Riemannian cubics and elastica in the manifold \begin{document}$ \operatorname{SPD}(n) $\end{document} of all \begin{document}$ n\times n $\end{document} symmetric positive-definite matrices

Abstract

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