On Realization of Gradient-like Flows on the Four-dimensional Projective-like Manifold
Manifolds admitting a Morse function with three critical points are called projective-like, by analogy with the projective plane. Eells and Kuiper showed that the dimension n of such manifolds takes on the values 2, 4, 8, and 16, and the critical points of the Morse function have indices 0, n / 2, and n. Zhuzhoma and Medvedev obtained a topological classification of gradient flows of such a function on manifolds of dimension 4. It is not difficult to construct a Morse function (and gradient-like flow) on a projective-like manifold, the set of critical points (equilibrium states) of which is not limited to three points with the listed indices. In this paper, we study the question of the relationship between the numbers of equilibrium states of different indices of a gradient-like flow, given on a projective-like manifold of dimension four, and give an algorithm for realization of such a flow for the given numbers of equilibrium states of indices 0,1,2,3,4.