New conformally Einstein metrics on Heisenberg group found.
problem Finding new conformally Einstein metrics on specific Lie groups.
method Described Lorentzian semi-direct extensions of the Heisenberg group.
result Classified Bach-flat left-invariant Lorentzian metrics.
Timelike minimal surfaces in Lorentzian Heisenberg group have singular points.
problem Characterizing singularities on timelike minimal surfaces.
method Constructing timelike minimal surfaces as Lorentzian harmonic maps and analyzing their singularities.
result Criteria for cuspidal edges, swallowtails, and cuspidal cross caps are provided.
Study on Heisenberg group's Lorentzian problems using Pontryagin's principle.
problem Lorentzian problems on the Heisenberg group.
method Applied Pontryagin's maximum principle to obtain extremal trajectories.
result Parameterization of abnormal and normal extremal trajectories, investigation of reachability sets and existence of optimal trajectories.
Study classifies helix surfaces in Lorentzian Heisenberg group.
problem Classifying helix surfaces in Lorentzian Heisenberg group.
method Complete description of ambient space geometry, classification of minimal and CMC helix surfaces, investigation of constant angle surfaces.
result Explicit parametrizations of minimal and CMC helix surfaces in $\htt$.
Optimal transport explored on a specific geometric space.
problem Optimal transport problem in sub-Lorentzian Heisenberg group.
method Synthetic metric spacetime structure analysis and sub-Lorentzian version of Brenier's theorem.
result Established sub-Lorentzian version of Brenier's theorem and derived Monge-Ampère equation.
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in sub-Lorentzian Heisenberg group.
method First-variation formula derivation and isoperimetric candidates classification.
result Characterization and conjecture of isoperimetric maximizers.
Study on Hausdorff dimension and curvature bounds in sub-Lorentzian Heisenberg group.
problem Hausdorff dimension and curvature bounds in sub-Lorentzian Heisenberg group.
method Elementary variational approach, Lorentzian isoperimetric problem, uniform estimate of causal diamonds.
result Heisenberg group has Lorentzian Hausdorff dimension 4 and satisfies neither timelike curvature-dimension nor measure contraction properties.
Study duality of zero mean curvature surfaces in Heisenberg group.
problem Understanding the duality of zero mean curvature surfaces in the Lorentzian Heisenberg group.
method Investigation of a transformation surface associated with zero mean curvature surfaces in the Heisenberg group under two metrics.
result Derivation of the Sym formula for the dual surface in both metric cases.
In this paper, we define and, then, we characterize constant angle spacelike and timelike surfaces in the three-dimensional Heisenberg group, equipped with a 1-parameter family of Lorentzian metrics. In particular, we give an explicit local parametrization of these surfaces and we produce some examples.
The paper classifies metrics on specific Lie groups and finds unique properties of these metrics.
problem Classifying left-invariant Lorentzian metrics on specific Lie groups.
method Analyzing the three-dimensional Heisenberg group and its direct product with Euclidean space.
result There are exactly six left-invariant Lorentzian metrics on the Lie group, one of which is flat and the others are Ricci solitons but not Einstein.
Study of maximal surfaces in a specific Heisenberg group with singularities.
problem Characterize maximal surfaces in the Lorentzian Heisenberg group with singularities.
method Use harmonic maps into the 2-sphere, loop group construction, and solve the Cauchy problem.
result Regular maximal discs must have at least two cuspidal cross-cap singularities on the boundary.
The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on $\z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces G/H such that the Lie algebra $\g$ of G admits a Γ-grading where Γ is a finite abelian group. In this work we study Rieman…
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 th…
Constructs Lorentzian harmonic maps and associated timelike surfaces.
problem Lorentzian harmonic maps and timelike surfaces properties.
method Constructs framed null curves and solves eigenvalue equation.
result Characterizes singularities on timelike minimal surfaces.
Study finds formulas for special curves in complex spaces.
problem Understanding special curves in complex spaces.
method Obtained explicit formulas for Killing magnetic curves.
result Explicit formulas for Killing magnetic curves in non-flat Lorentzian-Heisenberg spaces.
