A generalized Morse theory
Richard S. Palais, Stephen T. Smale · Bulletin of the American Mathematical Society · 1964
Let M be a C 2 -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 M p and we define a function || || on the tangent bundle T(M) by ||z>|| = (v, v) x J 2 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 2 function.Then df, the differential of/, is a C 1 cross section of the cotangent bundle of M, hence there is a uniquely determined C 1 vector field V/ on ikf, the gradient of/, such that df p (v) = (#, *7f(p))p for v(~M p .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 M P1 the Hessian of ƒ at p } such that H(f) p (u, v) = d 2 (/ o p(u), d b .Morse theory is concerned with relating the structure of the critical point set of ƒ in f a ' b with the homology, homotopy, homeomorphism, and diffeomorphism type of the pair (/ 6 , 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