Geodesic motion on $\mathsf{SL}(n)$ with the Hilbert-Schmidt metric

TL;DR

This paper explores the complex geometry of geodesics on the special linear group with the Hilbert-Schmidt metric, revealing new dynamics, classifications, and stability properties in higher dimensions.

Contribution

It generalizes known results from 2D and 3D to higher dimensions, introduces new solution classifications, and analyzes stability of specific geodesic flows.

Findings

Higher-dimensional geodesics exhibit complex dynamics.

Existence of purely rotational solutions in even dimensions.

Identification of stable and unstable geodesic families.

Abstract

We study the geometry of geodesics on $\mathsf{SL}(n)$, equipped with the Hilbert-Schmidt metric which makes it a Riemannian manifold. These geodesics are known to be related to affine motions of incompressible ideal fluids. The $n = 2$ case is special and completely integrable, and the geodesic motion was completely described by Roberts, Shkoller, and Sideris; when $n = 3$, Sideris demonstrated some interesting features of the dynamics and analyzed several classes of explicit solutions. Our analysis shows that the geodesics in higher dimensions exhibit much more complex dynamics. We generalize the Virial-identity-based criterion of unboundedness of geodesic given by Sideris, and use it to give an alternative proof of the classification of geodesics in 2D obtained by Roberts--Shkoller--Sideris. We study several explicit families of solutions in general dimensions that generalize those found by Sideris in 3D. We additionally classify all "exponential type" geodesics, and use it to demonstrate the existence of purely rotational solutions whenever $n$ is even. Finally, we study "block diagonal" solutions using a new formulation of the geodesic equation in first order form, that isolates the conserved angular momenta. This reveals the existence of a rich family of bounded geodesic motions in even dimension $n \geq 4$. This in particular allows us to conclude that the generalization of the swirling and shear flows of Sideris to even dimensions $n \geq 4$ are in fact dynamically unstable.