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.

scholarly journals Sectional and Ricci Curvature for Three-Dimensional Lie Groups

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Gerard Thompson ◽  
Giriraj Bhattarai
Keyword(s):  
Automorphism Group ◽  
Lie Groups ◽  
Ricci Curvature ◽  
Lie Group ◽  
Systematic Study ◽  
Three Dimensional ◽  
Invariant Metrics ◽  
Curvature Function ◽  
Invariant Metric ◽  
Curvature Tensors

Formulas for the Riemann and Ricci curvature tensors of an invariant metric on a Lie group are determined. The results are applied to a systematic study of the curvature properties of invariant metrics on three-dimensional Lie groups. In each case the metric is reduced by using the automorphism group of the associated Lie algebra. In particular, the maximum and minimum values of the sectional curvature function are determined.

scholarly journals Topological Loops with Decomposable Solvable Multiplication Groups

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Ameer Al-Abayechi ◽  
Ágota Figula
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Solvable Lie Groups ◽  
3 Dimensional ◽  
Multiplication Group ◽  
Simply Connected ◽  
Solvable Lie Group

AbstractIn this paper we deal with the class $$\mathcal {C}$$ C of decomposable solvable Lie groups having dimension six. We determine those Lie groups in $$\mathcal {C}$$ C and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.

Classification of 3-dimensional left-invariant almost paracontact metric structures

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Giovanni Calvaruso ◽  
Antonella Perrone
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Complete Classification ◽  
Natural Assumption ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Metric Structures

AbstractWe study left-invariant almost paracontact metric structures on arbitrary three-dimensional Lorentzian Lie groups. We obtain a complete classification and description under a natural assumption, which includes relevant classes as normal and almost para-cosymplectic structures, and we investigate geometric properties of these structures.

Three-Dimensional Anatomic Characterization of the Canine Laryngeal Abductor and Adductor Musculature

2000 ◽  
Vol 109 (5) ◽  
pp. 505-513 ◽  
Author(s):  
Corey W. Mineck ◽  
Roger Chan ◽  
Niro Tayama ◽  
Ingo R. Titze
Keyword(s):  
Vocal Fold ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Cross Sectional ◽  
3 Dimensional ◽  
Moment Arms ◽  
Intrinsic Laryngeal Muscle ◽  
Posterior Cricoarytenoid ◽  
Intrinsic Laryngeal Muscles

The biomechanics of vocal fold abduction and adduction during phonation, respiration, and airway protection are not completely understood. Specifically, the rotational and translational forces on the arytenoid cartilages that result from intrinsic laryngeal muscle contraction have not been fully described. Anatomic data on the lines of action and moment arms for the intrinsic laryngeal muscles are also lacking. This study was conducted to quantify the 3-dimensional orientations and the relative cross-sectional areas of the intrinsic abductor and adductor musculature of the canine larynx. Eight canine larynges were used to evaluate the 3 muscles primarily responsible for vocal fold abduction and adduction: the posterior cricoarytenoid, the lateral cricoarytenoid, and the interarytenoid muscles. Each muscle was exposed and divided into discrete fiber bundles whose coordinate positions were digitized in 3-dimensional space. The mass, length, relative cross-sectional area, and angle of orientation for each muscle bundle were obtained to allow for the calculations of average lines of action and moment arms for each muscle. This mapping of the canine laryngeal abductor and adductor musculature provides important anatomic data for use in laryngeal biomechanical modeling. These data may also be useful in surgical procedures such as arytenoid adduction.

3D Laser Scanner and its Application in Engineering Survey

2013 ◽  
Vol 671-674 ◽  
pp. 2111-2114
Author(s):  
Yan Ping Feng ◽  
Wei Guo Li ◽  
Li Bing Yang ◽  
Yan Li Gao ◽  
Wen Bin Li
Keyword(s):  
Laser Scanning ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Laser Scanner ◽  
Target Object ◽  
Scanning System ◽  
3 Dimensional ◽  
Working Principle ◽  
Laser Scanning System ◽  
Three Dimensional Space

3D laser scanning system is to use laser ranging principle to record intensively the 3D coordinates, reflectivity and texture information on the surface of the target object. It makes a real record of the three-dimensional space, which makes traditional measurement be released from the limit that couldn’t be exceeded in the past, and let the measurement precision up to a new level. At the same time, it has provided extensive researches with better help than ever. This paper mainly discusses the characteristics, working principle, application and future development of the ground 3 dimensional laser scanner.

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