\Large Finsler Metrics with ${K}=0$ and ${S}=0$
Zhongmin Shen
{In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a technique to construct non-projectively flat Finsler metrics with zero curvature in each dimension. The technique can be used to construct many non-projectively flat Finsler metrics of constant curvature. }
RandersK=0.tex

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\title{\Large Finsler Metrics with ${\bf K}=0$ and ${\bf S}=0$}
\author{Zhongmin Shen}


\date{May,  2001; Revised in September, 2001}

\begin{document}
\maketitle

\begin{abstract}
{In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. 
A recent  example constructed by the author is  projectively flat 
 with zero curvature. 
In this paper, we introduce a technique to construct non-projectively flat  Finsler metrics with zero curvature in each dimension. The technique can be used to construct many non-projectively flat Finsler metrics of constant curvature.
}
\end{abstract}

\section{Introduction}

One of the fundamental problems in Riemann-Finsler geometry is  to study and characterize Finsler metrics of constant flag curvature. 
The flag curvature in Finsler geometry is an analogue of the sectional curvature in Riemannian geometry. 
Riemannian metrics of constant  sectional curvature were classified  by E. Cartan a long time ago.
There are only three local Riemannian metrics of constant sectional curvature, up to a scaling. 
However,  the local metric structure of a Finsler metric with constant flag curvature is much more complicated.
Mathematicians have discovered several important  Finsler metrics of constant flag curvature (e.g.,
the Funk metrics (${\bf K}=-1/4$), the Hilbert-Klein metrics (${\bf K}=-1$)\cite{BaChSh}\cite{Sh1} and the Bryant metrics (${\bf K}=1$)\cite{Br1}\cite{Br2}\cite{Br3}). All of them are locally projectively flat.


Recently, 
the author and D. Bao discover a family of non-projectively flat 
Finsler metrics of constant curvature ${\bf K}=1$ on $\Bbb S^3$ using the Hopf-fibration structure \cite{BaSh}.
We know that 
Minkowski spaces are  Finsler spaces with ${\bf K}=0$.
In \cite{Sh2}, we construct a positively complete  projectively flat spray with vanishing Riemann curvature (R-flat). 
Then we construct a positively complete  locally projectively flat
Finsler metric with ${\bf K}=0$. It is defined on the unit ball $\Bbb B^n\subset \Bbb R^n$ by
\be
F({\bf y}) := { \Big ( \sqrt{ |{\bf y}|^2 - (|{\bf x}|^2|{\bf y}|^2 
-\langle {\bf x}, {\bf y} \rangle^2) } + \langle {\bf x}, {\bf y} \rangle \Big )^2 
\over ( 1-|{\bf x}|^2)^2 \sqrt{|{\bf y}|^2 - (|{\bf x}|^2|{\bf y}|^2 
-\langle {\bf x}, {\bf y} \rangle^2)} }, \ \ \ \ \ \ {\bf y}\in T_{\bf x}\Bbb B^n=\R^n,\label{fromFunk}
\ee 
where $|\cdot |$ and $\langle \; , \; \rangle$ denote the standard Euclidean 
norm and inner product. According to \cite{AZ}, any positively complete Finsler metric with 
${\bf K}=0$ must be locally Minkowskian if the  first and second Cartan torsions are bounded (see also \cite{Sh1}). Thus the Finsler metric 
in (\ref{fromFunk}) does not have bounded Cartan torsions.


 In this paper, we are going to construct a non-projectively flat Finsler metrics with ${\bf K}=0$ in each dimension. Further, these Finsler metrics have bounded
first and second Cartan torsions. Thus, they are not positively complete.


\begin{thm}\label{thmK=0}
Let $n\geq 2$ and
\[ \Omega:=\Big \{ p=(x, y, \bar{p}) \in \R^2 \times \R^{n-2} 
\ \Big | \ x^2+y^2 < 1 \Big \} .\]
Define
\be
 F({\bf y}) := {\sqrt{ \Big ( -y u + x v \Big )^2 
+ |{\bf y}|^2 \Big ( 1- x^2 -y^2 \Big ) } - \Big ( - yu + x v \Big ) 
\over 1- x^2 - y^2 },\label{RandersExample}
\ee
where ${\bf y} = (u, v,\bar{\bf y} ) \in T_p \Omega = \R^n$ and $p= (x, y, \bar{p})\in \Omega$.
$F$ is a Finsler metric on $\Omega$ satisfying 
\[ {\bf K} =0, \ \ \ \ \ \ {\bf S}=0.\]
\end{thm}

Here the quantity ${\bf S}$ denotes the S-curvature which is a non-Riemannian quantity. It is known that every  Berwald metric satisfies  ${\bf S}=0$ (\cite{Sh1}\cite{Sh4}). 
Since every Berwald metric with ${\bf K}=0$ must be
locally Minkowskian (see \cite{AIM}\cite{BaChSh}), the Finsler metric in (\ref{RandersExample}) is not Berwaldian. 



The construction of the Finsler metric in (\ref{RandersExample}) is motivated by the shortest time problem (see Section 3 below). 


The Finsler metric in (\ref{RandersExample}) is in the following form
$
 F = \alpha + \beta, $
where $\alpha $ is a Riemannian metric and $\beta $ is a 1-form with $\|\beta\|_{\alpha}(x):=\sup_{y\in T_xM} \beta(y)/\alpha(y) <1$ for all $x\in M$. 
This type of Finsler metrics were first studied by G. Randers in 1941 \cite{Ra} from the standard point of general relativity (see also  \cite{AIM}). Therefore they are called  {\it Randers metrics}. 




The Randers metric in (\ref{RandersExample}) is not locally projectively 
flat, hence not locally Minkowskian. Theorem \ref{thmK=0} is inconsistent  with 
the main result in 
\cite{SSAY}, where they claim that every Randers metric with ${\bf K}=0$ must be locally Minkowskian. See also \cite{YaSh}\cite{Ma} for related discussion. 
 Nevertheless, if a Randers metric with ${\bf K}=0$ is positively complete, then it must be
locally Minkowskian.


\begin{thm}\label{thm1}
Let $F=\alpha+\beta$ be a positively complete Randers metric on a manifold $M$.
Then ${\bf K}=0$ if and only if it is locally Minkowskian.
In this case, $\alpha$ is flat and $\beta$ is parallel with respect to $\alpha$.
\end{thm}

Here  a Finsler metric is said to be positively complete if every geodesic defined on $(a, b)$ can be extended to a geodesic defined on $(a, \infty)$. Example (\ref{RandersExample}) shows that
the positive completeness can not be dropped. 
In \cite{Sh6}, we show that every locally projectively flat Randers metric 
with ${\bf K}=0$ must be locally Minkowskian. In 
this case, the positive completeness is not required. 




\section{Preliminaries}

In this section, we recall some basic definitions in Riemann-Finsler geometry \cite{BaChSh}\cite{Sh1}.

