Geodesic orbit metrics on the compact Lie group $G_2$

TL;DR

This paper classifies all left-invariant geodesic orbit metrics on the compact Lie group G_2, advancing understanding of geodesic structures on special Lie groups through representation theory.

Contribution

It provides a complete classification of g.o. metrics on G_2 using representation theoretic methods, aiding future classifications of g.o. Lie groups.

Findings

All left-invariant g.o. metrics on G_2 are classified.

Representation theory of regular subgroups is key to the classification.

Results facilitate further studies on g.o. structures in Lie groups.

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 $G_2$, 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.