When the Morse index is infinite

Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold, all of whose critical points have infinite Morse index and co-index. For any critical point x choose an integer a(x) arbitrarily. Then there exists a Riemannian structure on M such that the corresponding gradient flow of f has the following property: for any pair of critical points x, y, the unstable manifold of x and the stable manifold of y have a transverse intersection of dimension a(x)− a(y). Introduction Let f be a smooth Morse function on an infinite dimensional separable Hilbert manifold M . Let us denote by crit(f) the set of its critical points, and let us assume that each x ∈ crit(f) has finite Morse index i(x). A Riemannian structure g on M determines the vector field −gradf , whose local flow φt : M → M has the critical points of f as rest points. The Morse condition is translated into the fact that these rest points are hyperbolic. Their unstable and stable manifolds, W(x; f, g) = W(x) = { p ∈ M | lim t→−∞ φt(p) = x }