Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics
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| Title: | Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics |
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| Authors: | Bisson, Olivier, Pennec, Xavier |
| Contributors: | Bisson, Olivier |
| Source: | Comptes Rendus. Mathématique, Vol 362, Iss G12, Pp 1847-1856 (2024) |
| Publisher Information: | Cellule MathDoc/Centre Mersenne, 2024. |
| Publication Year: | 2024 |
| Subject Terms: | Quotient Geodesics, Polar Decomposition, Stretch Tensor, quotient Geodesics, QA1-939, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], Mathematics, stretch Tensor |
| Description: | The polar decomposition X=WR, with X∈GL(n,ℝ), W∈𝒮 + (n), and R∈𝒪 n , suggests a right action of the orthogonal group 𝒪 n on the general linear group GL(n,ℝ). Equipped with the Frobenius metric, the 𝒪 n -principal bundle π:X∈GL(n,ℝ)↦X𝒪 n ∈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∘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)/𝒪 n . |
| Document Type: | Article |
| File Description: | application/pdf |
| Language: | English |
| ISSN: | 1778-3569 |
| DOI: | 10.5802/crmath.692 |
| Access URL: | https://doaj.org/article/8c0eb5bb70254b238914f3dc898e5d48 https://hal.science/hal-04831093v1 https://doi.org/10.5802/crmath.692 https://hal.science/hal-04831093v1/document |
| Rights: | CC BY |
| Accession Number: | edsair.doi.dedup.....bcff328ee3afca859a727a63f4c724e5 |
| Database: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstract: | The polar decomposition X=WR, with X∈GL(n,ℝ), W∈𝒮 + (n), and R∈𝒪 n , suggests a right action of the orthogonal group 𝒪 n on the general linear group GL(n,ℝ). Equipped with the Frobenius metric, the 𝒪 n -principal bundle π:X∈GL(n,ℝ)↦X𝒪 n ∈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∘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)/𝒪 n . |
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| ISSN: | 17783569 |
| DOI: | 10.5802/crmath.692 |