Riemannian cubics and elastica in the manifold $ \operatorname{SPD}(n) $ of all $ n\times n $ symmetric positive-definite matrices
Erchuan Zhang, Lyle Noakes · The Journal of Geometric Mechanics · 2019
Left Lie reduction is a technique used in the study of curves in bi-invariant Lie groups [32, 33, 40]. Although the manifold $ \operatorname{SPD}(n) $ of all $ n\times n $ symmetric positive-definite matrices is not a Lie group with respect to the standard matrix multiplication, it is a symmetric space with a left action of $ GL(n) $ and an isotropy group $ SO(n) $ leaving the identity matrix fixed. The main purpose of this paper is to extend the method of left Lie reduction to $ \operatorname{SPD}(n) $ and use it to study two second order variational curves: Riemannian cubics and elastica. Riemannian cubics in $ \operatorname{SPD}(n) $ are reduced to so-called Lie quadratics in the Lie algebra $ \mathfrak{gl}(n) $ and geometric analyses are presented. Besides, by using the Frenet-Serret frames and the extended left Lie reduction separately, we investigate elastica in the manifold $ \operatorname{SPD}(n) $. The latter presents a comparatively simple form of the equations for elastica in $ \operatorname{SPD}(n) $.