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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 $k > 1$.

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    $\begingroup$ I think the product of two rank 1 spaces will have the property (if the geodesics in the factors are closed subsets), but need not be symmetric. $\endgroup$ – Dave Witte Morris Jul 19 '16 at 2:39
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    $\begingroup$ If you assume, in addition that your manifold has nonpositive curvature, then the celebrated 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
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    $\begingroup$ The product of two Riemannian manifolds has this property: each geodesic projects to a geodesic or point in each factor, so lies in a product of geodesics. (David Witte Morris is not quite right: the product of symmetric spaces is symmetric.) $\endgroup$ – Ben McKay Jul 19 '16 at 13:13
  • $\begingroup$ @BenMcKay. Yes, thanks. I can then add the product of Zoll spheres between themselves and with compact symmetric spaces to the compact symmetric spaces of rank > 1 to my list (I really want tori). $\endgroup$ – alvarezpaiva Jul 19 '16 at 19:56
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    $\begingroup$ Perhaps this can be useful: ams.org/journals/proc/2001-129-12/S0002-9939-01-06008-7/… $\endgroup$ – Holonomia Jul 22 '16 at 18:21
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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).

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