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

Isolda Cardoso, Ana Cosgaya, Silvio Reggiani

TL;DR

This work classifies left-invariant metrics on 6-dimensional characteristically solvable triangular (CSLAT) nilmanifolds, determining the full automorphism groups, describing the moduli space of metrics up to isometric automorphism via a submanifold $\Sigma\subset T_6^+$, and computing the full isometry groups. It reveals that, for most CSLATs, every left-invariant metric has trivial index of symmetry, providing the first examples of Lie groups with no left-invariant metric of positive index, while identifying five distinguished algebras with nontrivial symmetry index and explicit distributions. The paper further connects to nilsoliton metrics, proving that a nilsoliton on a CSLAT has positive index of symmetry whenever the underlying algebra admits nontrivial symmetry, and supplies concrete diagonal models consistent with Will's nilsoliton classification. All results are supported by extensive SageMath computations, with tables and notebooks documenting the automorphism groups, moduli descriptions, and isotropy data, offering a comprehensive framework for the geometry of left-invariant metrics on low-dimensional nilpotent Lie groups.

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.

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

TL;DR

This work classifies left-invariant metrics on 6-dimensional characteristically solvable triangular (CSLAT) nilmanifolds, determining the full automorphism groups, describing the moduli space of metrics up to isometric automorphism via a submanifold , and computing the full isometry groups. It reveals that, for most CSLATs, every left-invariant metric has trivial index of symmetry, providing the first examples of Lie groups with no left-invariant metric of positive index, while identifying five distinguished algebras with nontrivial symmetry index and explicit distributions. The paper further connects to nilsoliton metrics, proving that a nilsoliton on a CSLAT has positive index of symmetry whenever the underlying algebra admits nontrivial symmetry, and supplies concrete diagonal models consistent with Will's nilsoliton classification. All results are supported by extensive SageMath computations, with tables and notebooks documenting the automorphism groups, moduli descriptions, and isotropy data, offering a comprehensive framework for the geometry of left-invariant metrics on low-dimensional nilpotent Lie groups.

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 -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.
Paper Structure (8 sections, 13 theorems, 50 equations, 9 tables)

This paper contains 8 sections, 13 theorems, 50 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Characteristically solvable Lie algebras
  3. The full automorphism group of a CSLAT
  4. The moduli space of left-invariant metrics for CSLAT
  5. The full isometry group of a CSLAT
  6. The index of symmetry for CSLAT
  7. Application to nilsoliton metrics on CSLAT
  8. Tables

Key Result

Proposition 2.1

Let $\mathfrak h$ be a real nilpotent Lie algebra of dimension $6$. Then $\mathfrak h$ is a CSLA if and only if $\mathfrak h$ is isomorphic to one of the following Lie algebras: Moreover, each of the above CSLAs is a CSLAT except for $\mathfrak h_{15}$, $\mathfrak h_{19}^-$ and $\mathfrak h_{26}^+$.

Theorems & Definitions (34)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 4.1
  • proof
  • Example 4.2
  • ...and 24 more