Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics
The polar decomposition $X=WR$, with $X \in \mathrm{GL}(n, \mathbb{R})$, $W \in \mathcal{S}_+(n)$, and $R \in \mathcal{O}_n$, suggests a right action of the orthogonal group $\mathcal{O}_n$ on the general linear group $\mathrm{GL}(n, \mathbb{R})$. Equipped with the Frobenius metric, the $ \mathcal{O...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.692/ |
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| author | Bisson, Olivier Pennec, Xavier |
| author_facet | Bisson, Olivier Pennec, Xavier |
| author_sort | Bisson, Olivier |
| collection | DOAJ |
| description | The polar decomposition $X=WR$, with $X \in \mathrm{GL}(n, \mathbb{R})$, $W \in \mathcal{S}_+(n)$, and $R \in \mathcal{O}_n$, suggests a right action of the orthogonal group $\mathcal{O}_n$ on the general linear group $\mathrm{GL}(n, \mathbb{R})$. Equipped with the Frobenius metric, the $ \mathcal{O}_n $-principal bundle $\pi : X \in \mathrm{GL}(n, \mathbb{R}) \mapsto X\mathcal{O}_n \in \mathrm{GL} (n, \mathbb{R}) / \mathcal{O}_n$ 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 $\mathrm{GL} (n, \mathbb{R}) / \mathcal{O}_n$. |
| format | Article |
| id | doaj-art-8c0eb5bb70254b238914f3dc898e5d48 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-8c0eb5bb70254b238914f3dc898e5d482025-02-07T11:26:38ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G121847185610.5802/crmath.69210.5802/crmath.692Differential of the Stretch Tensor for Any Dimension with Applications to Quotient GeodesicsBisson, Olivier0Pennec, Xavier1Université Côte d’Azur, INRIA, FranceUniversité Côte d’Azur, INRIA, FranceThe polar decomposition $X=WR$, with $X \in \mathrm{GL}(n, \mathbb{R})$, $W \in \mathcal{S}_+(n)$, and $R \in \mathcal{O}_n$, suggests a right action of the orthogonal group $\mathcal{O}_n$ on the general linear group $\mathrm{GL}(n, \mathbb{R})$. Equipped with the Frobenius metric, the $ \mathcal{O}_n $-principal bundle $\pi : X \in \mathrm{GL}(n, \mathbb{R}) \mapsto X\mathcal{O}_n \in \mathrm{GL} (n, \mathbb{R}) / \mathcal{O}_n$ 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 $\mathrm{GL} (n, \mathbb{R}) / \mathcal{O}_n$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.692/Polar Decompositionstretch Tensorquotient Geodesics |
| spellingShingle | Bisson, Olivier Pennec, Xavier Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics Comptes Rendus. Mathématique Polar Decomposition stretch Tensor quotient Geodesics |
| title | Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics |
| title_full | Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics |
| title_fullStr | Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics |
| title_full_unstemmed | Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics |
| title_short | Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics |
| title_sort | differential of the stretch tensor for any dimension with applications to quotient geodesics |
| topic | Polar Decomposition stretch Tensor quotient Geodesics |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.692/ |
| work_keys_str_mv | AT bissonolivier differentialofthestretchtensorforanydimensionwithapplicationstoquotientgeodesics AT pennecxavier differentialofthestretchtensorforanydimensionwithapplicationstoquotientgeodesics |