TY - JOUR

T1 - Left Lie reduction for curves in homogeneous spaces

AU - Zhang, Erchuan

AU - Noakes, Lyle

PY - 2018/10

Y1 - 2018/10

N2 - 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 (Formula presented.) be the horizontal lifting of x with (Formula presented.), where e denotes the identity of G. When (G, H) is a Riemannian symmetric pair, we prove that the left Lie reduction(Formula presented.) of (Formula presented.) for t ∈ [a, b] can be identified with the parallel pullbackP(t) of the velocity vector (Formula presented.) 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.

AB - 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 (Formula presented.) be the horizontal lifting of x with (Formula presented.), where e denotes the identity of G. When (G, H) is a Riemannian symmetric pair, we prove that the left Lie reduction(Formula presented.) of (Formula presented.) for t ∈ [a, b] can be identified with the parallel pullbackP(t) of the velocity vector (Formula presented.) 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.

KW - Cubics

KW - Cubics in tension

KW - Elastica

KW - Homogeneous space

KW - Lie reduction

KW - Symmetric space

UR - http://www.scopus.com/inward/record.url?scp=85044740433&partnerID=8YFLogxK

U2 - 10.1007/s10444-018-9601-0

DO - 10.1007/s10444-018-9601-0

M3 - Article

AN - SCOPUS:85044740433

VL - 44

SP - 1673

EP - 1686

JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

SN - 1019-7168

IS - 5

ER -