In a symmetric space of rank $k$ (and I'll take $k > 1$) *every* geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$.

**Question.** Are there non-symmetric homogeneous spaces that share this property?

In this paper the author shows that if we also require that the isometry group act transitively on the set of pairs $(p, \Sigma)$, where $\Sigma$ is a flat and $p$ is a point in it, then the space is symmetric.

My main interest is having many examples, *homogeneous or not*, of compact
Riemannian manifolds for which every geodesic is contained in a totally geodesic, flat torus of dimension $k > 1$.

rank rigidity theorem(Ballmann-Brin-Eberlein, Burns-Spatzier) states that the universal cover of your manifold splits isometrically as a product of symmetric spaces and manifolds of negative curvature. I am not sure what happens if the curvature is allowed to be positive. $\endgroup$ – Moishe Kohan Jul 19 '16 at 4:53