end{proposition}
 
\begin{proof}
Consider the curve in $S_c$, $\q(t)=B(t)\q$, where $B(t)= \begin{bmatrix}
  \cos  t& -\sin  t&0\\
  \sin  t & \cos t&0\\
  0&0&1
  \end{bmatrix}$.  They are also central configurations in $S_c$ with the same value of $\lambda$. Thus we have the equation $\nabla_{\q_i} U(\q(t))= \lambda \nabla_{\q_i} I(\q(t))$, $i=1,...,N$. Taking the derivative with respect to $t$ at  $t=0$, we get
\[ (D^2 U -\lambda D^2I) \dot {B}(0) \q =0. \]  
Since $B(t)\q \in S_c$,
 we obtain that $\dot {B}(0) \q \in T_{\q} S_c$. Thus the nullity of the Hessian is at least one. 

\end{proof} 
 
For central configurations, it is natural to call a critical point nondegenerate if its nullity is as small as possible given the rotational symmetry.
\begin{definition}
 A central configuration on $\H^2$ is nondegenerate if the nullity is one; a central configuration on $\H^1$ is nondegenerate if the nullity is zero.
 \end{definition}   
 


\section{the geodesic central configurations and their indices}\label{sec:geo}

In \cite{DSZ-B}, we have proved that existence of $N!/2$ geodesic central configurations. We study their indices now. 


\begin{theorem}\cite[page 60]{DSZ-B} \label{thm:cc_h1}
  Given  masses $m_1,\dots,m_N>0$ on  $\H^1$, for any $c>0$,  there are exactly $N!/2$ 
 geodesic central configurations on $S_c$, one for each ordering of  the masses along $\H^1$.
  \end{theorem}
  
The number of ordering of $N$ masses are $N!$. Since a $180^\circ$ rotation changes the ordering, which means that we counted each case twice, so there are exactly $N!/2$ classes of geodesic central configurations.  
 
Now we study the Hessian of the $N!/2$ geodesic central configurations on $S_c=\{\q \in (\H^2)^N \setminus \D| I(\q)=c \}$. For our purpose, we use the coordinate system of $\H^2$,  
\[ (x,y,w)=(\sinh \th, \cosh \th \sinh \vp,\cosh \th \cosh \vp ), \ \th, \vp \in \R.  \]   
Then $\H^1$ corresponds to $\vp=0$.  This coordinates system gives a homemorphism between $\R^2=(\th, \vp)$ and $\H^2$. Then 
\[ U= \sum_{1\le i<j\le N} m_im_j \coth d_{ij},\ \ \  I= \sum_{i=1}^{N} m_i (x_i^2 +y_i^2)=   \sum_{i=1}^{N} m_i (\sinh^2 \th_i +\cosh^2 \th_i \sinh^2 \vp_i). \] 
Order the coordinates as  $(\th_1, ...,\th_N, \vp_1, ...,\vp_N )$. For a geodesic central configuration $(\th_1, ...,\th_N, 0, ...,0 )$,   direct computations lead to  
\[ H(\q)=D^2 U-\lambda D^2I= \begin{bmatrix}\frac{\partial^2 U}{\partial \th_i\partial\th_j}&0\\
0&\frac{\partial^2 U}{\partial \varphi_i\partial\varphi_j}\end{bmatrix} -\lambda \begin{bmatrix}\frac{\partial^2 I}{\partial \th_i\partial\th_j}&0\\
0&\frac{\partial^2 I}{\partial \varphi_i\partial\varphi_j}\end{bmatrix}.  \]
 Thus it is enough to study the upper left block $H_\th:=[\frac{\partial^2 U}{\partial \th_i\partial\th_j}-\lambda \frac{\partial^2 I}{\partial \th_i\partial\th_j}]_{N\times N}  $ and the lower right block $H_\vp:[\frac{\partial^2 U}{\partial \vp_i\partial\vp_j}-\lambda \frac{\partial^2 I}{\partial \vp_i\partial\vp_j}]_{N\times N}$. 
 
The upper left block $H_\th$ is positive-definite, see P.61 of \cite{DSZ-B}.  More precisely, this block acts on $T_\q (\H^1)^N$, which is spanned by the $N$ vectors $\frac{\partial }{\partial \th_1}, ..., \frac{\partial }{\partial \th_N}$.  
 In this $N$-dimensional space, there is a 1-dimensional subspace that is  not in $T_\q S_c$. It is generated by
 \[ \nabla I = \sum_{i=1}^{N}m_i \sinh 2\th_i \frac{\partial }{\partial \th_i},\] 
which is actually orthogonal to $S_c$. Thus we see that there is an $(N-1)$-dimensional subspace (in the geodesic directions) of $T_\q S_c$ on which $H(\q)$ is positive-definite. 




The lower right block acts on  the complementary subspace spanned by the $N$ vectors $\frac{\partial }{\partial \vp_1}, ..., \frac{\partial }{\partial \vp_N}$.  At the geodesic  central configuration $\q$, it is easy to see that $dI(\frac{\partial }{\partial \vp_i})=0$ for each $i$ since $\vp_i=0$.   Thus the $N$-dimensional subspace belongs to $T_\q S_c$. Explicitly, this block is 
\begin{align*}
  H_\vp =& \begin{bmatrix}
    \sum\limits_{\substack{j=1, j\ne 1}}^N \frac{-m_1m_j\cosh \th_1\cosh \th_j}{\sinh^3d_{1j}} & \frac{m_1m_2 \cosh \th_1\cosh \th_2}{\sinh^3d_{12}}& \cdots& \frac{m_1m_N \cosh \th_1\cosh \th_N}{\sinh^3d_{1N}}\\
    \frac{m_2m_1\cosh \th_1\cosh \th_2}{\sinh^3d_{12}}&  \sum\limits_{\substack{j=1,j\ne 2}}^N \frac{-m_2m_j\cosh \th_2\cosh \th_j}{\sinh^3d_{2j}}&\cdots& \frac{m_2m_N\cosh \th_2\cosh \th_N}{\sinh^3d_{2N}}\\
    \cdots&  \cdots& \cdots& \cdots\displaybreak[0]\\
    \frac{m_1m_N\cosh \th_1\cosh \th_N}{\sinh^3d_{1N}}&\cdots&\cdots&\sum\limits_{\substack{j=1, j\ne N}}^N \frac{-m_Nm_j\cosh \th_N\cosh \th_j}{\sinh^3d_{Nj}}
    \end{bmatrix}\\
  &-2\lambda \begin{bmatrix}
    m_1 \cosh^2\theta_1&0&\cdots&0\\
    0&m_2 \cosh^2\theta_2&\cdots&0\\
    \cdots&  \cdots& \cdots& \cdots\\
    0&\cdots &\cdots &m_N \cosh^2\theta_N\
   \end{bmatrix}.
   \end{align*}

