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

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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