Optimal paths for symmetric actions in the unitary group

JORGE ANTEZANA; GABRIEL LAROTONDA; ALEJANDRO VARELA

COMMUNICATIONS IN MATHEMATICAL PHYSICS

SPRINGER

Lugar: Berlin; Año: 2014 vol. 328 p. 481 - 497

0010-3616

Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval [a,b], we study the action defined in the Lie group of nxn unitary matrices U(n) by S(alpha)=int_a^b L( alpha´(t) ) dt, where alpha:[a,b]-->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 pi. 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.