Concerning a conjecture of Taketomi-Tamaru

TL;DR

This paper investigates the geometric properties of 2-step nilpotent Lie groups, providing a counterexample to a conjecture by Taketomi-Tamaru by analyzing orbit congruences and Ricci soliton metrics.

Contribution

It offers a counterexample to the local version of Taketomi-Tamaru's conjecture and studies orbit congruences in non-exceptional 2-step nilpotent Lie groups.

Findings

Orbits of $\\mathbb R^{>0}\times Aut_0$ are congruent in certain settings

Counterexample to the local version of Taketomi-Tamaru's conjecture

Analysis of Ricci soliton metrics in non-exceptional cases

Abstract

We study the setting of 2-step nilpotent Lie groups in the particular case that its type (p,q) is not exceptional. We demonstrate that, generically, the orbits of $\mathbb R^{>0}\times Aut_0$ in $GL(n)/O(n)$ are congruent even when a Ricci soliton metric does exists. In doing so, we provide a counterexample to the local version of a conjecture of Taketomi-Tamaru.