scholarly journals Gauss—Bonnet Theorems in the Lorentzian Heisenberg Group and the Lorentzian Group of Rigid Motions of the Minkowski Plane

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 173
Author(s):  
Sining Wei ◽  
Yong Wang
Keyword(s):  
Heisenberg Group ◽  
Gaussian Curvature ◽  
Minkowski Plane ◽  
Geodesic Curvature ◽  
Characteristic Points ◽  
Rigid Motions

The aim of this paper was to obtain Gauss–Bonnet theorems on the Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane. At the same time, the sub-Lorentzian limits of Gaussian curvature for surfaces which are C2-smooth in the Lorentzian Heisenberg group away from characteristic points and signed geodesic curvature for curves which are C2-smooth on surfaces are studied. Using a similar method, we also studied the corresponding contents on Lorentzian group of rigid motions of the Minkowski plane.

scholarly journals 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

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Jianyun Guan ◽  
Haiming Liu
Keyword(s):  
Smooth Surface ◽  
Minkowski Plane ◽  
Model Space ◽  
Geodesic Curvature ◽  
Invariant Metric ◽  
Smooth Curves ◽  
Characteristic Points ◽  
Rigid Motions ◽  
Curves On Surfaces ◽  
Bonnet Theorem

The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E 1 , 1 , g λ 1 , λ 2 , where λ 1 ≥ λ 2 > 0 . It provides a natural 2 -parametric deformation family of the Riemannian homogeneous manifold Sol 3 = E 1 , 1 , g 1 , 1 which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in E 1 , 1 , g L λ 1 , λ 2 away from characteristic points and signed geodesic curvature for the Euclidean C 2 -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.

scholarly journals Gauss-Bonnet theorems in the generalized affine group and the generalized BCV spaces

2021 ◽  
Vol 6 (11) ◽  
pp. 11655-11685
Author(s):  
Tong Wu ◽  
◽  
Yong Wang
Keyword(s):  
Gaussian Curvature ◽  
Smooth Surface ◽  
Geodesic Curvature ◽  
Affine Group ◽  
Smooth Curves ◽  
Characteristic Points ◽  
Curves On Surfaces

<abstract><p>In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $ C^2 $-smooth surface in the generalized affine group and the generalized BCV spaces away from characteristic points and signed geodesic curvature for Euclidean $ C^2 $-smooth curves on surfaces. We get Gauss-Bonnet theorems in the generalized affine group and the generalized BCV spaces.</p></abstract>

Estimates of the Green’s Function and its Regular Part on Heisenberg Group Domains

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
Najoua Gamara ◽  
Habiba Guemri
Keyword(s):  
Heisenberg Group ◽  
Green’S Function ◽  
Green's Function ◽  
Forthcoming Paper ◽  
Regular Part ◽  
Kohn Laplacian ◽  
Preliminary Work ◽  
Characteristic Points ◽  
The Given ◽  
Euclidean Domains

AbstractThis paper is a preliminary work on Heisenberg group domains, devoted to the study of the Green’s function for the Kohn Laplacian on domains far away from the set of characteristic points. We give some estimates of the Green’s function, its regular part and their derivatives analogous to those proved by A. Bahri, Y.Y. Li, O. Rey in [1], and O. Rey in [16] for Euclidean domains. While the study of such functions on the set of characteristic points of the given domain will be discussed in a forthcoming paper.

scholarly journals The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss–Bonnet Theorem in the Rototranslation Group

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Haiming Liu ◽  
Jiajing Miao ◽  
Wanzhen Li ◽  
Jianyun Guan
Keyword(s):  
Lie Group ◽  
Smooth Surface ◽  
Euclidean Plane ◽  
Approximation Scheme ◽  
Geodesic Curvature ◽  
Smooth Curves ◽  
3 Dimensional ◽  
Characteristic Points ◽  
Curves On Surfaces ◽  
Bonnet Theorem

The rototranslation group ℛ T is the group comprising rotations and translations of the Euclidean plane which is a 3-dimensional Lie group. In this paper, we use the Riemannian approximation scheme to compute sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. Based on these results, we obtain a Gauss–Bonnet theorem in the rototranslation group.

