\Large Two-Dimensional Finsler Metrics\\ with Constant Curvature
Zhongmin Shen
We construct infinitely many two-dimensional Finsler metrics on $\Bbb S^2$ and $\Bbb D^2$ with non-zero constant flag curvature. They are all not locally projectively flat.
2DRandersFromRiemann.tex

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\title{\Large Two-Dimensional Finsler Metrics\\
 with  Constant Curvature}
\author{Zhongmin Shen}


\date{July,  2001}

\begin{document}


\maketitle

\begin{abstract}
We construct infinitely many two-dimensional Finsler metrics on $\Bbb S^2$ and $\Bbb D^2$ with non-zero constant flag curvature. They are all not locally projectively flat.
\end{abstract}

\section{Introduction}

In Finsler geometry, the flag curvature is an analogue of sectional curvature in Riemannian geometry.  A natural problem is to study and characterize Finsler metrics of constant (flag) curvature. There are  only three local Riemannian metrics of constant curvature, up to a scaling. However there are lots of non-Riemannian Finsler metrics of constant curvature, due to the 
non-Riemannian features of general Finsler metrics. The first set of non-Riemannian Finsler metrics of constant curvature are 
the Hilbert-Klein metric and the Funk metric on a strongly convex domain. The Funk metric is  positively complete and non-reversible with ${\bf K}=-1/4$ and the Hilbert-Klein metric is  complete and reversible with ${\bf K}=-1$.
Both metrics are locally projectively flat \cite{Ok}\cite{Sh1}. 
P. Funk first completely determined the local structure of  two-dimensional projectively flat Finsler metrics with constant curvature
 \cite{Fk1}\cite{Fk2}. 
R. Bryant showed that up to diffeomorphism, there is exactly  a 2-parameter family of locally projectively flat Finsler metrics on $\Bbb S^2$ with ${\bf K}=1$  and the only reversible one is the standard Riemannian metric \cite{Br1}\cite{Br2}. Later on,  he extended his construction to higher dimensional spheres $\Bbb S^n$ \cite{Br3}. 
Recently, the author has completely determined the local  structure of projectively flat analytic Finsler metrics of constant curvature in higher dimensions \cite{Sh4}. Our method is different from Funk's. 
 

The next problem is to classify non-projectively flat
Finsler metrics of constant curvature. This problem turns out to be very 
difficult. The very first step might be  to construct as many examples as possible.
In 2000, D. Bao and the author first constructed a family of non-projectively flat Finsler metrics on $\Bbb S^3$ with ${\bf K}=1$ using the Lie group structure of ${\Bbb  S}^3$ \cite{BaSh}. 
Our examples are in the form $F= \alpha+\beta$, where $\alpha ({\bf y})
=\sqrt{a_{ij}(x)y^iy^j }$ is a Riemannian metric and $\beta({\bf y})
= b_i(x) y^i$ is a  1-form. Finsler metrics in this form are called Randers metrics \cite{Ra}.  
Recently, the author has just constructed 
an incomplete non-projectively flat  Randers metric with ${\bf K}=0$ in each dimension\cite{Sh3}. 

The main technique in \cite{Sh3} is described as follows. Given a Finsler metric 
$\Phi$  and a vector field ${\bf v}$ on a manifold $M$, define a function
$F: TM \to [0, \infty)$ by
\be
 \Phi \Big ( { {\bf y}\over F({\bf y})} -\e{\bf v}_p \Big ) = 1, \ \ \ \ \ \ {\bf y}\in T_pM.\label{eq}
\ee
where $\e $ is a constant.
$F$ is a Finsler metric when $\e$ is small.
An important relationship between $\Phi$ and $F$ is that their (Busemann-Hausdorff) volume forms are equal, $dV_{\Phi}=dV_F$ \cite{Sh3}.
By choosing an appropriate Finsler metric $\Phi$ and an appropriate  vector field 
${\bf v}$, one  obtains a Finsler metric $F$ with the same  curvature properties as $\Phi$. 

In this paper, we are going to employ this technique to construct a family 
of Randers metrics on $\Bbb S^2$ with ${\bf K}=1$ and a family of Randers metrics on 
a disk $\Bbb D^2(\rho)$ with ${\bf K}=-1$ or ${\bf K}=-1/4$. They  are all not locally projectively flat. These examples show that the classification problem of non-projectively flat  Finsler metrics 
of constant curvature is  very difficult.  





Now let us describe our examples.
Let $\Phi({\bf y}) = \sqrt{ h( {\bf y}, {\bf y}) }$ denote the
standard Riemannian metric on the unit sphere $\Bbb S^2$ and ${\bf v}$ denote the vector field on $\Bbb S^2$ defined by
\be
 {\bf v}_p = \Big (-y,\;  x, \; 0\Big ) \ \ \ \ {\rm at}  \ p = \Big (x, \; y, \; z\Big )\in \Bbb S^2,\label{external1}
\ee
Define $F: T{\Bbb S^2} \to [0, \infty)$ by (\ref{eq}). Then 
 $F= \alpha +\beta $ is a Randers metric, where $\alpha=\alpha({\bf y})$ and $\beta =\beta({\bf y})$ are given by
\be
\alpha: ={ \sqrt{ \e^2 h( {\bf v}, {\bf y} )^2 + 
h( {\bf y}, {\bf y} ) \Big ( 1- \e^2 h( {\bf v}, {\bf v} ) \Big ) }
\over 1 -\e^2 h( {\bf v}, {\bf v} ) }, \ \ \ \ 
\beta: =  - {\e h( {\bf v}, {\bf y} )\over 
1-\e^2h( {\bf v}, {\bf v} ) }. \label{Spherical}
\ee
$F$ is defined on the whole sphere for $|\e | < 1$
and it is defined only on the open  disks  around the north pole and south
pole  with radius $\rho =\sin^{-1}(1/|\e|) $
for $|\e | \geq 1$. Note that when $\e =0$, $F=\Phi$ is the standard Riemannian metric on $\Bbb S^2$.

\begin{thm}\label{thm1}
Let $F=\alpha+\beta$ be any Finsler metric on $\Bbb S^2$ given in (\ref{Spherical}).
It 
 has the following properties
\ben
\item[(a)] ${\bf K}= 1$;
\item[(b)] ${\bf S}=0$;
\item[(c)] $ F$ is not locally projectively flat unless $\e =0$;
\item[(d)] the Gauss curvature $\bar{\bf K}$ of $\alpha$
is not a constant  unless $\e=0, \pm 1$. When $\e=0$, 
$\bar{\bf K}=1$; When $\e=\pm 1$, 
$\bar{\bf K}= -4 .$
\een
\end{thm}

According to Yasuda-Shimada\cite{YaSh}, if a Randers metric 
$F=\alpha+\beta$ is of positive constant curvature , then $\beta$ must be a Killing form of constant length with respect to $\alpha$. However,  $\beta$ in (\ref{Spherical}) is not a Killing form when $\e \not=0$. Thus Theorem \ref{thm1} is inconsistent with  Yasuda-Shimada's result in dimension two. 


\bigskip

Similarly, 
let $\Phi ({\bf y}) =\sqrt{ h( {\bf y}, {\bf y} ) } $ denote the standard Klein metric on
the unit disk $\Bbb D^2$ and ${\bf v}$ denote the vector field on
$\Bbb D^2$ defined by
\be
 {\bf v}_p = (-y,\ x) \ \ \ \ {\rm at} \ p=(x, y)\in \Bbb D^2.\label{external2}
\ee
Define $F: T{\Bbb D}^2 \to [0, \infty)$ by (\ref{eq}). Then  
$F= \alpha+\beta$ is a Randers metric, where $\alpha=\alpha({\bf y})$ and $\beta =\beta({\bf y})$ are given by
\be
\alpha: = { \sqrt{ \e^2 h( {\bf v}, {\bf y} )^2 + 
h( {\bf y}, {\bf y} ) \Big ( 1- \e^2 h( {\bf v}, {\bf v} ) \Big ) }
\over 1 -\e^2 h( {\bf v}, {\bf v} ) }, \ \ \ \ \ 
\beta: = - {\e h( {\bf v}, {\bf y} )\over 
1-\e^2h( {\bf v}, {\bf v} ) }.\label{Hyperbolic}
\ee 
$F$ is a Finsler metric defined on the disk $\Bbb D^2(\rho)$ 
with radius $\rho=1/\sqrt{1+\e^2} $. Note that when $\e=0$,
$F$ is the Klein metric on the unit disk.



\begin{thm}\label{thm2} Let $F=\alpha+\beta$ be the  Finsler metric on 
the disk $\Bbb D^2(\rho)$ given in (\ref{Hyperbolic}). 
It has the following properties
\ben
\item[(a)] ${\bf K}= - 1$;
\item[(b)] ${\bf S}=0$;
\item[(c)] $ F$ is not locally projectively if $\e \not=0$;
\item[(d)] the Gauss curvature $\bar{\bf K}$ of $\alpha$ is not constant
unless $\e=0$. When $\e=0$, $\bar{\bf K}=-1$.
\een
\end{thm}




\bigskip
According to Yasuda-Shimada \cite{YaSh} if a Randers metric $F=\alpha+\beta$ 
is of negative constant curvature, then the Riemannian metric $\alpha$ is of negative constant curvature.  However, the Randers metric defined in (\ref{Hyperbolic}) do not have this property when $\e\not=0$. Thus
Theorem \ref{thm2} is inconsistent  with  Yasuda-Shimada's result  in dimension two. 



\bigskip

Besides the Klein metric, the hyperbolic metric can be expressed in many other forms, such as the Poincare metric and the one arising from the proof of Theorem \ref{thm1}. We can use them  to construct many  non-projectively flat
 Finsler metrics with negative  constant curvature. See Remark \ref{remK=-1} below. 



\bigskip
Finally, let $\Phi ({\bf y}) 
= \sqrt{h( {\bf y}, {\bf y}) } + h( {\bf u}, {\bf y}) $ denote the Funk metric on the unit disk $\Bbb D^2$,
where $h $ is the Klein metric on $\Bbb D^2$ and ${\bf u}= (1-x^2-y^2)(x {\pa \over \pa x} + y {\pa \over \pa y} )\in T_{(x, y)} \Bbb D^2$ is a vector field.
$\Phi({\bf y})$, ${\bf y}\in T_p\Bbb D^2$,  is defined by 
\be
 {{\bf y}\over \Phi({\bf y})} + p \in \pa \Bbb D^2.\label{Funkmetric}
\ee 
Let ${\bf v}$ denote the vector field on $\Bbb D^2$ defined by
(\ref{external2}). 
Define $F: T\Bbb D^2\to [0,\infty)$ by (\ref{eq}), i.e.,
\be
\sqrt{  h\Big (  {{\bf y}\over F ({\bf y})} -\e {\bf v}, \; {{\bf y}\over F ({\bf y})} -\e {\bf v}\Big  ) }
+ h\Big ( {\bf u},\;  {{\bf y}\over F ({\bf y})} -\e {\bf v}\Big  ) =1.\label{FunkFunk}
\ee
Then $F=\alpha+\beta$ is a Randers metric on $\Bbb D^2(\rho)$ where $\rho = 1/\sqrt{1+\e^2}$.


\begin{thm}\label{thm3} Let $F=\alpha+\beta$ be the Randers metric on $\Bbb D^2(\rho)$
defined in (\ref{FunkFunk}). It has the following properties:
\ben
\item[(a)] ${\bf K}= -1/4$;
\item[(b)] ${\bf S}= {3\over 2} F$;
\item[(c)] $F$ is not locally projectively flat unless $\e =0$;
\item[(d)] the Gauss curvature $\bar{\bf K}$ of $\alpha$ is not a constant unless $\e =0$. When $\e =0$, $\bar{\bf K}= -1$.
\een
\end{thm}

Again Theorem \ref{thm3} is inconsistent with Yasuda-Shimada's result in dimension two, since $\alpha$ does not have constant curvature when $\e \not=0$.

\bigskip

In a recent paper by Bao-Robles \cite{BaRo}, they characterize Randers metrics with constant curvature by three equations. Moreover, they use the technique in \cite{Sh3} to construct two-dimensional Randers metrics with the Gauss curvature ${\bf K}= {\bf K}(x)$ independent of the directions, a three-dimensional Randers metric on ${\rm S}^3$ with ${\bf K}=1$
and a three-dimensional Randers metric on ${\rm B}^3$ with ${\bf K}=-1$. 
We should point out that their example on ${\rm S}^3$ is not equivalent to that in \cite{BaSh}.
With the examples in \cite{BaSh}\cite{BaRo}\cite{Sh3}, now we have 
non-projectively flat Randers metrics with constant curvature of any sign in higher dimensions. 


