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Register a new userOn seven dimensional quaternionic contact solvable Lie groups
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We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.
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The paper is a continuation of the authors et al.s work in the first half of the year 2021. It has classified a special class of 7-dimensional real solvable Lie algebras such that the nilradical of each from them is well-known 5-dimensional nilpotent Lie algebra in that work. In this paper, we will consider exponential, connected and simply connected Lie groups which are corresponding to these Lie algebras. Namely, we will describe the geometry of generic (i.e. 6-dimensional) orbits in coadjoint representation of considered Lie groups. Next, we will prove that for each considered group, the family of generic coadjoint orbits forms a measurable foliation in the sense of Connes and give the topological classification of these foliations.
The systematic study of CR manifolds originated in two pioneering 1932 papers of Elie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR structures on $mathrm{SU}_2 = S^3$, deforming the standard `spherical structure. In this paper, mostly expository, we illustrate and clarify Cartans results and methods by providing detailed classification results in modern language for four 3-dimensional Lie groups. In particular, we find that $mathrm{SL}_2(mathbb{R})$ admits two one-parameter families of left-invariant CR structures, called the elliptic and hyperbolic families, characterized by the incidence of the contact distribution with the null cone of the Killing metric. Low dimensional complex representations of $mathrm{SL}_2(mathbb{R})$ provide CR embedding or immersions of these structures. The same methods apply to all other three-dimensional Lie groups and are illustrated by descriptions of the left-invariant CR structures for $mathrm{SU}_2$, the Heisenberg group, and the Euclidean group.
We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in seven dimensions, when this condition is equivalent to integrability in the sense of Duchemin. We prove that the choice of a reduction to Sp(n)H^* (or equivalently, a complement of the qc distribution) yields a unique K-connection satisfying natural conditions on torsion and curvature. We show that the choice of a compatible metric on the qc distribution determines a canonical reduction to Sp(n)Sp(1) and a canonical Sp(n)Sp(1)-connection whose curvature is almost entirely determined by its torsion. We show that its Ricci tensor, as well as the Ricci tensor of the Biquard connection, has an interpretation in terms of intrinsic torsion.
Let $G$ be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of $G$ with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the identity component of the isometry group is compact if $G$ is not simply-connected.
We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifold is compact and we give a sharp bound on its sub-Riemannian diameter.
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