arXiv research

Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable struc…

Proves local solvability for $G_2$-structures with Poisson equations.

A Levi-Malcev type decomposition for $2$-step solvable Lie algebras with a complex structure

Study on complex curves in hypercomplex nilmanifolds with quaternionic-solvable Lie algebras.

Counterexample found for Stein property of certain solvable Lie groups.

Method finds explicit solutions to certain PDEs.

Study finds closed G2-structures on non-solvable Lie groups.

Study on solvable Lie groups with specific Weyl connections.

A compact solvmanifold of completely solvable type, i.e. a compact quotient of a completely solvable Lie group by a lattice, has a Kähler structure if and only if it is a complex torus. We show more in general that a compact solvmanifold $M$ of completely solvable type endowed with an invariant complex structure $J$ ad…

Classifies two-step solvable Lie groups with SKT structures.

Classifies solvable symplectic Lie algebras via extensions and proves structural theorems.

A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable g…

Study SKT and Kähler structures on specific Lie algebras.

Study classifies LC Kahler structures on 4D solv Lie alg, with applications.

The paper classifies isometries on specific Lie groups.

New examples of Laplacian solitons found on solvable Lie groups.

The paper constructs Einstein Sasaki metrics on solvable Lie groups.

Study on symplectic Lie algebras with specific dimensions.

We consider the problem of computing the integrable sub-distributions of the non-integrable Vessiot distribution of multi-dimensional second order partial differential equations (PDEs). We use Vessiot theory and solvable structures to find the largest integrable distributions contained in the Vessiot distribution assoc…

Characterizes non-degenerate cyclic metric Lie algebras.

In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.

We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-…

Classifies Lie algebras actions on 3D spaces, focusing on solvable groups.

It is the aim of this work to study product structures on four dimensional solvable Lie algebras. We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal. These results are applied to the case of Manin triples and complex product structures. We also analyze the thr…

Shear construction builds solvable Lie groups from Abelian ones.

The paper solves Monge-Ampère equations on reflexive polytopes, linking solvability to SYZ conjecture and tropical geometry.

Study solvable Lie algebras linked to graph structures.

The paper constructs Levi flat structures using structure sheaves and differential complexes.

We study the subelliptic heat kernels of the CR three dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three dimensional solvable Lie groups and obtain representations of these groups. We give expressions for the heat kernels on these groups and obtain heat semigroup gr…

Proves conjecture about compatible SKT and balanced metrics on compact solvmanifolds.

Shear construction builds solvable Lie algebras from \(\mathbb{R}^n\).

Study Weyl-Einstein structures on conformal solvmanifolds, proving Einstein property and classifying metrics.

Let $ω_\mathfrak{g}$ be a Lie algebra valued differential $1$-form on a manifold $M$ satisfying the structure equations $d ω_\mathfrak{g} + \frac{1}{2} ω_\mathfrak{g}\wedge ω_\mathfrak{g}=0$ where $\mathfrak{g}$ is solvable. We show that the problem of finding a smooth map $ρ:M\to G$, where $G$ is an $n$-dimensional so…

Paper constructs solutions for a class of overdetermined systems.

Flat Hermitian Lie algebras are always Kähler.

Study LCP structures on solvmanifolds, complete list up to 5 dimensions.

In the present paper we study six dimensional solvable Lie algebras with special emphasis on those admitting a symplectic structure. We list all the symplectic structures that they admit and we compute their Betti numbers finding some properties about the codimension of the nilradical. Next, we consider the conjecture …

Study on G2-structures on solvmanifolds, focusing on Laplacian solitons and Ricci pinching.

Study compact symplectic solvmanifolds' hard Lefschetz property.

We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integra…

Characterizes 0-solvable links and their properties.

This work builds on the foundation laid by Gordon and Wilson in the study of isometry groups of solvmanifolds, i.e. Riemannian manifolds admitting a transitive solvable group of isometries. We restrict ourselves to a natural class of solvable Lie groups called almost completely solvable; this class includes the complet…

The present work is devoted to compact completely solvable solvmanifolds which admit Kahlerian metrics whose Kahler forms are homogeneous. In particular, we show that such manifolds are diffeomorphic to flat tori. Our proof is based on Dynkin diagrams associated to left invariant closed 2-forms in completely solvable L…

A solvmanifold is a compact differentiable manifold M on which a connected solvable Lie group G acts transitively. As the main result, we will see, applying a result of Arapura and Nori on solvable Kaehler groups and some of the author's previous results, that a compact solvmanifold admits a Kaehler structure if and on…

Paper solves Hessian equations on Kähler manifolds.

We study a type of left-invariant structure on Lie groups, or equivalently on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting g^*=V_1 + V_2, and the vanishing of certain associate…

New solitons found for G$_2$-Laplacian flow on Lie groups.

Characterizes connections on normal distributions manifold.