On geodesic orbit nilmanifolds (Q6596125)

A Riemannian manifold \((M,g)\) is called a manifold with homogeneous geodesics or a geodesic orbit manifold (or shortly \(GO\)-manifold) if any geodesic \(\gamma\) on \(M\) is an orbit of a \(1\)-parameter subgroup of the full isometry group of \((M,g)\). The author studies Riemannian geodesic orbit metrics on nilpotent Lie groups when these ones can be constructed by using nilpotent Lie algebras equipped with suitable inner products. He considers some natural examples of \(GO\)-nilmanifolds and \(GO\)-nilmanifolds of the centralizer type. To show that the set of nilpotent Lie groups admitting Riemannian geodesic orbit metrics is quite extensive, he constructs special families of nilpotent Lie groups. First, he constructs a \(1\)-parameter family of pairwise non-isomorphic connected and simply connected \(10\)-dimensional nilpotent Lie groups \(N_t\) such that each of them admits a \(3\)-parameter family of Riemannian geodesic orbit metrics. The generalization of this result is given for a \(k\)-family of pairwise non-isomorphic connected and simply connected nilpotent Lie groups of dimension \(4k+6\) such that each of them admits a \(3\)-parameter family of Riemannian geodesic orbit metrics.