A review of compact geodesic orbit manifolds and the g.o. condition for $\SU(5)/\s(\U(2)\times \U(2))$

TL;DR

This paper surveys recent advances in the classification of compact geodesic orbit manifolds and investigates the g.o. condition specifically for the space SU(5)/S(U(2)U(2)).

Contribution

It reviews recent results on compact g.o. manifolds and analyzes the g.o. condition for a specific homogeneous space, contributing to the classification problem.

Findings

Summarizes recent progress in classifying compact g.o. manifolds.

Examines the g.o. condition for SU(5)/S(U(2)U(2)).

Identifies open questions in the classification of g.o. manifolds.

Abstract

Geodesic orbit manifolds (or g.o. manifolds) are those Riemannian manifolds $(M,g)$ whose geodesics are integral curves of Killing vector fields. Equivalently, there exists a Lie group $G$ of isometries of $(M,g)$ such that any geodesic $\gamma$ has the simple form $\gamma(t)=e^{tX}\cdot p$, where $e$ denotes the exponential map on $G$. The classification of g.o. manifolds is a longstanding problem in Riemannian geometry. In this brief survey, we present some recent results and open questions on the subject focusing on the compact case. In addition we study the geodesic orbit condition for the space $\SU(5)/\s(\U(2)\times \U(2))$.