Left-invariant Riemannian metrics on four-dimensional unimodular Lie groups with zero-divergence Weyl tensor

2010 ◽  
Vol 81 (2) ◽  
pp. 298-300 ◽  
Author(s):  
O. P. Gladunova ◽  
V. V. Slavskii
Keyword(s):  
Lie Groups ◽  
Weyl Tensor ◽  
Riemannian Metrics ◽  
Invariant 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 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.

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