  A geometric characterization of transition amplitudes between coherent
states, or equivalently, of the hermitian scalar product of holomorphic cross
sections in the associated D - M tilda - module, in terms of the embedding of
the cohe- rent state manifold M-tilda into a projective Hilbert space is
proposed. Cohe- rent state manifolds endowed with a homogeneous kaehler
structure are conside- red. Using the coherent state approach, an effective
method to find the cut loci on symmetric manifolds and generalized symmetric
manifolds M-tilda is proposed. The CW - complex structure of coherent state
manifolds of flag type is discussed. Recent results of Anandan and Aharonov are
commented vis-a-vis of last century constructions in projective geometry.
Calculations with signi- ficance in coherent state approch furnish explicit
proofs of the results announ- ced by Y. C. Wong on conjugate locus in complex
Grassmann manifold.
