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Exact G$_2$-structures on compact quotients of Lie groups

Anna Fino, Lucía Martín-Merchán, Alberto Raffero

TL;DR

The paper proves that compact quotients of a seven-dimensional simply connected Lie group by a cocompact lattice do not admit invariant exact $G_2$-structures. It reduces the problem to Lie algebra level via the Chevalley–Eilenberg differential and treats separately non-solvable and solvable cases. For the non-solvable case, all closed $G_2$-structures fail to be exact under invariance, either by a lack of definiteness in the defining map $b_ heta$ or by lattice obstructions; for the solvable case, Boc’s classification and an SU(3)-structure analysis show that no seven-dimensional strongly unimodular solvable Lie algebra can support an invariant exact $G_2$-structure. The results extend previous special-case nonexistence results and establish a broad negative answer to the existence of invariant exact $G_2$-structures on compact quotients of Lie groups, supported by explicit computations (via Maple) across decomposable and indecomposable six-dimensional ideals.

Abstract

We show that the compact quotient $Γ\backslash\mathrm{G}$ of a seven-dimensional simply connected Lie group $\mathrm{G}$ by a co-compact discrete subgroup $Γ\subset\mathrm{G}$ does not admit any exact $\mathrm{G}_2$-structure which is induced by a left-invariant one on $\mathrm{G}$.

Exact G$_2$-structures on compact quotients of Lie groups

TL;DR

The paper proves that compact quotients of a seven-dimensional simply connected Lie group by a cocompact lattice do not admit invariant exact -structures. It reduces the problem to Lie algebra level via the Chevalley–Eilenberg differential and treats separately non-solvable and solvable cases. For the non-solvable case, all closed -structures fail to be exact under invariance, either by a lack of definiteness in the defining map or by lattice obstructions; for the solvable case, Boc’s classification and an SU(3)-structure analysis show that no seven-dimensional strongly unimodular solvable Lie algebra can support an invariant exact -structure. The results extend previous special-case nonexistence results and establish a broad negative answer to the existence of invariant exact -structures on compact quotients of Lie groups, supported by explicit computations (via Maple) across decomposable and indecomposable six-dimensional ideals.

Abstract

We show that the compact quotient of a seven-dimensional simply connected Lie group by a co-compact discrete subgroup does not admit any exact -structure which is induced by a left-invariant one on .

Paper Structure

This paper contains 5 sections, 13 theorems, 88 equations, 1 table.

Table of Contents

  1. Introduction
  2. The non-solvable case
  3. The solvable case
  4. The decomposable case
  5. The indecomposable case

Key Result

Theorem 1.1

A potential compact $7$-manifold $M$ with an exact ${\mathrm G}_2$-structure $\varphi$ cannot be of the form $M=\Gamma\backslash{\mathrm G}$, where ${\mathrm G}$ is a seven-dimensional simply connected Lie group, $\Gamma\subset{\mathrm G}$ is a cocompact discrete subgroup, and the exact ${\mathrm G}

Theorems & Definitions (28)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • ...and 18 more