Minimal Geodesics and Nilpotent Fundamental Groups

TL;DR

This paper extends the construction of Riemannian metrics with rare minimal geodesics from n-tori to all nilpotent fundamental groups, demonstrating the optimality of Bangert's existence results.

Contribution

It introduces new examples of Riemannian metrics on manifolds with nilpotent fundamental groups where minimal geodesics are scarce, generalizing previous work on tori.

Findings

Constructed metrics on nilpotent fundamental groups with few minimal geodesics

Showed Bangert's results are optimal for these groups

Extended known examples from tori to broader class of manifolds

Abstract

Hedlund constructed Riemannian metrics on n-tori, $n \geq 3$ for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics are optimal for nilpotent fundamental groups.