On the construction of perfect Morse functions on compact manifolds of coherent states

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Perfect Morse functions on the manifold of coherent states are effectively constructed. The case of a compact, connected, simply connected Lie group of symmetry, having the same rank as the stationary group of the manifold of coherent states, such that the manifold of coherent states is a Kählerian C-space, is considered. It is proved that the set of perfect Morse functions is dense in the set of energy functions for linear Hamiltonians in the elements of the Cartan algebra of the Lie algebra of the representation of the group considered. It is proved that the maximum number of orthogonal vectors on a coherent vector manifold is equal to the Euler–Poincaré characteristic of the manifold.