Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms

Consider the Lie group of n×n complex unitary matrices U(n) endowed with the bi-invariant Finsler metric given by the spectral norm,‖X‖U=‖U⁎X‖∞=‖X‖∞ for any X tangent to a unitary operator U. Given two points in U(n), in general there exist infinitely many curves of minimal length. In this paper we provide a complete description of such curves and as a consequence we give an equivalent condition for uniqueness. Similar studies are done for the Grassmann manifolds.