Morse theory on Banach manifolds

AbstractLet ƒ be a C2 function on a C2 Banach manifold. A critical point x of ƒ is said to be weakly nondegenerate if there exists a neighborhood U of x and a hyperbolic linear isomorphism Lx: Tx(M) → Tx(M) such that in the coordinate system of U, dƒx + v(Lxv) > 0 if v ≠ 0. Lx defines an index invariantly, and it is shown that this is an extension of the usual definition of nondegeneracy and index. It is shown that this weaker nondegeneracy can be used in place of the stronger nondegeneracy conditions in Morse theory. In addition, sufficient conditions for the critical points of variational problems to be weakly nondegenerate are given