Maria Laura Barberis

National University of Córdoba | UNC · Faculty of Mathematics, Astronomy and Physics (FAMAF)

We give a characterization of almost abelian Lie groups carrying left invariant hypercomplex structures and we show that the corresponding Obata connection is always flat. We determine when such Lie groups admit HKT metrics and study the corresponding Bismut connection. We obtain the classification of hypercomplex almost abelian Lie groups in dimen...

We give a characterization of almost abelian Lie groups carrying left invariant hypercomplex structures and we show that the corresponding Obata connection is always flat. We determine when such Lie groups admit HKT metrics and study the corresponding Bismut connection. We obtain the classification of hypercomplex almost abelian Lie groups in dimen...

Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on \({\mathbb {R}}^n\) and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of \(\mathrm {SU}(3)\)-stru...

Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $\mathbb{R}^n$ and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of $\mathrm{SU}(3)$-structures i...

We study 4-dimensional simply connected Lie groups $G$ with left-invariant Riemannian metric $g$ admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action, or the metric is half conformally flat. In the first case, the problem reduces to the study of invariant confo...

We search for invariant solutions of the conformal Killing-Yano equation on Lie groups equipped with left invariant Riemannian metrics, focusing on 2-forms. We show that when the Lie group is compact equipped with a bi-invariant metric or 2-step nilpotent, the only invariant solutions occur on the 3-dimensional sphere or on a Heisenberg group. We c...

Given an almost complex manifold (M, J), we study complex connections with trivial holonomy such that the corresponding torsion is either of type (2, 0) or of type (1, 1) with respect to J. Such connections arise naturally when considering Lie groups, and quotients by discrete subgroups, equipped with bi-invariant and abelian complex structures.

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of th...

In this paper we study 2-forms which are solutions of the Killing–Yano equation on Lie groups endowed with a left invariant metric having various curvature properties. We prove a general result for 2-step nilpotent Lie groups and as a corollary we obtain a nondegenerate solution of the Killing–Yano equation on the Iwasawa manifold with its half-fla...

Given an almost complex manifold (M, J), we study complex connections with trivial holonomy and such that the corresponding torsion is either of type (2,0) or of type (1,1) with respect to J. Such connections arise naturally when considering Lie groups, and quotients by discrete subgroups, equipped with bi-invariant and abelian complex structures.

We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.

Manifolds with special geometric structures play a prominent role in some branches of theoretical physics, such as string theory and supergravity. For instance, it is well known that supersymmetry requires target spaces to have certain special geometric properties. In many,cases these requirements can be interpreted as restrictions on the holonomy,...

We give a procedure for constructing an $8n$-dimensional HKT Lie algebra starting from a $4n$-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-K\"ahler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy...

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of $G$-invariant complex structur...

In this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie groups G whose corresponding duals G∗ are complex Lie groups. We also prove that a Hermitian structure on g with ad...

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on $G$. Usi...

In this paper we apply the hyper-Kähler quotient construction to Lie groups with a left invariant hyper-Kähler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known hyper-Kähler metrics can be recovered in this way by considering different Lie group structures on (: the quatern...

In this paper we apply the hyper-K\"ahler quotient construction to Lie groups with a left invariant hyper-K\"ahler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known hyper-K\"ahler metrics can be recovered in this way by considering different Lie group structures on $\H^p \t...

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 three dimensional subalgebras.

We study existence of complex structures on semidirect products g ⊕ρ v, where g is a real Lie algebra and ρ is a representation of g on v. Our first examples, the Euclidean algebra e(3) and the Poincaré algebra e(2, 1), carry complex structures obtained by deformation of a regular complex structure on sl (2, C). We also exhibit a complex structure...

Let g be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold (M,). We show that when the isometry group I(M,g) contains a subgroup G acting simply transitively on M by hypercomplex isometries, then the metric g is conformal to a hyper-Khler metric. We describe explicitely the corresponding hyper-Khler metrics, which are of coh...

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie groups are complex affine manifold...

It is the aim of this work to give a characterization of the two-step nilpotent Lie algebras carrying abelian hypercomplex structures. In the special case of trivial extensions of irreducible H-type Lie algebras this characterization is given in terms of the dimension of the commutator subalgebra. As a consequence, we obtain the corresponding theor...

Let $g$ be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold $M$. We show that when the isometry group $I(M,g)$ contains a subgroup acting simply transitively on $M$ by hypercomplex isometries then the metric $g$ is conformal to a hyper-K\"ahler metric. We describe explicitely the corresponding hyper-K\"ahler metrics and it...

Let g be a hyper-Hermitian metric on a simply connected hyper-complex four-manifold (M, H). We show that when the isometry group I(M, g) contains a subgroup acting simply transitively on M by hypercomplex isome-tries then the metric g is conformal to a hyper-Kähler metric. We describe explicitely the corresponding hyper-Kähler metrics and it follow...

It is the aim of this work to study affine connections whose holonomy group is contained in . These connections arise in the context of hypercomplex geometry. We study the case of homogeneous hypercomplex manifolds and introduce an affine connection which is closely related to the Obata connection [M. Obata, Japan J. Math. 26 (1956) 43–77]. We find...

The purpose of this paper is to classify invariant hypercomplex structures on a 4-dimensional real Lie group G. It is shown that the 4- dimensional simply connected Lie groups which admit invariant hypercom- plex structures are the additive group H of the quaternions, the multiplicative group H of nonzero quaternions, the solvable Lie groups acting...

Given a manifoldM, a Clifford structure of orderm onM is a family ofm anticommuting complex structures generating a subalgebra of dimension 2 m of End(T(M)). In this paper we investigate the existence of locally invariant Clifford structures of orderm2 on a class of locally homogeneous manifolds. We study the case of solvable extensions ofH-type gr...

. Let g be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold (M;H). We show that when the isometry group I(M;g) contains a subgroup acting simply transitively on M by hypercomplex isometries then the metric g is conformal to a hyper-Kahler metric. We describe explicitely the corresponding hyper-Kahler metrics and it follow...