Three-dimensional Heisenberg Group Research Articles

TSALLIS ENTROPY COMPOSITION AND THE HEISENBERG GROUP

We present an embedding of the Tsallis entropy into the three-dimensional Heisenberg group, in order to understand the meaning of generalized independence as encoded in the Tsallis entropy composition property. We infer that the Tsallis entropy composition induces fractal properties on the underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we justify why the underlying configuration/phase space of systems described by the Tsallis entropy has polynomial growth for both discrete and Riemannian cases. We provide a geometric framework that elucidates Abe's formula for the Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian spaces.

Ruled minimal surfaces in the three-dimensional Heisenberg group

It is shown that parts of planes, helicoids and hyperbolic paraboloids are the only minimal surfaces ruled by geodesics in the three dimensional Riemannian Heisenberg group. It is also shown that they are the only surfaces in the three dimensional Heisenberg group whose mean curvature is zero with respect to both of the standard Riemannian metric and the standard Lorentzian metric.

CONSEQUENCES OF THE FUNDAMENTAL CONJECTURE FOR THE MOTION ON THE SIEGEL–JACOBI DISK

We find the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi domain [Formula: see text] as the sum of the Kähler two-form on ℂ and the one on the Siegel ball [Formula: see text]. The classical motion and quantum evolution on [Formula: see text] determined by a linear Hamiltonian in the generators of the Jacobi group [Formula: see text] is described by a Riccati equation on [Formula: see text] and a linear first-order differential equation in z ∈ ℂ, where H1 denotes the three-dimensional Heisenberg group. When the transformation FC is applied, the first-order differential equation for the variable z ∈ ℂ decouples of the motion on the Siegel disk. Similar considerations are presented for the Siegel–Jacobi space [Formula: see text], where [Formula: see text] denotes the Siegel upper half-plane.

3-MANIFOLD GROUPS, KÄHLER GROUPS AND COMPLEX SURFACES

Let G be a Kähler group admitting a short exact sequence [Formula: see text] where N is finitely generated. (i) Then Q cannot be non-nilpotent solvable. (ii) Suppose in addition that Q satisfies one of the following: (a) Q admits a discrete faithful non-elementary action on ℍn for some n ≥ 2. (b) Q admits a discrete faithful non-elementary minimal action on a simplicial tree with more than two ends. (c) Q admits a (strong-stable) cut R such that the intersection of all conjugates of R is trivial. Then G is virtually a surface group. It follows that if Q is infinite, not virtually cyclic, and is the fundamental group of some closed 3-manifold, then Q contains as a finite index subgroup either a finite index subgroup of the three-dimensional Heisenberg group or the fundamental group of the Cartesian product of a closed oriented surface of positive genus and the circle. As a corollary, we obtain a new proof of a theorem of Dimca and Suciu in [Which 3-manifold groups are Kähler groups? J. Eur. Math. Soc.11 (2009) 521–528] by taking N to be the trivial group. If instead, G is the fundamental group of a compact complex surface, and N is finitely presented, then we show that Q must contain the fundamental group of a Seifert-fibered 3-manifold as a finite index subgroup, and G contains as a finite index subgroup the fundamental group of an elliptic fibration. We also give an example showing that the relation of quasi-isometry does not preserve Kähler groups. This gives a negative answer to a question of Gromov which asks whether Kähler groups can be characterized by their asymptotic geometry.

On constant mean curvature surfaces in the Heisenberg group

This work is devoted to the theory of surfaces of constant mean curvature in the three-dimensional Heisenberg group. It is proved that each surface of such a kind locally corresponds to some solution of the system of a sine-Gordon type equation and a first order partial differential equation.

Global $\widetilde{SL(2,R)}$ representations of the Schrödinger equation with singular potential

Abstract We study the representation theory of the solution space of the one-dimensional Schrödinger equation with singular potential V λ(x) = λx −2 as a representation of $\widetilde{SL(2,\mathbb{R})}$ . The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category. By studying the subspace of K-finite vectors in this space, a distinguished family of potentials, parametrized by the triangular numbers is shown to generate a global representation of $\widetilde{SL(2,\mathbb{R})}$ ⋉ H 3, where H 3 is the three-dimensional Heisenberg group.

