Homogeneous geodesics in solvable Lie groups (Q1882125)

Let \((G/H,g)\) be a (connected) homogeneous Riemannian manifold. A homogeneous geodesic through the origin of \(G/H\) is by definition a geodesic which is an orbit of a one-parameter subgroup of \(G\). The second author and \textit{J. Szenthe} [Geom. Dedicata 81, No. 1--3, 209--214 (2000; Zbl 0980.53061); erratum ibid. 84, No. 1--3, 331--332 (2001)] proved that any homogeneous Riemannian manifold admits at least one homogeneous geodesic. In the paper under review, the authors consider the class of solvable Lie groups \(M^{2n+1}\) equipped with left-invariant metrics defined by \textit{M. Bozek} [Čas. Pešt. Mat. 105, 368--384 (1980; Zbl 0475.53045)]. They prove that each \(M^{2n+1}\) admits \((2n+1)\)-linearly independent homogeneous geodesics and the maximal number of mutually orthogonal homogeneous geodesics is (1) \(n+1\) in case \(n+1\) is odd; (2) \(n+2\) in case \(n+1\) is even but not divisible by \(4\); (3) \(2n+1\) in case \(n+1\) is divisible by \(4\) and there exists a Hadamard matrix of order \(n+1\).