# Hilbert第四问题与射影平坦流形的分类

#### Hilbert度量[1]

$d_\Omega(x,y)=\frac{1}{2}\log[a,x,y,b]=\frac{1}{2}\log\frac{|y-a||x-b|}{|x-a||y-b|},\quad x,y\in\Omega,\: a,b\in\pt\Omega.$

$(\Omega,d_\Omega)=\begin{cases} \text{Minkowski geometry},&\Omega 中心对称\\ \text{Lobachevskii geometry},&\Omega是椭球\\ \text{hyperbolic geometry(Klein模型)},&\Omega是单位球B^n(1). \end{cases}$

Hilbert第四问题是说:

#### 局部射影平坦流形的分类

• $\alpha$是射影平坦的黎曼度量, 而$\beta$关于$\alpha$平行;
• 当常旗曲率$K<0$时, $F=\sqrt{\alpha^2+k\beta^2}+\eps\beta$, 这里$\eps\neq0$, $k$都是常数;
• 当常旗曲率$K=0$时, $F=\frac{\left(\sqrt{\alpha^2+k\beta^2}+\eps\beta\right)^2}{\sqrt{\alpha^2+k\beta^2}}$, 这里$\eps\neq0$, $k$都是常数.

$F(Ax,Ay)=F(x,y),\quad\forall x\in B^n(r),\quad \forall y\in T_xB^n(r),\quad \forall A\in O(n),$

$F(x,y)=|y|\phi\left(|x|,\langle x,y\rangle/|y|\right),\quad x,y\in TB^n(r)\setminus\set{0}.$

Hamel4在1903年给出了$\R^n$中局部凸区域上射影平坦的Finsler度量的刻画:

$F_{x^jy^i}y^j=F_{x^i},$

$\frac{\pt}{\pt y^i}\left(\frac{\pt{F}}{\pt x^j}y^j\right)=\frac{\pt F}{\pt x^i}.$

$s\phi_{bs}+b\phi_{ss}-\phi_s=0.$

• 当 $K>0$时, $F$是Bryant度量, 且其参数满足$p_1=p_2=\cdots=p_{n+1}$;
• 当$K=0$是, $F$是Berwald度量.

$F=\frac{1}{2}\left(\theta_c(x,y)-\eps\theta_c(\eps x,y)\right),\quad\eps<1,$

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3. R. Bryant, Some remarks on Finsler manifolds with constant flag curvature, Houston J. of Math. 28(2002), 221-262.
4. G. Hamel,Über die Geometrieen in denen die Geraden die Kürzestensind, Math. Ann. 57(1903), 231-264.
5. Zhou, Linfeng. “Projective spherically symmetric Finsler metrics with constant flag curvature in R n.” Geometriae Dedicata 158.1 (2012): 353-364.