  For simple Lie groups, the only homogeneous manifolds $G/K$, where $K$ is
maximal compact subgroup,for which the phase of the scalar product of two
coherent state vectors is twice the symplectic area of a geodesic triangle are
the hermitian symmetric spaces. An explicit calculation of the multiplicative
factor on the complex Grassmann manifold and its noncompact dual is
presented.It is shown that the multiplicative factor is identical with the
two-cocycle considered by A. Guichardet and D. Wigner for simple Lie groups.
