Closed G$_2$-structures with a transitive reductive group of automorphisms

TL;DR

The article completely classifies seven-dimensional manifolds with a closed non-parallel ${\mathrm G}_2$-structure admitting a transitive reductive group of automorphisms. The authors show that such a group has a one-dimensional center and that the manifold splits as a product of a flat factor and a non-compact homogeneous six-manifold with an invariant strictly symplectic half-flat ${\mathrm SU}(3)$-structure, realized as a coadjoint orbit with the induced ${\mathrm G}_2$-structure. The proof proceeds by ruling out all simple and semisimple-not-simple Lie-algebra cases, and in the non-semisimple case, derives the ${\mathrm SU}(3)$-structure on a six-dimensional orbit and identifies the possible groups as ${\mathrm SO}(4,1)$ or ${\mathrm SU}(2,1)$. This yields a complete geometric and algebraic description, linking the ${\mathrm G}_2$-structure to six-dimensional symplectic-half-flat data on a homogeneous space. The results connect the global ${\mathrm G}_2$-geometry to rigid six-dimensional SU(3) structures and provide a precise non-compact classification in terms of coadjoint orbits.

Abstract

We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel G$_2$-structure and admitting a transitive reductive group G of automorphisms. In particular, we show that the center of G 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 SU(3)-structure.

Theorems & Definitions (16)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof