Mechanics on flag manifolds

We study the connection between $\mathrm{SU}(n)$ 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 $\mathbb{CP}^1$ and $\mathcal{F}_3$, which is a manifold of complete flags in $\mathbb{C}^3$.