Comptes Rendus
Article de recherche - Géométrie et Topologie, Mécanique
Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics
[Différentielle du tenseur de déformation en toute dimension et applications aux géodésiques quotient]
Olivier Bisson1  ; Xavier Pennec1
1 Université Côte d’Azur, INRIA, France
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1847-1856.

La décomposition polaire X=WR, avec X∈GL(n,ℝ), W∈𝒮+(n) et R∈𝒪n, suggère une action à droite du groupe orthogonal 𝒪n sur le groupe général linéaire GL(n,ℝ). Équipé de la métrique de Frobenius, le fibré 𝒪n-principal π:X∈GL(n,ℝ)↦X𝒪n∈GL(n,ℝ)/𝒪n devient une submersion Riemannienne. Dans cet article, nous obtenons une expression pour la dérivée en toute dimension de son unique section symétrique s∘π, en termes d’une solution d’une équation de Sylvester. Nous discutons comment résoudre ce type d’équation et vérifions que notre formule coïncide avec celles dérivées dans la littérature en basses dimensions. Nous appliquons notre résultat à la caractérisation des géodésiques de la métrique de Frobenius dans l’espace quotient GL(n,ℝ)/𝒪n.

The polar decomposition X=WR, with X∈GL(n,ℝ), W∈𝒮+(n), and R∈𝒪n, suggests a right action of the orthogonal group 𝒪n on the general linear group GL(n,ℝ). Equipped with the Frobenius metric, the 𝒪n-principal bundle π:X∈GL(n,ℝ)↦X𝒪n∈GL(n,ℝ)/𝒪n becomes a Riemannian submersion. In this note, we derive an expression for the derivative of its unique symmetric section s∘π in any dimension, in terms of a solution to a Sylvester equation. We discuss how to solve this type of equation and verify that our formula coincides with those derived in the literature for low dimensions. We apply our result to the characterization of geodesics of the Frobenius metric in the quotient space GL(n,ℝ)/𝒪n.

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DOI : 10.5802/crmath.692
Classification : 53A17, 53C22, 53C80, 15A24
Keywords: Polar Decomposition, stretch Tensor, quotient Geodesics
Mots-clés : Décomposition polaire, tenseur de déformation, géodésiques quotient

Olivier Bisson 1 ; Xavier Pennec 1

1 Université Côte d’Azur, INRIA, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Olivier Bisson and Xavier Pennec},
     title = {Differential of the {Stretch} {Tensor} for {Any} {Dimension} with {Applications} to {Quotient} {Geodesics}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1847--1856},
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Olivier Bisson; Xavier Pennec. Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1847-1856. doi : 10.5802/crmath.692. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.692/

[1] Rajendra Bhatia; Tanvi Jain; Yongdo Lim On the Bures-Wasserstein distance between positive definite matrices, Expo. Math., Volume 37 (2019) no. 2, pp. 165-191 | DOI | MR | Zbl

[2] Rajendra Bhatia; Peter Rosenthal How and why to solve the operator equation AX-XB=Y, Bull. Lond. Math. Soc., Volume 29 (1997) no. 1, pp. 1-21 | DOI | MR | Zbl

[3] Yi-Chao Chen; Lewis Wheeler Derivatives of the stretch and rotation tensors, J. Elasticity, Volume 32 (1993) no. 3, pp. 175-182 | DOI | MR | Zbl

[4] Florian Jeremy Feppon Riemannian geometry of matrix manifolds for Lagrangian uncertainty quantification of stochastic fluid flows, Master thesis, Massachusetts Institute of Technology (2017)

[5] Sylvestre Gallot; Dominique Hulin; Jacques Lafontaine Riemannian geometry, Universitext, Springer, 1990, xiv+284 pages | DOI | MR | Zbl

[6] Evan S. Gawlik; Melvin Leok Iterative computation of the Fréchet derivative of the polar decomposition, SIAM J. Matrix Anal. Appl., Volume 38 (2017) no. 4, pp. 1354-1379 | DOI | MR | Zbl

[7] Zhong Heng Guo Rates of stretch tensors, J. Elasticity, Volume 14 (1984) no. 3, pp. 263-267 | DOI | MR | Zbl

[8] Gene H. Golub; Charles F. Van Loan Matrix Computations, Johns Hopkins University Press, 2013 | DOI | MR | Zbl

[9] Brian Hall Lie groups, Lie algebras, and representations, Graduate Texts in Mathematics, 222, Springer, 2015, xiv+449 pages (An elementary introduction) | DOI | MR | Zbl

[10] Anne Hoger; Donald E. Carlson On the derivative of the square root of a tensor and Guo’s rate theorems, J. Elasticity, Volume 14 (1984) no. 3, pp. 329-336 | DOI | MR | Zbl

[11] Shoshichi Kobayashi; Katsumi Nomizu Foundations of differential geometry. I, Interscience Publishers, 1963, xi+329 pages | MR | Zbl

[12] Estelle Massart; P.-A. Absil Quotient geometry with simple geodesics for the manifold of fixed-rank positive-semidefinite matrices, SIAM J. Matrix Anal. Appl., Volume 41 (2020) no. 1, pp. 171-198 | DOI | MR | Zbl

[13] Martyn P. Nash; Alexander V. Panfilov Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias, Prog. Biophys. Mol. Biol., Volume 85 (2004) no. 2, pp. 501-522 | DOI

[14] Luciano Rosati Derivatives and rates of the stretch and rotation tensors, J. Elasticity, Volume 56 (1999) no. 3, pp. 213-230 | DOI | MR | Zbl

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