On singular curves with Gaussian curvature bounds

Analysis ◽  
2018 ◽  
Vol 38 (3) ◽  
pp. 127-136
Author(s):  
Hao Fang ◽  
Weifeng Wo
Keyword(s):  
Gaussian Curvature ◽  
Short Note ◽  
Geodesic Curvature ◽  
Curvature Bounds ◽  
Strictly Convex ◽  
Convex Curves

Abstract In this short note, we consider piece-wise smooth strictly convex curves in {\mathbb{R}^{2}} with prescribed singular angles. Given some geodesic curvature bounds, we give the sharp length estimates.

scholarly journals Applications of Integral Geometry to Geometric Properties of Sets in the 3D-Heisenberg Group

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Yen-Chang Huang
Keyword(s):  
Variational Methods ◽  
Heisenberg Group ◽  
Convex Domain ◽  
Line Segment ◽  
Integral Geometry ◽  
Geometric Probability ◽  
Geometric Quantity ◽  
Classical Integral ◽  
Rigid Motions ◽  
The Relationship

AbstractBy studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H1,we define a density and a measure in the set of horizontal lines. We show that the volume of a convex domain D ⊂ H1 is equal to the integral of the length of chords of all horizontal lines intersecting D. As in classical integral geometry, we also define the kinematic density for PSH(1) and show that the measure of all segments with length l intersecting a convex domain D ⊂ H1 can be represented by the p-area of the boundary ∂D, the volume of D, and 2l. Both results show the relationship between geometric probability and the natural geometric quantity in [10] derived by using variational methods. The probability that a line segment be contained in a convex domain is obtained as an application of our results.

HELICOIDAL SURFACES WITH PRESCRIBED CURVATURES IN Nil3

2013 ◽  
Vol 24 (14) ◽  
pp. 1350107 ◽  
Author(s):  
DAE WON YOON ◽  
DONG-SOO KIM ◽  
YOUNG HO KIM ◽  
JAE WON LEE
Keyword(s):  
Heisenberg Group ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Smooth Functions ◽  
Constant Gaussian Curvature ◽  
3 Dimensional ◽  
Helicoidal Surfaces ◽  
Prescribed Gaussian Curvature

In the present paper, we study helicoidal surfaces in the 3-dimensional Heisenberg group Nil3. Also, we construct helicoidal surfaces in Nil3 with prescribed Gaussian curvature or mean curvature given by smooth functions. As the results, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature.

scholarly journals Biharmonic S-curves according to Sabban frame in Heisenberg group Heis³

2012 ◽  
Vol 31 (1) ◽  
pp. 205 ◽  
Author(s):  
Talat Körpınar ◽  
Essin Turhan
Keyword(s):  
Heisenberg Group ◽  
Geodesic Curvature ◽  
Biharmonic Curves

In this paper, we study biharmonic curves accordig to Sabban frame in the Heisenberg group Heis³. We characterize the biharmonic curves in terms of their geodesic curvature and we prove that all of  biharmonic curves are helices in the Heisenberg group Heis³. Finally, we find out their explicit parametric equations according to Sabban Frame.

scholarly journals Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tommaso Rossi
Keyword(s):  
Heisenberg Group ◽  
Mean Curvature ◽  
Smooth Surface ◽  
Characteristic Point ◽  
Analytic Surface ◽  
Characteristic Set ◽  
Characteristic Points ◽  
The Mean ◽  
Real Analytic ◽  
Real Analytic Surface

Abstract We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this paper is the introduction of a concept of a mildly degenerate characteristic point for a smooth surface of the Heisenberg group, in a neighborhood of which the sub-Riemannian mean curvature is integrable (with respect to the perimeter measure induced by the Euclidean structure). As a consequence, we partially answer to a question posed by Danielli, Garofalo and Nhieu in [D. Danielli, N. Garofalo and D. M. Nhieu, Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group, Proc. Amer. Math. Soc. 140 2012, 3, 811–821], proving that the mean curvature of a real-analytic surface with discrete characteristic set is locally integrable.

Sign in / Sign up

Export Citation Format

Share Document