arXiv research

Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.

The paper classifies invariant structures on complex almost Abelian groups.

Study of complex and Hermitian structures on specific Lie groups.

Study on CKY forms on almost abelian Lie groups, proving parallelism and characterizing non-parallel cases.

The paper constructs a new Ricci-flat metric on almost abelian Lie groups.

The paper studies a flow on complex Lie groups, showing convergence to solitons.

Characterizes hypercomplex Lie groups and their solvmanifolds.

Study locally conformally balanced metrics on specific Lie algebras.

New examples show non-abelian fundamental groups for positive Ricci curvature manifolds.

Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.

For certain manifolds, nonnegative Ricci curvature limits dimension and forces almost abelian fundamental group.

Characterizes almost Abelian Lie algebras with special $H$-structures.

We find explicit solutions of the Laplacian coflow of $G_2-$structures on seven-dimensional almost-abelian Lie groups. Moreover, we construct new examples of solitons for the Laplacian coflow which are not eigenforms of the Laplacian and we exhibit a solution, which is not a soliton, having a bounded interval of existe…

The paper classifies hypercomplex Lie algebras and solvmanifolds using quaternionic Jordan form.

Characterizes symmetric Killing tensors on specific Lie groups.

We study the existence of lattices in almost abelian Lie groups that admit left invariant locally conformal Kähler or locally conformal symplectic structures in order to obtain compact solvmanifolds equipped with these geometric structures. In the former case, we show that such lattices exist only in dimension $4$, whi…

Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.

The note confirms a conjecture for specific Lie groups.

New representation theory for surface groups to SO0(2,3).

We construct lattices on six dimensional not completely solvable almost abelian Lie groups, for which the Mostow condition does not hold. For the corresponding compact quotients, we compute the de Rham cohomology (which does not agree in general with the Lie algebra one) and a minimal model. We show that some of these …

Study pseudo-Kähler structures on almost abelian solvmanifolds.

Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.

We prove that a fundamental group of codimension one nonnegative Ricci curvature C2-foliation of a closed Riemannian manifold is finitely generated and almost abelian, i.e. it contains abelian subgroup of finite index. In particular, we confirm the Milnor conjecture for manifolds which are leaves of codimension one non…

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…

Classifies complex symplectic structures on Lie algebras with large abelian ideals.

The study examines weakly Einstein Lie groups and proves non-existence for certain types.

Unique complex structures on specific Lie algebras.

Study of presymplectic forms on almost abelian Lie algebras, determining moduli spaces and their finiteness.

Study characterizes $G_2$-structures on specific Lie groups and identifies harmonic conditions.

Characterizes almost abelian Lie algebras with integrable complex structure

This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.

We study the cohomology properties of the singular foliation $\F$ determined by an action $Φ\colon G \times M\to M$ where the abelian Lie group $G$ preserves a riemannian metric on the compact manifold $M$. More precisely, we prove that the basic intersection cohomology $\lau{\IH}{*}{\per{p}}{\mf}$ is finite dimensiona…

Classifies two-step solvable Lie groups with SKT structures.

Classifies algebraic concordance for almost classical knots.

We study the spinor flow on homogeneous spin manifolds. After providing the general setup we discuss the homogeneous spinor flow in dimension 3 and on almost abelian Lie groups in detail. As a further example the flag manifold in dimension 6 is treated.

Given a non compact semisimple Lie group $G$ we describe all homogeneous spaces $G/L$ carrying an invariant almost Kähler structure $(ω,J)$. When $L$ is abelian and $G$ is of classical type, we classify all such spaces which are Chern-Einstein, i.e. which satisfy $ρ= λω$ for some $λ\in\mathbb R$, where $ρ$ is the Ricci…

We apply the general Ansatz in geometric flows on homogeneous spaces proposed by Jorge Lauret for the Laplacian co-flow of invariant $G_2$-structures on a Lie group, finding an explicit soliton on a particular almost Abelian $7$-manifold.

A Theorem of Kirichenko states that the torsion 3-form of the characteristic connection of a nearly Kähler manifold is parallel. On the other side, any almost hermitian manifold of type $\mathrm{G}_1$ admits a unique connection with totally skew symmetric torsion. In dimension six, we generalize Kirichenko's Theorem an…

In this paper we study some properties of almost abelian solvmanifolds using minimal models associated to a fibration. In particular we state a necessary and sufficient condition to formality and a method for finding symplectic strucures of this kind of solvmanifolds.

Abstract classifies Lie algebras with complex or symplectic structures.

The paper proves a conjecture about the positivity of the Euler characteristic for complex projective manifolds.

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at infinity is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff…

The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.

Prove long-time existence of pluriclosed flow on certain fibrations

We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manif…

The paper shows that certain manifolds with nonnegative Ricci curvature have finitely generated fundamental groups.

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

In this article, we determine the seven-dimensional almost Abelian Lie algebras which admit calibrated or parallel G_2-/G_2^*-structures. Along the way, we show that certain well-established curvature restrictions for calibrated and parallel G_2-structures are not valid in the G_2^* case. In more detail, we provide the…