  On certain manifolds, the phase which appears in the scalar product of two
coherent state vectors is twice the symplectic area of the geodesic triangle
determined by the corresponding points on the manifold and the origin of the
system of coordinates. This result is proved for compact Hermitian symmetric
spaces using the generalization via coherent states of the shape invariant for
geodesic triangles and re-obtained on the complex Grassmannian by brute- force
calculation.
