Related papers: Some geometric vector fields on $5$-dimensional $2…
We give a geometric classification of complex $n$-dimensional $2$-step nilpotent (all, commutative and anticommutative) algebras. Namely, has been found the number of irreducible components and their dimensions. As a corollary, we have a…
We study geometric aspects of horizontal 2-plane distributions on the complement of the zero section in the 5-dimensional total space of a rank-3 vector bundle equipped with connection over a surface. We show that any surface in…
We classify the $5$-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 2 of 3) classifies those in which the linear isotropy representation is either irreducible or trivial. The $5$-dimensional geometries…
We provide a self contained, elementary, and geometrically-flavored classification of $8$-dimensional $2$-step nilpotent Lie algebras over algebraically closed fields of characteristic $\ne 2,3$, using the algebro-geometric arguments from…
The aim of this work is the study of symplectic structures on 2-step nilmanifolds. We concentrate in the closeness condition, proving that the existence of a closed 2-form of type II is necessary to get a symplectic structure. In low…
A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected…
The aim of this article is to study the existence of invariant SKT structures on nilmanifolds. More precisely, we give a negative answer to the question of whether there exist a $k$-step ($k>2$) complex nilmanifold admitting an invariant…
The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields…
The paper is devoted to give a complete classification of five-dimension nilpotent evolution algebras over an algebraically closed field. We obtained a list of 27 isolated non-isomorphic nilpotent evolution algebras and 2 families of…
We study left-invariant Killing forms of arbitrary degree on simply connected $2-$step nilpotent Lie groups endowed with left-invariant Riemannian metrics, and classify them when the center of the group is at most two-dimensional.
We investigate harmonic unit vector fields with totally geodesic integral curves on 3-manifolds. Under mild curvature assumptions, we classify both the vector fields and the manifolds that support them. Our results are inspired by…
In this note we study contact structures on 5-dimensional manifolds. We give a complete answer under the assumption that the Abundance conjecture holds in dimension 5.
In this paper we present new examples of homogeneous 2-nondegenerate CR-manifolds of dimension 5 and give, up to local CR-equivalence, a full classification of all CR-manifolds of this type.
A vector field on a Riemannian manifold is called geodesic if its integral curves are reparametrized geodesics. We classify compact K\"ahler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields that satisfy an…
In this work, we consider degenerations between 8-dimensional 2-step nilpotent Lie algebras over $\mathbb{C}$ and obtain the geometric classification of the variety $\mathcal{N}^2_8$.
We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine, twelve, fifteen and eighteen degree.
In this paper we study contact structure on 2-step nilpotent, Heisenberg type Lie groups. We decompose this Lie groups to center and orthogonal complement, then investigate properties of both orthogonal Lie subgroups. Finally, we provide a…
We show that holomorphic vector fields on (C^3,0) have separatrices provided that they are embedded in a rank 2 representation of a two-dimensional Lie algebra. In turn, this result enables us to show that the second jet of a holomorphic…
This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous…
In this paper we consider linear systems of $\mathbb{P}^2$ with all but one of the base points of multiplicity 5. We give an explicit way to evaluate the dimensions of such systems.