Homogeneous geodesics in homogeneous Riemannian manifolds -- examples (Q2735665)

A connected Riemannian manifold \((M,\langle , \rangle)\) is said to be homogeneous if its full isometry group \(I(M)\) acts transitively on \(M\). In this case \(M\) can always be written in the form \(M=G/H\) where \(G\subset I(M)\) is a connected Lie group acting transitively and effectively on \(M\) and \(H\) is the isotropy subgroup at some point \(0\in M\). The author considers two problems:NEWLINENEWLINENEWLINEProblem 1. Let \((M,\langle , \rangle)=G/H\) be a homogeneous Riemannian manifold where \(G\) denotes the largest connected group of isometries and \(\dim M\geq 3\). Does \(M\) always admit more than one homogeneous geodesic, up to a reparametrization?NEWLINENEWLINENEWLINEProblem 2. Suppose that \((M,\langle , \rangle)= G/H\) admits \(m=\dim M\) linearly independent homogeneous geodesics through 0. Does it admit mutually orthogonal homogeneous geodesics?NEWLINENEWLINENEWLINEIn this note the authors shows that the answers to both problems are negative.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00038].