Moduli of triangles in the Heisenberg group

We study triangles in the three-dimensional Heisenberg group, \({\mathbb{H}}\) , equipped with the Carnot-Caratheodory geometry. The set of ordered triangles in \({\mathbb{H}}\) (excluding certain degenerate triangles), up to congruence is naturally identified via a parametrization map to a fine moduli space of parameters. We determine the homeomorphism type of this moduli space and also that of the coarse moduli space of unordered triangles. We describe a boundary for the fine moduli space and construct a compactification for it, up to similarity under the non-isotropic dilation of \({\mathbb{H}}\) . Additionally, some trigonometric results for the Carnot-Caratheodory geometry of \({\mathbb{H}}\) are given: an angle deficit formula and an analog of the Law of Sines in Euclidean geometry.

Surfaces in the three-dimensional Heisenberg group on which the Gauss map has bounded Jacobian

Surfaces in the three-dimensional Heisenberg group on which the Gauss map has bounded Jacobian

Generalized contact structures

We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensional manifolds. We name the latter strong generalized contact structures. Using a Boothby-Wang construction bridging symplectic structures and contact structures, we find examples to demonstrate that, within the category of generalized contact structures, classical contact structures have non-trivial deformations. Using deformation theory of Lie bialgebroids, we construct new families of strong generalized contact structures on the three-dimensional Heisenberg group and its co-compact quotients.

Half-flat structures and special holonomy

It was proven by Hitchin that any solution of his evolution equations for a half-flat SU(3)-structure on a compact six-manifold M defines an extension of M to a seven-manifold with holonomy in G_2. We give a new proof, which does not require the compactness of M. More generally, we prove that the evolution of any half-flat G-structure on a six-manifold M defines an extension of M to a Ricci-flat seven-manifold N, for any real form G of SL(3,C). If G is noncompact, then the holonomy group of N is a subgroup of the noncompact form G_2^* of G_2^C. Similar results are obtained for the extension of nearly half-flat structures by nearly parallel G_2- or G_2^*-structures, as well as for the extension of cocalibrated G_2- and G_2^*-structures by parallel Spin(7)- and Spin(3,4)-structures, respectively. As an application, we obtain that any six-dimensional homogeneous manifold with an invariant half-flat structure admits a canonical extension to a seven-manifold with a parallel G_2- or G_2^*-structure. For the group H_3 \times H_3, where H_3 is the three-dimensional Heisenberg group, we describe all left-invariant half-flat structures and develop a method to explicitly determine the resulting parallel G_2- or G_2^*-structure without integrating. In particular, we construct three eight-parameter families of metrics with holonomy equal to G_2 and G_2^*. Moreover, we obtain a strong rigidity result for the metrics induced by a half-flat structure (\omega,\rho) on H_3 \times H_3 satisfying \omega(Z,Z)=0 where Z denotes the centre. Finally, we describe the special geometry of the space of stable three-forms satisfying a reality condition. Considering all possible reality conditions, we find four different special K\"ahler manifolds and one special para-K\"ahler manifold.

Lorentz Ricci Solitons on 3-dimensional Lie groups

The three-dimensional Heisenberg group H 3 has three left-invariant Lorentzian metrics g 1, g 2, and g 3 as in Rahmani (J. Geom. Phys. 9(3), 295–302 (1992)). They are not isometric to each other. In this paper, we characterize the left-invariant Lorentzian metric g 1 as a Lorentz Ricci Soliton. This Ricci Soliton g 1 is a shrinking non-gradient Ricci Soliton. We also prove that the group E(2) of rigid motions of Euclidean 2-space and the group E(1, 1) of rigid motions of Minkowski 2-space have Lorentz Ricci Solitons.

