Torsion-free $H$-structures on almost Abelian solvmanifolds
Marco Freibert
TL;DR
The paper develops a general linear-algebraic framework to characterize when almost Abelian solvmanifolds $\Gamma\backslash G$ admit torsion-free $H$-structures for broad classes of $H\le\mathrm{GL}(n,\mathbb{R})$. It introduces the subspace $\mathcal{F}_{\mathfrak{h}}\subseteq\mathrm{End}(\mathbb{R}^{n-1})$ as the image of a linear map attached to $H$-connections, and shows $\tilde{\mathfrak{k}}_{\mathfrak{h}}\subseteq\mathcal{F}_{\mathfrak{h}}$, with equality in many cases yielding left-invariant flatness as the torsion-free condition. The work systematically analyzes subalgebras commuting with a fixed endomorphism, complex vs totally real vs hyperparacomplex vs unitary cases, and subalgebras with special first prolongations $\mathcal{K}_{\mathfrak{h}}^{(1)}$, providing explicit descriptions of $\mathcal{F}_{\mathfrak{h}}$ in representative settings. Across these sections, the authors recover and extend known characterizations for structures such as complex, Kähler, complex-symplectic, hypercomplex, hyperkähler, and G2-related geometries, and they unify conditions under which torsion-free implies left-invariant flatness. The results enable a broad, unified criterion to identify torsion-free invariant geometries on almost Abelian solvmanifolds and illuminate when flatness and torsion-freeness coincide for large classes of $H$.
Abstract
In this article, we provide a general set-up for arbitrary linear Lie groups $H\leq \mathrm{GL}(n,\mathbb{R})$ which allows to characterise the almost Abelian Lie algebras admitting a torsion-free $H$-structure. In more concrete terms, using that an $n$-dimensional almost Abelian Lie algebra $\mathfrak{g}=\mathfrak{g}_f$ is fully determined by an endomorphism $f$ of $\mathbb{R}^{n-1}$, we give a description of the subspace $\mathcal{F}_{\mathfrak{h}}$ of all $f\in\mathrm{End}(\mathbb{R}^{n-1})$ for which $\mathfrak{g}_f$ admits a ``special'' torsion-free $H$-structure in terms of the image of a certain linear map. For large classes of linear Lie groups $H$, we are able to explicitly compute $\mathcal{F}_{\mathfrak{h}}$ and so give characterisations of the almost Abelian Lie algebras admitting a torsion-free $H$-structure. Our results reprove all the known characterisations of the almost Abelian Lie algebras admitting a torsion-free $H$-structure for different single linear Lie groups $H$ and extends them to big classes of linear Lie groups $H$. For example, we are able to provide characterisations in the case $n=2m$, $H\leq \mathrm{GL}(m,\mathbb{C})$ and $H$ either being a complex Lie group or being totally real, or in the case that $H$ preserves a pseudo-Riemannian metric. In many cases, we show that the space $\mathcal{F}_{\mathfrak{h}}$ coincides with what we call the \emph{characteristic subalgebra} $\tilde{\mathfrak{k}}_{\mathfrak{h}}$ associated to $\mathfrak{h}$, and that then the torsion-free condition is equivalent to the left-invariant flatness condition. In particular, we prove this to be the case if $H$ is a complex linear Lie group or if $\mathfrak{h}$ does not contain any elements of rank one or two and is either metric or totally real.