First, notice that there is a null vector of $H(\q)$ in this subspace. Proposition \ref{prop:null_occ} shows, by the $SO(2)$ symmetry,  that there is at least one null vector for the Hessian of any central configuration on $\H^2$. In the $xyw$-coordinates, obviously, the null vector is 
\[ \v= \sum_{i=1}^{N} -y_i \frac{\partial }{\partial x_i} +x_i \frac{\partial }{\partial y_i}= \sum_{i=1}^{N} x_i \frac{\partial }{\partial y_i}.  \] 
Expressed in the $\th, \vp$ coordinates, it is in the subspace  spanned by the $N$ vectors $\frac{\partial }{\partial \vp_1}, ..., \frac{\partial }{\partial \vp_N}$, and 
\begin{equation}\label{equ:null_h1}
\v= \sum_{i=1}^{N}-\frac{\sinh \th_i}{\cosh \th_i} \frac{\partial }{\partial \vp_i}= -\left(0,...,0, \frac{\sinh \th_1}{\cosh \th_1}, ...,\frac{\sinh \th_N}{\cosh \th_N} \right).
\end{equation}
Thus $\v$ is a null vector of $H_\vp$. 
 
  
 We will need the following inequalities on distance. 
 \begin{proposition} \label{prop:dis_ine_h1}
 On  $\H^1=(\sinh \th, \cosh \th)$, for $N$ distinct points with $\th_1< \th_2<\cdots <\th_N$, we have the following inequalities: 
  \begin{enumerate}
  \item if $k<i<j$, then $\frac{1}{\sinh^3 (\th_j-\th_k) \cosh \th_j} < \frac{1}{\sinh^3 (\th_i-\th_k) \cosh \th_i}$;
  \item if $i<j<k$, then $\frac{1}{\sinh^3 (\th_k-\th_j)\cosh \th_j} > \frac{1}{\sinh^3 (\th_k-\th_i)\cosh \th_i}$.
  \end{enumerate}
 \end{proposition}
 \begin{proof}
 If $k<i<j$, what we need to show  is 
 \[ \sinh^3 (\th_j-\th_k) \cosh \th_j-\sinh^3 (\th_i-\th_k) \cosh \th_i>0. \]
 
 View this as a function of $\th_j$, i.e., 
 $ f(x)=\sinh^3 (x-\th_k) \cosh x-\sinh^3 (\th_i-\th_k) \cosh \th_i, \ x>\th_i.  $
 Then $f(\th_i)=0$, and
  \begin{equation*} 
  \begin{split}
  f'(x)&=\sinh^3 (x-\th_k) \sinh x +3 \sinh^2 (x-\th_k) \cosh (x-\th_k) \cosh x\\
       &=\sinh^2 (x-\th_k) \left( \sinh(x-\th_k) \sinh x +3\cosh (x-\th_k) \cosh x  \right)\\
       &= \sinh^2 (x-\th_k)   \cosh (  2x-\th_k)+2 \sinh^2 (x-\th_k) \cosh (x-\th_k) \cosh x>0.
  \end{split}
  \end{equation*}
 So for $k<i<j$, we always have 
 $\frac{1}{\sinh^3 (\th_j-\th_k) \cosh \th_j} < \frac{1}{\sinh^3 (\th_j-\th_k) \cosh \th_i}.$
 The proof of the other inequality  is similar. 
 \end{proof} 
 
 
  
The following theorem extends the result on the indices of geodesic central configurations in $\R^3$ \cite{LMS15}. As done in \cite{Pac87},   the essential idea of the proof is due to Conley.
   
 \begin{theorem}\label{thm:cc_h1_Hes}
Every geodesic central configuration $\q$ on $\H^2$ is nondegenerate with ${\rm null}(\q)=1$ and ${\rm ind}(\q)= N-2$. In the geodesic tangent directions, which are $(N-1)$-dimensional,  $H(\q)$ is positive definite,  while in the normal directions it is positive on a 1-dimensional subspace, zero on another 1-dimensional subspace,  negative-definite on the rest  $(N-2)$-dimensional subspace.
  \end{theorem}
  
