Details

Autor(en) / Beteiligte

• Zhang, Erchuan
• Noakes, Lyle

Titel

Left Lie reduction for curves in homogeneous spaces

Ist Teil von

• Advances in computational mathematics, 2018-10, Vol.44 (5), p.1673-1686

Ort / Verlag

New York: Springer US

Erscheinungsjahr

2018

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• 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 reduction V ( t ) : = x ~ ( t ) − 1 x ~ ̇ ( t ) of x ~ ̇ ( t ) for t ∈ [ a , b ] can be identified with the parallel pullback P ( 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 S 3 .