Geodesic orbit metrics on the compact Lie group G₂

2025/07/15 by Nikolaos Panagiotis Souris, Souris, Nikolaos Panagiotis
Mathematics · Physics and Astronomy · #53C25 (Primary) #53C30 (Secondary) #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2507.11125

openalex publication_date 2025/07/15 · openalex created_date 2025/10/08 · openalex updated_date 2026/07/28

Abstract

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 G₂, 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.

Citations

• Geodesic orbit spaces of compact Lie groups of rank two
• On a class of geodesic orbit spaces with abelian isotropy subgroup
• Subalgebras of the rank two semisimple Lie algebras
• Invariant Einstein metrics on three-locally-symmetric spaces
• Clifford-Wolf homogeneous Riemannian manifolds
• Naturally reductive metrics and Einstein metrics on compact Lie groups

Related

• Geodesic Orbit Metrics on Compact Simple Lie Groups arising from Generalized Flag Manifolds
• Geodesic orbit spaces of compact Lie groups of rank two
• Geodesic orbit spaces in real flag manifolds
• On a class of geodesic orbit spaces with abelian isotropy subgroup
• The Riemannian geometry of orbit spaces. The metric, geodesics, and integrable systems