 \begin{proof}
 We first simplify the form of the matrix $H_\vp$. 
 Introduce the following three $N\times N$ matrices:
 \begin{align*}
 C:&={\rm diag}\{ \cosh \th_1, \cdots, \cosh \th_N \},\\
 \bar{M}:&={\rm diag}\{ m_1, \cdots, m_N \},\\
 A:&=\begin{bmatrix}
     \sum\limits_{\substack{j=1, j\ne 1}}^N \frac{-m_j\cosh \th_j}{cosh \th_1\sinh^3d_{1j}} & \frac{m_2}{\sinh^3d_{12}}& \cdots& \frac{m_N}{\sinh^3d_{1N}}\\
     \frac{m_1}{\sinh^3d_{12}}&  \sum\limits_{\substack{j=1,j\ne 2}}^N \frac{-m_j\cosh \th_j}{\cosh \th_2\sinh^3d_{2j}}&\cdots& \frac{m_N}{\sinh^3d_{2N}}\\
     \cdots&  \cdots& \cdots& \cdots\displaybreak[0]\\
     \frac{m_1}{\sinh^3d_{1N}}&\cdots&\cdots&\sum\limits_{\substack{j=1, j\ne N}}^N \frac{-m_j\cosh \th_j}{\cosh \th_N\sinh^3d_{Nj}}
     \end{bmatrix}.
  \end{align*}
 Then it is easy to check that $H_\vp=C\bar{M}(A-2\lambda)C$. Thus to study the eigenvalues of $H_\vp$ is equivalent to studying the eigenvalues of $A-2\lambda$. Precisely, note that $\bar{M}^{-\frac{1}{2}}$ and $C^{-\frac{1}{2}}$  are well defined. Then  $H_\vp$ is congruent to $ G_1:=(C^{-\frac{1}{2}})^T H_\vp C^{-\frac{1}{2}}$, which is similar to $C^{\frac{1}{2}} G_1  C^{-\frac{1}{2}}=H_\vp C^{-1}=C\bar{M}(A-2\lambda)$. Similarly, we can obtain rid of $C\bar{M}$.  By Sylvester's law of inertia \cite{Fuh12}, we have 
  \[n_0(H_\vp )=n_0(A-2\lambda), \ \ n_-(H_\vp )=n_-(A-2\lambda),\ n_+(H_\vp )=n_+(A-2\lambda),\  \]
  where $n_0(*)$ is the number of zero eigenvalues of matrix $*$, $n_-(*)$  the number of negative eigenvalues, and $n_+(*)$  the number of positive eigenvalues.
 
 To study the eigenvalues of $A-2\lambda$,  it is enough  to study the eigenvalues of $A$ and compare them with the negative number $2\lambda $. First, notice that there are two obvious eigenvectors of $A$:
 \begin{align*}
 \v_1&=(\cosh \th_1, \cdots, \cosh \th_N), \ \ & A\v_1&=0\v_1,\\
  \v_2&=(\sinh \th_1, \cdots, \sinh \th_N), \ \ & A\v_2&=2\lambda \v_2.
  \end{align*}
  The first vector can be obtained by inspecting the matrix $A$. The second vector $\v_2$ equals $-C \v$, where $\v= -(\frac{\sinh \th_1}{\cosh \th_1}, ...,\frac{\sinh \th_N}{\cosh \th_N} )$ is 
  the null vector of $\H_\vp$, see \eqref{equ:null_h1}. Since $H_\vp \v =C\bar{M}(A-2\lambda)C \v =0$, we have 
 \[ AC\v= 2\lambda C\v, \   \Rightarrow A \v_2= 2\lambda \v_2.\] 
 
 Now we employ the idea of Conley to show that all other eigenvalues of $A$ are smaller than $2\lambda$. 
 The idea is to consider the linear system in $\R^N$:
 \[  \dot{\u}= A\u,   \u=(u_1, \cdots, u_N)\in \R^N. \] 
 Conley observed that to show that all other eigenvalues of $A$ are smaller than $2\lambda$ is equivalent to showing that,  in the flow on $\R^N$, the line determined by $\v_2$ is an attractor. It is enough to find a ``cone'', $K$, around $\v_2$ that is carried strictly inside itself by the flow (except for the origin).
 \begin{figure}[!h]
 	\centering
   \begin{tikzpicture}
   \draw [->] (0, -2.5)--(0, 2.5) node (zaxis) [above] {$\R^{N-1}$};
   \draw [->](-2.5, 0)--(2.5, 0) node (yaxis) [right] {$u_1$};
          
    \draw [-](-2, -2)--(2, 2) node  [right] {$\v_1$};       
   \draw [thick][-](-2, 2)--(2, -2) node  [right] {$\v_2$};
   \draw [-](-0.7, 2)--(0.7, -2) ;
   \draw [-]( 2,-0.7)--( -2,0.7) ;
   
   
   \draw [-]( 0.2,-0.57)--( 0.57,-0.2) ;
    \draw [-]( 0.4,-1.14)--( 1.14,-0.4) ;
     \draw [-]( 0.6,-1.7)--( 1.7,-0.6) ;
  \draw [-]( -0.2,0.57)--( -0.57,0.2) ;
    \draw [-]( -0.4,1.14)--( -1.14,0.4) ;
     \draw [-]( -0.6,1.7)--( -1.7,0.6) node at (-1.6,1.7) [above] {$K$}; 
 
   \fill (0.3,0.3) circle (1.4pt) ; \fill (0.7,0.7) circle (1.4pt) ;\fill (1.3,1.3) circle (1.4pt) ;\fill (1.7,1.7) circle (1.4pt) ;
     \fill (-0.3,-0.3) circle (1.4pt) ; \fill (-0.7,-0.7) circle (1.4pt) ;\fill (-1.3,-1.3) circle (1.4pt) ;\fill (-1.7,-1.7) circle (1.4pt) ; 
     
   \draw [thick][->](-1.4, 1.4)--(-1, 1);
 \draw  [thick][->](1.4, -1.4)--(1,-1);
 
 \draw [thick][-](2.5,-0.3) to [out=200,in=315] (0,0);
 \draw [thick][->](2, -0.45)--(1.7,-0.5);
 
 \draw [thick][-](0.3,-2.5) to [out=70,in=315] (0,0);
 \draw [thick][->](0.45,-2)--(0.5,-1.7);
          
   \end{tikzpicture}
   \caption{The linear flow in $\R^N$}
   \label{fig:conley}
   \end{figure} 
 
 Suppose that  the ordering of the geodesic central configuration $\q$ is  $\th_1<\th_2<\cdots<\th_N$. Then let
 \[ K=\left\{ \u\in \R^N| \sum_{i=1} ^N m_i \cosh \th_i u_i=0, \frac{u_1}{\cosh \th_1}\le \frac{u_2}{\cosh \th_2}\le \cdots \le \frac{u_N}{\cosh \th_N} \right \}. \]
 Endow $\R^N$ with the metric by the matrix $\bar{M}$. Then the cone $K$ is in the $(N-1)$-dimensional subspace perpendicular to $\v_1$.  Note that $\v_2 \in K$. First, by equation \eqref{equ:mc}, we have $0= \sum_{i=1}^N m_i x_i w_i =\sum_{i=1}^N m_i \sinh\th_i \cosh \th_i.$ Second, since  $\tanh \th$ is an increasing function, we have $ \frac{\sinh \th_1}{\cosh \th_1}\le \frac{\sinh \th_2}{\cosh \th_2}\le \cdots \le \frac{\sinh \th_N}{\cosh \th_N}$. 
  The boundary $\partial K$ consists of points for which one or more equalities hold. However, except for the origin, at least one inequality must hold (otherwise $\u=k(\cosh \th_1, \cdots, \cosh \th_N)=k\v_1$). 
 
