Related papers: $G_2$-structures on flat solvmanifolds

We study how a gluing construction, which produces compact manifolds with holonomy G_2 from matching pairs of asymptotically cylindrical G_2-manifolds, behaves under deformations. We show that the gluing construction defines a smooth map…

This is a survey paper. We explain the known constructions for two geometrically different classes of examples of compact Riemannian 7-manifolds with holonomy G2. One method uses resolutions of singularities of appropriately chosen…

Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G_2-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow…

A flat solvmanifold is a compact quotient $\Gamma\backslash G$ where $G$ is a simply-connected solvable Lie group endowed with a flat left invariant metric and $\Gamma$ is a lattice of $G$. Any such Lie group can be written as…

These notes give an informal and leisurely introduction to $\mathrm{G}_2$ geometry for beginners. A special emphasis is placed on understanding the special linear algebraic structure in $7$ dimensions that is the pointwise model for…

Interest in Riemannian manifolds with holonomy equal to the exceptional Lie group $\mathrm{G}_2$ have spurred extensive research in geometric flows of $\mathrm{G}_2$-structures defined on seven-dimensional manifolds in recent years. Among…

We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G_2 structures is a smooth manifold (if non-empty), and study some of its local properties. We also…

This article is concerned with the study of the holonomy group of flat solvmanifolds. It is known that the holonomy group of a flat solvmanifold is abelian; we give an elementary proof of this fact and moreover we prove that any finite…

Based on a general formula due to R.Bryant, we work out the topological structure of the space of torsion-free $G_2$-structures generating the same associated Riemannian metric on a compact $7$-manifold. We also identify a corresponding Lie…

This paper is a review of current developments in the study of moduli spaces of G2 manifolds. G2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G2. Although they are odd-dimensional, in many ways they can be…

We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G_2 can admit non-integrable G_2 structures of type R + S^2_0(R^7) + R^7 in the sense of Fern\'andez and Gray. This relies on the construction…

We give a new construction of compact Riemannian 7-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $\iota$ preserving the…

This article classifies closed G2-structures such that the induced metric is conformally flat. It is shown that any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples. In particular, it…

G2-manifolds with a cohomogeneity-one action of a compact Lie group G are studied. For G simple, all solutions with holonomy G2 and weak holonomy G2 are classified. The holonomy G2 solutions are necessarily Ricci-flat and there is a…

Let G be one of the Ricci-flat holonomy groups SU(n), Sp(n), Spin(7) or G_2, and M a compact manifold of dimension 2n, 4n, 8 or 7, respectively. We prove that the natural map from the moduli space of torsion-free G-structures on M to the…

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…

We classify four-dimensional compact solvmanifolds up to diffeomorphism, while determining which of them have complex analytic structures. In particular, we shall see that a four-dimensional compact solvmanifold S can be written, up to…

There are two well-known parabolic split $G_2$-geometries in dimension five, $(2,3,5)$-distributions and $G_2$-contact structures. Here we link these two geometries with yet another $G_2$-related contact structure, which lives on a…

We construct new compact manifolds endowed with closed $\mathrm{G}_2$ structures that satisfy the topological properties found by Joyce and Baraglia for the existence of a torsion-free $\mathrm{G}_2$ structure in the same cohomology class.…

We consider the existence of cohomogeneity one solitons for the isometric flow of $G_2$-structures on the following classes of torsion-free $G_2$-manifolds: the Euclidean $R^7$ with its standard $G_2$-structure, metric cylinders over…