Flats and symmetry of homogeneous spaces (Q1568203)

In a symmetric space of rank \(k\) every geodesic is contained in a flat, totally geodesic, connected, closed, \(k\)-dimensional submanifold, a so-called \(k\)-flat. Furthermore, the isometry group acts transitively on the set of pairs \((p,\Sigma)\), where \(\Sigma\) is a \(k\)-flat and \(p\in\Sigma\). A connected Riemannian manifold \(M\) with these properties is called \(k\)-flat homogeneous. We prove that \(k\)-flat homogeneity characterizes the symmetric spaces.