A Finsler metric on a manifold $M$ is a function $F: TM \to [0, \infty)$ 
with the following properties:
\ben
\item[(a)] $F(\lambda {\bf y}) = \lambda F({\bf y}) $, $\lambda >0$;
\item[(b)] For any non-zero vector ${\bf y}\in T_xM$,
the induced bilinear form  $g_{\bf y}$ on $T_xM$ is an inner product, where
\[ g_{\bf y}({\bf u}, {\bf v}):= {1\over 2} {\pa^2\over\pa s\pa t}\Big [ F^2({\bf y}+s {\bf u}+t {\bf v}) \Big ] |_{s=t=0}, \ \ \ \ \ \ {\bf u}, {\bf v}\in T_xM.
\]
\een


\bigskip
Riemannian metrics are special Finsler metrics. Traditionally, a Riemannian metric is denoted by $ a_{ij}(x) dx^i \otimes dx^j$. It is a family of inner products on tangent spaces. Let
 $\alpha({\bf y}):= \sqrt{g_{ij}(x)y^iy^j}$,
${\bf y}= y^i\pxi|_x\in T_xM$. $\alpha$ is  a family of Euclidean norms on tangent spaces.   Throughout this 
paper, we also denote   a Riemannian metric by  $\alpha=\sqrt{a_{ij}(x)y^iy^j}$.

Let $\alpha=\sqrt{a_{ij}(x)y^iy^j}$ be a Riemannian metric  and $\beta =b_i(x)y^i$ a 1-form on a manifold $M$. Define
\[ F = \alpha +\beta.\]
Then $F$ satisfies (a). If we assume that 
\[ \|\beta\|_{\alpha}(x)
:= \sup_{y\in T_xM}{\beta ({\bf y})\over\alpha({\bf y}) }=\sqrt{ a^{ij}(x) b_i(x)b_j(x)} <1 , \ \ \ \ \ \ \ x\in M,\]
then $F$ satisfies (b). By definition, $F$ is a Finsler metric.
We call $F$ a Randers metric. 


\bigskip

Let $F$ be a Finsler metric. For a non-zero vector ${\bf y}\in T_pM \setminus\{0\}$,
define 
\[ {\bf C}_{\bf y} ({\bf u}, {\bf v}, {\bf w} ) 
:= {1\over 4 } {d^3 \over dt^3} \Big [ F^2 \Big ( {\bf y}+ s {\bf u}+ t {\bf v}+ r {\bf w} \Big ) \Big ]|_{s=t=r=0}\]
\[ \tilde{\bf C}_{\bf y}  (  {\bf u}, {\bf v}, {\bf w}, {\bf z} ) 
:= {1\over 4 } {d^4 \over dt^4} \Big [ F^2 \Big ( {\bf y}+ s {\bf u}+ t {\bf v}+ r {\bf w}+ h{\bf z} \Big ) \Big ]|_{s=t=r=h=0}.\]
By the homogeneity of $F$, we have
\[ {\bf C}_{\bf y} ({\bf u}, {\bf v}, {\bf y} ) =0, \ \ \ \ \
\tilde{\bf C}_{\bf y}  (  {\bf u}, {\bf v}, {\bf w}, {\bf y}  ) 
= - {\bf C}_{\bf y} ({\bf u}, {\bf v}, {\bf w} ).\]
We call ${\bf C}$ and $\tilde{\bf C}$ the first and second Cartan torsion respectively. We usually call ${\bf C}$ the Cartan torsion if the second one does not appear.
It is obvious that $F$ is Riemannian if and only if ${\bf C}=0$,
and ${\bf C}=0$ if and only if $\tilde{\bf C}=0$.

The essential bounds of ${\bf C}$ and $\tilde{\bf C}$ at $x\in M$ are defined by
\begin{eqnarray*}
\|{\bf C}\|_x : & = &  \sup_{ {\bf u} \in T_xM} F({\bf y}){ |{\bf C}_{\bf y} ({\bf u}, {\bf u}, {\bf u} \Big ) |\over  [g_{\bf y}({\bf u}, {\bf u} ) ]^{3/2} }\\
\|\tilde{\bf C}\|_x:& = &  \sup_{ {\bf u} \in T_xM}F^2({\bf y}) { |\tilde{\bf C}_{\bf y} ({\bf u}, {\bf u}, {\bf u}, {\bf u} \Big ) |\over  [g_{\bf y}({\bf u}, {\bf u} ) ]^2 },
\end{eqnarray*}
where the supremum is taken over all non-zero vectors ${\bf y}, {\bf u}\in T_xM$ with $g_{\bf y}({\bf y}, {\bf u}) =0$. 




\bigskip
For a Finsler metric $F$ on an $n$-dimensional manifold $M$, the (Busemann-Hausdorff) volume form $dV_F = \sigma_F(x) dx^1 \cdots dx^n$ is defined by
\be
\sigma_F(x) := {{\rm Vol} (\Bbb B^n(1)) \over {\rm Vol} \Big \{ (y^i)\in \R^n \ \Big | \ F \Big ( y^i \pxi|_x \Big ) < 1 \Big \} } .\label{dV}
\ee

In general, the local scalar function 
$\sigma_F(x)$ can not be expressed in terms of elementary functions, even $F$ is locally expressed by elementary functions. However, for Randers metrics, the volume form is expressed by a very simple formula, since each indicatrix 
$S_xM := \{ {\bf y}\in T_xM \ | \ F({\bf y}) = 1 \}$ is a shifted Euclidean sphere
in $(T_xM, \alpha_x)$.
More precisely,
for a Randers metric 
$ F = \alpha + \beta$, where $\alpha =\sqrt{a_{ij}(x)y^iy^j}$ 
and $\beta(y)=b_i(x)y^i$, its volume form $dV_F$ is given by
\be
dV_F = \Big ( 1 - \|\beta \|^2_{\alpha}(x) \Big )^{n+1\over 2} dV_{\alpha}.\label{dV_F}
\ee
 See \cite{Sh1} for more details.

\bigskip
Now we introduce the notion of S-curvature for Finsler metrics
(see \cite{Sh1}).
Let $F$ be a Finsler metric on a manifold $M$.
Let \[ g_{ij}(x,y):= g_{\bf y} \Big (\pxi|_x, \pxj|_x\Big )= {1\over 2}[F^2]_{y^iy^j}(x,y), \ \ \ \ \ {\bf y}= y^i \pxi|_x.\]
Express the volume form $dV_F$ by
\[ dV_F = \sigma_F(x) dx^1 \cdots dx^n,\]
where $\sigma_F(x)$ is defined in (\ref{dV}).
Set
\[ \mu({\bf y}):= \ln \Big [ {\sqrt{\det (g_{ij}(x, y)) }\over \sigma_F(x) }
\Big ].\]
The quantity $\mu$ is a scalar function on $TM\setminus\{0\}$.
$\mu$ is a non-Riemannian quantity. 

\bigskip

Locally minimizing constant speed curves (geodesics) are characterized by
\[ {d^2 x^i\over dt^2} + 2 G^i(x, {dx\over dt} ) =0,\]
where $G^i(x,y)$ are given by
\be
G^i := {1\over 4} g^{il} \Big \{ 2 {\pa g_{jl}\over \pa x^k} - {\pa g_{jk}
\over \pa x^l} \Big \} y^j y^k.
\ee
$G^i$ are called the geodesic coefficients in a local coordinate system.
If $F$ is Riemannian, then $G^i(x, y)= {1\over 2} \Gamma^i_{jk}(x)y^jy^k$ 
are quadratic in $(y^i)$ at every point $x\in M$.
A Finsler metric is called a {\it Berwald metric} if the geodesic coefficients
have this property. There are many non-Riemannian Berwald metrics.
The classification of Berwald metrics is done by Z.I. Szabo \cite{Sz}.




