Study sub-Riemannian geodesics on a Heisenberg 3D nil-manifold.
problem Characterize geodesics on a specific nil-manifold.
method Analyze the projection of sub-Riemannian structure, describe geodesic flow dynamics, and estimate sub-Riemannian geodesics.
result Sharp bounds and estimates for sub-Riemannian geodesics.
New proof of wave trace formula for 3D-contact manifolds.
problem Wave trace formula for 3D-contact manifolds.
method Normal form reduction to Heisenberg group.
result Extension of Chazarain-Duistermaat-Guillemin formula.
By using the support function on the xy-plane, we show the necessary and sufficient conditions for the existence of envelopes of horizontal lines in the 3D-Heisenberg group. A method to construct horizontal envelopes from the given ones is also derived, and we classify the solutions satisfying the construction.
The paper extends Pappus-Guldin theorems to 3D-Heisenberg group surfaces.
problem Extending classical theorems to a new geometric setting.
method Deriving formulas for p-areas and volumes in the Heisenberg group.
result Pappus-Guldin theorems hold for surfaces in the Heisenberg group.
Proves Steiner and tube formulae for 3D contact sub-Riemannian surfaces.
problem Calculating surface properties in complex geometric structures.
method Develops a local Steiner formula for regular surfaces in 3D contact sub-Riemannian manifolds.
result Establishes a formula for surface expansion in arbitrary regions of contact sub-Riemannian manifolds.
We study the local equivalence problems of curves and surfaces in three dimensional Heisenberg group via Cartans method of moving frames and Lie groups, and find a complete set of invariants for curves and surfaces. For surfaces, in terms of these invariants and their suitable derivatives, we also give a Gaussian curva…
Higher-dimensional Milnor frames are characterized and contrasted with 3D Heisenberg and 4D nilpotent Lie algebras.
problem Characterizing higher-dimensional Milnor frames and their properties.
method Definition and classification of higher-dimensional Milnor frames and their relationship to known Lie algebras.
result Higher-dimensional Milnor frames are isomorphic to direct sums of 3D Heisenberg and 4D nilpotent Lie algebras and an abelian Lie algebra.
By studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H1, we define the density and the measure for the sets of horizontal lines. We show that the volume of a convex domain D⊂H1 is equal to the integral of length of chord over all horizontal lines intersecting D. As the classical r…
The paper classifies 3D spherical Sasakian manifolds using geometric and algebraic methods.
problem Classifying 3D spherical Sasakian manifolds with specific properties.
method Establishing correspondence between different sets of parameters and geometrically describing the moduli space.
result Determination of Sasakian automorphism groups and detection of homogeneous Sasakian manifolds.
Study on CR structures on 3D Lie groups, focusing on equivalence and closed chains.
problem Characterizing CR structures on 3D Lie groups and their chains.
method Analyzing left-invariant CR structures on 3D Lie groups and their equivalence.
result All chains on G are closed if and only if G is CR equivalent to specific spherical CR structures.
We give definition of a holonomy flag in subRiemannian geometry --- a generalization of a Riemannian holonomy algebra --- and calculate it for the 3D subRiemannian Lie groups. We rewrite and give new interpretation for the Codazzi equations for the (2,3)-distributions on the SU(2) and the Heisenberg group.
Researchers find limits on curvature of certain 3D solitons.
problem Limits on curvature of 3D Heterotic solitons with parallel torsion.
method Rigidity result for compact 3D Heterotic solitons with parallel non-trivial torsion.
result Universal bound of -24 for scalar curvature of Heterotic solitons with parallel skew-symmetric torsion.
Study on curvatures of surfaces in specific Lie groups.
problem Analyzing curvatures of surfaces in 3D contact sub-Riemannian Lie groups.
method Riemannian approximation scheme to derive formulas for curvatures.
result Classification of surfaces with constant horizontal curvatures.
In this paper a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group H3, or its conformal classification coincides with the metric one…
Unified interpretation of sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.
problem Proving a sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.
method Measure-theoretic perspective, focusing on singular measures and characteristic points.
result Unified interpretation of previous results and natural geometric conditions for the theorem.
