On homogeneous geodesics and weakly symmetric spaces

被引：6

作者：

Berestovskii, Valerii Nikolaevich ^{[1
,2
]}

Nikonorov, Yurii Gennadievich ^{[3
]}

机构：

[1] Russian Acad Sci, Siberian Branch, Sobolev Inst Math, Acad Koptyug Ave 4, Novosibirsk 630090, Russia

[2] Novosibirsk State Univ, Mech Math Dept, Pirogov Str 1, Novosibirsk 630090, Russia

[3] Russian Acad Sci, Southern Math Inst, Vladikavkaz Sci Ctr, Markus Str 22, Vladikavkaz 362027, Russia

来源：

ANNALS OF GLOBAL ANALYSIS AND GEOMETRY | 2019年 / 55卷 / 03期

关键词：

Geodesic orbit Riemannian space; Homogeneous Riemannian manifold; Homogeneous space; Quadratic mapping; Totally geodesic torus; Weakly symmetric space;

D O I：

10.1007/s10455-018-9641-1

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

In this paper, we establish a sufficient condition for a geodesic in a Riemannian manifold to be homogeneous, i.e. an orbit of an 1-parameter isometry group. As an application of this result, we provide a new proof of the fact that every weakly symmetric space is a geodesic orbit manifold, i.e. all its geodesics are homogeneous. We also study general properties of homogeneous geodesics, in particular, the structure of the closure of a given homogeneous geodesic. We present several examples where this closure is a torus of dimension 2 which is (respectively, is not) totally geodesic in the ambient manifold. Finally, we discuss homogeneous geodesics in Lie groups supplied with left-invariant Riemannian metrics.

引用

页码：575 / 589

页数：15

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