\bigskip



For a vector ${\bf y}\in T_xM$
Let $c(t), -\e < t <\e $, denote the geodesic with $c(0)=x$ and $\dot{c}(0)={\bf y}$. Define
\[ {\bf S}({\bf y}):= {d \over dt} \Big [ \mu \Big (\dot{c}(t)\Big ) \Big ]|_{t=0}.\]
We call ${\bf S}$ the S-curvature. This quantity was introduced in \cite{Sh4}\cite{Sh1}. 


Let $G^i(x, y)$ denote the geodesic coefficients of $F$ in the same local coordinate system. The S-curvature is defined by
\be
{\bf S}({\bf y}) := {\pa G^i\over \pa y^i}(x,y) - y^i {\pa \over \pa x^i} \Big [ \ln \sigma_F (x)\Big ],\label{Slocal}
\ee
where ${\bf y}=y^i\pxi|_x\in T_xM$.
This quantity was first introduced in \cite{Sh4} for a volume comparison theorem.
 It is proved that ${\bf S}=0$ if $F$ is a Berwald metric \cite{Sh4}.
There are many non-Berwald metrics satisfying ${\bf S}=0$. 


\bigskip

Now, we  recall the definition of 
Riemann curvature. Let $F$ be a Finsler metric on an $n$-manifold
and $G^i$ denote the geodesic coefficients of $F$.
For a vector ${\bf y}=y^i\pxi|_x\in T_xM$, define
${\bf R}_{\bf y}= R^i_{\ k}(x, y) dx^k \otimes \pxi|_x: T_xM \to T_xM$ 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^2 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
The Ricci curvature is defined by
\[
{\bf Ric} ({\bf y}):= R^i_{\ i} (x, y).
\]
In dimension two, let $x:=x^1, y:=x^2,  u: =y^1, v:= y^2$.
We can express the Ricci curvature by
\begin{eqnarray}
{\bf Ric}({\bf y}) & = & 2 \Big \{ {\pa G^1 \over \pa x} + {\pa G^2 \over \pa y} + {\pa G^1 \over \pa u} {\pa G^2 \over \pa v} 
- {\pa G^1 \over \pa v} {\pa G^2 \over \pa u} \Big \}\nonumber\\
&&  - S^2 -\Big ( u {\pa \over x} + v {\pa \over \pa y} - 2 G^1 {\pa \over \pa u} - 2 G^2 {\pa \over \pa v} \Big ) (S), \label{RicRic}
\end{eqnarray}
where $S = {\pa G^1 \over \pa u} + {\pa G^2 \over \pa v}$.

\bigskip

For a two-dimensional tangent plane $P\subset T_xM$ and a non-zero vector 
${\bf y}\in P$, define
\be
{\bf K}(P, {\bf y}):= { g_{\bf y} \Big ( {\bf R}_{\bf y}({\bf u}),\; {\bf u} \Big )\over g_{\bf y}({\bf y}, {\bf y}) g_{\bf y}({\bf u}, {\bf u})
- g_{\bf y}({\bf y}, {\bf u}) g_{\bf y} ({\bf y}, {\bf u})},\label{flag}
\ee
where $P= {\rm span}\{ {\bf y}, {\bf u}\}$.  ${\bf K}$ is called the flag curvature. 
Usually, ${\bf K}(P, {\bf y})$ depends on the direction ${\bf y}\in P$.
If $F$ is Riemannian, then ${\bf K}(P, {\bf y})$ is independent of ${\bf y}\in P$. 
The flag curvature is an analogue of the sectional curvature in Riemannian geometry. 
It is obvious that ${\bf K}=0$ if and only if ${\bf R}=0$.



The following theorem is due to Akbar-Zadeh \cite{AZ}.
\bigskip
\begin{thm}\label{thmAZ} {\rm (\cite{AZ})}
Let $(M, F)$ be a positively complete Finsler manifold with ${\bf K}=0$.
Assume that 
\[ \sup_{x\in M} \|{\bf C}\|_x <\infty, \ \ \ \ \ \sup_{x\in M} \|\tilde{\bf C}\|_x < \infty.\]
Then $F$ is locally Minkowskian.
\end{thm}

See also Theorem 10.3.7 in \cite{Sh1} for a proof.




\bigskip

Let $F$ be a Finsler metric on a manifold $M$ and $G^i (x, y)$ denote the geodesic coefficients of $F$. 
Let $\tilde{F}$ be another metric and $\tilde{G}^i(x,y)$ denote the geodesic coefficients of $\tilde{F}$. To find the relationship between
the Riemann curvature $R^i_{\ k}(x,y)dx^k \otimes \pxi $ of $F$ and the Riemann curvature $\tilde{R}^i_{\ k}(x,y) dx^k \otimes \pxi$ of $\tilde{F}$, we introduce 
\[ H^i(x, y): = G^i(x, y) - \tilde{G}^i(x,y).\]
Define
\[ H^i_{\ | k}: = {\pa H^i\over \pa x^k} + H^j {\pa^2 \tilde{G}^i\over \pa y^j \pa y^k}
- {\pa H^i\over \pa y^j} {\pa \tilde{G}^j \over \pa y^k}.\]
We have the following useful formula
\be
 R^i_{\ k} = \tilde{R}^i_{\ k} + 2 H^i_{\; |k} - y^j (H^i_{\; |j})_{y^k} + 2 H^j (H^i)_{y^jy^k} - (H^i)_{y^j}(H^j)_{y^k}.\label{RRH}
\ee
The proof is straight forward, so is omitted. See \cite{Sh1}.


\section{Shortest Time Problem}
Let $(M, F )$ be a Finsler space.
Suppose that an object on $(M, F)$ is pushed by an internal force ${\bf u}$  
with constant length,  $F({\bf u}) =1$. 
Due to the friction, the object moves on $M$ at a constant speed, but it can change direction freely.  
Without  external force acting on the object, any path of shortest time is a shortest 
path of $F$.




Now given an external force ${\bf v}$ pushing the object.  We assume  that $F(-{\bf v}) < 1$, otherwise, the object can not move forward in the direction  $-{\bf v}$.
Due to friction, the speed of the object is proportional  to
the length of the combined force 
\[ {\bf t}:= {\bf v} + {\bf u}.\]
For simplicity, we may assume that 
the speed of the object  is equal to $F({\bf t})$. 
That is, ${\bf t}:={\bf v}+{\bf u}$ is the velocity vector once the direction of the internal force is chosen.

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Express the velocity vector in the following  form
\[ {\bf t} = F({\bf t}) {\bf y},\]
where ${\bf y}$ is a unit vector with respect to $F$. Since $F({\bf u})=1$,
$F({\bf t})$ is determined by
\be
 F \Big ( F ({\bf t}) {\bf y} - {\bf v} \Big ) =F({\bf u}) = 1.\label{eqv}
\ee

Now we are going to find the Finsler metric $\tilde{F}$
such that the $\tilde{F}$-length of any  curve is equal to 
the time for which the object travels along it.