\section{Preliminaries}\label{sectionpre}

Let $F$ be a Finsler metric on a manifold $M$. 
In a standard local coordinate system $(x^i, y^i)$ in $TM$, 
$F= F(x, y)$ is a function of $(x^i, y^i)$. Let
\[ g_{ij} = {1\over 2} [F^2]_{y^iy^j}\]
and $(g^{ij}):=(g_{ij})^{-1}$.
The geodesics of $F$ are characterized locally by
\[ {d^2 x^i\over dt^2} + 2 G^i \Big ( x, {dx\over dt} \Big )=0,
\]
where
\[ G^i = {1\over 4} g^{ik} \Big \{ 2 {\pa g_{pk}\over \pa x^q} 
- {\pa g_{pq}\over \pa x^k} \Big \} y^py^q.\]
The coefficients of the Riemann curvature ${\bf R}_{\bf y}= R^i_{\ k} dx^k 
\otimes \pxi$ are given by
\be
R^i_{\ k} = 2 {\pa G^i \over \pa x^k}
- y^j {\pa^2 G^i\over \pa x^j \pa y^k} + 2 G^j {\pa G^i\over \pa y^j \pa y^k} - {\pa G^i\over \pa y^j} {\pa G^j \over \pa y^k}. \label{Rik}
\ee
$F$ is said to be of constant curvature  ${\bf K}=\lambda$, if 
\[ R^i_{\ k} = \lambda \Big \{ F^2 \delta^i_k - F F_{y^k} y^i \Big \}.\]

\bigskip
When $F=\sqrt{a_{ij}(x)y^iy^j}$ is a Riemannian metric,
$R^i_{\ k}= R^{\ i}_{j \ kl}(x) y^j y^l$, where $R^{\ i}_{j\ kl}(x)$ denote the coefficients of the usual Riemannian curvature tensor.
Thus the quantity ${\bf R}_{\bf y}$ in Finsler geometry is  still called the Riemann curvature.


There are many interesting non-Riemannian quantities in Finsler geometry.
In this paper, we will only discuss the  S-curvature \cite{Sh1}.
Express the (Busemann-Hausdorff) volume form of $F$ by
\[ dV_F = \sigma(x) dx^1 \cdots dx^n.\]
The S-curvature is defined by
\be
{\bf S}({\bf y}) := {\pa G^i \over \pa y^i} (x,y)- { y^i \over \sigma(x) } {\pa \sigma \over \pa x^i}(x).\label{localS}
\ee
See \cite{Sh1} for a related discussion on the S-curvature. 

\bigskip
Randers metrics are among the simplest non-Riemannian Finsler metrics, so that many well-known geometric quantities are computable.


Let  $F = \alpha+\beta$ be a Randers metric on a manifold $M$,
where 
\[\alpha (y)=\sqrt{a_{ij}(x) y^iy^j}, \ \ \ \ \ \beta(y)= b_i(x) y^i\]
with 
$\|\beta\|_x :=\sup_{y \in T_xM} \beta(y)/\alpha(y) < 1$. 
Define $b_{i|j}$ by
\[ b_{i|j} \theta^j := db_i -b_j \theta_i^{\ j},\]
where 
$\theta^i :=dx^i$ and 
$\theta_i^{\ j} :=\tilde{\Gamma}^j_{ik} dx^k$ denote the Levi-Civita  connection forms of $\alpha$.
Let
\[ r_{ij} :={1\over 2} \Big ( b_{i|j}+b_{j|i}\Big ) , \ \ \ \ \ 
s_{ij}:= {1\over 2} \Big (b_{i|j}-b_{j|i} \Big ),\]
\[s^i_{\ j}:= a^{ih}s_{hj}, \ \ \ \ \  s_j:=b_i s^i_{\ j}, \ \ \ \ \   e_{ij} := r_{ij}
+ b_i s_j + b_j s_i.\]
Then $G^i$  are given by
\be
 G^i = \bar{G}^i + {e_{00} \over 2F} y^i -s_0y^i+ \alpha s^i_{\ 0},\label{Gi}
\ee
where $e_{00}:= e_{ij}y^iy^j$, $s_0:=s_iy^i$, $s^i_{\ 0}:=s^i_{\ j} y^j$
and $\bar{G}^i$ denote the geodesic coefficients of $\alpha$.
See \cite{AIM}. 

According to Lemma 3.1 in \cite{ChSh}, 
\be
{\bf S}= c (n+1) F  \ \ \ \ \Longleftrightarrow \ \ \ \ e_{00} = 2c (\alpha^2-\beta^2). \label{Se}
\ee
where $c = c(x)$ is a scalar function. See also Proposition 5.1 in \cite{Sh3} in the case
when $c=0$.


Assume that  
${\bf S} = c (n+1) F$ for some constant $c$. 
Then
\be
G^i = \bar{G}^i + c (\alpha-\beta) y^i - s_0 y^i + \alpha s^i_{\ 0}.\label{Gi2}
\ee
By a direct computation, one obtains a formula for  the Riemann curvature 
is given by
\begin{eqnarray}
R^i_{\ k} &=& \bar{R}^i_{\ k}+3c^2\Big (\alpha^2 \delta^i_k- y^i y_k\Big )  -c^2\beta \Big ( \beta \delta^i_k -b_ky^i \Big )\nonumber\\
&& 
 +\Big ( s_{0|0} \delta^i_k -s_{0|k}  y^i\Big )  + s_0\Big ( s_0 \delta^i_k -s_k y^i \Big )  +\Big ( s_{k|0} - s_{0|k}  \Big ) y^i \nonumber \\
&& -\Big (  \alpha^2s^i_{\ j}s^j_{\ k} -y_k s^i_{\ j}s^j_{\ 0} \Big )   +6 c s_{k0} y^i +3 s_{k0} s^i_{\ 0}\nonumber\\
&& - \Big \{(c^2 \beta + 2 cs_0 +s_js^j_{\ 0}) \Big ( \alpha^2 \delta^i_k -y_k y^i\Big )  + c^2 \alpha^2 \Big ( \beta \delta^i_k -b_k y^i \Big )\nonumber\\
&&  +2c \alpha^2 \Big ( s_0 \delta^i_k -s_k y^i \Big )  - \Big ( \alpha^2 s^i_{\ 0|k} - y_k s^i_{\ 0|0} \Big )\nonumber\\
&& + \alpha^2 \Big ( s_j s^j_{\ 0} \delta^i_k - s_j s^j_{\ k} y^i \Big ) +\alpha^2 \Big ( s^i_{\ k|0} - s^i_{\ 0|k} \Big )   \Big \}\alpha^{-1}.
\label{localRik}
\end{eqnarray}

\bigskip
Taking the trace of $R^i_{\ k}$, we  obtain a formula for the Ricci curvature ${\bf Ric}$ of $F$ which is expressed in terms of the Ricci curvature $\overline{\bf Ric}$ of $\alpha$ and the covariant derivatives of $\beta$ with respect to $\alpha$. 
\begin{eqnarray}
{\bf Ric} & = & \overline{\bf Ric} 
+ (n-1) \Big \{ c^2 (\alpha^2+\beta^2) + 2c^2 (\alpha^2-\beta^2)  + s_{0|0} + s_0s_0 \Big \}\nonumber \\
&& + 2 s_{k0}s^k_{\ 0} -\alpha^2 s^k_{\ j} s^j_{\ k} \nonumber\\
&&  + \Big \{ 2 s^k_{\ 0|k} - (n-1) \Big ( 4c s_0 + 2 s_js^j_{\ 0}+   2c^2 \beta\Big )   \Big \}\alpha. \label{RikRik}
\end{eqnarray}




\section{Proof of Theorem \ref{thm1}}

The Finsler metric in Theorem \ref{thm1} is constructed 
by solving  the equation (\ref{eq}), i.e.,
\be 
 \Phi \Big ({{\bf y}\over F({\bf y})} - \e {\bf v}\Big ) = \sqrt{h\Big ( { {\bf y}\over F({\bf y})} - \e {\bf v} ,\;{ {\bf y}\over F({\bf y})} - \e {\bf v} \Big ) }= 1.\label{eq1}
\ee  


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Let $\psi: \R^2 \to \Bbb S^2_{+}$ by
\[ \psi (x, y) := \Big ( {x\over \sqrt{ 1+x^2+y^2}},\ {y\over \sqrt{ 1+ x^2+y^2}},\ { 1\over \sqrt{1+x^2+y^2} }\Big ).\]
With this map, the standard Riemannian metric $\Phi$ on 
$\Bbb S^2$ can be expressed on $\R^2$ by
\[ \Phi({\bf y}) = {\sqrt{ (u^2+v^2) + (xv-yu)^2  }\over 1+x^2 +y^2},\]
where ${\bf y}= u{\pa \over \pa x} + v {\pa \over \pa y} \in T_{(x, y)}\R^2$. The Finsler metric  defined by (\ref{eq1}) is a 
 Randers metric $F=\alpha+\beta$, where 
$\alpha=\alpha({\bf y})$ and $\beta =\beta({\bf y})$ are given by
 \begin{eqnarray*}
{\alpha} : & = & {\sqrt{ \Big ( 1+(1-\e^2)(x^2+y^2)\Big )(u^2+v^2)
 + \Big ( 1+\e^2 + x^2+y^2\Big ) (xv-yu)^2  }
\over \Big ( 1+(1-\e^2)(x^2+y^2) \Big ) \sqrt{ 1+x^2+y^2} }\\
{\beta} : & = & - { \e (xv-yu)\over 1+(1-\e^2) (x^2+y^2) }.
\end{eqnarray*}
Note that when $|\e|>1$,  $F$ is defined only on the open disk $\Bbb D^2(r)$ of radius 
$r = 1/\sqrt{\e^2-1}$. The corresponding domain  on $\Bbb S^2$ 
is a metric disk $B(\rho)$ around the north pole with radius $\rho= \sin^{-1} (1/|\e|)$.


To compute the curvatures of $F$, we express it in a polar coordinate system,
$ x = r \cos(\theta), \ y = r \sin(\theta).$
For ${\bf y}= \mu {\pa \over \pa r} + \nu {\pa \over \pa \theta}$,
$\alpha=\alpha({\bf y})$ and $\beta =\beta({\bf y})$ are given by
\begin{eqnarray*}
\alpha & = & {\sqrt{\Big (1+(1-\e^2) r^2\Big )\mu^2+ r^2\Big (1+r^2\Big )^2\nu^2}\over 
\Big (1+(1-\e^2)r^2\Big )\sqrt{1+r^2}}\\
\beta& = & - { \e r^2 \nu\over 1+ (1-\e^2)r^2}
\end{eqnarray*}
Express $\alpha =\sqrt{ a_{11}\mu^2+ a_{12} \mu\nu + a_{21} \nu \mu+ a_{22}\nu^2}$
and $\beta= b_1 \mu +b_2 \nu$, where
\[ a_{11}= {1\over (1+r^2)(1+(1-\e^2)r^2)}, 
\ \ \ \ a_{12}=0 = a_{21}, \ \ \ \ a_{22} = { r^2 (1+r^2)\over (1+(1-\e^2)r^2)^2},\]
\[ b_1 =0, \ \ \ \ \  b_2 = - {\e r^2 \over 1 + (1-\e^2) r^2 }.\]
The geodesic coefficients $\bar{G}^1 $ and $\bar{G}^2$  of $\alpha$ are given by
\begin{eqnarray*}
\bar{G}^1 & = &{ \Big (1 +(1  -\e^2)(1 + 2  r^2  ) \Big )r \over 2(1+r^2)\Big (1+(1-\e^2)r^2\Big )}\; \mu^2  - { (1+r^2)\Big (1+2 r^2-(1-\e^2)r^2\Big  ) r \over 2\Big (1+(1-\e^2)r^2\Big )^2}\;\nu^2\\
\bar{G}^2 & = &{ 1+2 r^2- (1-\e^2) r^2 \over (1+r^2)\Big (1+(1-\e^2)r^2\Big ) r }\; \mu\nu
\end{eqnarray*}
We immediately obtain the Gauss curvature $\bar{\bf K}$ of $\alpha$,
\[ \bar{\bf K} ={1-5 \e^2 + (1-\e^4) r^2 \over 1+ (1-\e^2)r^2 } .\]
Note that for $\e = \pm 1$, $\alpha$ has negative constant Gauss curvature,
\[ \bar{\bf K}= - 4.\]

Now we are going to find the geodesic coefficients $G^1$ and $G^2$ of $F$.
By (\ref{Gi}), we first compute $r_{ij}, s^i_{\ j}$ and $s_i $, etc.
A direct computation yields that
\begin{eqnarray*}
r_{11} & = & 0 \ \ = \ \ r_{22}\\
r_{12} & = & { \e^3 r^3\over (1+r^2)\Big (1+ (1-\e^2)r^2\Big )^2} \ \ = \ \ r_{21}\\
s_{11} & = & 0 \ \ = \ \ s_{22}\\
s_{12} & = & { \e r \over \Big ( 1+ (1-\e^2)r^2\Big  )^2 } = - s_{21}\\
s^1_{\ 1} & = & 0 \ \ = \ \ s^2_{\ 2}\\
s^1_{\ 2} & = & { \e r (1+r^2)\over 1 + (1-\e^2)r^2 }\\
s^2_{\ 1} & = & - {\e \over r (1+r^2) }\\ 
s_1 & = & { \e^2 r\over (1+r^2) \Big ( 1+ (1-\e^2)r^2\Big ) }\\
s_2 & = & 0.
\end{eqnarray*}
We obtain that
\[ e_{ij} := r_{ij} + b_i s_j + b_j s_i= 0\]
This is equivalent to that ${\bf S}=0$.  By (\ref{Gi2}) and the above identities, we obtain
\begin{eqnarray*}
G^1 & = & \bar{G}^1   -{\e^2 r  \over (1+r^2)\Big (1+(1-\e^2)r^2\Big )}\; \mu^2 + {\e r (1+r^2)\over 1+(1-\e^2)r^2 } \; \alpha \; \nu \\
G^2 & = &\bar{G}^2 -{\e^2 r  \over (1+r^2)(1+(1-\e^2)r^2)}\; \mu \nu
 - {\e \over r (1+r^2) }\; \alpha\; \mu
\end{eqnarray*}
Plugging them into (\ref{Rik}), we obtain
\be
 R^i_{\ k} = F^2 \Big \{ \delta^i_k -  {F_{y^k}\over F}  y^i\Big \}.\label{Rik=1}
\ee
We conclude that  the Gauss curvature ${\bf K}=1$.


