论文标题
$ sl（2，\ mathbb {r}）$上的sub-riemannian Geodesics $
Sub-Riemannian Geodesics on $SL(2, \mathbb{R})$
论文作者
论文摘要
我们明确地描述了在$ sl（2，\ mathbb {r}）$上定义的椭圆类型的子里曼尼亚式结构的长度最小化的大地测量学结构。我们的方法使用对称性减少，将问题转化为二维商空间上的Riemannian问题，在该空间上可以很容易地在其上可视化地理物的投影。作为副产品，我们获得了在\ cite {boscarossi}中获得的cut-locus表征的替代推导。我们使用分类结果对谎言组上的三维右不变次 - 里工结构进行\ cite {agbd}，\ cite {biggs}，\ cite {hb2}，以确定我们的结果适用于我们的结果。
We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus obtained in \cite{BoscaRossi}. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups \cite{AGBD}, \cite{Biggs}, \cite{HB2} to identify exactly automorphic structures on which our results apply.