\begin{lem} Let $(M, F)$ and ${\bf v}$ be a vector field on $M$ with $F(-{\bf v}) < 1$.
 Define $\tilde{F}: TM \to [0, \infty)$ by
\be
F \Big ( {{\bf y}\over \tilde{F}({\bf y}) } - {\bf v} \Big ) =1,
\ \ \ \ \ \ {\bf y}\in T_xM\setminus\{0\}.
\label{tF}
\ee
 For any curve $C$ in $M$, the $\tilde{F}$-length of $C$ is equal to
the time for which the object travels along it.
\end{lem}
{\it Proof}: Take an arbitrary curve $C$ from $p$ to $q$,  
and a coordinate map $c: [0, T] \to C$ such that 
$c(0)=p$,  $c(T)=q$ and the velocity vector $\dot{c}(t)$ is equal to the 
combined force  at $c(t)$. 
Express $\dot{c}(t) =F(\dot{c}(t)) {\bf y}(t)$,
where ${\bf y}(t)$ is tangent to $C$ at $c(t)$ with $F({\bf y}(t))=1$.
By (\ref{eqv}),
\be 
F \Big ( F (\dot{c}(t)) {\bf y}(t) - {\bf v}(t) \Big ) =1,\label{lam1}
\ee
where ${\bf v}(t):= {\bf v}_{c(t)}$. 
Consider the following equation
\[ F\Big ( \lambda {\bf y}(t) - {\bf v}(t) \Big ) =1, \ \ \ \ \ \lambda >0.\]
Since $F(-{\bf v}(t)) < 1$, the above equation has a unique solution, that is $F(\dot{c}(t))$. By the definition of $\tilde{F}$, we obtain
\be
\tilde{F}({\bf y}(t)) = {1\over F ( \dot{c}(t) ) }. \label{lam2}
\ee
From (\ref{lam1}) and (\ref{lam2}), we obtain
\[ \tilde{F}(\dot{c}(t)) 
= F (\dot{c}(t)) \tilde{F}({\bf y}(t)) = 1.\]
This implies 
\[  T = \int_0^T \tilde{F}(\dot{c}(t)) dt.\]
Thus $\tilde{F}$ is the desired Finsler metric.
\qed
\bigskip


The relation between $F$ and $\tilde{F}$ is actually very simple. 
For a point $x\in M$, the indicatrix of $F$ at $x$ is related to 
that of $\tilde{F}$ at $x$, 
\[ \Big \{ {\bf y}\in T_xM \ | \ \tilde{F}({\bf y}) = 1 \Big \}
= \Big \{ {\bf y}\in T_xM \ | \ F({\bf y} ) = 1 \Big \} + {\bf v}_x.\]
This leads to the following 


\begin{prop}\label{lemdvdv} Let 
$(M, F)$ be a Finsler manifold and ${\bf v}$ be a vector field on $M$ with
$F(-{\bf v}) < 1$. Let  $\tilde{F}: TM \to [0, \infty)$ denote the
Finsler metric defined by (\ref{tF}).
The volume form of $F$ equals that of $\tilde{F}$, 
\[ dV_F = dV_{\tilde{F}}.\]
\end{prop}
{\it Proof}: 
Let $(x^i)$ be a local coordinate system at $x\in M$
and 
\begin{eqnarray*}
{\cal U}_F : & = & \Big \{ (y^i)\in \R^n \ | \ F\Big ( y^i\pxi|_x \Big ) < 1 \Big \},\\
{\cal U}_{\tilde{F}} : & = & \Big \{ (y^i)\in \R^n \ | \ F\Big ( y^i\pxi|_x \Big ) < 1 \Big \}.
\end{eqnarray*}
From (\ref{tF}), we have
\[ {\cal U}_{\tilde{F}} = {\cal U}_F + (v^i),\]
where ${\bf v} = v^i\pxi|_x$. 
This gives that 
\[ dV_F = {{\rm Vol}(\Bbb B^n(1))\over {\rm Vol} ({\cal U}_F )}\; dx^1\cdots dx^n = {{\rm Vol} (\Bbb B^n(1)) \over {\rm Vol} ({\cal U}_{\tilde{F}})}\; dx^1\cdots dx^n = dV_{\tilde{F}}.\]
\qed 

\bigskip


\begin{rem}
{\rm Consider the shortest time problem on a Riemannian manifold 
$(M, \alpha)$ with an external force field ${\bf v}$. 
Denote by $\langle \; , \; \rangle_{\alpha}$  the family of  inner products on tangent spaces, which are  determined by $\alpha$,
\[ \alpha({\bf y}) =\sqrt{\langle{\bf y}, {\bf y}\rangle_{\alpha}}, \ \ \ \ \ {\bf y}\in T_pM.\]
Solving the following equation for $\tilde{F}({\bf y})$,
\[\alpha \Big ( {{\bf y}\over \tilde{F}({\bf y}) } - {\bf v} \Big ) = 1,\]
we obtain
\be
 \tilde{F} ({\bf y}) =\tilde{\alpha}({\bf y}) + \tilde{\beta}({\bf y}), \ \ \ \ \ \ {\bf y}\in T_pM\label{Randersmetric}
\ee
where
\[
\tilde{\alpha}({\bf y}):=
{ \sqrt{\langle {\bf v}, {\bf y}\rangle_{\alpha}^2 + \alpha({\bf y})^2 
(1-\alpha({\bf v})^2) }\over 1 -\alpha({\bf v})^2 },
\ \ \ \ \ \tilde{\beta}({\bf y}) := - {\langle {\bf v}, {\bf y}\rangle_{\alpha}
\over 1-\alpha({\bf v})^2 }. 
\]
By Proposition \ref{lemdvdv}, we have
\[ dV_{\tilde{F}} = dV_{\alpha}.\]
}
\end{rem}

The Finsler metric $\tilde{F}$ in (\ref{Randersmetric}) is called a {\it Randers metric} in Finsler geometry. 





\bigskip


\begin{ex}{\rm
Let $\alpha$
denote the standard Euclidean metric on the unit ball $\Bbb B^n$ and
 ${\bf v}$ denote  the radial  vector field on
$\Bbb B^n$, which is given by
\[ {\bf v}_p = - (x^i), \ \ \ \ \ \ \ \ p=(x^i)\in \Bbb B^n.\]
The Randers metric  associated with $(\alpha, {\bf v})$ as defined in (\ref{Randersmetric}) is given by
\[ \tilde{F} ({\bf y}) = { \sqrt{ \langle {\bf v}, {\bf y} \rangle^2 + |{\bf y}|^2 (1-|{\bf v}|^2 ) } + \langle {\bf v}, {\bf y} \rangle \over 1- |{\bf v}|^2}, 
\ \  \ \ {\bf y}\in T_p\Bbb B^n,\]
where $|\cdot |$ and $\langle\; , \; \rangle$ denote the standard
Euclidean norm and inner product.
This is just the Funk metric on $\Bbb B^n$. 
Geodesics of $\tilde{F}$ are straight lines. Moreover, the flag curvature is a
negative constant, ${\bf K}= -1/4$. 
If an object moves away from the center, it takes infinite time to reach the boundary. However, it  takes finite time to reach the center along any shortest path.  Thus $\tilde{F}$ is positively complete, but not negatively complete.
}
\end{ex}

\begin{ex}\label{exBaSh} {\rm 
(\cite{BaSh}) Let $\Bbb S^3$ be the standard unit sphere in $\Bbb R^4$.
Let $\alpha$ denote the standard Riemannian metric on $\Bbb S^3$ and 
 ${\bf v}: = \e {\bf w}$, where $ |\e | < 1$ and ${\bf w}$ is  a left-invariant unit vector field on $\Bbb S^3$. 
The Randers metric associated with $(\alpha, {\bf v})$ as defined in (\ref{Randersmetric})
is given by
\be
\tilde{F} ({\bf y} )
= { \sqrt{ \e^2 \langle {\bf w}, {\bf y} \rangle^2_{\alpha} + (1-\e^2) \alpha ({\bf y})^2    } - \e \langle {\bf w}, {\bf y} \rangle_{\alpha} \over 1- \e^2 },
\ \ \ \ \ \ \ {\bf y}\in T_p\BBb S^3.
\ee
We have shown that ${\bf K}=1$ for any $\e$ with $|\e | < 1$. 
}
\end{ex}

Other examples will be discussed in Section 7 below  and \cite{Sh5}.
\bigskip


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

According to Theorem \ref{thmAZ}, 
to prove Theorem \ref{thm1}, it suffices to prove that the first and second Cartan torsions have uniform upper bounds. 



