On homogeneous geodesics and weakly symmetric spaces
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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 \(\ge 2\) 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.
KeywordsGeodesic orbit Riemannian space Homogeneous Riemannian manifold Homogeneous space Quadratic mapping Totally geodesic torus Weakly symmetric space
Mathematics Subject Classification53C20 53C25 53C35
The first author was supported by the Ministry of Education and Science of the Russian Federation (Grant 1.3087.2017/4.6). The authors are indebted to Prof. Andreas Arvanitoyeorgos for helpful discussions concerning this paper. The authors are grateful to the referee for helpful comments and suggestions that improved the presentation of this paper.
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