 Consider a boundary point with 
 \[ \frac{u_1}{\cosh \th_1}\le \cdots \frac{u_i}{\cosh \th_i}=\cdots = \frac{u_j}{\cosh \th_j}\le \cdots \le \frac{u_N}{\cosh \th_N}. \]
To prove that at this point  the flow is pointing inwards, see Figure \ref{fig:conley},  we need to show $\frac{ \dot{u}_j}{\cosh \th_j}-\frac{ \dot{u}_i}{\cosh \th_i}>0$. Direct computation shows that $\frac{ \dot{u}_j}{\cosh \th_j}-\frac{ \dot{u}_i}{\cosh \th_i}$ is
\begin{equation*} 
 \begin{split}
 &\sum _{k=1,k\ne j}^N \frac{m_k}{\cosh \th_j\sinh^3 d_{kj}}\left(u_k- \frac{u_j \cosh \th_k}{\cosh \th_j}\right)- \sum _{k=1,k\ne i}^N \frac{m_k}{\cosh \th_i\sinh^3 d_{ki}}\left( u_k- \frac{u_i \cosh \th_k}{\cosh \th_i}\right)\\
&= \sum _{k=1,k\ne i,j}^N   m_k\left( \frac{u_k}{\sinh^3 d_{kj} \cosh \th_j}- \frac{u_j \cosh \th_k}{\cosh^2 \th_j} - \frac{u_k}{\sinh^3 d_{ki} \cosh \th_i}+ \frac{u_i \cosh \th_k}{\cosh^2 \th_i}      \right) \\
 & + \frac{m_i}{\sinh^3 d_{ij}\cosh \th_j}\left( u_i - \frac{u_j \cosh \th_i}{\cosh \th_j}\right)-\frac{m_j}{\sinh^3 d_{ij}\cosh \th_i}\left( u_j - \frac{u_i \cosh \th_j}{\cosh \th_i}\right).
 \end{split}
 \end{equation*}
Since $\frac{u_i}{\cosh \th_i} = \frac{u_j}{\cosh \th_j}$, the last two terms are zero, and the first part can be nicely written as 
\[ \sum _{k=1,k\ne i,j}^N   m_k \left(   u_k-\frac{u_i \cosh \th_k}{ \cosh \th_i}  \right) \left(   \frac{1}{\sinh^3 d_{kj} \cosh \th_j}-\frac{1}{\sinh^3 d_{ki} \cosh \th_i} \right).\]
Every term in this sum is non-negative:
\begin{enumerate}
\item  If $k<i$, then $   u_k-\frac{u_i \cosh \th_k}{ \cosh \th_i}  \le 0$   and $\frac{1}{\sinh^3 d_{kj} \cosh \th_j}-\frac{1}{\sinh^3 d_{ki} \cosh \th_i} <0$ by Proposition  \ref{prop:dis_ine_h1}.
\item If $i\le k\le j$, then $   u_k-\frac{u_i \cosh \th_k}{ \cosh \th_i} =0$.
\item If $i<k$, then $ u_k-\frac{u_i \cosh \th_k}{ \cosh \th_i}  \ge0$  and $   \frac{1}{\sinh^3 d_{kj} \cosh \th_j}-\frac{1}{\sinh^3 d_{ki} \cosh \th_i} >0$ by Proposition  \ref{prop:dis_ine_h1}.
\end{enumerate}
Moreover, at least one term is strictly positive since at least one inequality in the definition of the cone must hold.  Thus we have proved $\frac{ \dot{u}_j}{\cosh \th_j}-\frac{ \dot{u}_i}{\cosh \th_i}>0$ on $\partial K$ and the boundary point moves into the interior of the cone as required.  This proves that all the other eigenvalues of $A$ are smaller than $2\lambda$, thus the $N$ eigenvalues of $A-2\lambda$ are 
$-2\lambda>0, 0, \lambda_3<0, \cdots, \lambda_N<0.  $
Therefore we get
\[n_0(H_\vp )=n_0(A-2\lambda)=1, \ \ n_-(H_\vp )=n_-(A-2\lambda)=N-2,\ n_+(H_\vp )=n_+(A-2\lambda)=1.\  \]
This remark completes the proof. 
 \end{proof}  




   
 \section{ the compactness of central configurations and A lower bound for the Number of central configurations } \label{sec:morse}
 In this section, we employ Morse inequality to obtain a lower bound for the number of central configurations.
 The Morse inequality holds for compact manifolds. But the manifold we got is $S_c$, which is not compact. In $\R^3$, this difficulty is overcame by a result known as Shub's Lemma, which  shows that there are no central configurations near the singularity set $\D$ in $S_c$ for given masses. We first need to extend this result.   
  
