标题： 测地线轨道度量在紧致李群$G_2$上 显示英文标题

标题： Geodesic orbit metrics on the compact Lie group $G_2$

摘要： 黎曼流形上的测地线是局部长度最小化的曲线，但通过简单函数进行显式描述通常是不可能的。 最简单的测地线形式，如欧几里得空间中的直线和球面中的大圆，通常作为等距变换的一参数群在李群作用下的轨道出现。 所有测地线都是这些轨道的流形称为测地线轨道流形（或g.o.流形），对其理解和分类在黎曼几何中有着相当长且持续的历史。 在本文中，我们利用一类称为（弱）正则的李子群的出色表示理论行为，对紧致李群$G_2$上的左不变g.o.度量进行分类。 我们期望这里讨论的主要工具和见解将有助于进一步分类g.o.李群，特别是低秩的情况。 显示英文摘要

摘要： Geodesics on Riemannian manifolds are precisely the locally length-minimizing curves, but their explicit description via simple functions is rarely possible. Geodesics of the simplest form, such as lines on Euclidean space and great circles on the round sphere, usually arise as orbits of one-parameter groups of isometries via Lie group actions. Manifolds where all geodesics are such orbits are called geodesic orbit manifolds (or g.o. manifolds), and their understanding and classification spans a quite long and continuous history in Riemannian geometry. In this paper, we classify the left-invariant g.o. metrics on the compact Lie group $G_2$, using the nice representation theoretic behaviour of a class of Lie subgroups called (weakly) regular. We expect that the main tools and insights discussed here will facilitate further classifications of g.o. Lie groups, particularly of lower ranks.