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In this paper, we examine the geodesics on adjoint orbits of SL(n, ℝ) that are equipped with SO(n)-invariant metrics, where SO(n) is the maximal compact subgroup. Our primary technique involves translating this problem into a geometric problem in the tangent bundle of certain 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 SL(2, ℝ). [ABSTRACT FROM AUTHOR]

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible subrepresentations. In this situation, one can have equivalent submodules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds. [ABSTRACT FROM AUTHOR]