   Recall that $S_c=\{ \q\in (\H^2)^N\setminus \D| I(\q)=c  \}$.  Let $X$ be a  point in $\D$,  $X=(\q'_1, ..., \q'_{k_1}, \q'_{k_1+1}, ..., \q'_{k_2}, ..., \q'_*, ..., \q'_N)$, where 
     $\q'_1= ...= \q'_{k_1}$, $\q'_{k_1+1}= ...= \q'_{k_2}$, $\q'_*= ...=\q'_N$, $\q'_i=(x_i', y_i', w_i')$.  Assume that $\sum_{i=1}^N m_i (x_i'^2 +y_i'^2)=c$, i.e., $X$ belongs to $I^{-1}(c)$. 
      Our
    purpose is to show that there is some neighbourhood $\mc U$ of $X$ in $I^{-1}(c)$, such that   there are no central configurations in  $S_c \cap \mc U$.
     We represent a point in such a  neighbourhood of $X$  by $\q=(\q_1, \q_2, ...\q_N)$,
    \[ \q_i=(x_i' + \delta_{i1},y_i' + \delta_{i2},w_i' + \delta_{i3}),  \]
    with $\q_i\in \H^2$, $\q \notin \D$,  and $I(\q)=c$. 
    The configuration $X$ defines a partition of the bodies into \emph{clusters}, where $m_i, m_j$ are in the same cluster if there is an $l$ such that $k_l < i<j\le k_{l+1}$ or $*\le i<j\le N$.  
    
    We can assume that there are at least two clusters away from $(0,0,1)$. If there is no such cluster, then $\q_1'=...=\q_N'=(0,0,1)$. This contradicts with the fact that $\sum_{i=1}^N m_i (x_i'^2 +y_i'^2)=c$. If there is only one such cluster, say, $(x_1',y_1',w_1')\ne (0,0,1)$, and $\q_{k_1+1}'=...=\q_N'=(0,0,1)$,  then equation \eqref{equ:mc},  $\sum_{i=1}^Nm_ix_iw_i=0$,  can't be satisfied for $\q$ sufficiently close to $X$, since 
    \[ \sum_{i=1}^Nm_ix_iw_i\approx \sum_{i=1}^{k_1}m_ix'_iw'_i + \sum_{i=k_1+1}^{N}m_ix'_iw'_i= x'_1w'_1(\sum_{i=1}^{k_1} m_i).  \]
    Thus $\q$ can't be a central configuration. 
    
    \begin{theorem}\label{thm:cc_compact}
     For fixed masses $m_1, ...m_N$ on $\H^2$, there is a neighbourhood of $\D$ in $S_c$ that contains no central configurations.
     \end{theorem}   
     
    \begin{proof}
     We need to show that the  function $U=\sum _{1\le i< j\le N} m_im_j \coth d_{ij}$ restricted to the $(2N-1)$-sphere $I^{-1}(c)$
     has no critical points in a neighbourhood of $X\in \D \cap I^{-1}(c)$. Let $X$ be the point as defined  above.  We have showed that we can assume that there are at least two clusters away from $(0,0,1)$. Thus, we may require $k_1\ge 2$, $\q_N'\ne \q_1'$, and $\q_N'\ne (0,0,1)$. 
     
     We use $(x,y)$ as the local coordinates of $\H^2$ and let $\bar \q_i=(x_i, y_i)$. We proceed as follows: It is easy to find that the differential of $U$ is 
     \[ dU= \sum _{i=1} ^N \frac{\partial U}{\partial x_i} dx_i+ \frac{\partial U}{\partial y_i} dy_i   = \sum _{i=1} ^N  \left( \sum _{j=1, j\ne i}^N  m_im_j\frac{x_j -\frac{w_j}{w_i}x_i  }{ \sinh^3 d_{ij}} dx_i+ m_im_j\frac{y_j -\frac{w_j}{w_i}y_i  }{ \sinh^3 d_{ij}} dy_i  \right). \] 
    For a point $\q\in S_c$ that approaches $X$, we will pick a bounded vector $\v(\q)=(\v_1, \v_2,...\v_N)\in T_\q S_c$ such that 
    \[    dU \v\to -\infty. \] 
    If this is done, then we can conclude that any point $\q\in S_c$ sufficiently close to $X$ can't  be a critical point of $U|_{S_c}$. Let $\v_i=(v_{i1},v_{i2})=v_{i1} \frac{\partial}{\partial x_i}+v_{i2} \frac{\partial}{\partial y_i}.$ 
    We do this by letting 
    \begin{enumerate}
    \item $\v_i =w_i^2 \bar \q_i=w_i^2 (x_i,y_i), 1\le i\le k_1$,
    \item $\v_{k_1+1}=\v_{k_1+2}= ...=\v_{*-1}=0$,
    \item $\v_i= w_i \v_0, *\le i\le N$, 
    \end{enumerate}
    where $v_{01} (\sum_{i=*}^N m_i w_i x_i)+ v_{02} (\sum_{i=*}^N m_i w_i y_i)=-  \sum_{1}^{k_1} m_i w^2_i r^2_i.$  Note that this is a linear equation of $(v_{01}, v_{02})$ for any given $\q$ and that the coefficients have the property  
    \[   (\sum_{i=*}^N m_i w_i x_i, \sum_{i=*}^N m_i w_i y_i )\approx (\sum_{i=*}^N m_i)(w_N'x_N',w_N'y_N')\ne (0,0),  \]
   for $\q$ sufficiently close to $X$. So we can always find such a $\v_0= (v_{01}, v_{02})$.  
     The vector $\v$ constructed in this way is bounded and  it is in $ T_\q S_c,$ since
    \begin{equation*}
    \begin{split}
    dI \v= \sum _{i=1}^N m_i (x_i v_{i1} + y_i v_{i2})
    = \sum_{1}^{k_1} m_i w^2_i r^2_i + v_{01} (\sum_{i=*}^N m_i w_i x_i)+ v_{02} (\sum_{i=*}^N m_i w_i y_i)=0.
    \end{split}
    \end{equation*}
    
