The symmetry of 2-flat homogeneous spaces (Q1357629)

Riemannian symmetric spaces of rank \(k\) enjoy the following property: Any geodesic lies in a \(k\)-dimensional totally geodesic flat subspace (so-called \(k\)-flat), and any two pointed \(k\)-flats are congruent under the isometry group. In [\textit{E. Heintze, R. Palais, C.-L. Terng} and \textit{G. Thorbergsson}, J. Reine Angew. Math. 454, 163-179 (1994; Zbl 0804.53074)], this property was called \(k\)-flat homogeneity, and it was shown that this property characterizes symmetric spaces: there are no compact \(k\)-flat homogeneous spaces other than rank-\(k\) symmetric spaces. A classification free proof was given later by \textit{J.-H. Eschenburg} and \textit{C. Olmos} [Comment. Math. Helv. 69, 483-499 (1994; Zbl 0822.53032)]. It was conjectured that the same statement is true without the compactness assumption. For \(k = 1\) this is the classical characterization of 2-point homogeneous spaces due to H.-C. Wang and J. Tits. In the present paper, the case \(k = 2\) is proven. Similar to the compact case, the main idea is to show that the action of the isometry group \(G\) on the full tangent bundle with its Sasaki or connection metric is polar, i.e., it has a totally geodesic submanifold (namely any tangent space of a \(2\)-flat) which intersects all \(G\)-orbits perpendicularly. Two main facts make the proof work: One may assume that the the isotropy representation is reducible (a noncompact isotropy irreducible space is known to be symmetric). Then the 2-flat homogeneity forces the isotropy action to be polar. Furthermore, one needs the fact that any transitive group of isometries on a 2-flat contains the translation group (which is no longer true for higher dimension). Recently, the same author was able to prove this statement for general \(k\) [Thesis, University of Augsburg].