The coherent states: old geometrical methods in new quantum clothes

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.