The moduli space of left-invariant metrics on six-dimensional characteristically solvable nilmanifolds

TL;DR

This paper explicitly characterizes the moduli space of left-invariant metrics on certain 6-dimensional nilmanifolds with characteristically solvable Lie algebras, and explores their symmetry properties and isometry groups.

Contribution

It provides the first explicit description of the moduli space for these nilmanifolds and analyzes their symmetry indices, including examples without positive symmetry index.

Findings

Identified the moduli space of metrics up to isometry for these nilmanifolds.

Computed the full isometry groups for each metric.

Discovered examples of Lie groups lacking metrics with positive symmetry index.

Abstract

A real Lie algebra is said to be characteristically solvable if its derivation algebra is solvable. We explicitly determine the moduli space of left-invariant metrics, up to isometric automorphism, for $6$-dimensional nilmanifolds whose associated Lie algebra is characteristically solvable of triangular type. We also compute the corresponding full isometry groups. For each left-invariant metric on these nilmanifolds we compute the index and distribution of symmetry. In particular, we find the first known examples of Lie groups which do not admit a left-invariant metric with positive index of symmetry. As an application we study the index of symmetry of nilsoliton metrics. We prove that nilsoliton metrics detect the existence of left-invariant metrics with positive index of symmetry.