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>

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 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.

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 Solitons of the midpoint mapping and affine curvature

2021 ◽  
Vol 112 (1) ◽  
Author(s):  
Christine Rademacher ◽  
Hans-Bert Rademacher
Keyword(s):  
Differential Equation ◽  
Large Class ◽  
Affine Group ◽  
Arc Length ◽  
Smooth Curves ◽  
Affine Map ◽  
Affine Curvature

AbstractFor a polygon $$x=(x_j)_{j\in \mathbb {Z}}$$ x = ( x j ) j ∈ Z in $$\mathbb {R}^n$$ R n we consider the midpoints polygon $$(M(x))_j=\left( x_j+x_{j+1}\right) /2.$$ ( M ( x ) ) j = x j + x j + 1 / 2 . We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on $$\mathbb {R}^n.$$ R n . These smooth curves are also characterized as solutions of the differential equation $$\dot{c}(t)=Bc (t)+d$$ c ˙ ( t ) = B c ( t ) + d for a matrix B and a vector d. For $$n=2$$ n = 2 these curves are curves of constant generalized-affine curvature $$k_{ga}=k_{ga}(B)$$ k ga = k ga ( B ) depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.

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 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.

Wavelet analysis on some smooth surface with nonzero constant Gaussian curvature

2018 ◽  
Vol 16 (01) ◽  
pp. 1850007
Author(s):  
Xiaohui Zhou ◽  
Baoqin Wang
Keyword(s):  
Wavelet Transform ◽  
Wavelet Analysis ◽  
Gaussian Curvature ◽  
Smooth Surface ◽  
Translation Operator ◽  
Discrete Wavelet ◽  
General Area ◽  
Constant Gaussian Curvature ◽  
Nonzero Constant ◽  
Area Preserving

According to the theory of wavelet analysis on [Formula: see text], the wavelet analysis on a smooth surface [Formula: see text] with nonzero constant Gaussian curvature will be discussed systematically in this paper. First, a general area-preserving projection from a smooth surface to the plane will be presented by the Gaussian projection and the area-preserving projection on the sphere. Then the continuous wavelet transform and its inverse transform on a smooth surface [Formula: see text] with nonzero constant Gaussian curvature will be discussed by a general area-preserving projection, relative dilation operator and translation operator. Further, according to the multi-resolution analysis on a smooth surface, the discrete wavelet transform and relative properties will be investigated systematically, including the two-scale equations of the wavelet function, orthogonality and so on. Finally, two numerical examples will be given.

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