We can also use (\ref{RikRik}) and the above identities to verify 
that ${\bf K}=1$.
To do so, it suffices to compute $s_{0|0}$ and $s^k_{\ 0|k}$. They are given by
\begin{eqnarray*}
s_{0|0} & = & { \e^2 \Big ( 1 - (1-\e^2)r^4 \Big ) \over 
\Big ( 1+r^2 \Big )^2 \Big ( 1 + (1-\e^2) r^2 \Big )^2 }\mu^2
+ { \e^2 r^2 \Big ( 1+ (1+\e^2)r^2 \Big ) \over \Big ( 1 + (1-\e^2)r^2 \Big )^3 }\nu^2 \\
s^k_{\ 0|k} & = & - {\e (1-\e^2) r^2 ( 1+r^2) \nu \over \Big ( 1+(1-\e^2)r^2 \Big )^2 }.
\end{eqnarray*}
Plugging them into (\ref{RikRik}) gives 
\[ {\bf Ric} =  F^2 .\]
We conclude that ${\bf K}={\bf Ric}/F^2 = 1$. 


\bigskip

\begin{rem}\label{remK=1}
{\rm
Express the spherical metric in a radial form 
\[ \Phi ({\bf y}) 
= \sqrt{ u^2 + \sin^2(r) v^2 },
\]
where ${\bf y}= u {\pa \over \pa r} + v {\pa \over \pa \theta}\in T_{(r, \theta)}((0, \infty)\times {\rm S}^1 )$.
 Take 
${\bf v}= {\pa \over \pa \theta}\in T_{(r, \theta)}((0, \infty)\times {\rm S}^1 )$ and define $F$ by (\ref{eq}). We obtain
\be
F = {\sqrt{ \Big ( 1-\e^2 \sin^2(r) \Big ) u^2 + \sin^2(r) v^2 } -\e \sin^2(r) v\over 1-\e^2 \sin^2(r) }.
\ee
$F$ satisfies that ${\bf K}=1$ and ${\bf S}=0$, but it is not locally projectively flat.
}
\end{rem}


\section{Proof of Theorem \ref{thm2}}

The Finsler metric in Theorem \ref{thm2} is also constructed 
by solving the equation (\ref{eq}), i.e.,
\be 
 \Phi \Big ({{\bf y}\over F({\bf y})} - \e {\bf v}\Big ) = \sqrt{h\Big ( { {\bf y}\over F({\bf y})} - \e {\bf v} ,\;{ {\bf y}\over F({\bf y})} - \e {\bf v} \Big ) }= 1.\label{eq2}
\ee

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The Klein metric $\Phi$  on $\Bbb D^2$ is given by
\[ \Phi ({\bf y}) = {\sqrt{  (u^2+v^2) -(xv-yu)^2 }\over 1- (x^2+y^2) },\]
where 
${\bf y}=(u, v)\in T_{(x, y)}\R^2$. 
The Finsler  metric defined by 
(\ref{eq2}) is a Randers metric $F=\alpha+\beta$,
where 
$\alpha =\alpha({\bf y})$ and $\beta=\beta({\bf y})$ are given by
\begin{eqnarray*}
{\alpha} : & = & {\sqrt{ \Big(1-(1+\e^2)(x^2+y^2)\Big )(u^2+v^2)
-\Big ( 1-\e^2 -(x^2+y^2) \Big ) (xv-yu)^2 }
\over \Big ( 1- (1+\e^2)(x^2+y^2) \Big ) \sqrt{ 1-x^2-y^2} }\\
{\beta} : & = & - { \e (xv-yu)\over 1- (1+\e^2) (x^2+y^2) }.
\end{eqnarray*}

To compute the curvatures of $F$, we take a polar coordinate system,
$x = r \cos (\theta), y = r\sin(\theta)$.
For a vector ${\bf y} = \mu {\pa \over \pa r} +\nu {\pa \over \pa \theta}$,
$\alpha =\alpha({\bf y})$ and $\beta =\beta({\bf y})$ are given by
\begin{eqnarray*}
\alpha & = & {\sqrt{\Big (1-(1+\e^2)r^2\Big ) \mu^2 + r^2\Big (1-r^2\Big )^2 \nu^2}\over \Big (1-(1+\e^2)r^2\Big )\sqrt{1-r^2}}\\
\beta & = & - {\e r^2 \nu\over 1 - (1+\e^2)r^2 }.
\end{eqnarray*}
Express $\alpha =\sqrt{ a_{11}\mu^2 + a_{12} \mu \nu + a_{21}\nu \mu + a_{22} \nu^2 }$
and $\beta = b_1 \mu + b_2 \nu$, where
\[a_{11} = { 1 \over \Big ( 1-(1+\e^2)r^2 \Big ) (1-r^2)}\
\ \ \ a_{12} =0 = a_{21}, \ \ \ \ a_{22} = { r^2 (1-r^2) \over \Big ( 1-(1+\e^2)r^2 \Big )^2} ,\]
\[ b_1 =0, \ \ \ \ \ b_2 = - {\e r^2 \over 1- (1+\e^2) r^2 }.\]
The geodesic coefficients $\bar{G}^1$ and $\bar{G}^2$ of $\alpha$ are given by
\begin{eqnarray*}
\bar{G}^1 & = & { \Big (1 + (1 +\e^2)(1-2r^2)  \Big ) r \over 
2(1-r^2) \Big (1-(1+\e^2)r^2\Big )}\; \mu^2
- {\Big (1-2r^2+(1+ \e^2) r^2\Big )(1-r^2) r \over 2\Big (1-(1+\e^2)r^2\Big )^2 }\; \nu^2\\
\bar{G}^2 & = & {  1-2r^2 +(1+\e^2) r^2 \over (1-r^2)\Big (1-(1+\e^2)r^2\Big ) r }\; \mu \nu
\end{eqnarray*}
The Gauss curvature $\bar{\bf K}$ of $\alpha$ is given by
\be
 \bar{\bf K} = { -1 -5 \e^2 + (1-\e^4) r^2 \over 1 - (1+\e^2) r^2 }.\label{K1}
\ee
We see that $\bar{\bf K}$ is not a constant unless $\e =0$.

Now we are going to find the geodesic coefficients $G^1$ and $G^2$ of $F=\alpha+\beta$. Let $r_{ij}, s_{ij}, s^i_{\ j}, s_j $ and $e_{ij}$ as above. 
A direct computation yields that 
\begin{eqnarray*}
r_{11} & = & 0 \ \ = \ \ r_{22}\\
r_{12} & = &  {\e^3 r^3\over (1-r^2)\Big (1-(1+\e^2)r^2\Big )^2} \ \ = \ \ r_{21}\\
s_{11} & = & 0 \ \ =\ \  s_{22}\\
s_{12} & = & {\e r \over \Big ( 1 - (1+\e^2)r^2 \Big )^2 } \ \ =  \ \ -s_{21}\\
s^1_{\ 1} & = &  0 \ \ = \ \ s^2_{ \ 2} \\
s^1_{\ 2} & = & { \e r (1-r^2) \over 1- (1+\e^2)r^2 }\\
s^2_{\ 1} & = & - {\e\over (1-r^2) r }\\
s_1 & = & {\e^2 r \over (1-r^2)\Big (1- (1+\e^2) r^2\Big  ) }\\
s_2 & = & 0.
\end{eqnarray*}
We immediately see that 
\[ e_{ij} :=r_{ij}+b_i s_j +b_j s_i=0.\]
Thus the S-curvature vanishes, ${\bf S}=0$.  By (\ref{Gi2}) and  the above identities, we obtain
\begin{eqnarray*}
G^1 & = & \bar{G}^1 - {\e^2 r \over (1-r^2) (1- (1+\e^2)r^2 ) } \; \mu^2 
+  { \e r (1-r^2) \over 1-(1+\e^2)r^2 }\; \alpha\nu\\
G^2 & = & \bar{G}^2 - {\e^2 r  \over (1-r^2) (1- (1+\e^2)r^2 ) }\;\mu\nu
- {\e  \over (1-r^2) r }\; \alpha \mu.
\end{eqnarray*}
Plugging them into (\ref{Rik}), we immediately obtain
\be
R^i_{\ k} = - \Big \{ F^2 \delta^i_k - F F_{y^k} y^i\Big \}.\label{Rik=-1}
\ee
Thus the Gauss curvature ${\bf K}=-1$. 

We can also use (\ref{RikRik}) and the above identities to verify that ${\bf K}=-1$. To do so, it suffices to compute 
$s_{0|0}$ and $s^k_{\ 0|k}$. They are given by
\begin{eqnarray*}
s_{0|0} & = & { \e^2 \Big ( 1 - (1+\e^2) r^4 \Big ) \over \Big ( 1-r^2 \Big )^2 \Big ( 1 - (1+\e^2)r^2 \Big )^2 }\; \mu^2 
+ { \e^2 r^2 \Big ( 1 - (1-\e^2 ) r^2 \Big ) \over \Big (1 - (1+\e^2) r^2 \Big )^3 }\;\nu^2,\\
s^k_{\ 0|k} & = & { \e (1+\e^2) r^2 ( 1 -r^2) \over \Big ( 1 - (1+\e^2)r^2 \Big )^2 } \; \nu.
\end{eqnarray*}
Plugging them into (\ref{RikRik}), we obtain
\[ {\bf Ric} =- F^2.\]
Again, we conclude that ${\bf K}={\bf Ric}/F^2 = -1$.


\bigskip
\begin{rem}\label{remK=-1}
{\rm 
Express the Klein metric in the radial form,
\[ \Phi({\bf y})
= \sqrt{u^2 + \sinh^2 (r) v^2} , 
\]
where ${\bf y}= u {\pa \over \pa r} + v {\pa \over \pa \theta} \in T_{(r, \theta)} ((0, \infty) \times {\rm S}^1 )$.
Take ${\bf v}= {\pa \over \pa \theta} \in T_{(r, \theta)}(\R \times {\rm S}^1 )$ and define $F$ by (\ref{eq}).  We obtain
\be
F= {\sqrt{ \Big ( 1 - \e^2 \sinh^2(r) \Big ) u^2 
+ \sinh^2(r) v^2 } -\e \sinh^2(r) v \over 1-\e^2 \sinh^2(r) },
\ee
where ${\bf y}= u {\pa \over \pa r}+ v {\pa \over \pa \theta}\in T_{(r, \theta)} ((0, \infty)\times {\rm S}^1)$.
$F$ satisfies ${\bf K}=-1$ and ${\bf S}=0$, but it is not locally projectively flat.





\bigskip
The  Poincare metric on the disk $\Bbb D^2$ is given by 
\be
 \Phi ({\bf y} ) = { 2 \sqrt{u^2+v^2}\over 1-x^2-y^2}, 
\ee
where ${\bf y}= u {\pa \over \pa x}+ v{\pa \over \pa y}\in T_{ (x,y)}\Bbb D^2.$
The Poincare metric has negative constant curvature ${\bf K}=-1$.
Take ${\bf v}=-y {\pa \over \pa x}+ x{\pa \over \pa y}\in T_{(x,y)}\Bbb D^2$ and define $F$ by (\ref{eq}). We obtain
\be
F =
{ \sqrt{\e^2 (xv-yu)^2+(u^2+v^2) \Big ( {1\over 4} (1-x^2-y^2)^2 -\e^2 (x^2+y^2)   \Big )   } -\epsilon ( xv-yu)
\over {1\over 4} (1-x^2-y^2)^2 -\e^2 (x^2+y^2)    }.
\ee 
$F$  satisfies ${\bf K}=-1$ and ${\bf S}=0$, but it is not locally projectively flat.




