Left Lie reduction for curves in homogeneous spaces

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Abstract

Let H be a closed subgroup of a connected finite-dimensional Lie group G, where the canonical projection π : GG/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 \(\tilde {x}:[a,b]\rightarrow G\) be the horizontal lifting of x with \(\tilde {x}(a)=e\), where e denotes the identity of G. When (G, H) is a Riemannian symmetric pair, we prove that the left Lie reduction\(V(t):=\tilde x(t)^{-1}\dot {\tilde x}(t)\) of \(\dot {\tilde x}(t)\) for t ∈ [a, b] can be identified with the parallel pullbackP(t) of the velocity vector \(\dot {x}(t)\) 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.

Keywords

Lie reduction Homogeneous space Symmetric space Cubics Cubics in tension Elastica 

Mathematics Subject Classification 2010

53A17 58E50 53C35 53B50 

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Notes

Acknowledgments

The authors wish to express their appreciation to the editor and anonymous reviewers for their helpful comments and suggestions which proved the presentation of this paper. The first author would like to express sincere thanks to China Scholarship Council (CSC) and The university of Western Australia (UWA) for finical support.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Western AustraliaPerthAustralia

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