A simply connected nilpotent Lie group with a closed geodesic

Abstract

We give a six-dimensional simply connected nilpotent Lie group together with a left-invariant Riemannian metric admitting a nonconstant closed geodesic. The example gives a positive answer to Böhm--Lafuente's question asking whether a homogeneous Riemannian manifold diffeomorphic to Euclidean space can have a closed geodesic.