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
2007 ◽  
Vol 83 (1) ◽  
pp. 105-124 ◽  
Author(s):  
Tomasz Popiel ◽  
Lyle Noakes

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.


2016 ◽  
Vol 13 (04) ◽  
pp. 1650039 ◽  
Author(s):  
M. Parhizkar ◽  
D. Latifi

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.


2017 ◽  
Vol 28 (06) ◽  
pp. 1750048 ◽  
Author(s):  
Takahiro Hashinaga ◽  
Hiroshi Tamaru

In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.


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