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](http://link.springer.com/article/10.1007/s002090050507) 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 $k > 1$. 



k-flats in homogeneous spaces
If the k-flats are compact, then the space must be symmetric
(Molina-Olmos, J. Differential geometry
45 (1997) 575-592; see also Proc. Amer. Math. Soc. 129 (2001), 3701-3709).


Homogeneous spaces (non-symmetric and irreducible) with the property that every geodesic is contained in a k-flat ($k\geq 2$) can be constructed as follows 
(due to Ernst Heintze): choose a simple compact Lie group $G$ and a compact 
subgroup $H$ such that $\mathrm{rank} (G)\geq \mathrm{dim} (H) +2$. Then $G/H$, with the normal hogeneous metric is a desired example (see also Spatzier-Strake, Comment. Math. Helv. 65 (1990) 299-317). Observe that the projection from $G$, with the bi-invariant metric, onto $G/H$, with the normal homogeneous metric, is a Riemannian submersion.}


(The known fact that $G/H$ is irreducible, as a Riemannian manifold, can be found in Olmos-Reggiani-Tamaru, Math.Z. 277 (2014), 611-628). 