\bigskip
The Riemannian metric ${\alpha}$ from Theorem \ref{thm1} is given by
\[ \Phi({\bf y}) := \sqrt{ { u^2+v^2 + (xv-yu)^2\over 1+x^2+y^2}  + (xv-yu)^2 },\]
where ${\bf y}= u {\pa \over \pa x} + v {\pa \over \pa y}\in T_{(x, y)} \R^2$. $\Phi$  has constant curvature ${\bf K}=-4$.
Take ${\bf v}_p = -y {\pa \over \pa x}+ x {\pa \over \pa y}$ at $p=(x,y)$
and define $F$ by (\ref{eq}).
We obtain a Randers metric $F=\alpha+\beta$, where 
$\alpha=\alpha({\bf y})$ and $\beta=\beta({\bf y})$ are given by
\begin{eqnarray*}
\alpha : & = & {\sqrt{  u^2+v^2 + (2+x^2+y^2)(xv-yu)^2 -\e^2 (xu+yv)^2 (1+x^2+y^2)} \over \sqrt{1+x^2+y^2} \Big ( 1-\e^2 (x^2+y^2)(1+x^2+y^2) \Big ) }\\
\beta: & = & - {\e (1+x^2+y^2) (xv-yu) \over 1-\e^2 (x^2+y^2)(1+x^2+y^2)}.
\end{eqnarray*}
 $F$ satisfies ${\bf K}=-4$ and ${\bf S}=0$, but it is not locally projectively flat when $\e \not=0$.
}
\end{rem}


\section{Proof of Theorem \ref{thm3}}

Let 
$\Phi$ denote 
the Funk metric on $\Bbb D^2$. It is given by
\[ \Phi ({\bf y}) 
= {\sqrt{ (u^2+v^2) - (xv-yu)^2} + xu+yv \over 1-x^2-y^2},\]
where ${\bf y} = u {\pa \over \pa x} + v {\pa \over \pa y} \in T_{(x, y)} \Bbb D^2$. 
The Finsler metric in Theorem \ref{thm3} is defined by (\ref{FunkFunk}).
Solving the equation (\ref{FunkFunk}), we obtain
\be
F:=  {\sqrt{ u^2+v^2 - \Big ( \e (xu+yv)+ (xv-yu) \Big )^2 }+ (xu+yv) -\e (xv-yu ) \over 1- (1+\e^2)(x^2+y^2) }\\
. \label{Funktypemetric}
\ee
where  ${\bf y}
= u {\pa \over \pa x} + v {\pa \over \pa y}\in T_{(x, y)}\R^2$.
$F =\alpha+\beta $ is a Randers metric on the disk $\Bbb D^2(\rho)$ with 
$\rho =1/\sqrt{1+\e^2}$, where 
 $\alpha$ and $\beta$  are given by
\begin{eqnarray*}
\alpha & = & {\sqrt{ u^2+v^2 - \Big ( \e (xu+yv) + (xv-yu) \Big )^2 }\over 1- (1+\e^2) (x^2+y^2) }\\
\beta & = & { (xu+yv)-\epsilon (xv-yu)\over 1- (1+\e^2)(x^2+y^2) }
\end{eqnarray*}
To compute the curvatures of $F=\alpha+\beta$, we express the Randers metric in a polar coordinate system $x =r \cos\theta, y =r \sin \theta$.
For a vector ${\bf y} = \mu {\pa \over \pa r} +\nu {\pa \over \pa \theta}$, 
$\alpha=\alpha({\bf y})$ and $\beta = \beta({\bf y})$ are given by
\begin{eqnarray*}
\alpha & = & {\sqrt{ \mu^2 + r^2 \nu^2 -r^2 \Big ( r\nu + \e  \mu \Big )^2}\over 1 - (1+\e^2)r^2 }\\
\beta & = & { r \mu -\epsilon r^2 \nu \over 
1 - (1+\e^2) r^2 }.
\end{eqnarray*}
Express $\alpha =\sqrt{ a_{11}\mu^2 + a_{12} \mu \nu + a_{21}\nu \mu + a_{22} \nu^2 }$
and $\beta = b_1 \mu + b_2 \nu$, where
\begin{eqnarray*}
a_{11} & = &  { 1-\e^2 r^2 \over \Big ( 1-(1+\e^2)r^2 \Big )^2},\\
a_{12} & = & -{\e r^3 \over \Big ( 1-(1+\e^2)r^2\Big )^2}\ =\ a_{21},\\
  a_{22} & =&  { r^2 (1-r^2) \over \Big ( 1-(1+\e^2)r^2 \Big )^2} ,
\end{eqnarray*}
\[ b_1 ={r\over 1- (1+\e^2)r^2}, \ \ \ \ \ b_2 = - {\e r^2 \over 1- (1+\e^2) r^2 }.\]
The geodesic coefficients $\bar{G}^1$ and $\bar{G}^2$ of $\alpha$ are given by
\begin{eqnarray*}
\bar{G}^1 & = & { \Big (\e^2 -5 \e^2 r^2 -\e^4 r^2 +2-2r^2\Big ) r \over 
2 \Big (1-(1+\e^2)r^2\Big )^2}\; \mu^2
+ {\e \Big ( 1-r^2+\e^2 r^2 \Big ) r^2\over \Big ( 1-(1+\e^2)r^2 \Big )^2 } \; \mu \nu\\
&& - {\Big (1-r^2+\e^2 r^2\Big )(1-r^2) r \over 2\Big (1-(1+\e^2)r^2\Big )^2 }\; \nu^2\\
\bar{G}^2 & = & {  \e \Big ( -3 +r^2+3\e^2 r^2 \Big )   \over 2\Big (1-(1+\e^2)r^2\Big )^2 }\; \mu^2
+ { (1-\e^2r^2)\Big (1-r^2+\e^2r^2 \Big ) \over \Big ( 1- (1+\e^2)r^2 \Big )^2 r } \; \mu \nu \\
&&
 - {\e \Big ( 1-r^2 +\e^2 r^2 \Big ) r^2 \over 
2 \Big ( 1- (1+\e^2)r^2 \Big )^2 }\; \nu^2
\end{eqnarray*}
The Gauss curvature $\bar{\bf K}$ of $\alpha$ is given by
\be
 \bar{\bf K} = { -1 -5 \e^2 + (1-\e^4) r^2 \over 1 - (1+\e^2) r^2 }.\label{K2}
\ee
We see that $\bar{\bf K}$ is not a constant unless $\e =0$.

Now we are going to find the geodesic coefficients $G^1$ and $G^2$ of $F=\alpha+\beta$. Let $r_{ij}, s_{ij}, s^i_{\ j}, s_j $ and $e_{ij}$ as above. 
A direct computation yields that 
\begin{eqnarray*}
r_{11} & = & { 1-r^2 - 3\e^2 r^2 \over 
\Big ( 1-(1+\e^2)r^2 \Big )^2 }\\
r_{12} & = & - {\e (1-\e^2) r^3\over \Big (1-(1+\e^2)r^2\Big )^2} \ \ = \ \ r_{21}\\
r_{22} & = & { \Big ( 1-r^2 +\e^2 r^2 \Big ) r^2 
\over \Big ( 1-(1+\e^2)r^2 \Big )^2}\\
s_{11} & = & 0 \ \ =\ \  s_{22}\\
s_{12} & = & {\e r \over \Big ( 1 - (1+\e^2)r^2 \Big )^2 } \ \ =  \ \ -s_{21}\\
s^1_{\ 1} & = &  -{\e^2r^2 \over 1- (1+\e^2) r^2 }\ \ = \ \ - s^2_{ \ 2} \\
s^1_{\ 2} & = & { \e r (1-r^2) \over 1- (1+\e^2)r^2 }\\
s^2_{\ 1} & = & - {\e ( 1-\e^2 r^2 ) \over (1- (1+\e^2) r^2) r }\\
s_1 & = & {\e^2 r \over 1- (1+\e^2) r^2 }\\
s_2 & = & {\e r^2 \over 1- (1+\e^2) r^2 }.
\end{eqnarray*}
We immediately see that 
\be
 e_{ij} :=r_{ij}+b_i s_j +b_j s_i= a_{ij}- b_i b_j.\label{eij}
\ee
By Lemma 3.1 in \cite{ChSh}, 
(\ref{eij}) is equivalent to that
\[ {\bf S}={3\over 2} F.\]
 By (\ref{Gi2}) and  the above identities, we obtain
\begin{eqnarray*}
G^1 & = & \bar{G}^1+{1\over 2} (\alpha -\beta)\mu - {\e r \Big ( \e \mu + r \nu \Big ) 
\over 1 - (1+\e^2) r^2 } \; \mu -  {\e r \Big ( \e r \mu -\nu + r^2 \nu \Big ) 
\over 1 - (1+\e^2) r^2 } \; \alpha \\
G^2 & = & \bar{G}^2 + {1\over 2} (\alpha -\beta)\nu -{\e r \Big ( \e \mu + r \nu \Big ) 
\over 1 - (1+\e^2) r^2 }\; \nu  - { \e \Big ( \mu -\e^2 r^2 \mu - \e r^3 \nu \Big ) \over r \Big ( 1- (1+\e^2) r^2 \Big ) }\; \alpha  
\end{eqnarray*}
Plugging them into (\ref{Rik}), we immediately obtain
\[ R^i_{\ k} = -{1\over 4} \Big \{ F^2 \delta^i_k - F F_{y^k} y^i\Big \}.\]
Thus the Gauss curvature ${\bf K}=-1/4$.

We can also use (\ref{RikRik}) to verify that ${\bf K}= -1/4$. 
To do so, it suffices to compute $s_{0|0}$ and $s^k_{\ 0|k}$. They are given by
\begin{eqnarray*}
s_{0|0} & = & { \e^2 ( 1 + r^2 -\e^2 r^2 )\over \Big ( 1- (1+\e^2) r^2\Big  )^3    }\; \mu^2 - { 4 \e^3 r^3 \over \Big ( 1- (1+\e^2) r^2 \Big )^3 }\; \mu\nu 
+ { \e^2 r^2 ( 1- r^2 + \e^2 r^2)   \over \Big ( 1- (1+\e^2) r^2 \Big )^3 }\; \nu^2  \\
s^k_{\ 0|k} & = & - { \e^2 (1+\e^2)  r^3 \over\Big  ( 1- (1+\e^2) r^2\Big  )^2 }\; \mu 
+ {\e (1+\e^2)  r^2 ( 1 - r^2  ) \over\Big  ( 1- (1+\e^2) r^2\Big  )^2 }\; \nu.
\end{eqnarray*} 
Plugging $c= 1/2$ and the above identities into (\ref{RikRik}) gives 
\[ {\bf Ric} = - {1\over 4} F^2.\]
We conclude that ${\bf K} = {\bf Ric}/F^2 = -1/4$.

\bigskip


\begin{rem}
{\rm Below is a byproduct.
Let
\begin{eqnarray*}
\alpha : & = & {\sqrt{ u^2+v^2 - \Big ( \e (xu+yv) + (xv-yu) \Big )^2 }\over 1- (1+\e^2) (x^2+y^2) }\\
\tilde{\alpha} : & = & {\sqrt{ \Big(1-(1+\e^2)(x^2+y^2)\Big )(u^2+v^2)
-\Big ( 1-\e^2 -(x^2+y^2) \Big ) (xv-yu)^2 }
\over \Big ( 1- (1+\e^2)(x^2+y^2) \Big ) \sqrt{ 1-x^2-y^2} }.\end{eqnarray*}
$\alpha$ and $\tilde{\alpha}$ are two Riemannian metrics
on $\Bbb D^2(\rho)$ with radius $\rho = 1/\sqrt{1+\e^2}$.
According to (\ref{K1}) and (\ref{K2}), 
The Gauss curvatures  of $\alpha$ and $\tilde{\alpha}$
are equal and given by
\[ \bar{\bf K}=
{ -1 -5 \e^2 + (1-\e^4) (x^2+y^2)
\over 1- (1+\e^2) (x^2+y^2) }.
\]
}
\end{rem}





\begin{thebibliography}{AnBnCng3}

\bibitem[AIM]{AIM} P.L. Antonelli, R.S. Ingarden, and M. Matsumoto, 
{\it The theory of sprays and Finsler spaces with applications in physics and biology}, FTPH 58, Kluwer Academic Publishers, 1993.


\bibitem[AZ]{AZ} H. Akbar-Zadeh, {\it Sur les espaces de Finsler \'{a} courbures sectionnelles constantes}, Bull. Acad. Roy. Bel. Cl, Sci, 5e S\'{e}rie - Tome LXXXIV (1988), 281-322.

\bibitem[BaChSh]{BaChSh} D. Bao, S.S. Chern and Z. Shen, 
{\it  An introduction to Riemann-Finsler geometry}, Springer, 2000.

\bibitem[BaRo]{BaRo} D. Bao and C. Robles, {\it On Randers metrics of constant curvature}, preprint,   2001.

\bibitem[BaSh]{BaSh} D. Bao and Z. Shen, {\it Finsler metrics of constant curvature on the Lie group $S^3$}, preprint (2000).

\bibitem[Br1]{Br1}  {R. Bryant, \textit{Finsler structures on the
2-sphere satisfying $K=1$}, Finsler Geometry, Contemporary Mathematics {\bf 196},
Amer. Math. Soc., Providence, RI, 1996, 27-42. }

\bibitem[Br2]{Br2}  {R. Bryant, \textit{Projectively flat Finsler $2$-spheres of constant curvature}, Selecta Math., New Series, 
{\bf 3}(1997), 161-204. }

\bibitem[Br3]{Br3} R. Bryant, {\it Finsler manifolds 
with constant curvature},  Talk at the 1998 Geometry Festival in Stony Brook.  



\bibitem[ChSh]{ChSh} X. Chen and Z. Shen, 
{\it Randers metrics with special curvature properties}, preprint (2001).

\bibitem[Fk1]{Fk1} P. Funk, {\it \"{U}ber Geometrien, bei denen die Geraden die K\"{u}rzesten sind}, Math. Annalen {\bf 101}(1929), 226-237.