\bigskip

Consider a Randers metric $ F =\alpha +\beta$,
where $ \alpha =\sqrt{a_{ij}y^iy^j}$ and $\beta=b_i y^i$ 
with $ \|\beta\|_{\alpha}  =\sqrt{a^{ij}b_i b_j} < 1$.
It is proved that the first Cartan torsion of $F$ satisfies the bound
\be
 \|{\bf C}\|_x \leq {3\over \sqrt{2}} \sqrt{ 1-\sqrt{ 1-\|\beta\|^2(x)}} < {3\over \sqrt{2}}, \ \ \ \ \  \ x\in M.\label{CBound}
\ee
This is verified by B. Lackey in dimension two \cite{BaChSh}, and can be extended to higher dimensions with a
simple argument \cite{Sh3}. 

Now we are going to find an upper bound on the second Cartan torsion for Randers metrics.
First, we consider a special two-dimensional case.

Let  $p\in M$
and $\kappa:=\|\beta\|_{\alpha}(p) < 1$. There is an  orthonormal basis $\{ {\bf e}_1, {\bf e}_2 \}$ for $T_pM$ such that 
\[ F ({\bf y})=\sqrt{u^2+v^2} + \kappa u, \ \  \ \ \ \ {\bf y}= u{\bf e}_1 + v {\bf e}_2 \in T_pM.\]


\begin{lem}\label{lemsecondCartan}
Let $\kappa$ be a constant with $ 0 \leq \kappa < 1$ and
\be
F := \sqrt{u^2+v^2} + \kappa u.
\ee
The second Cartan torsion of $F$ satisfies the following bound
\[ \|\tilde{\bf C}\| \leq {27\over 2} \kappa <  {27\over 2} .\]
\end{lem}
{\it Proof}: Let ${\bf y}=  r (\cos \theta, \sin \theta) $, where $ r >0$, and ${\bf y}^{\bot}$ denote 
the vector perpendicular to ${\bf y}$ with respect to $g_{\bf y}$,
\[ g_{\bf y} ({\bf y}, {\bf y}^{\bot} ) =0, \ \ \ \ \ \
g_{\bf y} ({\bf y}^{\bot}, {\bf y}^{\bot} ) = F^2({\bf y}).\]
We have
\[ {\bf y}^{\bot}= {r\over \sqrt{ 1+\kappa \cos \theta} } ( -\sin \theta, \ \kappa + \cos\theta ).\]
By a direct computation, we obtain 
\[ \tilde{\bf C}_{\bf y} ({\bf y}^{\bot},{\bf y}^{\bot},{\bf y}^{\bot},{\bf y}^{\bot}) = F^2 ({\bf y})\Big \{ 
6 \kappa { \kappa +\cos \theta \over 1+ \kappa \cos \theta } - {15\over 2} \kappa \cos \theta \Big \}.\]
This gives
\begin{eqnarray*} \|\tilde{\bf C}\|
& = &  \max_{ 0 \leq \theta \leq 2\pi}  \Big |6 \kappa { \kappa +\cos \theta \over 1+ \kappa \cos \theta } - {15\over 2} \kappa \cos \theta \Big | \\
&  \leq & \max_{ 0 \leq \theta \leq 2\pi}\Big \{ 6 \kappa\Big | { \kappa +\cos \theta \over  1 + \kappa \cos \theta } \Big | + {15\over 2} \kappa \Big |\cos \theta \Big |\Big \}\\
& \leq & 6 \kappa + {15\over 2}\kappa = {27\over 2} \kappa .
\end{eqnarray*}
This proves the lemma. \qed


\bigskip
We can extend Lemma \ref{lemsecondCartan} to higher dimensions. 
\begin{lem}\label{lem3.3}
Let $F= \alpha +\beta$ be a Randers metric on an $n$-manifold $M$. 
The second Cartan torsion of $F$ satisfies
\[ \|\tilde{\bf C}\|\leq {27\over 2} \|\beta\|_{\alpha} < {27\over 2}.\]
\end{lem}
{\it Proof}:
At a point $p\in M$, there are two vectors ${\bf y}, {\bf u}\in T_pM$
with 
\[ F({\bf y}) =1, \ \ \ \ g_{\bf y}({\bf y}, {\bf u}) =0, \ \ \ \ g_{\bf y}({\bf u}, {\bf u}) =1\]
 such that
\[ \|\tilde{\bf C}\|_p = |\tilde{\bf C}_{\bf y} ({\bf u}, {\bf u}, {\bf u}, {\bf u})|
.\]
Let $V:={\rm span} \{ {\bf y}, {\bf u} \}$ and 
\[ \kappa := \sup_{{\bf v}\in V} {\beta({\bf v})\over \alpha({\bf v})}.\]
By Lemma \ref{lemsecondCartan}, 
\[ |\tilde{\bf C}_{\bf y} ({\bf u}, {\bf u}, {\bf u}, {\bf u})| \leq {27\over 2} \kappa.\]
Note that 
\[ \kappa =\sup_{{\bf v}\in V} {\beta({\bf v})\over \alpha({\bf v})}\leq \sup_{{\bf v}\in T_pM}{\beta({\bf v})\over \alpha({\bf v})} = : \|\beta\|_{\alpha} (p).\]
We obtain 
\[ \|\tilde{\bf C}\|_p \leq {27\over 2} \|\beta \|_{\alpha}(p).\]
\qed

\bigskip
\noindent
{\it Proof of Theorem \ref{thm1}}: By (\ref{CBound}),
we know that
$\|{\bf C}\| < 3/\sqrt{2}$. By (\ref{lem3.3}), we know that
$\|\tilde{\bf C}\| < 13.5$.  Then Theorem \ref{thm1} follows from Theorem \ref{thmAZ}.
\qed






\section{Randers Metrics with ${\bf S}=0$}


In this section, we are going to  find a sufficient and necessary condition on $\alpha$ and $\beta$ for ${\bf S}=0$.
In particular, we will show that if
$\beta$ is a Killing form of constant length, then 
${\bf S}=0$. 


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 
$\|\beta\|_x :=\sup_{y \in T_xM} \beta(y)/\alpha(y) < 1$. 