Weierstrass representation for surfaces in the three-dimensional Heisenberg group

The authors define the Gauss map of surfaces in the three-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean curvature. Furthermore, a second order partial differential equation for the Gauss map is obtained, and it is shown that this equation is the complete integrability condition of the representation.

The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

When equipped with the natural topology first defined by Chabauty, the closed subgroups of a locally compact group G form a compact space C(G). We analyse the structure of C(G) for some low-dimensional Lie groups, concentrating mostly on the 3-dimensional Heisenberg group H. We prove that C(H) is a 6-dimensional space that is path-connected but not locally connected. The lattices in H form a dense open subset L(H) ?C(H) that is the disjoint union of an infinite sequence of pairwise homeomorphic aspherical manifolds of dimension six, each a torus bundle over (S3 T) × R, where T denotes a trefoil knot. The complement of L(H) in C(H) is also described explicitly. The subspace of C(H) consisting of subgroups that contain the centre Z(H) is homeomorphic to the 4-sphere, and we prove that this is a weak retract of C(H).

Higher order parallel surfaces in the Heisenberg group

We give a classification of k-parallel surfaces in the three-dimensional Heisenberg group. In particular, we prove that every k-parallel surface in the Heisenberg group is a vertical cylinder over a polynomial spiral of degree at most k − 1 .

Conservations laws for critical Kohn-Laplace equations on the Heisenberg group

Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group ℍ, carried out in a preceding work, we establish the conservation laws for the critical Kohn-Laplace equations via the Noether's Theorem.

A criterion for global hyperbolicity of left-invariant Lorentz metrics on Lie groups

We show that every left-invariant Lorentz metric on a non-abelian simply connected Lie group is globally hyperbolic whenever its restriction to the commutator ideal of the Lie algebra is positive definite. We also show that a left-invariant Lorentz metric on the three-dimensional Heisenberg group is globally hyperbolic if and only if its restriction to the center of the Lie algebra is positive definite or degenerate.

Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

We study the generalization of the Willmore functional for surfaces in the three-dimensional Heisenberg group. Its construction is based on the spectral theory of the Dirac operator entering into theWeierstrass representation of surfaces in this group. Using the surfaces of revolution we demonstrate that the generalization resembles the Willmore functional for the surfaces in the Euclidean space in many geometrical aspects. We also observe the relation of these functionals to the isoperimetric problem.

Inversion of the Radon transform on the product Laguerre hypergroup by using generalized wavelets

Let H 1 be the three-dimensional Heisenberg group. The fundamental manifold of the radial function space for H 1 can be denoted by [0, ∞)×R, which is just the Laguerre hypergroup. Naturally, K n =[0, ∞) n ×R n is the product Laguerre hypergroup. In this paper, we give the theory of continuous wavelet analysis and the Radon transform on K n , and devise a subspace 𝒮ℛ(K n ) of 𝒮(K n ) (Schwartz space) on which the Radon transform is a bijection. Also, we give two equivalent characterizations on 𝒮(K n ) for the Radon transform. By using the inverse wavelet transform we establish an inversion formula of the Radon transform on K n in the weak sense.

K– andL–theory of the semi-direct product of the discrete 3–dimensional Heisenberg group by ℤ∕4

We compute the group homology, the topological K-theory of the reduced C^*-algebra, the algebraic K-theory and the algebraic L-theory of the group ring of the semi-direct product of the three-dimensional discrete Heisenberg group by Z/4. These computations will follow from the more general treatment of a certain class of groups G which occur as extensions 1-->K-->G-->Q-->1 of a torsionfree group K by a group Q which satisfies certain assumptions. The key ingredients are the Baum-Connes and Farrell-Jones Conjectures and methods from equivariant algebraic topology.

The Classification of Complex Factor-Representations of the Three-Dimensional Heisenberg Group over a Countable Group Field of Finite Characteristic

We consider a field F that is a direct limit of an increasing chain of finite fields, and describe the Bratteli diagram, complex factor-representations, and projective moduli of the Heisenberg group of 3 × 3 upper-triangular matrices with elements from F. Bibliography: 3 titles.