Mechanics on flag manifolds

We study the connection between $\mathrm{S}\mathrm{U}(n)$ spin chains and one-dimensional sigma models on flag manifolds. Using this connection, we calculate the spectrum of the Laplace-Beltrami operator and geodesics for a particular class of metrics on ${\mathbb{C}\mathbb{P}}^1$ and ${\mathcal{F}}_3$, which is a manifold of complete flags in ${\mathbb{C}}^3$.