Study null curves and their motion in 3D flat space-time, leading to integrable hierarchies.
problem Understanding null curves and their motion in 3D flat space-time.
method Analyzing the motion of null curves and their surfaces, deriving integrability conditions and hierarchies.
result Obtained one- and two-soliton surfaces associated with the MKdV equation, showing singularities in finite time.
This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using KK-theory.
problem Analyzing the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators.
method Applying Kasparov's methodology and examining specific conditions using Fourier transform of the nilpotent group C∗-algebra. result Demonstrated enhanced methods for analyzing hypoellipticity and defined transversal Heisenberg ellipticity in a KK-theoretic context. This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, or Heisenberg calculus. The Heisenberg manifolds generalize CR and contact manifolds and in this context the main differential operators at stake include the Hörmander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian and its conf…
Study eigenvalues and functions on specific Heisenberg manifolds.
problem Eigenvalues and eigenfunctions of a specific operator on Heisenberg Bieberbach manifolds.
method Analysis of eigenvalues and eigenfunctions of the Folland-Stein operator on Heisenberg Bieberbach manifolds.
result Characterized eigenvalues and eigenfunctions of the Folland-Stein operator on specific manifolds.
Clarifies a trace for Heisenberg operators on contact manifolds.
problem Calculating the index of Heisenberg elliptic operators on contact manifolds.
method Introduced a new trace on Heisenberg pseudodifferential operators and constructed a cocycle in periodic cyclic cohomology.
result Simplified the construction of the trace on Heisenberg pseudodifferential operators.
In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point q0∈M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C0∞(M∖{q0}) is essentially self-adjoint. A particular …
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.
Extends Heisenberg homology to ribbon graphs.
problem Configurations in bounded surfaces.
method Regular thickening of ribbon graphs.
result Heisenberg homology applied to ribbon graphs.
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. 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.
We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenberg invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as wel…
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.
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…
Compact metrics on Heisenberg manifolds have a specific condition for being relatively compact.
problem Conditions for relatively compact sets of left invariant metrics on Heisenberg manifolds.
method Necessary and sufficient condition for relatively compact sets of left invariant metrics.
result A condition for a set of left invariant metrics to be relatively compact in the moduli space.
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.
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.
In this paper we study the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group. We prove that all of the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group are helices. Moreover, we obtain explicit parametric equations for non-geodesic non-null biharmon…
Study on Killing magnetic curves in Heisenberg group geometry.
problem Understanding Killing magnetic curves in Heisenberg group.
method Presentation of Heisenberg group geometry and geodesics, study of Killing magnetic curves with explicit formulas.
result Explicit formulas for Killing magnetic curves in Heisenberg group.
In this paper, we study the Ricci-Bourguignon flow on higher dimensional classical Heisenberg nilpotent Lie groups and construct a solution of this flow on Heisenberg and quaternion nilpotent Lie groups. In the end, we investigate the deformation of spectrum and length spectrum on compact nilmanifolds obtained of Heise…
Sharp stability of isometries on Heisenberg group proven.
problem Quantitative stability of isometries on the Heisenberg group.
method Proving quasi-isometries close to isometries with specific closeness orders.
result Quasi-isometries of Heisenberg group are close to isometries with specific closeness orders.
Study shows compact Sasakian manifolds are locally Heisenberg up to deformation.
problem Characterizing compact Sasakian manifolds.
method Analyzing basic Chern classes and using left invariant Sasakian structures.
result Compact Sasakian manifolds are locally isomorphic to the real Heisenberg group.
We study immersed, connected, umbilic hypersurfaces in the Heisenberg group Hn with n ≥ 2. We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigm…
Develops analysis of Hölder continuous mappings on Heisenberg groups.
problem Analyzing Hölder continuous mappings on Heisenberg groups.
method Theory of distributional Jacobians and pullbacks of differential forms.
result Simple proof of a generalization of the Gromov non-embedding theorem and new results about Hölder homotopy groups.
Study on curvature exponent of sub-Finsler Heisenberg groups, proving N_min ≥ 5.
problem Determining the curvature exponent of sub-Finsler Heisenberg groups.
method Analyzing the measure contraction property and constructing sub-Finsler structures.
result Proved that curvature exponent N_min ≥ 5, with equality if sub-Riemannian.