Differential of the Stretch Tensor for Any Dimension with Applications to Quotient Geodesics - Bisson, Olivier; Pennec, Xavier (Comptes Rendus. Mathématique)
The polar decomposition $X=WR$, with $X \in GL(n, \mathbb{R})$, $W \in \mathcal{S}_+(n)$, and $R \in \mathcal{O}_n$, suggests a right action of the orthogonal group $\mathcal{O}_n$ on the general linear group $GL(n, \mathbb{R})$. Equipped with the Frobenius metric, the $ \mathcal{O}_n $-principal bundle $π : X \in GL(n, \mathbb{R}) \mapsto X\mathcal{O}_n \in GL (n, \mathbb{R}) / \mathcal{O}_n$ becomes a Riemannian submersion. In this note, we derive an expression for the derivative of its unique symmetric section $ s ° 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 $GL (n, \mathbb{R}) / \mathcal{O}_n$.