A Generalized Morse Theory

1. Abstract theory. Let M be a C-Riemannian manifold without boundary modeled on a separable Hubert space (see Lang [3]). For pÇzM we denote by ( , )p the inner product in the tangent space Mp and we define a function || || on the tangent bundle T(M) by ||z>|| = (v, v)J for vÇzMp. Given p and q in the same component of M we define p(p, q)==lnïfl\\ikf such that a(0)=p and cr(l)=g. Just as in the finite dimensional case one shows that p is a metric on each component of M which is consistent with the manifold topology. If each component of M is complete in this metric M is called a complete Riemannian manifold and we assume this in all that follows. Let ƒ: M—>R be a C function. Then df, the differential of/, is a C cross section of the cotangent bundle of M, hence there is a uniquely determined C vector field V/ on ikf, the gradient of/, such that dfp(v) = (#, *7f(p))p for v(~Mp. We denote by $* the maximum local oneparameter group generated by — V/. A critical point of ƒ is a point where V/ vanishes; equivalently a stationary point of <£*. At a critical point p of ƒ there is a uniquely determined continuous bilinear form H(f)p on MP1 the Hessian of ƒ at p} such that H(f)p(u, v) = d 2 ( / o p(u), d. Morse theory is concerned with relating the structure of the critical point set of ƒ in f' with the homology, homotopy, homeomorphism, and diffeomorphism type of the pair (/, ƒ*). We shall be concerned with the Morse theory of pairs {My ƒ) as above which satisfy at least the following extra condition : (C) If 5 is a subset of M on which | ƒ) is bounded but on which || V/1| is not bounded away from zero, then there is a critical point of f in the closure of 5. Note that if ƒ is proper (which implies that M is finite dimensional) and in particular if M is compact then condition (C) is automatically satisfied. More interesting though is the fact, which we will make