In a standard local coordinate system $(x^i, y^i)$ in $TM$,
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 
\[ r_{ij}:= {1\over 2} \Big ( b_{i|j} + b_{j|i} \Big ), \ \ \ \ 
s_{ij} := {1\over 2} \Big ( b_{i|j}-b_{j|i} \Big ),\]
\[ s^i_{\ j} =a^{ip}s_{\ pj} \ \ \ \ s_j:= b_is^i_{\ j} .\]
The geodesic  coefficients $G^i$ of $F$ are related to the geodesic coefficients $\tilde{G}^i$ of $\alpha$ by 
\be
G^i = \tilde{G}^i + P y^i + Q^i ,\label{Gi}
\ee
where 
\begin{eqnarray*}
P: & =& {1\over 2F} \Big \{ r_{ij} y^iy^j -2\alpha  s_i y^i \Big \}\\
Q^i: & = & \alpha s^i_{\ j}y^j.
\end{eqnarray*}
See \cite{AIM}.
Observe that  
\[ {\pa Q^i \over \pa  y^i} = \alpha^{-1} y_i s^i_{\ j}y^j+ \alpha s^i_{\ i} =\alpha^{-1}s_{ij}y^iy^j + \alpha a^{ij}s_{ij}=
0,\]
where $y_i :=a_{ij}y^j$.
Thus
\[ {\pa G^i\over \pa y^i} = {\pa \tilde{G}^i\over \pa y^i}  + (n+1) P.\]
Put 
\[ dV_F :=\sigma_F(x) dx^1 \cdots dx^n, \ \ \ \ dV_{\alpha} = \sigma_{\alpha}(x) dx^1\cdots dx^n.\]
 According to (\ref{Gi}), 
we have 
\[ \sigma_F = \Big ( 1 -\|\beta\|^2_{\alpha} \Big ) ^{n+1\over 2} \sigma_{\alpha}.\]
Note that 
\[ d \Big [ \ln \sigma_{\alpha} \Big ] ={\pa \tilde{G}^i\over \pa y^i}.\]
By (\ref{Slocal}), we obtain a formula for ${\bf S}$,
\be
{\bf S} = (n+1) \Big \{ P - d \Big [ \ln \sqrt{ 1-\|\beta\|^2_{\alpha} } \Big ] \Big \}. \label{SRandersS}
\ee

We have the following
\begin{prop}\label{propS=c}
Let $F=\alpha+\beta$ be a Randers metric on an $n$-manifold $M$,
where $\alpha=\sqrt{a_{ij}(x)y^iy^j}$ and $\beta =b_i(x) y^i$.
Then
\be
{\bf S}=0\label{ScF}
\ee 
 if and only if 
\be
r_{ij} +b_is_j + b_j s_i = 0.\label{eqS=0}
\ee
\end{prop}
{\it Proof}: 
For the sake of simplicity, we
choose an orthonormal basis for $T_xM$ such that $a_{ij}=\delta_{ij}$.
Let
\[ \rho := \ln\sqrt{1-\|\beta\|_{\alpha}^2 }.\]
and
$d\rho =\rho_i dx^i$, i.e.,
\be
\rho_i = - { b_j b_{j|i}\over 1-\|\beta\|_{\alpha}^2 }.\label{rhoi}
\ee
By (\ref{SRandersS}), 
${\bf S}=0$ if and only if  
\be
r_{ij} y^i y^j - 2\alpha s_i y^i = 2 (\alpha+\beta) \rho_i y^i.
\label{eqS=0*}
\ee
(\ref{eqS=0*}) is equivalent to 
the following 
equations
\begin{eqnarray}
&& r_{ij} = b_j \rho_i + b_i \rho_j  \label{eqrij}\\
&& -s_i = \rho_i \label{eqspl}
\end{eqnarray}

First we assume that 
${\bf S}=0$. Then (\ref{eqrij}) and (\ref{eqspl}) hold.
Plugging (\ref{eqspl}) into (\ref{eqrij}) gives (\ref{eqS=0}).

Now we assume that (\ref{eqS=0}) holds. 
Note that $s_jb_j = b_i s_{ij} b_j =0$. 
Contracting (\ref{eqS=0}) with $b_j$ yields
\be
 b_j r_{ij} =  - \|\beta\|^2_{\alpha} s_i , \label{eqsss}
\ee
that is,
\[ b_j b_{i|j} + b_j b_{j|i} =- \|\beta \|^2 (b_j b_{j|i} - b_j b_{i|j} ).\]
We obtain
\be
 b_j b_{i|j} =  - {1+\|\beta\|^2_{\alpha} 
\over 1-\|\beta\|^2_{\alpha} } \; b_j b_{j|i}.\label{bijij}
\ee
It follows from  (\ref{rhoi}) and (\ref{bijij}) that
\be
 s_i =  {1\over 2} \Big ( b_j b_{j|i}
+ {1+\|\beta\|^2_{\alpha} 
\over 1-\|\beta\|^2_{\alpha} } \; b_j b_{j|i} \Big ) =  - \rho_i.
\label{s_i}
\ee
Thus
\begin{eqnarray*}
r_{ij} y^iy^j - 2\alpha s_i y^i & = &
 - 2 \beta s_i y^i - 2\alpha s_i y^i\\
& = & - 2 (\alpha+\beta ) s_i y^i \\
& = &  2 F \rho_i y^i.
\end{eqnarray*}
We obtain
\[
{\bf S}= (n+1) \Big \{\rho_i y^i - \rho_i y^i \Big \}
= 0.
\]
This gives (\ref{ScF}).\qed


\section{Randers Metrics with ${\bf K}=0$ and ${\bf S}=0$}

In this section, we are going to compute the Riemann curvature
of a Randers metric satisfying ${\bf S}=0$.



Let $F=\alpha+\beta$ be a Randers metric on a manifold $M$. 
Denote by $b_{i|j}$, $b_{i|j|k}$, etc the coefficients of the covariant derivatives of $\beta$ with respect to $\alpha$. Set
$r_{ij}:= (b_{i|j}+b_{j|i})/2$, $s_{ij}:=(b_{i|j}-b_{j|i})/2$, $s^i_{\ j}=a^{ik}s_{kj}$ 
and $s_j := b_is^i_{\ j}$. Further,  we set $s_0:=s_py^p, s^i_{\ 0}:=s^i_{\ p} y^p $, $s_{0|j}=s_{p|j}y^p$  and $s_{0|0}=s_{p|q}y^py^q$, etc.
Denote the Riemann curvature of $\alpha$ by 
$\tilde{R}^i_{\ k} dx^k \otimes \pxi$. We have the following 


