Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group

DOI10.1007/s00209-016-1815-6zbMath1375.53042OpenAlexW2549831894WikidataQ125994121 ScholiaQ125994121MaRDI QIDQ2412523

Zoltán M. Balogh, Jeremy T. Tyson, Eugenio Vecchi

Publication date: 23 October 2017

Published in: Mathematische Zeitschrift (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00209-016-1815-6

zbMATH Keywords

Heisenberg groupGauss-Bonnet theoremsub-Riemannian geometrySteiner formulaRiemannian approximation

Mathematics Subject Classification ID

Mixed volumes and related topics in convex geometry (52A39) Non-Euclidean differential geometry (53A35) Sub-Riemannian geometry (53C17)

Related Items (28)

Gauss-Bonnet theorem in lorentzian Sasakian space forms ⋮ Gauss-Bonnet theorems in the generalized affine group and the generalized BCV spaces ⋮ Gauss-Bonnet theorem in sub-Riemannian Heisenberg space \(\mathbb H^1\) ⋮ On sub-Riemannian geodesic curvature in dimension three ⋮ Sub-Lorentzian geometry of curves and surfaces in a Lorentzian Lie group ⋮ Heat content asymptotics for sub-Riemannian manifolds ⋮ Gauss–Bonnet theorems and the Lorentzian Heisenberg group ⋮ Gauss-Bonnet theorem in the universal covering group of Euclidean motion group \(E(2)\) with the general left-invariant metric ⋮ Gauss-Bonnet theorems for surfaces in sub-Riemannian three-dimensional manifolds ⋮ Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group ⋮ Intrinsic sub-Laplacian for hypersurface in a contact sub-Riemannian manifold ⋮ Affine generalized Ricci solitons of three-dimensional Lorentzian Lie groups associated to Yano connection ⋮ Affine algebraic Ricci solitons associated to the Yano connections on three-dimensional Lorentzian Lie groups ⋮ Surfaces of genus g ≥ 1 in 3D contact sub-Riemannian manifolds ⋮ Correction to: ``Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group ⋮ Gauss-Bonnet theorems in the BCV spaces and the twisted Heisenberg group ⋮ Gauss-Bonnet theorems for Lorentzian and spacelike surfaces associated to canonical connections in the Lorentzian Heisenberg group ⋮ Generalized Ricci solitons of three-dimensional Lorentzian Lie groups associated canonical connections and Kobayashi-nomizu connections ⋮ The sub-Riemannian limit of curvatures for curves and surfaces and a Gauss-Bonnet theorem in the rototranslation group ⋮ The sub-Riemannian curvature of curves in the Borel subgroup of the group \(\operatorname{SL}(2,\mathbb{R})\) ⋮ The sub-Riemannian limit of curvatures for curves and surfaces and a Gauss-Bonnet theorem in the group of rigid motions of Minkowski plane with general left-invariant metric ⋮ Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane ⋮ Heat content and horizontal mean curvature on the Heisenberg group ⋮ Left-invariant geometries on \(\operatorname{SU}(2)\) are uniformly doubling ⋮ The sub-Riemannian curvature of curves in the group of semiaffine transformations of the Euclidean plane ⋮ A metric interpretation of the geodesic curvature in the Heisenberg group ⋮ Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds ⋮ On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Gauss-Bonnet theorem in sub-Riemannian Heisenberg space \(\mathbb H^1\)
• The fundamental theorem for hypersurfaces in Heisenberg groups
• Steiner's formula in the Heisenberg group
• Regularity of mean curvature flow of graphs on Lie groups free up to step 2
• An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem
• Constant mean curvature surfaces in sub-Riemannian geometry
• A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds
• A ``Gauss-Bonnet formula for contact sub-Riemannian manifolds.
• Minimal surfaces in the Heisenberg group
• A note on the Gauss-Bonnet theorem for Finsler spaces
• Removable sets for Lipschitz harmonic functions on Carnot groups
• Sub-Riemannian calculus on hypersurfaces in Carnot groups
• Notion of convexity in Carnot groups
• Metric normal and distance function in the Heisenberg group
• Uniform estimates of the fundamental solution for a family of hypoelliptic operators
• Sub-Riemannian constant mean curvature surfaces in the Heisenberg group as limits
• Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group \(\mathbb H^n\)
• Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group
• Properly embedded and immersed minimal surfaces in the Heisenberg group
• A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group
• Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups
• Size of characteristic sets and functions with prescribed gradient
• A partial solution of the isoperimetric problem for the Heisenberg group

This page was built for publication: Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group