Transitivity theorems in Lie geometry

1997 ◽  
Vol 58 (1-2) ◽  
pp. 87-94
Author(s):  
Yi Chen
Keyword(s):  
Lie Geometry

scholarly journals Isothermic surfaces in sphere geometries as Moutard nets

2007 ◽  
Vol 463 (2088) ◽  
pp. 3171-3193 ◽  
Author(s):  
Alexander I Bobenko ◽  
Yuri B Suris
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Gauss Map ◽  
Point Property ◽  
Property A ◽  
Lie Geometry ◽  
Möbius Geometry ◽  
Isothermic Surfaces ◽  
Three Dimensional Space

We give an elaborated treatment of discrete isothermic surfaces and their analogues in different geometries (projective, Möbius, Laguerre and Lie). We find the core of the theory to be a novel characterization of discrete isothermic nets as Moutard nets. The latter are characterized by the existence of representatives in the space of homogeneous coordinates satisfying the discrete Moutard equation. Moutard nets admit also a projective geometric characterization as nets with planar faces with a five-point property: a vertex and its four diagonal neighbours span a three-dimensional space. Restricting the projective theory to quadrics, we obtain Moutard nets in sphere geometries. In particular, Moutard nets in Möbius geometry are shown to coincide with discrete isothermic nets. The five-point property, in this particular case, states that a vertex and its four diagonal neighbours lie on a common sphere, which is a novel characterization of discrete isothermic surfaces. Discrete Laguerre isothermic surfaces are defined through the corresponding five-plane property, which requires that a plane and its four diagonal neighbours share a common touching sphere. Equivalently, Laguerre isothermic surfaces are characterized by having an isothermic Gauss map. S-isothermic surfaces as an instance of Moutard nets in Lie geometry are also discussed.

scholarly journals Lie geometry of 2×2 Markov matrices

2013 ◽  
Vol 327 ◽  
pp. 88-90 ◽  
Author(s):  
Jeremy G. Sumner
Keyword(s):  
Lie Geometry

Lie Geometry

2021 ◽  
pp. 71-80
Author(s):  
Alexander I. Bobenko ◽  
Carl O. R. Lutz ◽  
Helmut Pottmann ◽  
Jan Techter
Keyword(s):  
Lie Geometry

AFFINE COVERS OF LIE GEOMETRIES AND THE AMALGAMATION PROPERTY

2002 ◽  
Vol 85 (3) ◽  
pp. 513-563 ◽  
Author(s):  
AMBAR CHOWDHURY ◽  
BRADD HART ◽  
ŽELJKO SOKOLOVIĆ
Keyword(s):  
Amalgamation Property ◽  
Subject Classification ◽  
Lie Geometry ◽  
Small Index Property ◽  
Small Index

Certain classes of smoothly approximable structures — the class of affine covers of Lie geometries — are shown to have the amalgamation property. In particular, this shows that any affine cover of a Lie geometry has the small index property.2000 Mathematical Subject Classification: 03C45.

scholarly journals An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library

2019 ◽  
Vol 11 (3) ◽  
Author(s):  
Vladimir Kisil
Keyword(s):  
Continued Fractions ◽  
Linear Equations ◽  
Three Dimensional ◽  
Conformal Invariant ◽  
System Of Linear Equations ◽  
Quadratic Relation ◽  
Conformally Invariant ◽  
Lie Geometry ◽  
Symbolic Data

We propose to consider ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. ``to be orthogonal'', ``to be tangent'', etc.), as new objects in an extended M\"obius--Lie geometry. It was recently demonstrated in several related papers, that such ensembles of cycles naturally parameterize many other conformally-invariant families of objects, e.g. loxodromes or continued fractions. The paper describes a method, which reduces a collection of conformally in\-vari\-ant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation. To show its usefulness, the method is implemented as a {\CPP} library. It operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures. Numeric calculations can be done in exact or approximate arithmetic. In the two- and three-dimensional cases illustrations and animations can be produced. An interactive {\Python} wrapper of the library is provided as well.

scholarly journals Lie geometry of flat fronts in hyperbolic space

2010 ◽  
Vol 348 (11-12) ◽  
pp. 661-664 ◽  
Author(s):  
Francis E. Burstall ◽  
Udo Hertrich-Jeromin ◽  
Wayne Rossman
Keyword(s):  
Hyperbolic Space ◽  
Lie Geometry

scholarly journals Surfaces in Möbius geometry

1992 ◽  
Vol 125 ◽  
pp. 53-72 ◽  
Author(s):  
Changping Wang
Keyword(s):  
Transformation Group ◽  
Möbius Transformation ◽  
Lie Geometry ◽  
Möbius Geometry ◽  
Lie Sphere Geometry ◽  
Lie Transformation ◽  
Unit Tangent Bundle ◽  
Sphere Geometry ◽  
Mobius Transformation ◽  
Lie Sphere

Our purpose in this paper is to give a basic theory of Möbius differential geometay. In such geometry we study the properties of hypersurfaces in unit sphere Sn which are invariant under the Möbius transformation group on Sn.Since any Möbius transformation takes oriented spheres in Sn to oriented spheres, we can regard the Möbius transformation group Gn as a subgroup MGn of the Lie transformation group on the unit tangent bundle USn of Sn. Furthermore, we can represent the immersed hypersurfaces in Sn by a class of Lie geometry hypersurfaces (cf. [9]) called Möbius hypersurfaces. Thus we can use the concepts and the techniques in Lie sphere geometry developed by U. Pinkall ([8], [9]), T. Cecil and S. S. Chern [2] to study the Möbius differential geometry.

Fault Detection and Isolation of Nonlinear Systems: An Unknown Input Observer Approach With Sum-of-Squares Techniques

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Jun Xu ◽  
Kai-Yew Lum ◽  
Lihua Xie ◽  
Ai-Poh Loh
Keyword(s):  
Nonlinear Systems ◽  
Fault Detection ◽  
Design Method ◽  
Design Procedure ◽  
Sum Of Squares ◽  
Fault Detection And Isolation ◽  
Unknown Input ◽  
Unknown Input Observer ◽  
Lie Geometry ◽  
Rank Constraint

This paper presents a novel nonlinear unknown input observer (UIO) design method for fault detection and isolation (FDI) of a class of nonlinear affine systems. By using sum-of-squares (SOS) theory and Lie geometry as the main tools, we demonstrate how to relax the rank constraint in the traditional UIO approach and simplify the design procedure, especially for the polynomial nonlinear systems. Meanwhile, we show that the detection and isolation thresholds based on the L2 gains can be easily obtained via optimization formulated in terms of SOS. Simulation examples are given to illustrate the design procedure and the advantages.

Lie geometry of linear Weingarten surfaces

2012 ◽  
Vol 350 (7-8) ◽  
pp. 413-416 ◽  
Author(s):  
Francis E. Burstall ◽  
Udo Hertrich-Jeromin ◽  
Wayne Rossman
Keyword(s):  
Weingarten Surfaces ◽  
Lie Geometry
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