     Let us show that $ dU \v\to -\infty$ for $\q$ sufficiently close to $X$. Note that we can write $(\frac{\partial U}{\partial x_i},\frac{\partial U}{\partial y_i})=\sum _{j=1, j\ne i}^N \frac{ m_im_j }{ \sinh^3 d_{ij}}(\bar\q_j -\frac{w_j}{w_i}\bar \q_i)$. Let $\cdot$ be the inner product in $\R^2$. Then
     \[ dU \v = \sum_{i=1}^{k_1}(\frac{\partial U}{\partial x_i},\frac{\partial U}{\partial y_i}) \cdot \v_i +\sum_{i=k_1+1}^{*-1}(\frac{\partial U}{\partial x_i},\frac{\partial U}{\partial y_i}) \cdot \v_i+ \sum_{i=*}^{N}(\frac{\partial U}{\partial x_i},\frac{\partial U}{\partial y_i}) \cdot \v_i.     \] 
    The first sum goes to $-\infty$ when $\q\to X$. Explicitly, $\sum_{i=1}^{k_1}(\frac{\partial U}{\partial x_i},\frac{\partial U}{\partial y_i}) \cdot \v_i$ is 
    \begin{equation*}
    \begin{split}
    &  \ \ \ \ \ \sum_{i=1}^{k_1} \sum_{j=1,j\ne i}^{k_1} \frac{ m_im_j }{ \sinh^3 d_{ij}}(\bar\q_j -\frac{w_j}{w_i}\bar \q_i)\cdot \v_i  + \sum_{i=1}^{k_1} \sum_{j=k_1+1}^N \frac{ m_im_j }{ \sinh^3 d_{ij}}(\bar\q_j -\frac{w_j}{w_i}\bar \q_i)\cdot \v_i\\
    &=\sum_{1\le i<j\le k_1}  \frac{ m_im_j }{ \sinh^3 d_{ij}}\left((\bar\q_j -\frac{w_j}{w_i}\bar \q_i)\cdot \v_i  + (\bar\q_i -\frac{w_i}{w_j}\bar \q_j)\cdot \v_j  \right) + O(1)\\
    &=\sum_{1\le i<j\le k_1}  \frac{ m_im_j }{ \sinh^3 d_{ij}}\left((w_i^2+ w_j^2) \bar \q_i\cdot \bar\q_j -w_iw_j (\bar \q_i \cdot \bar \q_i +\bar \q_j \cdot \bar \q_j )\right) + O(1),\\ 
    \end{split}
    \end{equation*} 
    where $O(1)$ means a bounded term. Note that $(w_i^2+ w_j^2) \bar \q_i\cdot \bar\q_j -w_iw_j (\bar \q_i \cdot \bar \q_i +\bar \q_j \cdot \bar \q_j )$ is
     \begin{equation*} 
     \begin{split}
     & (x_ix_j +y_iy_j )(w_i^2+w_j^2)-w_iw_j(r_i^2 +r_j^2)\\
     & =  (w_iw_j-\cosh d_{ij} )(w_i^2+w_j^2)-w_iw_j(w_i^2 +w_j^2-2)\\
     &= -\cosh d_{ij}(w_i^2+w_j^2) + (w_i^2+w_j^2) -(w_i^2+w_j^2) +2w_iw_j\\
     &=  (w_i^2+w_j^2)(1-\cosh d_{ij}) - (w_i-w_j)^2\le  (w_i^2+w_j^2)(1-\cosh d_{ij}). 
     \end{split}
     \end{equation*}
     When $\q$ approaches $X$,  $d_{ij}$ approaches $0$ for $1\le i<j\le k_1$. Thus 
     \begin{equation*} 
     \begin{split}
     \sum_{i=1}^{k_1}(\frac{\partial U}{\partial x_i},\frac{\partial U}{\partial y_i}) \cdot \v_i & \le\sum_{1\le i<j\le k_1}  \frac{ m_im_j }{ \sinh^3 d_{ij}}(w_i^2+w_j^2)(1-\cosh d_{ij}) + O(1) \\
     & \le\sum_{1\le i<j\le k_1}   -\frac{ m_im_j }{ \sinh^3 d_{ij}}  ( w_1^2 +w_2^2) (\frac{d^2_{12}}{2}+\frac{d^4_{12}}{4!} +...)\to -\infty.
     \end{split}
     \end{equation*} 
     The second sum is obviously zero. The third sum is bounded. Explicitly, $\sum_{i=*}^{N}(\frac{\partial U}{\partial x_i},\frac{\partial U}{\partial y_i}) \cdot \v_i$ is 
     \begin{equation*}
     \begin{split}
     & \sum_{i=*}^{N} \sum_{j=1}^{*-1} \frac{ m_im_j }{ \sinh^3 d_{ij}}(\bar\q_j -\frac{w_j}{w_i}\bar \q_i)\cdot \v_i  + \sum_{*\le i<j\le N}  \frac{ m_im_j }{ \sinh^3 d_{ij}}[(\bar\q_j -\frac{w_j}{w_i}\bar \q_i)\cdot  \v_i  + (\bar\q_i -\frac{w_i}{w_j}\bar \q_j)\cdot \v_j ]\\
     & =O(1) +  \sum_{*\le i<j\le N}  \frac{ m_im_j }{ \sinh^3 d_{ij}}\left( w_i\bar\q_j \cdot \v_0 -w_j\bar \q_i\cdot  \v_0   + w_j \bar\q_i \cdot \v_0 -w_i\bar \q_j\cdot \v_0 \right)\\
     & =O(1).
     \end{split}
     \end{equation*}  
     
     We have shown that  $ dU \v\to -\infty$ for $\q$ sufficiently close to $X$, where $\v \in T_{\q}S_c$ is bounded. Thus $\q$ can't be a critical point of $U|_{S_c}$ when $\q$ is  sufficiently close to $X$, a remark that completes the proof. 
      \end{proof}  
     
     
     We can now extend the result of Smale \cite{Sma70-3} and Palmore \cite{Pal73} on the number of central configurations in the Newtonian  $N$-body problem. We will make use of Morse theory and assume that,  for generic masses, the central configurations has nullity $1$, the minimum value compatible with the symmetry, see Proposition \ref{prop:null_occ}. 
     
