  We show that the holonomy of a connection defined on a principal composite
bundle is related by a non-abelian Stokes theorem to the composition of the
holonomies associated with the connections of the component bundles of the
composite. We apply this formalism to describe the non-abelian geometric phase
(when the geometric phase generator does not commute with the dynamical phase
generator). We find then an assumption to obtain a new kind of separation
between the dynamical and the geometric phases. We also apply this formalism to
the gauge theory of gravity in the presence of a Dirac spinor field in order to
decompose the holonomy of the Lorentz connection into holonomies of the linear
connection and of the Cartan connection.
