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 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>

scholarly journals Correction: Çakmak, A. New Type Direction Curves in 3-Dimensional Compact Lie Group Symmetry 2019, 11, 387

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 284
Author(s):  
Ali Çakmak
Keyword(s):  
Lie Group ◽  
Group Symmetry ◽  
Compact Lie Group ◽  
3 Dimensional ◽  
New Type ◽  
Lie Group Symmetry

The authors wish to make the following corrections to their paper [...]

Curves on Surfaces of Constant Width

1966 ◽  
Vol 9 (1) ◽  
pp. 15-22
Author(s):  
William W. Armstrong
Keyword(s):  
Convex Set ◽  
Dimensional Space ◽  
Constant Width ◽  
Smoothness Condition ◽  
3 Dimensional ◽  
Curves On Surfaces ◽  
Class C

A surface S of constant width is the boundary of a convex set K of constant width in euclidean 3-dimensional space E3. (See [l] pp. 127–139. )Our first result concerns the interdependence of five properties which a curve on such a surface may possess. Let S be a surface of constant width D > 0 which satisfies the smoothness condition that it be a 2-dimensional submanifold of E3 of class C2.

scholarly journals Lie Supergroups Obtained from 3-Dimensional Lie Superalgebras Associated to the Adjoint Representation and Having a 2-Dimensional Derived Ideal

2008 ◽  
Vol 2008 ◽  
pp. 1-9
Author(s):  
I. Hernández ◽  
R. Peniche
Keyword(s):  
Lie Group ◽  
Lie Superalgebra ◽  
Adjoint Representation ◽  
Lie Superalgebras ◽  
Base Manifold ◽  
3 Dimensional ◽  
Lie Supergroups

We give the explicit multiplication law of the Lie supergroups for which the base manifold is a 3-dimensional Lie group and whose underlying Lie superalgebrag=g0⊕g1which satisfiesg1=g0,g0acts ong1via the adjoint representation andg0has a 2-dimensional derived ideal.

An Iterative Monte Carlo Scheme for Generating Lie Group-Valued Random Variables

1994 ◽  
Vol 26 (03) ◽  
pp. 616-628
Author(s):  
Mauro Piccioni ◽  
Sergio Scarlatti
Keyword(s):  
Monte Carlo ◽  
Lie Group ◽  
Wide Class ◽  
Approximation Scheme ◽  
Random Variables ◽  
Time Discretization ◽  
Compact Lie Group ◽  
Discretization Step ◽  
Ergodic Diffusion ◽  
Natural Way

In this paper a simple approximation scheme is proposed for the problem of generating and computing expectations of functionals of a wide class of random variables with values in a compact Lie group. The algorithm is suggested by the time-discretization of an ergodic diffusion leaving invariant the distribution of interest. It is shown to converge as the discretization step goes to zero with the iterations in a natural way.

scholarly journals The index of symmetry of three-dimensional Lie groups with a left-invariant metric

2018 ◽  
Vol 18 (4) ◽  
pp. 395-404 ◽  
Author(s):  
Silvio Reggiani
Keyword(s):  
Lie Groups ◽  
Constant Curvature ◽  
Lie Group ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Invariant Metric ◽  
3 Dimensional ◽  
Space Of Constant Curvature ◽  
Positive Index

Abstract We determine the index of symmetry of 3-dimensional unimodular Lie groups with a left-invariant metric. In particular, we prove that every 3-dimensional unimodular Lie group admits a left-invariant metric with positive index of symmetry. We also study the geometry of the quotients by the so-called foliation of symmetry, and we explain in what cases the group fibers over a 2-dimensional space of constant curvature.

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