     Recall that a central configuration on $\H^2$ is a critical point of $U: S_c\to \R$ and that $S_c=\{\q\in (\H^2)^N \setminus\D| I(\q)=c \}$. The group $SO(2)$ acts freely on $S_c$, which reduces $U$ as a smooth function on the quotient manifold 
     \[ \mc M =(S_c)/SO(2). \]
     
     A {\it Morse function} is such a function that all its critical points are nondegenerate. 
    Thus assuming that the all central configurations for generic masses are nondegenerate is the same as assuming that all  critical points of $U: \mc M \to \R$ are nondegenerate, that is, $U: \mc M \to \R$ is a Morse function. Recall that the critical points of $U: \mc M\to \R$ correspond to classes of central configurations in a $1$-$1$ manner. Thus the counting of central configurations is the same as the counting of critical points of $U$ restricted to the quotient manifold.  
     
     The Morse inequality is  most easily expressed in terms of polynomial generating functions. Define a {\it Morse polynomial}
     \[ M(t)= \sum _k \gamma _k t^k, \ \ \ \gamma _k= \ {\rm number \ of\ critical\ points\ of \ index  }\ k,   \]
     and the {\it Poincar\'{e}   polynomial}
     $ P(t)= \sum _k \beta _k t^k$, where $\beta _k$ is the  $k$-th  Betti number of the  manifold.
     By the Betti numbers, we mean the ranks of the homology groups $H_k(\mc M, \R)$ with real coefficients. Then the Morse inequalities can be written as
     \begin{equation*}
     M(t)=P(t)+(1+t)R(t), 
     \end{equation*}  
     where $R(t)$ is some polynomial with non-negative integer coefficients \cite{Hir76}. Thus the Poincar\'{e}  polynomial can be used to obtain an estimate of the number of critical points.  
     
     The above Morse inequality holds for a compact manifold. The manifold we are interested is $S_c$, a non-compact manifold. Recall that Proposition \ref{thm:cc_compact} shows that the critical point set of $U|_{S_c}$ is compact and that,  near the boundary of $S_c$, $U$ approaches $+\infty$.  Thus we can restrict to a compact set of the form $\mc K=\{ \q\in S_c: U(\q)\le U_0 \}$ for some sufficiently large $U_0$. Therefore, Morse inequality applies.  
      
         
     
     
     \begin{proposition}
     For the curved $N$-body problem in $\H^2$, the Poincar\'{e}   polynomial of $S_c/SO(2)$ is 
     \[ P(t)=(1+2t)...(1+(N-1)t).   \]
     \end{proposition}
    \begin{proof}
     Consider  the homemorphism  $\pi: \R^2 \to \H^2$, $\pi (x,y)= (x,y, \sqrt{x^2+y^2+1})$. The map induces a homemorphism from $(\R^2)^N$ to $(\H^2)^N$, which we still denote by $\pi$. Thus the function $I: (\H^2)^N \to \R $ induces a function $\bar I: (\R^2)^N \to \R $ by $\bar I (\bar \q)= I (\pi \bar \q)$, where $\bar \q$ is a point in $(\R^2)^N$. Note that $\pi$ is also a  homemorphism between the singularity set $\D$ in $(\H^2)^N$ and $\bar \D= \cup_{1\le i<j\le N}\{\bar \q\in (\R^2)^N\ \! ; \ \! \bar \q_i=\bar \q_j\}$. Then $S_c \simeq \bar I ^{-1}(c)\setminus \bar \D$. Also $\pi$ commutes with the $SO(2)$ action on $\R^2$. Thus we obtain
     \[ S_c/SO(2)=\mc M \simeq  (\bar I ^{-1}(c)\setminus \bar \D) /SO(2). \] 
     
     It has been proved that the Poincar\'{e} polynomial of $(\bar I ^{-1}(c)\setminus \bar \D) /SO(2)$ is $(1+2t)...(1+(N-1)t)$ \cite{LMS15, Pal73}, thus we see that the Poincar\'{e} polynomial of  $S_c/SO(2)$ is $P(t)=(1+2t)...(1+(N-1)t).$
     \end{proof} 
     
    The following proof is   the same as in the Newtonian  $N$-body problem case \cite{LMS15}, we reproduce it here just for completeness. 
     
     \begin{theorem}\label{thm:est_h2}
      Suppose that all of the central configurations  are nondegenerate for a certain choice of masses in the curved $N$-body problem in $\H^2$.  Then on $S_c$ $(c>0)$, there are at least  
           \[  \frac{(3N-4)(N-1)!}{2}\]
           central configurations, of which at least 
           \[  \frac{(2N-4)(N-1)!}{2}\]
           are non-geodesic. 
     \end{theorem}
     
 \begin{proof}
 Obviously, the compact subset $\mc K=\{ \q\in S_c: U(\q)\le U_0 \}$ is homotopic to $S_c$, where $U_0$ is sufficiently large. Thus the Poincar\'{e} polynomial of $\mc K/SO(2)$ is 
 $ P(t)=(1+2t)...(1+(N-1)t). $ 
 We have assumed that $U$ is a Morse function on $\mc K/SO(2)$. We further assume that its Morse polynomial is 
 \[ M(t)= \sum_{k=0}^{2N-2} \gamma _k t^k,\ \ \ \gamma _k= \ {\rm number \ of\ critical\ points\ of \ index  }\ k.\]
 Thus the Morse inequality implies that there is some $R(t)$ with non-negative integer coefficients such that 
$ M(t)= P(t) +(1+t) R(t).  $
 The simplest estimate is obtained by setting $t=1$, 
 \[ \sum_{k=0}^{2N-2} \gamma _k\ge P(1)\ge  N!/2,   \]
which predicts the existence of the $N!/2$ geodesic central configurations. Theorem \ref{thm:cc_h1_Hes} shows that the index of each  geodesic central configuration  in $\mc K/SO(2)$ is $N-2$. Then $\gamma_{N-2}$ is at least $N!/2$. On the other hand, the coefficient of $t^{N-2}$ in $P(t)$ is $2\cdot 3...(N-1)=(N-1)!$. 

