A simply connected nilpotent Lie group with a closed geodesic

Glen Wheeler

A simply connected nilpotent Lie group with a closed geodesic

Differential Geometry 2026-07-06 v1

Authors: Glen Wheeler

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 B\"ohm--Lafuente's question asking whether a homogeneous Riemannian manifold diffeomorphic to Euclidean space can have a closed geodesic.

Keywords

nilpotent Lie group closed geodesic left-invariant metric homogeneous Riemannian manifold

Cite

@article{arxiv.2607.04678,
  title  = {A simply connected nilpotent Lie group with a closed geodesic},
  author = {Glen Wheeler},
  journal= {arXiv preprint arXiv:2607.04678},
  year   = {2026}
}

Comments

5 pages