Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics

Bisson, Olivier; Pennec, Xavier

Comptes Rendus. Mathématique, Tome 362 (2024) no. G12, pp. 1847-1856

The polar decomposition $X=WR$, with $X\in \mathrm{GL}(n,ℝ)$, $W\in {𝒮}_{+}\left(n\right)$, and $R\in {𝒪}_n$, suggests a right action of the orthogonal group ${𝒪}_n$ on the general linear group $\mathrm{GL}(n,ℝ)$. Equipped with the Frobenius metric, the ${𝒪}_n$-principal bundle $\pi :X\in \mathrm{GL}(n,ℝ)\mapsto X{𝒪}_n\in \mathrm{GL}(n,ℝ)/{𝒪}_n$ becomes a Riemannian submersion. In this note, we derive an expression for the derivative of its unique symmetric section $s\circ \pi$ 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 $\mathrm{GL}(n,ℝ)/{𝒪}_n$.

La décomposition polaire $X=WR$, avec $X\in \mathrm{GL}(n,ℝ)$, $W\in {𝒮}_{+}\left(n\right)$ et $R\in {𝒪}_n$, suggère une action à droite du groupe orthogonal ${𝒪}_n$ sur le groupe général linéaire $\mathrm{GL}(n,ℝ)$. Équipé de la métrique de Frobenius, le fibré ${𝒪}_n$-principal $\pi :X\in \mathrm{GL}(n,ℝ)\mapsto X{𝒪}_n\in \mathrm{GL}(n,ℝ)/{𝒪}_n$ devient une submersion Riemannienne. Dans cet article, nous obtenons une expression pour la dérivée en toute dimension de son unique section symétrique $s\circ \pi$, 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 $\mathrm{GL}(n,ℝ)/{𝒪}_n$.

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

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