\bibitem[Fk2]{Fk2} P. Funk, {\it \"{U}ber zweidimensionale Finslersche R\"{a}ume, insbesondere \"{u}ber solche mit geradlinigen Extremalen und positiver konstanter Kr\"{u}mmung}, Math. Zeitschr. {\bf 40}(1936), 86-93.


\bibitem[Ma]{Ma} M. Matsumoto,
{\it Randers spaces of constant curvature}, Reports on Mathematical Physics,
{\bf 28}(1989), 249-261.


\bibitem[Ok]{Ok} T. Okada, {\it On models of projectively flat Finsler spaces of constant negative curvature}, Tensor, N. S. {\bf 40}(1983), 117-123.


\bibitem[Ra]{Ra} G. Randers, {\it On an asymmetric metric in the four-space of general relativity}, Phys. Rev. {\bf 59}(1941), 195-199.


\bibitem[Sh1]{Sh1} Z. Shen, {\it Differential Geometry of Spray and Finsler Spaces}, Kluwer Academic Publishers, Dordrecht, 2001.

\bibitem[Sh2]{Sh2} Z. Shen, {\it Funk metrics and R-flat sprays},
preprint.

\bibitem[Sh3]{Sh3} Z. Shen, {\it Finsler metrics with ${\bf K}=0$ and ${\bf S}=0$}, preprint, May, 2001.

\bibitem[Sh4]{Sh4} Z. Shen, {\it Projectively flat Finsler metrics of constant curvature}, preprint, 2001.



\bibitem[YaSh]{YaSh} H. Yasuda and H. Shimada, {\it On Randers spaces of scalar curvature}, Rep. on Math. Phys. {\bf 11}(1977), 347-360.

\end{thebibliography}


\noindent
Math Dept, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202-3216, USA.  \\
zshen@math.iupui.edu



\end{document}



texdraw.tex
% save it as texdraw.tex


 
\def\setRevDate $#1 #2 #3${#2}
\def\TeXdrawId{\setRevDate $Date: 1995/12/19 16:40:42 $ TeXdraw V2R0}

% ===============================================================

% The TeXdraw macros allow PostScript line drawings and such to be
% generated from within TeX and LaTeX.
% (1) TeXdraw allows TeX text (either horizontal or rotated) to be
%     superimposed on the figure.
% (2) TeXdraw implements a \bsegment-\esegment environment which allows
%     parameter changes and coordinate changes to be kept local.  In
%     effect, these segments are self-contained relocatable procedures.
% (3) TeX's macro facility can be used to modularize drawing units and
%     produce more complex entities from simple elements.
% (4) The drawing can be positioned on the page like any other TeX box.

% TeXdraw coordinate units have positive X to the right and positive Y up.
% The drawing units can be selected (initially inches).  In addition, two
% scaling parameters - unit scale and segment scale - are available.  Their
% effects are multiplicative.

% Segments allow for relocatable drawing units.  Inside each segment the
% coordinates are relative to the initial point, which becomes (0 0).
% Scaling is local to segments.  Each segment inherits the unit scale
% scaling from outside the segment, but any changes apply to that segment
% and inferior segments.  The segment scale factor is reset to unity on
% entry to each segment.

% The coordinates given as command arguments are used to determine
% the size of the drawing.  The width of the plot line, sizes of
% arrowheads, arcs or text do not affect the size of the drawing.

% TeXdraw writes PostScript commands to an intermediate file.  After the
% drawing is finished, the PostScript file is included in the document as a
% PostSript include file.

%  This file is divided into 4 parts,
%  - TeXdraw user interface
%  - Utility definitions
%  - Low level definitions
%  - PostScript file macros


\chardef\catamp=\the\catcode`\@
\catcode`\@=11

% ===============================================================
% ===== TeXdraw user interface ==================================
\long                              % \centertexdraw needs to be very \long
\def\centertexdraw #1{\hbox to \hsize{\hss
                                      \btexdraw #1\etexdraw
                                      \hss}}

% ====== Begin TeXdraw
% Inside the texdraw box:
%   The \vbox should be of zero size; none of the TeXdraw drawing commands
%   generate text, the TeXdraw text commands generate zero size boxes.

\def\btexdraw {\x@pix=0             \y@pix=0
               \x@segoffpix=\x@pix  \y@segoffpix=\y@pix
% Set the default values (define outside of the group so that \etexdraw can
% see the scaling parameters)
               \t@exdrawdef
               \setbox\t@xdbox=\vbox\bgroup\offinterlineskip
                   \global\d@bs=0           % pending segments
                   \global\t@extonlytrue    % no PS commands yet
                   \global\p@osinitfalse
                   \s@avemove \x@pix \y@pix % capture the initial position
                   \m@pendingfalse
                   \global\p@osinitfalse    % capture the next move
                   \p@athfalse
                   \the\everytexdraw}

% ====== End TeXdraw
% Write out a trailer, close the file, bring in the PostScript code as
% a \special include file.  The \special is offset to position on the page
% to be (llx,ury) in PostScript coordinates.

% The drawing is placed in a \vbox of appropriate size (zero depth).  The
% temporary PostScript file is superimposed with offsets such that the
% lower lefthand corner of the drawing is aligned with the lower lefthand of
% the box.  Then, the position is temporarily offset to that corresponding
% to (0 0) to place the TeX text.  Note that TeX text goes on top of the
% PostScript drawing.
\def\etexdraw {\ift@extonly \else
                 \t@drclose      % close the PostScript file
               \fi
               \egroup           % ends the \vbox \bgroup
               \ifdim \wd\t@xdbox>0pt
                 \t@xderror {TeXdraw box non-zero size,
                             possible extraneous text}%
               \fi
               \vbox {\offinterlineskip
                      \pixtobp \xminpix \l@lxbp  \pixtobp \yminpix \l@lybp
                      \pixtobp \xmaxpix \u@rxbp  \pixtobp \ymaxpix \u@rybp
                      \hbox{\t@xdinclude 
                        [{\l@lxbp},{\l@lybp}][{\u@rxbp},{\u@rybp}]{\p@sfile}}%
                      \pixtodim \xminpix \t@xpos  \pixtodim \yminpix \t@ypos
                      \kern \t@ypos
                      \hbox {\kern -\t@xpos
                             \box\t@xdbox       % TeX text
                             \kern \t@xpos}%
                      \kern -\t@ypos\relax}}

% ===== Drawing scaling
% The units in any segment may be scaled arbitrarily.  A unit scale is
% local to a segment but affects enclosed segments unless specifically
% overridden in that segment.  In addition there is a segment scale.  The
% overall scale is the product of the two scaling factors.
%
% Scaling is handled entirely on the TeX side, the PostScript side gets
% absolute pixel coordinates.

% Drawing units, e.g. "in" or "cm"
\def\drawdim #1 {\def\d@dim{#1\relax}}

% \u@nitsc - unit scale
% \s@egsc  - segment scale
% \d@sc    - drawing scale, product of the unit scale and segment scale

% Note that successive application of relative scale factors can lead
% to poor accuracy of the final scale factor.  Each scale factor is
% represented to about 5 decimal digits after the decimal point.
\def\setunitscale #1 {\edef\u@nitsc{#1}%
                      \realmult \u@nitsc \s@egsc \d@sc}
\def\relunitscale #1 {\realmult {#1}\u@nitsc \u@nitsc
                      \realmult \u@nitsc \s@egsc \d@sc}
\def\setsegscale #1 {\edef\s@egsc {#1}%
                     \realmult \u@nitsc \s@egsc \d@sc}
\def\relsegscale #1 {\realmult {#1}\s@egsc \s@egsc
                     \realmult \u@nitsc \s@egsc \d@sc}

% ===== Drawing segments
% The position is restored after a segment.
% Segments use TeX grouping on the TeX side and gsave/grestore on the
% PostScript side to keep changes local.  On the TeX side segments have
% (0 0) as the initial point, while the PostScript side sees no scale
% changes or translations.
\def\bsegment {\ifp@ath
                 \f@lushbs
                 \f@lushmove
               \fi
               \begingroup
               \x@segoffpix=\x@pix
               \y@segoffpix=\y@pix
               \setsegscale 1
               \global\advance \d@bs by 1\relax}
\def\esegment {\endgroup
               \ifnum \d@bs=0
                 \writetx {es}%
               \else
                 \global\advance \d@bs by -1
               \fi}

% Save a position
% Save each coordinate as the macro "*name".  The macro is defined to
% be the pixel coordinate value.
\def\savecurrpos (#1 #2){\getsympos (#1 #2)\a@rgx\a@rgy
                         \s@etcsn \a@rgx {\the\x@pix}%
                         \s@etcsn \a@rgy {\the\y@pix}}
\def\savepos (#1 #2)(#3 #4){\getpos (#1 #2)\a@rgx\a@rgy
                            \coordtopix \a@rgx \t@pixa
                            \advance \t@pixa by \x@segoffpix
                            \coordtopix \a@rgy \t@pixb
                            \advance \t@pixb by \y@segoffpix
                            \getsympos (#3 #4)\a@rgx\a@rgy
                            \s@etcsn \a@rgx {\the\t@pixa}%
                            \s@etcsn \a@rgy {\the\t@pixb}}

% ===== Line parameters
% The following parameters apply to subsequent lines.  Each of these
% commands invokes a stroke to draw the previous line segments,
% establishes the current point and then changes the line parameter.
% The parameters are kept local by the PostScript gsave/grestore
% mechanism.  We use \writetx here, instead of \writeps, since we
% do not want to flush any moves.
\def\linewd #1 {\coordtopix {#1}\t@pixa
                \f@lushbs
                \writetx {\the\t@pixa\space sl}}
\def\setgray #1 {\f@lushbs
                 \writetx {#1 sg}}
\def\lpatt (#1){\listtopix (#1)\p@ixlist
                \f@lushbs
                \writetx {[\p@ixlist] sd}}

% ===== Line drawing
% PostScript uses the concept of a path consisting of line segments.
% In this interface to PostScript, paths are continuous across the
% beginning of segments.  Paths terminate at the end of a segment with
% an implicit move.  In addition, paths are both terminated and started
% with a move.  There is a current point at all times, starting with
% initial position (0,0).
\def\lvec (#1 #2){\getpos (#1 #2)\a@rgx\a@rgy
                  \s@etpospix \a@rgx \a@rgy
                  \writeps {\the\x@pix\space \the\y@pix\space lv}}
\def\rlvec (#1 #2){\getpos (#1 #2)\a@rgx\a@rgy
                   \r@elpospix \a@rgx \a@rgy
                   \writeps {\the\x@pix\space \the\y@pix\space lv}}
\def\move (#1 #2){\getpos (#1 #2)\a@rgx\a@rgy
                  \s@etpospix \a@rgx \a@rgy
                  \s@avemove \x@pix \y@pix}
\def\rmove (#1 #2){\getpos (#1 #2)\a@rgx\a@rgy
                   \r@elpospix \a@rgx \a@rgy
                   \s@avemove \x@pix \y@pix}

% ===== Circles, ellipses and arcs
% Note that arcs do not update the size of the drawing.
% \lcir, stroked circle
%   r:#1 -  radius
% \fcir, filled circle
%   f:#1 - fill gray level, 0 is black, 1 is white
%   r:#2 - radius
% \ellip, stroked ellipse
%   rx:#1 - x radius
%   ry:#2 - y radius
% \fellip, filled ellipse
%   f:#1 - fill gray level, 0 is black, 1 is white
%   rx:#1 - x radius
%   ry:#2 - y radius
% \larc, stroked counterclockwise arc, with the present position being
%        the center of the arc.  Only the arc is drawn (not the line
%        joining the center to the beginning of the arc)
%   r:#1  - radius
%   sd:#2 - start angle (degrees)
%   ed:#3 - end angle (degrees)
\def\lcir r:#1 {\coordtopix {#1}\t@pixa
                \writeps {\the\t@pixa\space cr}%
                \r@elupd \t@pixa \t@pixa
                \r@elupd {-\t@pixa}{-\t@pixa}}
\def\fcir f:#1 r:#2 {\coordtopix {#2}\t@pixa
                     \writeps {\the\t@pixa\space #1 fc}%
                     \r@elupd \t@pixa \t@pixa
                     \r@elupd {-\t@pixa}{-\t@pixa}}
\def\lellip rx:#1 ry:#2 {\coordtopix {#1}\t@pixa
                     \coordtopix {#2}\t@pixb
                     \writeps {\the\t@pixa\space \the\t@pixb\space el}%
                     \r@elupd \t@pixa \t@pixb
                     \r@elupd {-\t@pixa}{-\t@pixb}}
\def\fellip f:#1 rx:#2 ry:#3 {\coordtopix {#2}\t@pixa
                     \coordtopix {#3}\t@pixb
                     \writeps {\the\t@pixa\space \the\t@pixb\space #1 fe}%
                     \r@elupd \t@pixa \t@pixb
                     \r@elupd {-\t@pixa}{-\t@pixb}}
\def\larc r:#1 sd:#2 ed:#3 {\coordtopix {#1}\t@pixa
                            \writeps {\the\t@pixa\space #2 #3 ar}}

% ===== Fill commands
% The form here completes a path with a closepath, applies the fill,
% starts a newpath and moves to the current point.  The gray level has
% 0 as black and 1 as white.  The current path is terminated.

