Minimal matrices and the corresponding minimal curves on flag manifolds in low dimension

In general C∗-algebras, elements with minimal norm in some equivalence class are introduced and characterized. We study the set of minimal hermitian matrices, in the case where the C∗-algebra consists of 3×3 complex matrices, and the quotient is taken by the subalgebra of diagonal matrices. We thoroughly study the set of minimal matrices particularly because of its relation to the geometric problem of finding minimal curves in flag manifolds. For the flag manifold of ‘four mutually orthogonal complex lines’ in C4, it is shown that there are infinitely many minimal curves joining arbitrarily close points. In the case of the flag manifold of ‘three mutually orthogonal complex lines’ in C3, we show that the phenomenon of multiple minimal curves joining arbitrarily close points does not occur.