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

Zhang Erchuan; Noakes Lyle

doi:10.3934/jgm.2019015

Article Contents

2019, Volume 11, Issue 2: 277-299. Doi: 10.3934/jgm.2019015

This issue Previous Article Dual pairs for matrix groups Next Article

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

Zhang Erchuan and
Noakes Lyle^,

Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia

^* Corresponding author
^* Corresponding author

To Darryl Holm, with warm greetings and deep respect, on his 70th birthday

Received: February 28, 2018

Revised: January 31, 2019

Published: May 2019

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Abstract

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) $.

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Mathematics Subject Classification: Primary: 58E50, 58E40; Secondary: 53C35, 53A35.

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Figure 1. On the left, the red curve represents $ u $, blue denotes $ \nu $ and green one shows $ \theta $. On the right we plot coordinates $ (x_1, x_2, x_3) $ of the non-null Riemannian cubic $ x $

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Figure 2. On the left, the blue curve represents $ \kappa^2 $ and the red one shows $ \tau^2 $. On the right we plot coordinates $ (x_1, x_2, x_3) $ of the elastic curve $ x = [x_1 x_3;x_3 x_2] $

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