A generalized Morse theory

Abstract

Let M be a C 2 -Riemannian manifold without boundary modeled on a separable Hubert space (see Lang [3]). For pzM 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 vzMp. Given p and q in the same component of M we define p(p, q)==lnfl\\<r'(t)\\dt t where the Inf is over all C 1 paths <r\ [0, l]->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 : 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 is a point where V/ vanishes; equivalently a stationary point of <*. At a critical point p of there is a uniquely determined continuous bilinear form H(f) p on M P1 the Hessian of at p } such that H(f) p (u, v) = d 2 (/ o <jr l )(d<}>p(u), d<p p (v)) if < is any chart at p. The supremum of the dimensions of subspaces on which H(f) p is negative (positive) definite is called the index (coindex) of at p. H(f) p is symmetric, hence there is a uniquely determined bounded self-ad joint operator A on M p such that H{f) v {u 1 v) = (Au, v) p . The critical point p is called nondegenerate if A has a bounded inverse. In this case p is isolated in the set of critical points.