\begin{thm}\label{thmK=0S=0}
Let $F = \alpha +\beta$ be a Randers metric satisfying ${\bf S}=0$. Then 
${\bf K}=0$   if and only if the following two equations hold,
\begin{eqnarray}
\bar{R}^i_{\ k} & = & -\Big ( s_{0|0} \delta^i_k -s_{0|k} y^i \Big )
- \Big ( s_{k|0}-s_{0|k} \Big )y^i \nonumber\\
& & - s_0\Big ( s_0\delta^i_k- s_k y^i\Big ) +\Big (\alpha^2 s^i_{\ j}s^j_{\ k}-  s^i_{\ j}s^j_{\ 0} y_k \Big ) -3 s_{k0} s^i_{\ 0}, \label{K=0C}\\
0 & = & s_js^j_{\ 0} \Big ( \alpha^2 \delta^i_k - y_k y^i \Big )+
\alpha^2 \Big ( s_j s^j_{\ 0} \delta^i_k - s_j s^j_{\ k} y^i \Big )\nonumber\\
&& + \alpha^2 \Big ( s^i_{\ k|0}-s^i_{\ 0|k} \Big )
- \Big ( \alpha^2 s^i_{\ 0|k}-s^i_{\ 0|0} y_k  \Big ). \label{K=0B} 
\end{eqnarray}
\end{thm}
{\it Proof}:
By assumption ${\bf S}=0$, we obtain from (\ref{SRandersS}) and (\ref{eqspl}), we obtain
\[ P= - s_0.\] Thus
$ G^i = \tilde{G}^i + H^i,$
where 
\[ H^i = -s_0 y^i + \alpha s^i_{\ 0}.\]
By a direct computation, we obtain
\begin{eqnarray*}
H^i_{\; |k} & = & -s_{0|k} y^i +\alpha s^i_{\ 0|k}  \\
\Big ( H^i_{\; |j} \Big )_{y^k} & = & -s_{k|j} y^i -s_{0|j} \delta^i_k  + \alpha^{-1} y_k s^i_{\ 0|j}  + \alpha s^i_{\ k|j}\\
y^j \Big ( H^i_{\; |j} \Big )_{y^k}
& = & -s_{k|0} y^i -s_{0|0} \delta_k^i   + \alpha^{-1} y_k s^i_{\ 0|0} +\alpha s^i_{\ k|0} \\
\Big (H^i\Big )_{y^j} & = & 
-s_j y^i -s_0\delta^i_j 
+ \alpha^{-1} y_j s^i_{\ p}y^p +\alpha s^i_{\ j}\\
\Big ( H^i\Big )_{y^jy^k} & = & -s_j \delta^i_k -s_k \delta^i_j 
+\alpha^{-3}(a_{jk}\alpha^2 - y_j y_k) s^i_{\ 0} 
 + \alpha^{-1}( y_j s^i_{\ k} 
+  y_k s^i_{\ j}),
\end{eqnarray*}
$y_k :=a_{jk}y^j$.
Plugging them into (\ref{RRH}), we obtain
\begin{eqnarray}
R^i_{\ k} &=& \bar{R}^i_{\ k}+\Big ( s_{0|0} \delta^i_k -s_{0|k} y^i \Big )
+ \Big ( s_{k|0}-s_{0|k} \Big )y^i
\nonumber\\
&&+ s_0\Big ( s_0\delta^i_k- s_k y^i\Big ) -\Big (\alpha^2 s^i_{\ j}s^j_{\ k}-  s^i_{\ j}s^j_{\ 0} y_k \Big ) +3 s_{k0} s^i_{\ 0}\nonumber\\
&& - \Big \{ s_js^j_{\ 0} \Big ( \alpha^2 \delta^i_k - y_k y^i \Big )
+\alpha^2 \Big ( s_j s^j_{\ 0} \delta^i_k - s_j s^j_{\ k} y^i \Big )\nonumber\\
&& +\alpha^2 \Big ( s^i_{\ k|0}-s^i_{\ 0|k} \Big )
- \Big ( \alpha^2 s^i_{\ 0|k}-s^i_{\ 0|0} y_k  \Big ) \Big \}\alpha^{-1}
\label{localRik}
\end{eqnarray}

According to (\ref{localRik}), the coefficients of the Riemann curvature
of $F$ are in the following form
\[ R^i_{\ k} = A + B\alpha^{-1} ,\]
where $A$ and $B$ are polynomials of $y^i$ at each point $x\in M$. 
Thus $R^i_{\ k}=0$ if and only if $A=0$ and $B=0$.
This proves Theorem \ref{thmK=0S=0}.
\qed


\bigskip


Taking the trace of $R^i_{\; k}$ in (\ref{localRik}), we obtain 
\begin{eqnarray}
{\bf Ric} & = & \overline{\bf Ric} 
+ (n-1) \Big \{s_{0|0} + s_0s_0 \Big \} + 2 s_{k0}s^k_{\ 0} -\alpha^2 s^k_{\ j} s^j_{\ k} \nonumber \\
&& + 2\Big \{  s^k_{\ 0|k} - (n-1)  s_js^j_{\ 0}   \Big \}\alpha\label{Ricik}
\end{eqnarray}

By (\ref{Ricik}), we immediately obtain the following

\begin{prop}
Let $F = \alpha+\beta$ be a Randers metric with ${\bf S}=0$.
Then $F$ is of zero Ricci curvature,
${\bf Ric} = 0$, 
 if and only if 
\be
s^k_{\ 0|k} = (n-1) s_j s^j_{\ 0}\ee
\be
\overline{\bf Ric}
=- (n-1) \Big (s_{0|0}+s_0s_0 \Big )
+\alpha^2 s^k_{\ j}s^j_{\ k}- 2 s_{k0}s^k_{\ 0} .
\ee
\end{prop}


\bigskip

We do not know if there is a three-dimensional Randers metric
satisfying ${\bf Ric}=0, \ {\bf S}=0$ and ${\bf K}\not=0$.
Such examples, if exist, are of interest. 


\section{Examples}

In this section, we will construct two interesting examples, one in dimension two and the other in dimension three. Both metrics satisfy 
that ${\bf S} =0$ and ${\bf K}=0$. Yet, they are not locally projectively flat. More precisely, they are not Douglas metrics. 

\bigskip
\noindent{\bf Examples in Dimension two}:
Let $\alpha=\sqrt{u^2+v^2}$ denote the standard Euclidean metric on $\Bbb R^2$.
Take a vector field ${\bf v}$ on the unit disk $\Bbb D^2$ given by
\[ {\bf v} =(-y, x), \ \ \ \ \ \ p=(x, y) \in \Bbb D^2.\]
The Finsler metric associated with $(\alpha, {\bf v})$ is a Randers metric $F=\tilde{\alpha} +\tilde{\beta}$, where
\begin{eqnarray*}
 \tilde{\alpha} :& = &  {\sqrt{ (-yu+xv)^2  + (u^2+v^2) (1-x^2-y^2) } \over 1-x^2-y^2},\\
\tilde{\beta} :& = &- {-yu+xv\over 1- x^2-y^2 }.
\end{eqnarray*}


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\[a_{11}= {1-x^2\over \Big (1-x^2-y^2\Big )^2},
\ \ \ \ a_{12} = - {xy\over \Big ( 1-x^2-y^2\Big )^2} = a_{21},
\ \ \ \ a_{22} = { 1-y^2\over \Big (1-x^2-y^2\Big )^2 }\]
\[ b_1 = {y\over 1-x^2-y^2}, \ \ \ \ b_2 = - {x\over 1-x^2-y^2}.\]
The geodesic coefficients $\tilde{G}^1$ and $\tilde{G}^2$ of $\tilde{\alpha}$ are given by
\begin{eqnarray*}
\tilde{G}^1 & = & -{x(u^2+v^2)\over 2(1-x^2-y^2)} 
- { y(xu+yv)-v \over 1-x^2-y^2} 
\;\tilde{\beta} + {xu+yv\over 1-x^2-y^2} \; u 
\\
\tilde{G}^2 & = & - { y(u^2+v^2)\over 2(1-x^2-y^2)} 
+ {x(xu+yv)-u \over 1-x^2-y^2} \; \tilde{\beta} + {xu+yv\over 1-x^2-y^2} \;  v .
\end{eqnarray*}
The Gauss curvature $\tilde{\bf K}$ of $\tilde{\alpha}$ is given by
\[ \tilde{\bf K} =-{5+x^2+y^2\over 1-x^2-y^2}    .\]
By a direct computation, we obtain
\begin{eqnarray*}
r_{11} & = & - {2xy\over \Big (1-x^2-y^2\Big)^2}\\
r_{12} & = & {x^2-y^2\over \Big (1-x^2-y^2\Big )^2 } =r_{21}\\
r_{22} & = & {2xy\over \Big ( 1-x^2-y^2 \Big )^2 }.
\end{eqnarray*}
\begin{eqnarray*}
s_{11} & = & 0\\
s_{12} & = & {1\over \Big (1-x^2-y^2\Big )^2 } = - s_{21}\\
s_{22} & = & 0.
\end{eqnarray*}
From the above equations, we obtain the following formulas for $s_i:=b_r a^{rh}s_{hi}$.
\begin{eqnarray*}
s_1 & = & { x\over 1-x^2-y^2}\\
s_2 & = & { y\over 1-x^2-y^2}
\end{eqnarray*}
We immediately see that the following identity holds.
\[ r_{ij}= - b_is_j - b_j s_i.\]
By Proposition \ref{propS=c}, we conclude that ${\bf S}=0$.

