Geodesics and curvatures of left-invariant sub-Riemannian metrics on Lie groups

2018 ◽  
Author(s):  
Valerii Nikolaevich Berestovskii
Keyword(s):  
Lie Groups ◽  
Riemannian Metrics

scholarly journals Milnor-type theorems for left-invariant Riemannian metrics on Lie groups

2016 ◽  
Vol 68 (2) ◽  
pp. 669-684 ◽  
Author(s):  
Takahiro HASHINAGA ◽  
Hiroshi TAMARU ◽  
Kazuhiro TERADA
Keyword(s):  
Lie Groups ◽  
Riemannian Metrics ◽  
Invariant Riemannian Metrics

scholarly journals LEFT-INVARIANT ALMOST PARA-COMPLEX EINSTEINIAN STRUCTURES ON SIX-DIMENSIONAL NILPOTENT LIE GROUPS

2017 ◽  
pp. 88-95
Author(s):  
Nikolay Smolentsev ◽  
Nikolay Smolentsev
Keyword(s):  
Lie Groups ◽  
Complex Structure ◽  
New Left ◽  
Riemannian Metrics ◽  
Nilpotent Lie Groups ◽  
Complex Structures ◽  
Structural Group ◽  
Almost Complex ◽  
Flat Structures ◽  
Simply Connected

As is well known, there are 34 classes of isomorphic simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes admit left-invariant symplectic structures and only 18 admit left-invariant complex structures. There are five six-dimensional nilpotent Lie groups G , which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo- Kӓhlerian, nor almost Hermitian. In this work, these Lie groups are being studied. The aim of the paper is to define new left-invariant geometric structures on the Lie groups under consideration that compensate, in some sense, the absence of symplectic and complex structures. Weakening the closedness requirement of left-invariant 2-forms ω on the Lie groups, non-degenerated 2-forms ω are obtained, whose exterior differential dω is also non-degenerated in Hitchin sense [6]. Therefore, the Hitchin’s operator K dω is defined for the 3-form dω . It is shown that K dω defines an almost complex or almost para-complex structure for G and the couple ( ω, dω ) defines pseudo-Riemannian metrics of signature (2,4) or (3,3), which is Einsteinian for 4 out of 5 considered Lie groups. It gives new examples of multiparametric families of Einstein metrics of signature (3,3) and almost para-complex structures on six-dimensional nilmanifolds, whose structural group is being reduced to SL (3 , R) SO (3 , 3). On each of the Lie groups under consideration, compatible pairs of left-invariant forms (ω, Ω), where Ω = d ω, are obtained. For them the defining properties of half-flat structures are naturally fulfilled: d Ω = 0 and ωΩ = 0. Therefore, the obtained structures are not only almost Einsteinian para-complex, but also pseudo- Riemannian half-flat.

scholarly journals Length spectra of sub-Riemannian metrics on compact Lie groups

2018 ◽  
Vol 296 (2) ◽  
pp. 321-340
Author(s):  
András Domokos ◽  
Matthew Krauel ◽  
Vincent Pigno ◽  
Corey Shanbrom ◽  
Michael VanValkenburgh
Keyword(s):  
Lie Groups ◽  
Riemannian Metrics ◽  
Compact Lie Groups

scholarly journals Elastica in SO(3)

2007 ◽  
Vol 83 (1) ◽  
pp. 105-124 ◽  
Author(s):  
Tomasz Popiel ◽  
Lyle Noakes
Keyword(s):  
Riemannian Manifold ◽  
Computer Graphics ◽  
Variational Problem ◽  
Lie Groups ◽  
Real Line ◽  
Lagrange Equation ◽  
Riemannian Metrics ◽  
Geodesic Curvature ◽  
Unit Speed ◽  
Invariant Riemannian Metrics

AbstractIn a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group SO(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = SO(3), we solve the Euler-Lagrange equation by quadratures.

On the flag curvature of invariant (α,β)-metrics

2016 ◽  
Vol 13 (04) ◽  
pp. 1650039 ◽  
Author(s):  
M. Parhizkar ◽  
D. Latifi
Keyword(s):  
Lie Groups ◽  
Explicit Formula ◽  
Vector Fields ◽  
Homogeneous Spaces ◽  
Riemannian Metrics ◽  
Flag Curvature ◽  
Invariant Vector ◽  
Invariant Riemannian Metrics ◽  
Randers Metrics ◽  
Invariant Vector Fields

In this paper, we consider invariant [Formula: see text]-metrics which are induced by invariant Riemannian metrics [Formula: see text] and invariant vector fields [Formula: see text] on homogeneous spaces. We study the flag curvatures of invariant [Formula: see text]-metrics. We first give an explicit formula for the flag curvature of invariant [Formula: see text]-metrics arising from invariant Riemannian metrics on homogeneous spaces and Lie groups. We then give some explicit formula for the flag curvature of invariant Matsumoto metrics, invariant Kropina metrics and invariant Randers metrics.

scholarly journals Left invariant Randers metrics of Berwald type on tangent Lie groups

2017 ◽  
Vol 15 (01) ◽  
pp. 1850015
Author(s):  
Farhad Asgari ◽  
Hamid Reza Salimi Moghaddam
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Riemannian Metrics ◽  
Randers Metric ◽  
Tangent Lie Group ◽  
Left Invariant Riemannian Metric ◽  
Simply Connected ◽  
Randers Metrics ◽  
Tangent Bundles

Let [Formula: see text] be a Lie group equipped with a left invariant Randers metric of Berward type [Formula: see text], with underlying left invariant Riemannian metric [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are lifted Randers and Riemannian metrics arising from [Formula: see text] and [Formula: see text] on the tangent Lie group [Formula: see text] by vertical and complete lifts. In this paper, we study the relations between the flag curvature of the Randers manifold [Formula: see text] and the sectional curvature of the Riemannian manifold [Formula: see text] when [Formula: see text] is of Berwald type. Then we give all simply connected three-dimensional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors.

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