On seven-dimensional quaternionic contact solvable Lie groups

Annales de l'Institut Fourier, 2003

L'infini conforme d'une metrique quaternion-kahlerienne complete sur une variete de dimension 4n est une distribution de contact quaternionienne de codimension 3. En dimension 4n - 1 >7, une telle structure de contact quaternionienne est toujours l'infini conforme d'une metrique quaternion-kahlerienne. Cependant, nous decrivons dans cet article une condition necessaire et suffisante pour qu'une telle distribution sur une variete de dimension 7 soit l'infini conforme d'une metrique quaternion-kalerienne. Ceci nous permet de trouver les structures de contact quaternioniennes sur la sphere de dimension 7, proches de la structure standard et qui sont les infinis conformes de metriques quaternion-kahleriennes completes sur la boule de dimension 8. Nous construisons ensuite une famille a 25 parametres de metriques quaternion-kahleriennes Sp(1)-invariantes et completes sur la boule de dimension 8.

arXiv (Cornell University), 2010

We construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional manifold, equipped with a certain qc structure, has a quaternionic Kähler metric as well as a metric with holonomy contained in Spin(7). As a consequence we determine explicit quaternionic Kähler metrics and Spin(7)-holonomy metrics which seem to be new. Moreover, we give explicit non-compact eight dimensional almost quaternion hermitian manifolds with either a closed fundamental four form or fundamental two forms defining a differential ideal that are not quaternionic Kähler.

arXiv: Differential Geometry, 2017

We exploit the Cartan-K\"ahler theory to prove the local existence of real analytic quaternionic contact structures for any prescribed values of the respective curvature functions and their covariant derivatives at a given point on a manifold. We show that, in a certain sense, the different real analytic quaternionic contact geometries in $4n+3$ dimensions depend, modulo diffeomorphisms, on $2n+2$ real analytic functions of $2n+3$ variables.

Annals of Global Analysis and Geometry, 2017

Following the Cartans's original method of equivalence supported by methods of parabolic geometry, we provide a complete solution for the equivalence problem of quaternionic contact structures, that is, the problem of finding a complete system of differential invariants for two quaternionic contact manifolds to be locally diffeomorphic. This includes an explicit construction of the corresponding Cartan geometry and detailed information on all curvature components. Contents 1. Introduction 1 2. Preliminaries 4 2.1. Conventions concerning the use of complex tensors and indices 4 2.2. Quaternionic contact manifolds 5 3. Solution to the quaternionic contact equivalence problem 6 4. The Curvature and the Bianchi identities 18 5. The associated Cartan geometry 30 5.1. A few algebraic constructions 30 5.2. The normal Cartan connection 33 6. Appendix 34 6.1. Cartan geometries 34 6.2. Parabolic geometries 35 6.3. Regular filtrations on manifolds 35 6.4. The curvature 36 References 36

2008

A result from Gromov ensures the existence of a contact struc ture on any connected non-compact odd dimensional Lie group. But in general such structures are no t invariant under left translations. The problem of finding which Lie groups admit a left invariant contact struc ure (contact Lie groups), is then still wide open. We perform a ‘contactization’ method to construct, in every od d dimension, many contact Lie groups with a discrete centre and discuss some applications and consequences of su ch a construction. We give classification results in low dimensions. In any dimension ≥ 7, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also classify contact Lie groups having a prescri bed Riemannian or semi-Riemannian structure and derive obstructions results .

The Quarterly Journal of Mathematics, 2012

We show that the Cauchy-Riemann structure on the twistor space of a quaternionic contact structure described by O. Biquard Biquard is normal if and only if the Ricci curvature of the Biquard connection commutes with the endomorphisms in the quaternionic structure of the contact distribution.

Differential Geometry and its Applications, 2008

Amongst other results, we perform a ‘contactization’ method to construct, in every odd dimension, many contact Lie groups with a discrete center, unlike the usual (classical) contactization which only produces Lie groups with a non-discrete center. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. We give classification results in low dimensions. A complete list is supplied in dimension 5. In any odd dimension greater than 5, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also characterize solvable contact Lie algebras whose derived ideal has codimension one. For simplicity, most of the results are given in the Lie algebra version.

2006

In this text, we prove that every quaternionic-contact structure can be embedded in a quaternionic manifold.

Physics Letters B, 2005

We show that in N = 2 supergravity, with a special quaternionic manifold of (quaternionic) dimension h 1 + 1 and in the presence of h 2 vector multiplets, a h 2 + 1 dimensional abelian algebra, intersecting the 2h 1 + 3 dimensional Heisenberg algebra of quaternionic isometries, can be gauged provided the h 2 + 1 symplectic charge-vectors V I , have vanishing symplectic invariant scalar product V I × V J = 0. For compactifications on Calabi-Yau three-folds with Hodge numbers (h 1 , h 2 ) such condition generalizes the half-flatness condition as used in the recent literature. We also discuss non-abelian extensions of the above gaugings and their consistency conditions.

Differential Geometry and its Applications, 2004