The polar decomposition , with , , and , suggests a right action of the orthogonal group on the general linear group . Equipped with the Frobenius metric, the -principal bundle becomes a Riemannian submersion. In this note, we derive an expression for the derivative of its unique symmetric section 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 .
La décomposition polaire , avec , et , suggère une action à droite du groupe orthogonal sur le groupe général linéaire . Équipé de la métrique de Frobenius, le fibré -principal devient une submersion Riemannienne. Dans cet article, nous obtenons une expression pour la dérivée en toute dimension de son unique section symétrique , 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 .
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Keywords: Polar Decomposition, stretch Tensor, quotient Geodesics
Mots-clés : Décomposition polaire, tenseur de déformation, géodésiques quotient
Olivier Bisson 1; Xavier Pennec 1
@article{CRMATH_2024__362_G12_1847_0, 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}, publisher = {Acad\'emie des sciences, Paris}, volume = {362}, year = {2024}, doi = {10.5802/crmath.692}, language = {en}, }
TY - JOUR AU - Olivier Bisson AU - Xavier Pennec TI - Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics JO - Comptes Rendus. Mathématique PY - 2024 SP - 1847 EP - 1856 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.692 LA - en ID - CRMATH_2024__362_G12_1847_0 ER -
%0 Journal Article %A Olivier Bisson %A Xavier Pennec %T Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics %J Comptes Rendus. Mathématique %D 2024 %P 1847-1856 %V 362 %I Académie des sciences, Paris %R 10.5802/crmath.692 %G en %F CRMATH_2024__362_G12_1847_0
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/
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