On the automorphism group of a closed G$_2$-structure

TL;DR

This paper investigates the automorphism group of compact 7-manifolds with closed non-parallel G$_2$-structures, establishing bounds on its dimension and implications for the existence of certain homogeneous manifolds.

Contribution

It provides new bounds on the automorphism group's dimension and shows that compact homogeneous manifolds with invariant closed non-parallel G$_2$-structures cannot exist.

Findings

The identity component of the automorphism group is abelian.

Dimension of the automorphism group is bounded by min{6, b_2(M)}.

No compact homogeneous manifold admits an invariant closed non-parallel G$_2$-structure.

Abstract

We study the automorphism group of a compact 7-manifold $M$ endowed with a closed non-parallel G$_2$-structure, showing that its identity component is abelian with dimension bounded by min$\{6,b_2(M)\}$. This implies the non-existence of compact homogeneous manifolds endowed with an invariant closed non-parallel G$_2$-structure. We also discuss some relevant examples.