\def\ifill f:#1 {\writeps {#1 fl}}     % Fill only
\def\lfill f:#1 {\writeps {#1 fp}}     % Stroke and fill

% ===== Text
% TeX text superimposed on the drawing
%  \htext (x y){text} or \htext {text}
%  \vtext (x y){text} or \vtext {text}
%  \rtext td:angle (x y){text} or \rtext td:angle {text}
%  \textref h:#1 v:#2

% The TeX text (or whatever) is placed in an \hbox.  The box is rotated
% for vertical text.  The text is placed on the drawing at the specified
% location (coordinates specified) or the current location (coordinates
% not specified).  The text reference point is placed at that location.
% For the purposes of determining the drawing size, the text box is of
% zero size.

% Horizontal text
% Check if the argument starts with a left parenthesis
\def\htext #1{\def\testit {#1}%
              \ifx \testit\l@paren
                \let\next=\h@move
              \else
                \let\next=\h@text
              \fi
              \next {#1}}

% Rotated text
\def\rtext td:#1 #2{\def\testit {#2}%
                    \ifx \testit\l@paren
                      \let\next=\r@move
                    \else
                      \let\next=\r@text
                    \fi
                    \next td:#1 {#2}}

% Vertical text
\def\vtext {\rtext td:90 }

% Text reference point
%  h:#1  text horizontal reference point - L, C or R
%  v:#2  text vertical reference point - T, C or B
\def\textref h:#1 v:#2 {\ifx #1R%
                          \edef\l@stuff {\hss}\edef\r@stuff {}%
                        \else
                          \ifx #1C%
                            \edef\l@stuff {\hss}\edef\r@stuff {\hss}%
                          \else  % default L
                            \edef\l@stuff {}\edef\r@stuff {\hss}%
                          \fi
                        \fi
                        \ifx #2T%
                          \edef\t@stuff {}\edef\b@stuff {\vss}%
                        \else
                          \ifx #2C%
                            \edef\t@stuff {\vss}\edef\b@stuff {\vss}%
                          \else  % default B
                            \edef\t@stuff {\vss}\edef\b@stuff {}%
                          \fi
                        \fi}

% ===== Arrow vectors
\def\avec (#1 #2){\getpos (#1 #2)\a@rgx\a@rgy
                  \s@etpospix \a@rgx \a@rgy
                  \writeps {\the\x@pix\space \the\y@pix\space (\a@type) %
                            \the\a@lenpix\space \the\a@widpix\space av}}

\def\ravec (#1 #2){\getpos (#1 #2)\a@rgx\a@rgy
                   \r@elpospix \a@rgx \a@rgy
                   \writeps {\the\x@pix\space \the\y@pix\space (\a@type) %
                             \the\a@lenpix\space \the\a@widpix\space av}}

% Arrowhead size
%  l:#1 - length of the arrowhead
%  w:#2 - width of the base of the arrowhead
\def\arrowheadsize l:#1 w:#2 {\coordtopix{#1}\a@lenpix
                              \coordtopix{#2}\a@widpix}
% Arrowhead type
%  t:#1 - arrowhead type, F  filled triangle (using current gray level)
%                         T  empty closed triangle
%                         W  white filled triangle
%                         V  Vee shape, at the end of the vector
%                         H  (or other character) Vee shape, vector stops
%                            short of the Vee
\def\arrowheadtype t:#1 {\edef\a@type{#1}}

% ===== Bezier curve
% The initial point is assumed to be the current point.  Only the last
% coordinate affects the size of the plot.
\def\clvec (#1 #2)(#3 #4)(#5 #6)%
           {\getpos (#1 #2)\a@rgx\a@rgy
            \coordtopix \a@rgx\t@pixa
            \advance \t@pixa by \x@segoffpix
            \coordtopix \a@rgy\t@pixb
            \advance \t@pixb by \y@segoffpix
            \getpos (#3 #4)\a@rgx\a@rgy
            \coordtopix \a@rgx\t@pixc
            \advance \t@pixc by \x@segoffpix
            \coordtopix \a@rgy\t@pixd
            \advance \t@pixd by \y@segoffpix
            \getpos (#5 #6)\a@rgx\a@rgy
            \s@etpospix \a@rgx \a@rgy
            \writeps {\the\t@pixa\space \the\t@pixb\space
                      \the\t@pixc\space \the\t@pixd\space
                      \the\x@pix\space \the\y@pix\space cv}}

% ===== Draw the bounding box
\def\drawbb {\bsegment
               \drawdim bp
               \linewd 0.24       % line width 1/300 inch = 0.24 bp
               \setunitscale {\p@sfactor}
               \writeps {\the\xminpix\space \the\yminpix\space mv}%
               \writeps {\the\xminpix\space \the\ymaxpix\space lv}%
               \writeps {\the\xmaxpix\space \the\ymaxpix\space lv}%
               \writeps {\the\xmaxpix\space \the\yminpix\space lv}%
               \writeps {\the\xminpix\space \the\yminpix\space lv}%
             \esegment}


% ===============================================================
% ===== Utility macros used by TeXdraw ==========================

% ===== Decode coordinates
% Get coordinates
% This macro is used to get two arguments separated by a blank, with
% possible leading and trailing blanks.  Symbolic coordinates are
% converted to user coordinates.
%  (#1 #2) - coordinates
%  #3 - macro name to receive the x-coordinate value
%  #4 - macro name to receive the y-coordinate value
\def\getpos (#1 #2)#3#4{\g@etargxy #1 #2 {} \\#3#4%
                        \c@heckast #3%
                        \ifa@st
                          \g@etsympix #3\t@pixa
                          \advance \t@pixa by -\x@segoffpix
                          \pixtocoord \t@pixa #3%
                        \fi
                        \c@heckast #4%
                        \ifa@st
                          \g@etsympix #4\t@pixa
                          \advance \t@pixa by -\y@segoffpix
                          \pixtocoord \t@pixa #4%
                        \fi}

% Get symbolic coordinate names
%  (#1 #2) - symbolic coordinates
%  #3 - macro name to receive the symbolic x coordinate name
%  #4 - macro name to receive the symbolic y coordinate name
\def\getsympos (#1 #2)#3#4{\g@etargxy #1 #2 {} \\#3#4%
                     \c@heckast #3%
                     \ifa@st \else
                       \t@xderror {TeXdraw: invalid symbolic coordinate}%
                     \fi
                     \c@heckast #4%
                     \ifa@st \else
                       \t@xderror {TeXdraw: invalid symbolic coordinate}%
                     \fi}

% ===== Convert a list of values to pixel values
%  (#1) - blank separated list of values in user coordinates
%  #2 - macro name to receive the blank separated list of pixel values
\def\listtopix (#1)#2{\def #2{}%
                      \edef\l@ist {#1 }%    % append a blank to the string
                      \m@oretrue
                      \loop
                        \expandafter\g@etitem \l@ist \\\a@rgx\l@ist
                        \a@pppix \a@rgx #2%
                        \ifx \l@ist\empty
                          \m@orefalse
                        \fi
                      \ifm@ore
                      \repeat}

% ===== Real multiplication
% This function uses the property that a box dimension may be scaled by
% a real value.  The values are converted to dimensions in units of pt.
% This choice gives us a reasonable dynamic range.  The final step is to
% clean off the "pt" on the resulting dimension.  Note that these are fixed
% point operations with each operand represented to an accuracy of about 5
% decimal places.

% Note we must use magnified points not "true" points, since the answer is
% expressed in magnified points.  The result will be calculated in the same
% manner no matter what the magnification is.
%  #1 and #2 are multiplicands
%  #3 macro name to capture the real result
\def\realmult #1#2#3{\dimen0=#1pt
                     \dimen2=#2\dimen0
                     \edef #3{\expandafter\c@lean\the\dimen2}}

% ===== Divide integers, real result
%  #1 integer numerator value
%  #2 integer denominator (divisor) value
%  #3 macro name to capture the real result
\def\intdiv #1#2#3{\t@counta=#1
                   \t@countb=#2
%  Limitations: #1 must be negatable, i.e. it must not be the largest
%                  magnitude negative number
%               #2 must be able to be multiplied by 2 without overflow
%  Calculate a*65536/b  where the factor 65536 converts from pt to sp.
%  This operation can also be interpretated as an extended precision
%  numerator divided by the denominator.  The scheme used is basically a
%  long division, except that it is bootstrapped by an integer divide.
%  The computations are carried out with positive numerator and
%  denominator, with the appropriate restoration of sign at the end.
%    \t@counta == remainder, r, initially set to a
%    \t@countb == denominator, b
%    \t@countc == quotient, q
%    \t@countd == +1, a and b have the same sign
%                 -1, a and b have opposite signs
%    \t@counte == temporary register
                   \ifnum \t@countb<0
                      \t@counta=-\t@counta
                      \t@countb=-\t@countb
                   \fi
                   \t@countd=1                    % record the sign
                   \ifnum \t@counta<0
                      \t@counta=-\t@counta
                      \t@countd=-1
                   \fi
%                                                 % q=a/b, r=a-q*b
                   \t@countc=\t@counta  \divide \t@countc by \t@countb
                   \t@counte=\t@countc  \multiply \t@counte by \t@countb
                   \advance \t@counta by -\t@counte
                   \t@counte=-1
                   \loop
                     \advance \t@counte by 1
                   \ifnum \t@counte<16                  % loop 16 times
                       \multiply \t@countc by 2           % q=2q
                       \multiply \t@counta by 2           % r=2r
                       \ifnum \t@counta<\t@countb \else   % if ( r >= b )
                         \advance \t@countc by 1          %   q=q+1
                         \advance \t@counta by -\t@countb %   r=r-b
                       \fi
                   \repeat
                   \divide \t@countb by 2         % rounding
                   \ifnum \t@counta<\t@countb     % if ( r >= b/2 ) q=q+1
                     \advance \t@countc by 1
                   \fi
                   \ifnum \t@countd<0             % restore the sign
                     \t@countc=-\t@countc
                   \fi
                   \dimen0=\t@countc sp           % express as a dimension
                   \edef #3{\expandafter\c@lean\the\dimen0}}

% ===============================================================
% ===== Internal TeXdraw macros =================================

% ===== Macros for converting between dimensions and units
% Convert drawing units (coordinate value, scaled by the unit scale and
% segment scale) to pixels.  We use rounding to get more accurate results.
%  #1 dimension in drawing units
%  #2 count in pixels (returned into a count)
\def\coordtopix #1#2{\dimen0=#1\d@dim
                     \dimen2=\d@sc\dimen0
                     \t@counta=\dimen2          % scaled dimension in sp
                     \t@countb=\s@ppix
                     \divide \t@countb by 2
                     \ifnum \t@counta<0         % rounding
                       \advance \t@counta by -\t@countb
                     \else
                       \advance \t@counta by \t@countb
                     \fi
                     \divide \t@counta by \s@ppix
                     #2=\t@counta}

% Convert from absolute pixels to relative scaled coordinates
%  #1 - input integer pixel value
%  #2 - macro name to receive the character string corresponding to the
%       floating point coordinate value
\def\pixtocoord #1#2{\t@counta=#1%
                     \multiply \t@counta by \s@ppix
                     \dimen0=\d@sc\d@dim
                     \t@countb=\dimen0
                     \intdiv \t@counta \t@countb #2}

% Convert pixels to TeX dimensions.
%  #1 - input integer pixel value
%  #2 - returned dimension (returned into a dimension register)
\def\pixtodim #1#2{\t@countb=#1%
                   \multiply \t@countb by \s@ppix
                   #2=\t@countb sp\relax}

% Convert pixels to (integer) bp units
%  #1 - input pixel value
%  #2 - integer value, returned as a macro definition
\def\pixtobp #1#2{\dimen0=\p@sfactor pt
                  \t@counta=\dimen0
                  \multiply \t@counta by #1%
                  \ifnum \t@counta < 0             % rounding
                    \advance \t@counta by -32768
                  \else
                    \advance \t@counta by 32768
                  \fi
                  \divide \t@counta by 65536
                  \edef #2{\the\t@counta}}
                  
% ===== Allocations for registers and counts
% == Temporary count registers
\newcount\t@counta    \newcount\t@countb   % Use at lowest levels
\newcount\t@countc    \newcount\t@countd
\newcount\t@counte
\newcount\t@pixa      \newcount\t@pixb     % Use for pixel values
\newcount\t@pixc      \newcount\t@pixd

% == Temporary dimension registers
\newdimen\t@xpos      \newdimen\t@ypos

% == Position and parameter registers
% The minimum and maximum extent in the X and Y direction in pixel units
% (updated globally to reach outside segments)
\newcount\xminpix      \newcount\xmaxpix
\newcount\yminpix      \newcount\ymaxpix

% == Arrowhead parameters
\newcount\a@lenpix     \newcount\a@widpix

% == Absolute pixel positions
\newcount\x@pix        \newcount\y@pix
\newcount\x@segoffpix  \newcount\y@segoffpix
\newcount\x@savepix    \newcount\y@savepix

% == Conversion factor
\newcount\s@ppix       % sp/pixel 

% == Pending segments count
\newcount\d@bs

% == Counter to form unique file names
\newcount\t@xdnum
\global\t@xdnum=0

% == TeXdraw box
\newbox\t@xdbox

% == Output stream number for the PostScript file
\newwrite\drawfile

% == \newif
\newif\ifm@pending
\newif\ifp@ath
\newif\ifa@st
\newif\ifm@ore
\newif \ift@extonly
\newif\ifp@osinit

% == \newtoks
\newtoks\everytexdraw

% ===== Character definitions
\def\l@paren{(}
\def\a@st{*}

% ===== Special character macros
% Need to be able to insert "%", "{" and "}" characters into the
% PostScript file.
% Define macros which have these characters with category "other".
% We will assume, that these characters have the standard meanings --
% after all, we use comments and braces in this code.
\catcode`\%=12
  \def\p@b {%!}  \def\p@p {%%}
\catcode`\%=14
\catcode`\{=12  \catcode`\}=12  \catcode`\u=1 \catcode`\v=2
  \def\l@br u{v  \def\r@br u}v
\catcode `\{=1  \catcode`\}=2   \catcode`\u=11 \catcode`\v=11

% ===== Pixel conversion factors
% The position is kept as an integer value (count).  It is set to a
% resolution corresponding to 300 units/inch.  We refer to them as pixels,
% but in fact the resolution is just that: movements are quantized to lie
% on a grid with that resolution.

