RIEMANNIAN CUBICS AND ELASTICA IN THE MANIFOLD SPD(n) OF ALL n x n SYMMETRIC POSITIVE-DEFINITE MATRICES

Left Lie reduction is a technique used in the study of curves in bi-invariant Lie groups [32, 33, 40]. Although the manifold SPD(n) of all n x 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 SPD(n) and use it to study two second order variational curves: Riemannian cubics and elastica. Riemannian cubics in SPD(n) are reduced to so-called Lie quadratics in the Lie algebra 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 SPD(n). The latter presents a comparatively simple form of the equations for elastica in SPD(n).