On the automorphism group of a closed G$_2$-structure
Fabio Podestà, Alberto Raffero
TL;DR
This work analyzes the automorphism group of a compact 7-manifold $M$ endowed with a closed non-parallel ${\\mathrm G}_2$-structure $\\varphi$. By examining the Lie algebra ${\\mathfrak g}=\\{X\\in\\mathfrak{X}(M):\\mathcal{L}_X\\varphi=0\\}$ and its action on $M$, the authors embed ${\\mathfrak g}$ into the space of harmonic $2$-forms via $X\\mapsto\\iota_X\\varphi$, establishing $\\dim({\\mathfrak g})\\le b_2(M)$ and proving ${\\mathfrak g}$ is abelian; they further show $\\dim({\\mathfrak g})\\le 6$ unless $M$ is the flat $7$-torus. Isotropy considerations imply $\\dim({\\mathfrak g}_p)\\le 2$ and that the action is free for $\\dim({\\mathfrak g})\\ge 5$, which together yield a negative answer to the existence of compact homogeneous manifolds with invariant closed non-parallel $\\varphi$. The paper also discusses examples and contrasts with non-compact homogeneous cases, highlighting sharpness of the bounds and connections to related geometric structures.
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.