% Using pixel units which correspond to the actual resolution of the device
% has advantages in that all horizontal and vertical lines then will be
% drawn with the same line thickness.  In addition the coordinates are
% then integer values (no decimal point or leading zeros) which leads
% to a more compact PostScript file.

% The following macro sets the conversion from PostScript units (bp) to the
% integer units (pixels).  The file inclusion \special environment in the
% PostScript driver restores the context to default PostScript values
% (bp or 72/in and origin in the lower lefthand corner).  A scaling value
% of 0.24 converts to 300/inch.  Note that the PostScript commands written
% to the temporary PostScript file do not depend on the TeX magnification
% in effect.  Magnification should be handled by the dvi to PostScript
% driver at the time that the file is included in the output.

% Calculate the conversion factors
% Let s@ppix = sp/pixel = u / p, where u = sp/unit and p = pix/unit (both
% integer values).
% We calculate s@ppix as
%         s@ppix = [ (u+0.5p)/p ]
% We also calculate the PostScript scale factor bp/pixel
% Let b = sp/bp.  We want p@sfactor = s@ppix/b.  For 300 pixels/inch, this
% gives p@sfactor=0.24.  Using rounding
%         p@sfactor = [ (s@ppix+0.5b)/b ] .
% To carry out the arithmetic, we will operate in sp units (integers) and
% generate the answer in pt units (multiplying by sp/pt).  This result will
% expressed as a character string representing a real number after the "pt"
% designator is stripped off.

{\catcode`\p=12 \catcode`\t=12
 \gdef\c@lean #1pt{#1}}

\def\sppix#1/#2 {\dimen0=1#2 \s@ppix=\dimen0
                 \t@counta=#1%
                 \divide \t@counta by 2
                 \advance \s@ppix by \t@counta
                 \divide \s@ppix by #1%             % \s@ppix available
                 \t@counta=\s@ppix
                 \multiply \t@counta by 65536       % 1 pt = 65536 sp
                 \advance \t@counta by 32891        % 0.5 bp = 32890.88 sp
                 \divide \t@counta by 65782         % 1 bp = 65781.76 sp
                 \dimen0=\t@counta sp
                 \edef\p@sfactor {\expandafter\c@lean\the\dimen0}}

% ===== Low level coordinate decoding macros
% Get two values, separated by a blank
% Invoke as \g@etargxy <stuff> {} \\\ma\mb
\def\g@etargxy #1 #2 #3 #4\\#5#6{\def #5{#1}%
                           \ifx #5\empty
                             \g@etargxy #2 #3 #4 \\#5#6%  leading blank
                           \else
                             \def #6{#2}%
                             \def\next {#3}%
                             \ifx \next\empty \else
                               \t@xderror {TeXdraw: invalid coordinate}%
                             \fi
                           \fi}

% Check for a leading asterisk
% Sets \a@stfalse or \a@sttrue, test with \ifa@st
\def\c@heckast #1{\expandafter
                  \c@heckastll #1\\}
\def\c@heckastll #1#2\\{\def\testit {#1}%
                        \ifx \testit\a@st
                          \a@sttrue
                        \else
                          \a@stfalse
                        \fi}

% Decode a symbolic coordinate
% Pixel value returned to a count
\def\g@etsympix #1#2{\expandafter
               \ifx \csname #1\endcsname \relax
                 \t@xderror {TeXdraw: undefined symbolic coordinate}%
               \fi
               #2=\csname #1\endcsname}

% Set a macro named #1 to have value #2
\def\s@etcsn #1#2{\expandafter
                  \xdef\csname#1\endcsname {#2}}

% ===== Low level list decoding macros
% Pick off the first item -> #3, rest of string -> #4
\def\g@etitem #1 #2\\#3#4{\edef #4{#2}\edef #3{#1}}
\def\a@pppix #1#2{\edef\next {#1}%
                  \ifx \next\empty \else
                    \coordtopix {#1}\t@pixa
                    \ifx #2\empty
                      \edef #2{\the\t@pixa}%
                    \else
                      \edef #2{#2 \the\t@pixa}%
                    \fi
                  \fi}

% ===== Macros for updating the position
% Calculate the position in pixels and update the maximum excursions
\def\s@etpospix #1#2{\coordtopix {#1}\x@pix
                     \advance \x@pix by \x@segoffpix
                     \coordtopix {#2}\y@pix
                     \advance \y@pix by \y@segoffpix
                     \u@pdateminmax \x@pix \y@pix}

\def\r@elpospix #1#2{\coordtopix {#1}\t@pixa
                     \advance \x@pix by \t@pixa
                     \coordtopix {#2}\t@pixa
                     \advance \y@pix by \t@pixa
                     \u@pdateminmax \x@pix \y@pix}

\def\r@elupd #1#2{\t@counta=\x@pix
                  \advance\t@counta by #1%
                  \t@countb=\y@pix
                  \advance\t@countb by #2%
                  \u@pdateminmax \t@counta \t@countb}

\def\u@pdateminmax #1#2{\ifnum #1>\xmaxpix
                          \global\xmaxpix=#1%
                        \fi
                        \ifnum #1<\xminpix
                          \global\xminpix=#1%
                        \fi
                        \ifnum #2>\ymaxpix
                          \global\ymaxpix=#2%
                        \fi
                        \ifnum #2<\yminpix
                          \global\yminpix=#2%
                        \fi}

% =====  Save moves / flush moves
% A TeXdraw segment which generates only TeX text uses only move, begin
% segment and end segment commands.  The goal is to avoid writing out
% empty segments for such cases.  To this end, moves are held back and
% only written out if necessary to set the position or terminate a path.
% Also in this way, a TeXdraw drawing which generates only TeX text will
% not generate a PostScript file.

% Two flags are used.  Both flags are local to a segment.
% - move pending:  Set when a move has been invoked but the move command
%   has not been written out to the PostScript file
% - path in progress:  Set when a PostScript path has been started but the
%   path has not been terminated and stroked.
% (1) Moves are kept back.  Using TeX's groups, a local flag and local
%     position registers are used to keep track of whether the latest
%     move applicable to a given segment has been written out or not.  In
%     effect there is a stack of pending moves, one for each level of
%     segment nesting.
% (2) At the beginning of a segment, if a PS path is in progress and a
%     a move is pending, the move is written out, terminating the path
%     and stroking the path.  This is done to ensure that the path is
%     stroked before lines and/or fills are executed in the segment.
% (3) At the beginning of a segment, if a PS path is not in progress,
%     any pending moves are kept back.  Effectively, the move will be
%     transferred into the segment.  It will be written out only when
%     the position needs to be updated for some other command.  Such
%     moves which are transferred into segments may have to be repeated
%     outside the segment.  The move pending flag will be restored to
%     the value outside the segment on exit from the segment.
% (4) A begin segment command is not written out, but instead a global
%     segment backlog counter is incremented.  The backlog of begin
%     segment commands is written out when a drawing command is
%     encountered.
% The effect of the above on the TeXdraw commands is as follows.
% (a) move:
%     - set the current position
%     - record the position of the saved move
%     - set the move pending flag
% (b) begin segment:
%     - if a path is in progress
%         - if a move is pending
%           - if there is a backlog of segments
%             - write out enough begin segments to clear the backlog
%           - write out the pending move
%           - reset the move pending flag (local to the containing segment,
%             but affects inferior segments)
%           - reset the path in progress flag
%     - increment the segment backlog counter
%     - begin a group
% (c) end segment:
%     - end a group
%     - if there is no backlog of segments
%       - write an end segment command
%     - if there is a backlog of segments
%       - decrement the backlog counter, thereby omitting an empty
%         empty segment.
%     - the move pending flag and path in progress flag are automatically
%       restored on leaving the TeX group
% (d) text:
%     - create a text box
% (e) line parameters:
%     - if there is a backlog of segments
%        - write out enough begin segments to clear the backlog
%     - clear the path in progress flag
%     - write the PS command changing the line parameter
% (f) other drawing commands:
%     - update the current position
%     - if there is a backlog of segments
%        - write out enough begin segments to clear the backlog
%     - if there is a pending move
%        - write out the pending move
%        - reset the move pending flag (local to this segment, but affects
%          inferior segments)
%     - set the path in progress flag
%     - write the drawing command
% Notes:
% (1) The es PS command strokes the path at the end of a segment to
%     ensure that the correct line parameters are used for the segment.
%     The path before the corresponding bs command is restored and
%     continued.
% (2) The \f@lushbs and \f@lushmove commands must be invoked before each
%     drawing command written to the PS file.  The macro \writeps includes
%     these operations.

% Another awkward business has to do with initialization.  We want a
% default (0 0) initial position so that the user can draw vectors
% immediately.  However, if the user specifies another move before
% beginning to draw, that position should be the initial position.  The
% importance of this initial position is that the determination of the
% maximum excursion must take this value into account.  We handle the
% initialization in the \s@avemove and \f@lushmove macros.  The macro
% \ifp@osinit indicates whether the next move should be captured as the
% initial values for \xminpix, \yminpix, \xmaxpix, and \ymaxpix.  However,
% if a \f@lushmove is invoked, then we assume that the appropriate initial
% values has already been set.  The "mv" command in PostScript is defined
% to stroke the current path (if any) and move to the pixel coordinates
% specified.

% Note that \m@pendingtrue and \m@pendingfalse define the flag locally.
% In addition, \x@savepix and \y@savepix are local variables.  We make
% use of the fact that the values of the flag and positions propagate
% down to inferior segments but not up to superior segments.  This
% behaviour Is consistent with the gsave/grestore operation on the
% PostScript side.
\def\s@avemove #1#2{\x@savepix=#1\y@savepix=#2%
                    \m@pendingtrue
                    \ifp@osinit \else
                      \global\p@osinittrue
                      \global\xminpix=\x@savepix \global\yminpix=\y@savepix
                      \global\xmaxpix=\x@savepix \global\ymaxpix=\y@savepix
                    \fi}

\def\f@lushmove {\global\p@osinittrue
                 \ifm@pending
                   \writetx {\the\x@savepix\space \the\y@savepix\space mv}%
                   \m@pendingfalse
                   \p@athfalse
                 \fi}

% =====  Flush begin segment
% \f@lushbs flushes any saved up \bsegments.  Some of these may be
% redundant, but we cannot know without looking ahead beyond the
% \esegment.
\def\f@lushbs {\loop
                 \ifnum \d@bs>0
                   \writetx {bs}%
                   \global\advance \d@bs by -1
               \repeat}
               
% ===== Internal text macros
% Horizontal text, use only 3 levels of box nesting here
\def\h@move #1#2 #3)#4{\move (#2 #3)%
                       \h@text {#4}}
\def\h@text #1{\pixtodim \x@pix \t@xpos
               \pixtodim \y@pix \t@ypos
               \vbox to 0pt{\normalbaselines
                            \t@stuff
                            \kern -\t@ypos
                            \hbox to 0pt{\l@stuff
                                         \kern \t@xpos
                                         \hbox {#1}%
                                         \kern -\t@xpos
                                         \r@stuff}%
                            \kern \t@ypos
                            \b@stuff\relax}}

% Rotated text
% Uses 5 levels of box nesting here (so that the text reference point
% is that <before> rotation).  This was done so that the reference point
% definition makes sense with arbitrary angle rotation.  The text is
% rotated with respect to the text reference point.  The result is zero
% sized.  These macros generate in-line PostScript.
% #1 - rotation angle in degrees
% #2 - text to be rotated
\def\r@move td:#1 #2#3 #4)#5{\move (#3 #4)%
                             \r@text td:#1 {#5}}
\def\r@text td:#1 #2{\vbox to 0pt{\pixtodim \x@pix \t@xpos
                                  \pixtodim \y@pix \t@ypos
                                  \kern -\t@ypos
                                  \hbox to 0pt{\kern \t@xpos
                                               \rottxt {#1}{\z@sb {#2}}%
                                               \hss}%
                                  \vss}}
\def\z@sb #1{\vbox to 0pt{\normalbaselines
                          \t@stuff
                          \hbox to 0pt{\l@stuff \hbox {#1}\r@stuff}%
                          \b@stuff}}

