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 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 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 A simple differential geometry for complex networks

2020 ◽  
pp. 1-28
Author(s):  
Emil Saucan ◽  
Areejit Samal ◽  
Jürgen Jost
Keyword(s):  
Differential Geometry ◽  
Complex Networks ◽  
Ricci Curvature ◽  
Real World ◽  
Large Scale ◽  
Metric Spaces ◽  
Geodesic Curvature ◽  
Higher Dimensional ◽  
Large Scale Networks ◽  
Bonnet Theorem

Abstract We introduce new definitions of sectional, Ricci, and scalar curvatures for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in general metric spaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicable to unweighted or weighted and undirected or directed networks and are more intuitive and easier to compute than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature allow us to give a network analogue of the classical local Gauss–Bonnet theorem. Furthermore, we propose even simpler and more intuitive proxies for the Haantjes curvature that allow for even faster and easier computations in large-scale networks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures. Lastly, we also investigate the behavior, both on model and real-world networks, of the curvatures introduced herein with more established notions of Ricci curvature and other widely used network measures.

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.

Spectral asymmetry and Riemannian Geometry. I

1975 ◽  
Vol 77 (1) ◽  
pp. 43-69 ◽  
Author(s):  
M. F. Atiyah ◽  
V. K. Patodi ◽  
I. M. Singer
Keyword(s):  
Riemannian Geometry ◽  
Integral Formula ◽  
Geodesic Curvature ◽  
Gauss Curvature ◽  
Boundary Integral ◽  
Manifolds With Boundary ◽  
Cohomological Invariants ◽  
Significant Difference ◽  
Image Position ◽  
Bonnet Theorem

1. Introduction. The main purpose of this paper is to present a generalization of Hirzebruch's signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper. To explain this difference let us begin by recalling that the classical Gauss–Bonnet theorem for a surface X with boundary Y asserts that the Euler characteristic E(X) is given by a formula:where K is the Gauss curvature of X and σ is the geodesic curvature of Y in X. In particular if, near the boundary, X is isometric to the product Y x R+, the boundary integral in (1.1) vanishes and the formula is the same as for closed surfaces. Similar remarks hold in higher dimensions. Now if X is a closed oriented Riemannian manifold of dimension 4, there is another formula relating cohomological invariants with curvature in addition to the Gauss–Bonnet formula. This expresses the signature of the quadratic form on H2(X, R) by an integral formulawhere p1 is the differential 4-form representing the first Pontrjagin class and is given in terms of the curvature matrix R by p1 = (2π)−2Tr R2. It is natural to ask if (1.2) continues to hold for manifolds with boundary, provided the metric is a product near the boundary. Simple examples show that this is false, so that in general

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