Let $R(t)=\sum_k r_k t^k$. Then the coefficient of $t^{N-2}$ in $(1+t)R(t)$ is $r_{N-2} + r_{N-3}$. So
\[ r_{N-2} + r_{N-3}+(N-1)!\ge N!/2.  \] 
Setting $t=1$ in the Morse inequality now gives 
\[ \sum_{k=0}^{2N-2} \gamma _k \ge  N!/2 + 2(r_{N-2} + r_{N-3})  \ge 3N!/2 -2(N-1)! = \frac{(3N-4)(N-1)!}{2}. \] 
 Subtracting $N!/2$ gives the non-geodesic estimate. This remark completes the proof. 
 \end{proof}
        

\section{acknowledgement}
The author  thanks Florin Diacu and Jialong Deng for discussion and constructive suggestions.    


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\end{document}


1702.05535.extracted_bib
{Alb13 A. Albouy, {There is a Projective Dynamics, Eur. Math. Soc. Newsl. No. 89 (2013), 37-43.
1702.05535.extracted_bib
{Dia11F. Diacu, On the Singularities of the Curved $N$-Body Problem, Trans. Amer. Math. Soc. 363 (2011), no. 4, 2249-2264.
1702.05535.extracted_bib
{DP11 F. Diacu, E. P\'erez-Chavela, Homographic Solutions of the Curved 3-Body Problem, J. Differential Equations 250 (2011), no. 1, 340-366.
1702.05535.extracted_bib
{Dia12-1 F. Diacu, Relative Equilibria of the Curved N-Body Problem, Atlantis Press, 2012.
1702.05535.extracted_bib
{Dia12-3 F. Diacu, Polygonal Homographic Orbits of the Curved N-Body Problem, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2783-2802.
1702.05535.extracted_bib
{Dia13-1 F. Diacu, Relative Equilibria of the 3-Dimensional Curved $N$-Body Problem, Memoirs Amer. Math. Soc. 228 (2013), no. 1071.
1702.05535.extracted_bib
{DK13 F. Diacu, S. Kordlou, Rotopulsators of the Curved $N$-Body Problem, J. Differential Equations 255 (2013), no. 9, 2709-2750.
1702.05535.extracted_bib
{DMPS13F. Diacu, R. Mart\'{\inez, E. P\'erez-Chavela, J.C. Sim\'{o, On the Stability of Tetrahedral Relative Equilibria in the Positively Curved 4-Body Problem, Phys. D 256/257 (2013), 21-35.
1702.05535.extracted_bib
{DS14F. Diacu, P. Sergiu, All the Lagrangian Relative Equilibria of the Curved 3-Body Problem Have Equal Masses, J. Math. Phys. 55 (2014), no. 11.
1702.05535.extracted_bib
{DSZ16 F. Diacu, J.M. S\'{anchez-Cerritos, and S. Zhu, Stability of Fixed Points and Associated Relative Equilibria of the 3-Body Problem on $S^1$ and $S^2$, J. Dynam. Differential Equations (2016). doi:10.1007/s10884-016-9550-6
1702.05535.extracted_bib
{Dia-A F. Diacu, The Classical N-Body Problem in the Context of Curved Space, arXiv:1405.0453.
1702.05535.extracted_bib
{DSZ-B F. Diacu, C. Stoica, and S.\ Zhu, Central Configurations of the Curved $N$-Body Problem, arXiv:1603.03342.
1702.05535.extracted_bib
{Fuh12 P.A. Fuhrmann, A Polynomial Approach to Linear Algebra, Second edition. Universitext. Springer, New York, 2012.
1702.05535.extracted_bib
{Hir76 M.W. Hirsch, {Differential Topology, Graduate Texts in Mathematics, vol.\ 33, Springer-Verlag, 1976.
1702.05535.extracted_bib
{Kil98 A. A. Kilin, Libration Points in Spaces $S^2$ and $L^2$, Regul. Chaotic Dyn. 4 (1999), no. 1, 91-103.
1702.05535.extracted_bib
{KH92 V.V. Kozlov, A.O. Harin, Kepler's Problem in Constant Curvature Spaces, Celestial Mech. Dynam. Astronom. 54 (1992), no. 4, 393-399.
1702.05535.extracted_bib
{LMS15 J. Llibre, R. Moeckel, C. Sim\'{o, Central Configurations, Periodic Orbits, and Hamiltonian Systems, Advanced Courses in Mathematics, CRM Barcelona, Birkh\"{auser/Springer, Basel, 2015.
1702.05535.extracted_bib
{Mou10 F.R.\ Moulton, {The Straight Line Solutions of $N$ Bodies, Ann. of Math. (2) 12 (1910), no. 1, 1-17.
1702.05535.extracted_bib
{Pac87 F.\ Pacella, {Central Configurations of the $N$-Body Problem via Equivariant Morse Theory, Arch. Rational Mech. Anal. 97 (1987), no. 1, 59-74.
1702.05535.extracted_bib
{Pal73J.I. Palmore, {Classifying Relative Equilibria. I, Bull. AMS 79(1973), 904-908, II, Bull. AMS 81(1975), 489-491, III, Lett. Math. Phys. 1(1975), 71-73.
1702.05535.extracted_bib
{Saa80 D. Saari, {On the Role and Properties of Central Configurations, Celestial Mech. 21 (1980), 9-20.
1702.05535.extracted_bib
{Sch40 E. Schr\"{odinger, A Method for Determining Quantum-Mechanical Eigenvalues and Eigenfunctions, Proc. Royal Irish Acad., Sec. A 46 (1940), 9-16.
1702.05535.extracted_bib
{Shc06 A.V. Shchepetilov, Nonintegrability of the Two-Body Problem in Constant Curvature Spaces, J. Phys. A 39 (2006), no. 20, 5787-5806.
1702.05535.extracted_bib
{Shc06-1 A.V. Shchepetilov, {Calculus and Mechanics on Two-Point Homogeneous Riemannian Spaces, Lecture Notes in Physics, 707. Springer, Berlin, 2006.
1702.05535.extracted_bib
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