Isabel G. Dotti

Classification of abelian complex structures on 6-dimensional Lie algebras

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

Double Products and Hypersymplectic Structures on ℝ4n

In this paper we give a procedure to construct hypersymplectic structures on ℝ4n beginning with affine-symplectic data on ℝ2n. These structures are shown to be invariant by a 3-step nilpotent double Lie group and the resulting metrics are complete and not necessarily flat. Explicit examples of this construction are exhibited.

Abelian Hypercomplex 8-Dimensional Nilmanifolds

We study invariant Abelian hypercomplex structures on 8-dimensional nilpotent Lie groups. We prove that a group N admitting such a structure is either Abelian or an Abelian extension of a group of type H. We determine the Poincaré polynomials of the associated nilmanifolds and study the existence of symplectic and quaternionic structures on such spaces.

Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

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. Using this correspondence and results of (8) and (10), it turns out that when G is nilpotent and four or six dimensional, the cotangent bundle T G always has a hermitian structure. However, we prove that if G is a four dimensional solvable Lie group admitting neither complex nor symplectic structures, then T G has no hermitian structure or, equivalently, G has no left invariant generalized complex structure.

Negatively curved homogeneous Osserman spaces

Abstract We show that solvable Lie groups of Iwasawa type satisfying the Osserman condition are symmetric spaces of noncompact type and rank one. As a consequence, the Osserman conjecture holds for non-flat homogeneous manifolds of nonpositive curvature.

Hypercomplex eight-dimensional nilpotent Lie groups

Abstract In this paper we give a description of all eight-dimensional simply connected nilpotent Lie groups carrying a left invariant hypercomplex structure.

Corrigendum: Classification of abelian complex structures on six-dimensional Lie algebras

It has been pointed to us by E. Rodŕıguez Valencia that the complex structures J t and J 2 t on the Lie algebra n4, appearing in Theorem 3.3, are in fact equivalent. These structures are introduced in the proof of Theorem 3.2, which is used later in the paper to determine the moduli spaces of abelian complex structures. However, in that proof, statement (J) is in fact impossible, since it implies that the Lie algebra g is abelian. Indeed, if ker(adx |v) were J-stable for any x ∈ v, then we would have [x, Jx] = 0 for any x ∈ g. Therefore, for any x, y ∈ g,

COMPLEX CONNECTIONS WITH TRIVIAL HOLONOMY

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.

Quaternion Kahler flat manifolds

Abstract We give a general method to construct hyperkahler and quaternion Kahler flat manifolds with holonomy Z 2 k . We give many families of quaternion Kahler flat manifolds of dimension ≥8 which admit no Kahler structures

On the curvature of certain extensions of -type groups

We show that a one-dimensional solvable extension of an H -type group is symmetric if and only if it has negative curvature. H -type groups are 2-step nilpotent Lie groups which are natural generalizations of the Iwasawa N-groups associated to semisimple Lie groups of real rank one. They were introduced by A. Kaplan and, together with certain natural 1-dimensional solvable extensions, have been studied by various authors in connection with a number of interesting questions in geometry and analysis (see [K], [B], [C], [D], [DR], [G]). If N is an H -type group, let S = AN be its canonical one-dimensional solvable extension, endowed with the natural left invariant metric (see Section 2). It is well known that this class of solvable groups include rank one symmetric spaces of noncompact type. Furthermore, such S has always nonpositive curvature ([B], [D]). On the other hand, the following result was stated in [B]: Theorem. S has negative sectional curvature if and only if S is symmetric. F. Tricerri contacted the author about some difficulties found following the argument in the proof of this result. In particular, the plane exhibited on p. 541 of [B] does not have zero curvature as claimed. The main purpose of this note is to give an independent proof of the above theorem. We shall express the negative curvature property on S by an algebraic condition on the Lie algebra n ofN (we call it the NC-condition) and will prove that the only H -type algebras satisfying this condition are the Iwasawa n algebras. The basic idea in the proof is to use the NC-condition to define a bilinear multiplication without zero divisors on R+ z (z the center of n) and then use the classification of H -type algebras with center of dimension 0, 1, 3 or 7. L. Vanhecke, jointly with J. Berndt and F. Tricerri, have proved the above result in the case when the H -groupN has an even-dimensional center (see [BTV], Section 4.2). Their proof uses the explicit computation of the eigenvalues of the curvature operator in S and its relationship with the eigenvalues of the KY operator studied by Szabo ([S]). Also, M. Lanzendorf ([L]) has given, by different methods, an alternative proof of the theorem. Received by the editors February 13, 1995 and, in revised form, August 16, 1995. 1991 Mathematics Subject Classification. Primary 53C25. Partially supported by grants from Conicor, SECYTUNC (Argentina), and I.C.T.P. (Trieste). c ©1997 American Mathematical Society 573 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use