Left lie reduction for curves in homogeneous spaces
Abstract
Let H be a closed subgroup of a connected finite-dimensional Lie group G, where the canonical projection π : G → G/H is a Riemannian submersion with respect to a bi-invariant Riemannian metric on G. Given a C∞ curve x : [a, b] → G/H, let x~:[a,b]→G be the horizontal lifting of x with x~(a)=e, where e denotes the identity of G. When (G, H) is a Riemannian symmetric pair, we prove that the left Lie reductionV(t):=x~(t)−1x~˙(t) of x~˙(t) for t ∈ [a, b] can be identified with the parallel pullbackP(t) of the velocity vector x˙(t) from x(t) to x(a) along x. Then left Lie reductions are used to investigate Riemannian cubics, Riemannian cubics in tension and elastica in homogeneous spaces G/H. Simplifications of reduced equations are found when (G, H) is a Riemannian symmetric pair. These equations are compared with equations known for curves in Lie groups, focusing on the special case of Riemannian cubics in the 3-dimensional unit sphere S3.
RAS ID
36837
Document Type
Journal Article
Date of Publication
2018
Funding Information
China Scholarship Council University of Western Australia
School
School of Science
Copyright
subscription content
Publisher
Springer
Comments
Zhang, E., & Noakes, L. (2018). Left Lie reduction for curves in homogeneous spaces. Advances in Computational Mathematics, 44(5), 1673-1686. https://doi.org/10.1007/s10444-018-9601-0