Homogeneous Riemannian almost P-manifolds (Q803961)

A Riemannian homogeneous space is called a P-manifold if all its geodesics have a positive common period, and an almost P-manifold if all proper totally geodesic submanifolds of dimension \(\geq 2\) are isometric to a P-manifold. The following theorem is proved. Let \(G\) be a compact Lie group and \(M=G/H\) a simply connected P-manifold which is not isometric to a compact Lie group endowed with a left-invariant metric. If \(\dim M\) is even then \(M\) is isometric to a symmetric space of rank 1, while for odd \(\dim M\) we have \(\operatorname{rk}G=\operatorname{rk}H+1\). The proof makes use of the following fact: for any homogeneous Riemannian manifold \(G/H\), the submanifold \(C(T)/C(T)\cap H\), where \(C(T)\) is the centralizer of a maximal torus \(T\) of \(H\), is totally geodesic in \(M\).