Asian Journal of Mathematics

Volume 25 (2021)

Number 6

Closed $\mathrm{G}_2$-structures with a transitive reductive group of automorphisms

Pages: 897 – 910



Fabio Podestà (Dipartimento di Matematica e Informatica, Università degli Studi di Firenze, Italy)

Alberto Raffero (Dipartimento di Matematica, Università degli Studi di Torino, Italy)


We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel $\mathrm{G}_2$-structure and admitting a transitive reductive group $\mathrm{G}$ of automorphisms. In particular, we show that the center of $\mathrm{G}_2$ is one-dimensional and the manifold is the Riemannian product of a flat factor and a non-compact homogeneous six-dimensional manifold endowed with an invariant strictly symplectic half-flat $\mathrm{SU}(3)$-structure.


closed $\mathrm{G}_2$-structure, automorphism

2010 Mathematics Subject Classification

53C10, 57S20

Dedicated to Dmitri V. Alekseevsky on the occasion of his 80th birthday

The authors were supported by GNSAGA of INdAM and by the project PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and Holomorphic Dynamics”.

Received 2 July 2020

Accepted 16 August 2021

Published 24 October 2022