Optimal Paths for Symmetric Actions in the Unitary Group

Abstract

Given a positive and unitarily invariant Lagrangian L defined in the algebra of matrices, and a fixed time interval [0, t0 ] ⊂ ℝ, we study the action defined in the Lie group of n × n unitary matrices U(n) by, where α: [0, t0] → U(n) is a rectifiable curve. We prove that the one-parameter subgroups of U(n) are the optimal paths, provided the spectrum of the exponent is bounded by π. Moreover, if L is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in U(n) as well as angular metrics in the Grassmann manifold.