Geodesics on adjoint orbits of $SL(n, \mathbb{R})$

TL;DR

This paper investigates geodesics on adjoint orbits of SL(n, R) with SO(n)-invariant metrics, translating the problem into tangent bundle geometry of SO(n)-flag manifolds and providing explicit descriptions, especially for SL(2, R).

Contribution

It introduces a geometric approach using tangent bundles and Lie theory to explicitly describe geodesics on adjoint orbits of SL(n, R).

Findings

Geodesic equations are described via the Sasaki metric.

Explicit solutions are obtained for SL(2, R).

The approach links Lie theory with differential geometry of flag manifolds.

Abstract

In this paper we study geodesics on adjoint orbits of $SL(n,\mathbb{R})$ equipped with $SO(n)$-invariant metrics (maximal compact subgroup). Our main technique is translate this problem into a geometric problem in the tangent bundle of certain $SO(n)$-flag manifolds and describe the geodesics equations with respect to the Sasaki metric on tangent bundle. We also use tools of Lie Theory in order to obtain some explicit description of families of geodesics. We deal with the case of $SL(2,\mathbb{R})$ in full details.