Left Lie reduction for curves in homogeneous spaces

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.