  The aim of this article is to give a rigorous although simple treatment of
the geometric notions around parallel transport in quantum mechanics. I start
by defining the teleparallelism (or generalized Pancharatnam connection)
between n-dimensional vector subspaces (or n-planes) of the whole Hilbert
space. It forms the basis of the concepts of parallel transport and of both
cyclic and non-cyclic holonomies in the Grassmann manifold of n-planes. They
are introduced in the discrete case (broken lines) before being rendered
'continuous' (smooth curves) and the role of the geodesics is stressed. Then, I
discuss the interest of such a construction to geometrize a part of the
dynamics when a (quasi-)dynamical invariant is known, especially in the
adiabatic limit. Finally, I illustrate the general theory with a three-state
toy model allowing for non-Abelian adiabatic transports.
