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

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 $\SU(5)/\s(\U(2)\times \U(2))$.