Geodesics on adjoint orbits of SL(n, ℝ)

In this paper, we examine the geodesics on adjoint orbits of $\mathrm{\text{SL}}(n,ℝ)$ that are equipped with $\mathrm{\text{SO}}(n)$-invariant metrics, where $\mathrm{\text{SO}}(n)$ is the maximal compact subgroup. Our primary technique involves translating this problem into a geometric problem in the tangent bundle of certain $\mathrm{\text{SO}}(n)$-flag manifolds and describing the geodesic equations with respect to the Sasaki metric on the tangent bundle. Additionally, we utilize tools from Lie Theory to obtain explicit descriptions of families of geodesics. We provide a detailed analysis of the case for $\mathrm{\text{SL}}(2,ℝ)$.