Minimal surfaces in Heisenberg group have null curves and lines.
problem Characterizing timelike minimal surfaces in the Heisenberg group.
method Characterization through null curves and lines with prescribed curvatures.
result Minimal surfaces are defined by the multiplication of null curves and affine null lines.
The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on Z2-grading of Lie algebras. In our case, we consider homogeneous spaces G/H such that the Lie algebra $\g$ of G admits a Γ-grading where Γ is a finite abelian group. In this work we study Rieman…
Study of Lorentzian manifolds with specific transformations.
problem Characterizing Lorentzian manifolds with essential pseudo-groups of local conformal transformations.
method Generalizing recent results, using Gromov's theory of rigid transformations.
result Locally conformally homogeneous Lorentzian manifolds are either conformally flat or locally conformally equivalent to homogeneous plane waves.
We consider the three-dimensional Heisenberg group, equipped with any left-invariant metric, either Lorentzian or Riemannian. We completely classify their affine vector fields and investigate their relationship with Killing vector fields and their casual character. We also classify their Ricci, curvature and matter col…
We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the 2k+1-dimensional Heisenberg Lie group H2k+1 carries a Ricci flat left invariant Lorentzian metric if and only if k=1. We show also that for any 2≤q≤k, H2k+1 carries a R…
Mathematical treatment of plane waves, proving their inextendibility and completeness.
problem Completeness and inextendibility of homogeneous plane waves.
method Cohomogeneity one Heisenberg actions, isometry group analysis.
result Proof of C2-inextendibility and geodesic completeness of non-flat homogeneous plane waves. We compute the full holonomy group of compact Lorentzian manifolds with parallel Weyl tensor, which are neither conformally flat nor locally symmetric, for the case where the fundamental group is contained in a distinguished subgroup G of the isometry group of the universal cover. To prove this, we show that every such…
We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces L(κ,τ) with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts E(κ,τ), and obtain some consequence…
The paper analyzes the spectra of compact quotients of the oscillator group.
problem Computing spectra of compact solvmanifolds.
method Classification of lattices, decomposition of representations, explicit computation of spectra.
result Explicit computation of the spectrum of the wave operator on compact locally-symmetric Lorentzian manifolds.
The three-dimensional Heisenberg group H3 has three left-invariant Lorentz metrics g1, g2 and g3. They are not isometric each other. In this paper, we characterize the left-invariant Lorentzian metric g1 as a Lorentz Ricci soliton. This Ricci soliton g1 is a shrinking non-gradient Ricci soliton. Likew…
Based on the work of Adams and Stuck as well as on the work of Zeghib, we classify the Lie groups which can act isometrically and locally effectively on Lorentzian manifolds of finite volume. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebr…
This work concerns the non-flat metrics on the Heisenberg Lie group of dimension three $\Heis_3(\RR)$ and the bi-invariant metrics on the solvable Lie groups of dimension four. On $\Heis_3(\RR)$ we prove that the property of the metric being naturally reductive is equivalent to the property of the center being non-dege…
The main aim of this survey paper is to gather together some results concerning the Calabi type duality discovered by Hojoo Lee between certain families of (spacelike) graphs with constant mean curvature in Riemannian and Lorentzian homogeneous 3-manifolds with isometry group of dimension 4. The duality is conformal an…
We exploit the correspondence between the three-dimensional Lorentzian Einstein-Weyl geometries of the hyper-CR type, and the Veronese webs to show that the former structures are locally given in terms of solutions to the dispersionless Hirota equation. We also demonstrate how to construct hyper-CR Einstein--Weyl struc…
Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalisation of the la…
In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special li…
Study of contact whirl curves in Sasakian Lorentzian 3-manifolds.
problem Understanding the geometric properties of curves in Lorentzian contact manifolds.
method Introducing and analyzing contact whirl curves, deriving differential equations, and proving rigidity phenomena.
result Every non-geodesic Legendre Frenet curve is a contact whirl curve with constant torsion τ=1.
Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.
problem Understanding submanifolds with boundary in sub-Riemannian Heisenberg groups.
method Introduced examples and proved Stokes' Theorem involving Rumin's differential forms.
result Stokes' Theorem for submanifolds with boundary in Heisenberg groups.
Classifies geodetically convex sets and functions on Heisenberg group.
problem Characterizing geodetically convex sets and functions in the Heisenberg group.
method Classification through mathematical analysis.
result Geodetically convex sets and functions defined on Heisenberg group Hn classified. Homogeneous magnetic paths found in Heisenberg space.
problem Understanding magnetic geodesics in the Heisenberg group.
method Proving homogeneity of magnetic geodesics derived from the canonical contact structure.
result Magnetic geodesics in the Heisenberg group are homogeneous.
Study lifts plane mappings to Heisenberg group.
problem Contact quasiconformal mappings in hyperbolic Heisenberg group.
method Lifting Theorem for symplectic mappings.
result Symplectic mappings lifted to Heisenberg group.
Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.
problem Understanding submanifolds with boundaries in Heisenberg groups.
method Introduce and study CH1-regular submanifolds with boundaries, prove Stokes' Theorem for them. result Prove a version of Stokes' Theorem for submanifolds with boundaries in Heisenberg groups.
Find conditions for starshapedness of level sets in Heisenberg group.
problem Ensure starshapedness of level sets of p-capacitary potentials. method Examine horizontally p-harmonic functions in the Heisenberg group. result Sharp conditions for strictly starshaped level sets.
Defines contact structures on Heisenberg groups for geometric interpretation.
problem Finding a geometric interpretation for the Yamabe equation on Heisenberg groups.
method Defines contact structures of Heisenberg type, introduces a natural connection, and computes conformal scalar curvature.
result Establishes equivalence between contact Riemannian manifolds and contact structures of Heisenberg type.
Computes minimal polynomials for generalized Heisenberg groups.
problem None explicitly stated; focus on method.
method Computes minimal polynomials for generalized Heisenberg groups.
result Explicit minimal polynomials for generalized Heisenberg groups.
The paper describes geodesics on a Kähler cone of the Heisenberg group.
problem Understanding geodesics on the Kähler cone of the Heisenberg group.
method Analyzing the Heisenberg group to describe geodesics.
result It is not a complete manifold.
Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
In Heisenberg group, bisectors are spinal spheres with specific curvature.
problem Understanding bisectors in the Heisenberg group.
method Showed bisectors are spinal spheres and calculated their curvature.
result Metric bisectors in Heisenberg group are spinal spheres with specific curvature.
Criterion for surfaces in Heisenberg group to be graphs using flat cones.
problem Characterizing surfaces in Heisenberg group as graphs.
method Using planar cones to define intrinsic rectifiability.
result Criterion for topological surfaces to be intrinsic Lipschitz graphs.
Translation surfaces in Heisenberg group classified by Gauss map determinant.
problem Classifying translation surfaces with zero intrinsic curvature in Heisenberg group.
method Constructed surfaces as product of planar curves, classified by Gauss map determinant.
result Surfaces with vanishing intrinsic curvature identified.
Study maps surface configurations to Heisenberg homologies for mapping class groups.
problem Understanding Mapping Class Groups of punctured surfaces.
method Action of mapping classes on Heisenberg homologies of surface configurations.
result Representations of Mapping Class Groups derived from Heisenberg homologies.
Among eight possible geometric structures on three-dimensional manifolds less studied from the differential geometric point of view are those modelled on the Heisenberg group Heis3. We consider the Heisenberg left-invariant metric and use some results on Levi-Civita connection and curvature tensor to present solutio…
Study shows only hyperplanes in Heisenberg groups have zero curvature.
problem Understanding Bernstein problem in higher dimensional Heisenberg groups.
method Sub-Riemannian characterization of ruling property and study of geodesics.
result Only hyperplanes have zero horizontal symmetric second fundamental form in Heisenberg groups.