By the above equations, we obtain the following formulas for 
$H^i := -s_j y^j y^i + \tilde{\alpha} a^{ir}s_{rl} y^l$:
\begin{eqnarray*}
H^1 & = & - { xu+yv\over 1-x^2-y^2} u - { y (xu+yv)-v \over 1-x^2-y^2} \tilde{\alpha} \\
H^2 & = & - {xu+yv\over 1-x^2-y^2} v + 
{ x (xu+yv)-u\over 1-x^2-y^2} \tilde{\alpha}.
\end{eqnarray*}
Then we obtain the following formulas for the geodesic coefficients $G^1=\tilde{G}^1+H^1$ and $G^2=\tilde{G}^2+ H^2$, 
\begin{eqnarray}
G^1 & = & - {x(u^2+v^2)\over 2( 1-x^2-y^2)}
- { y(xu+yv)- v \over 1-x^2-y^2} F \label{G1local}\\
G^2 & = & - { y (u^2+v^2) \over 2(1-x^2-y^2)} 
+ { x(xu+yv)-u \over 1-x^2-y^2} F. \label{G2local} 
\end{eqnarray}

One can easily verify that $G^1$ and $G^2$ satisfy 
\be
 S:={\pa G^1\over\pa  u} + {\pa G^2 \over\pa  v} =0.\label{GuHv}
\ee
By Proposition \ref{lemdvdv}, we know that the area form of $F$ is equal to the Euclidean area form 
\[dA_{F} =dA_{\alpha}= dx dy.\]
 By (\ref{Slocal}),  we conclude that
 ${\bf S}=0$ again.

A direct computation yields
\be
{\pa G^1 \over \pa x} + {\pa G^2\over \pa y} + {\pa G^1 \over \pa u}
{\pa G^2 \over \pa v} - {\pa G^1 \over \pa v} {\pa G^2 \over \pa u} =0.
\label{GHGH}
\ee
Plugging (\ref{GuHv}) and (\ref{GHGH}) into (\ref{RicRic}), we obtain that
${\bf Ric}=0$, hence ${\bf K}=0$.



\bigskip
One can also plug
the formulas of $G^1$ and $G^2$  in (\ref{G1local}) and (\ref{G2local}) directly into (\ref{Rik}) to verify 
that  $R^1_{\ 1}=0, R^1_{\ 2}=0, R^2_{\ 1}=0$ and $R^2_{\ 2}=0$. 



\bigskip
\noindent{\bf Examples in Dimension Three or Higher}:

We first construct a Randers metric in dimension three. One can easily extend it to higher dimensions.
Let $\alpha :=\sqrt{u^2+v^2+w^2}$ denote the canonical Euclidean metric on 
$\Bbb R^3$. Take a vector ${\bf v}$ in the cylinder $\Omega:= \{ (x, y, z) \ |  \ x^2+y^2 < 1 \}$ given by
\[ {\bf v}_p : = (-y, x, 0), \ \ \ \ \  \ p=(x,y, z)\in \Omega.\]
We consider the shortest time problem under the influence of ${\bf v}$.
Image that the water in a cylindrical fish tank is rotating around the central axis, and the fishes in  the tank see the food hanging near  the water surface. Each fish has to figure out the path of  the shortest time to reach the food. The path of  shortest time 
is the shortest path of the Randers metric,  $F =\tilde{\alpha} +\tilde{\beta}$, where 
\begin{eqnarray*}
\tilde{\alpha} :& = &  { \sqrt{ (-yu+xv)^2 + (u^2+v^2+w^2)(1-x^2-y^2) }\over 1-x^2-y^2 },\\
\tilde{\beta}: & = & - {-yu+xv
\over 1-x^2-y^2 }.\\
\end{eqnarray*}






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By a similar argument, we obtain the following formulas for the geodesic coefficients $G^i$ of $F$,
\begin{eqnarray*}
G^1 & = & - {x(u^2+v^2+w^2)\over 2( 1-x^2-y^2)}
- { y(xu+yv)- v \over 1-x^2-y^2} F\\
G^2 & = & - { y (u^2+v^2+w^2) \over 2(1-x^2-y^2)} 
+ { x(xu+yv)-u \over 1-x^2-y^2} F\\
G^3 & = & 0.
\end{eqnarray*}


First, one can easily verify  that
\[ {\pa G^1 \over \pa u} + {\pa G^2\over \pa v} + {\pa G^3\over \pa w}=0.\]
By Proposition \ref{lemdvdv}, the volume form of $dV_F$ is given by
\[dV_F = dV_{\alpha} = dxdydz.\] 
By (\ref{Slocal}), we conclude that 
${\bf S}=0$.


Plugging the above formulas into (\ref{Rik}), we obtain 
$R^i_{\ k}=0$. Thus ${\bf K}=0$.

Certainly, one can extend the above Randers metric to higher dimensions with a slight modification. The formulas for  geodesic coefficients and the volume form remain same. 
Thus ${\bf S}=0$.  

Since $G^3=0$, from (\ref{Rik}), we immediately see that $R^3_{\ 1}=0, R^3_{\ 2}=0$ and $R^3_{\ 3}=0$. 
Plugging the formulas of $G^1, G^2, G^3$ into (\ref{Rik}), one can verify  that 
$R^1_{\ 1}=0, R^1_{\ 2}=0, R^1_{\ 3}=0, R^2_{\ 1}=0, R^2_{\ 2} =0$ and $R^2_{\ 3}=0$. 




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

\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, revised in June, 2001.

\bibitem[Sh3]{Sh3} Z. Shen, {\it On R-quadratic Finsler spaces},
Publicationes Math. Debrecen, {\bf 58} (2001), 263-274

\bibitem[Sh4]{Sh4} Z. Shen, {\it  Volume comparison and its applications in Riemann-Finsler geometry}, Advances in Math. {\bf 128}(1997), 306-328.


\bibitem[Sh5]{Sh5} Z. Shen, {\it Two-dimensional Finsler metrics with constant  curvature}, preprint, July, 2001.

\bibitem[Sh6]{Sh6} Z. Shen, {\it Projectively flat Randers metrics with constant curvature}, preprint, June 2001.





\bibitem[SSAY]{SSAY}
C. Shibata, H. Shimada, M. Azuma and H. Yasuda,
{\it On Finsler spaces with Randers' metric}, Tensor, N.S. 
{\bf 31}(1977), 219-226.


\bibitem[Sz]{Sz} Z.I. Szab\'{o}, {\it Positive definite Berwald spaces
(structure theorems on Berwald spaces)}, Tensor, N.S. {\bf 35}(1981), 25-39.


\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}


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                     \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

math0109060.extracted_bib
\begin{thebibliography}{AnBnCng3}

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\bibitem[SSAY]{SSAY}
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