% ===== Rotate text, in-line PostScript code
\ifx \rotatebox\@undefined
  \def\rottxt #1#2{\bgroup
                     \special {ps: gsave currentpoint currentpoint translate
                               #1\space neg rotate
                              neg exch neg exch translate}%
                     #2%
                     \special {ps: currentpoint grestore moveto}%
                   \egroup}
\else
  \let\rottxt=\rotatebox
\fi

% ===== Error message
% If not defined, use the plain TeX errmessage macro
\ifx \t@xderror\@undefined
  \let\t@xderror=\errmessage
\fi

% ===== Default values
% These are reset each time TeXdraw is invoked
\def\t@exdrawdef {\sppix 300/in            % 300 pixels/inch
                  \drawdim in              % drawing units are inches
                  \edef\u@nitsc {1}%       % unit scale 1 (has to be set
                                           % before invoking \setsegscale)
                  \setsegscale 1           % segment scale 1
                  \arrowheadsize l:0.16 w:0.08
                  \arrowheadtype t:T
                  \textref h:L v:B }


% ===============================================================
% ===== PostScript file macros ==================================

% ===== Include the TeXdraw graphics
% The drawing in a box of appropriate size will be placed such that its
% lower left hand corner will be at the current TeX position.
\ifx \includegraphics\@undefined
  \def\t@xdinclude [#1,#2][#3,#4]#5{%
    \begingroup                           % keep definitions local
      \message {<#5>}%
      \leavevmode
      \t@counta=-#1%                      % integer bounding box coordinates
      \t@countb=-#2%
      \setbox0=\hbox{%
        \special {PSfile="#5"\space
                  hoffset=\the\t@counta\space voffset=\the\t@countb}}%
      \t@ypos=#4 bp%
        \advance \t@ypos by -#2 bp%
      \t@xpos=#3 bp%
        \advance \t@xpos by -#1 bp%
      \dp0=0pt \ht0=\t@ypos  \wd0=\t@xpos
      \box0%
    \endgroup}
\else
  \let\t@xdinclude=\includegraphics
\fi

% ===== Write to the PostScript file
% Macro to write PostScript commands to the temporary PostScript file
% To decrease the size of the PostScript file, moves are kept back to
% allow redundant multiple moves to be removed.  In addition empty gsave/
% grestore pairs are not written.  The PostScript file is not opened if
% TeXdraw has not generated any PostScript commands, i.e. it has produced
% only TeX text.
% \writeps : flushes the pending move to make sure things are positioned
%            correctly and flushes pending begin segments before calling
%            \writetx to write to the PostScript file
% \writetx : writes directly to the PostScript file.  This version is used
%            only for those commands which just change line parameters
%            without drawing.  This routine opens the file and writes the
%            PS file header the first time it is called.
% \w@rps :   lowest level direct write to the PostScript file
\def\writeps #1{\f@lushbs
                \f@lushmove
                \p@athtrue
                \writetx {#1}}
\def\writetx #1{\ift@extonly
                  \global\t@extonlyfalse
                  \t@xdpsfn \p@sfile
                  \t@dropen \p@sfile
                \fi
                \w@rps {#1}}
\def\w@rps #1{\immediate\write\drawfile {#1}}

% ===== Open a PostScript file
% Open a PostScript file, write the definitions used by TeXdraw.
\def\t@xdpsfn #1{%
  \global\advance \t@xdnum by 1
  \ifnum \t@xdnum<10
    \xdef #1{\jobname.ps\the\t@xdnum}
  \else
    \xdef #1{\jobname.p\the\t@xdnum}
  \fi
}
\def\t@dropen #1{%
  \immediate\openout\drawfile=#1%
  \w@rps {\p@b PS-Adobe-3.0 EPSF-3.0}%
  \w@rps {\p@p BoundingBox: (atend)}%
  \w@rps {\p@p Title: TeXdraw drawing: #1}%
  \w@rps {\p@p Pages: 1}%
  \w@rps {\p@p Creator: \TeXdrawId}%
  \w@rps {\p@p CreationDate: \the\year/\the\month/\the\day}%
  \w@rps {50 dict begin}%
  \w@rps {/mv {stroke moveto} def}%
  \w@rps {/lv {lineto} def}%
  \w@rps {/st {currentpoint stroke moveto} def}%
  \w@rps {/sl {st setlinewidth} def}%
  \w@rps {/sd {st 0 setdash} def}%
  \w@rps {/sg {st setgray} def}%
  \w@rps {/bs {gsave} def /es {stroke grestore} def}%
  \w@rps {/fl \l@br gsave setgray fill grestore}%
  \w@rps    { currentpoint newpath moveto\r@br\space def}%
  \w@rps {/fp {gsave setgray fill grestore st} def}%
  \w@rps {/cv {curveto} def}%
  \w@rps {/cr \l@br gsave currentpoint newpath 3 -1 roll 0 360 arc}%
  \w@rps    { stroke grestore\r@br\space def}%
  \w@rps {/fc \l@br gsave setgray currentpoint newpath}%
  \w@rps    { 3 -1 roll 0 360 arc fill grestore\r@br\space def}%
  \w@rps {/ar {gsave currentpoint newpath 5 2 roll arc stroke grestore} def}%
  \w@rps {/el \l@br gsave /svm matrix currentmatrix def}%
  \w@rps    { currentpoint translate scale newpath 0 0 1 0 360 arc}%
  \w@rps    { svm setmatrix stroke grestore\r@br\space def}%
  \w@rps {/fe \l@br gsave setgray currentpoint translate scale newpath}%
  \w@rps    { 0 0 1 0 360 arc fill grestore\r@br\space def}%
  \w@rps {/av \l@br /hhwid exch 2 div def /hlen exch def}%
  \w@rps    { /ah exch def /tipy exch def /tipx exch def}%
  \w@rps    { currentpoint /taily exch def /tailx exch def}%
  \w@rps    { /dx tipx tailx sub def /dy tipy taily sub def}%
  \w@rps    { /alen dx dx mul dy dy mul add sqrt def}%
  \w@rps    { /blen alen hlen sub def}%
  \w@rps    { gsave tailx taily translate dy dx atan rotate}%
  \w@rps    { (V) ah ne {blen 0 gt {blen 0 lineto} if} {alen 0 lineto} ifelse}%
  \w@rps    { stroke blen hhwid neg moveto alen 0 lineto blen hhwid lineto}%
  \w@rps    { (T) ah eq {closepath} if}%
  \w@rps    { (W) ah eq {gsave 1 setgray fill grestore closepath} if}%
  \w@rps    { (F) ah eq {fill} {stroke} ifelse}%
  \w@rps    { grestore tipx tipy moveto\r@br\space def}%
  \w@rps {\p@sfactor\space \p@sfactor\space scale}%
  \w@rps {1 setlinecap 1 setlinejoin}%
  \w@rps {3 setlinewidth [] 0 setdash}%
  \w@rps {0 0 moveto}%
}

% Notes:
% - mv (move to) This command includes a stroke before the moveto.  The
%      stroke terminates a path and the move begins another path.
% - bs (begin segment) encloses a segment in a gsave/grestore to keep
%      changes to line parameters local.
% - es (end segment) does a "stroke grestore" to make sure lines inside
%      the segment use the line parameters local to that segment
% - ar (arc) The path is generated and stroked inside a gsave/grestore,
%      leaving the current path intact.
% - cr (circle) The path is generated and stroked inside a gsave/grestore,
%      leaving the current path intact.
% - fc (filled circle) The path is generated and filled inside a gsave/
%      grestore, keeping the fill level local to the circle.  The current
%      path is left intact.
% - el (ellipse) The path is generated and stroked inside a gsave/grestore,
%      leaving the current path intact.  The elliptical path is defined
%      with different x and y scaling, then stroked with default scaling
%      to give a constant line thickness.
% - fe (filled ellipse) The path is generated and filled inside a gsave/
%      grestore, leaving the current path intact.
% - fl (fill) The current path is closed and filled inside a gsave/restore,
%      keeping the fill level local.  A newpath terminates the path.
% - fp (fill path) The current path is closed and then filled inside a
%      gsave/grestore.  Finally the closed path is stroked, implicitly
%      terminating the path.
% - av (arrow vector) The arrow vector is drawn inside a gsave/grestore.
%      The line width and type are those currently in effect.  After the
%      grestore, the current path is continued with a move to the tip of
%      the vector.

% ===== Close the PostScript file
% Write a trailer with the BoundingBox, close the file.  Note that the
% BoundingBox may be larger than the commands in the PostScript file
% indicate.  This is due to the fact that multiple move commands in
% a row are collapsed into a single move.  The BoundingBox information
% includes the effect of the moves which were expunged.
\def\t@drclose {%
  \bgroup
    \w@rps {stroke end showpage}%
    \w@rps {\p@p Trailer:}%
    \pixtobp \xminpix \l@lxbp  \pixtobp \yminpix \l@lybp
    \pixtobp \xmaxpix \u@rxbp  \pixtobp \ymaxpix \u@rybp
    \w@rps {\p@p BoundingBox: \l@lxbp\space \l@lybp\space
                              \u@rxbp\space \u@rybp}%
    \w@rps {\p@p EOF}%
  \egroup
  \immediate\closeout\drawfile
}

% ===============================================================
\catcode`\@=\catamp

math0109097.extracted_bib
\begin{thebibliography}{AnBnCng3}

\bibitem[AIM]{AIM} P.L. Antonelli, R.S. Ingarden, and M. Matsumoto, 
{\it The theory of sprays and Finsler spaces with applications in physics and biology}, FTPH 58, Kluwer Academic Publishers, 1993.


\bibitem[AZ]{AZ} H. Akbar-Zadeh, {\it Sur les espaces de Finsler \'{a} courbures sectionnelles constantes}, Bull. Acad. Roy. Bel. Cl, Sci, 5e S\'{e}rie - Tome LXXXIV (1988), 281-322.

\bibitem[BaChSh]{BaChSh} D. Bao, S.S. Chern and Z. Shen, 
{\it  An introduction to Riemann-Finsler geometry}, Springer, 2000.

\bibitem[BaRo]{BaRo} D. Bao and C. Robles, {\it On Randers metrics of constant curvature}, preprint,   2001.

\bibitem[BaSh]{BaSh} D. Bao and Z. Shen, {\it Finsler metrics of constant curvature on the Lie group $S^3$}, preprint (2000).

\bibitem[Br1]{Br1}  {R. Bryant, \textit{Finsler structures on the
2-sphere satisfying $K=1$}, Finsler Geometry, Contemporary Mathematics {\bf 196},
Amer. Math. Soc., Providence, RI, 1996, 27-42. }

\bibitem[Br2]{Br2}  {R. Bryant, \textit{Projectively flat Finsler $2$-spheres of constant curvature}, Selecta Math., New Series, 
{\bf 3}(1997), 161-204. }

\bibitem[Br3]{Br3} R. Bryant, {\it Finsler manifolds 
with constant curvature},  Talk at the 1998 Geometry Festival in Stony Brook.  



\bibitem[ChSh]{ChSh} X. Chen and Z. Shen, 
{\it Randers metrics with special curvature properties}, preprint (2001).

\bibitem[Fk1]{Fk1} P. Funk, {\it \"{U}ber Geometrien, bei denen die Geraden die K\"{u}rzesten sind}, Math. Annalen {\bf 101}(1929), 226-237.

\bibitem[Fk2]{Fk2} P. Funk, {\it \"{U}ber zweidimensionale Finslersche R\"{a}ume, insbesondere \"{u}ber solche mit geradlinigen Extremalen und positiver konstanter Kr\"{u}mmung}, Math. Zeitschr. {\bf 40}(1936), 86-93.


\bibitem[Ma]{Ma} M. Matsumoto,
{\it Randers spaces of constant curvature}, Reports on Mathematical Physics,
{\bf 28}(1989), 249-261.


\bibitem[Ok]{Ok} T. Okada, {\it On models of projectively flat Finsler spaces of constant negative curvature}, Tensor, N. S. {\bf 40}(1983), 117-123.


\bibitem[Ra]{Ra} G. Randers, {\it On an asymmetric metric in the four-space of general relativity}, Phys. Rev. {\bf 59}(1941), 195-199.


\bibitem[Sh1]{Sh1} Z. Shen, {\it Differential Geometry of Spray and Finsler Spaces}, Kluwer Academic Publishers, Dordrecht, 2001.

\bibitem[Sh2]{Sh2} Z. Shen, {\it Funk metrics and R-flat sprays},
preprint.

\bibitem[Sh3]{Sh3} Z. Shen, {\it Finsler metrics with ${\bf K}=0$ and ${\bf S}=0$}, preprint, May, 2001.

\bibitem[Sh4]{Sh4} Z. Shen, {\it Projectively flat Finsler metrics of constant curvature}, preprint, 2001.



\bibitem[YaSh]{YaSh} H. Yasuda and H. Shimada, {\it On Randers spaces of scalar curvature}, Rep. on Math. Phys. {\bf 11}(1977), 347-360.

\end{thebibliography}