Morse-Smale Functions

Abstract

In this chapter we introduce the Morse-Smale transversality condition for gradient vector fields, and we prove the Kupka-Smale Theorem (Theorem 6.6) which says that the space of smooth Morse-Smale gradient vector fields is a dense subspace of the space of all smooth gradient vector fields on a finite dimensional compact smooth Riemannian manifold (M,g) [92] [135]. We also prove Palis’ λ-Lemma (Theorem 6.17) following [114] and [143], and we derive several important consequences of the λ-Lemma. These consequences include transitivity for Morse-Smale gradient flows (Corollary 6.21), a description of the closure of the stable and unstable manifolds of a Morse-Smale gradient flow (Corollary 6.27), and the fact that for any Morse-Smale gradient flow there are only finitely many gradient flow lines between critical points of relative index one (Corollary 6.29). In the last section of this chapter we present a couple of results due to Franks [59] that relate the stable and unstable manifolds of a Morse-Smale function f: M → ℝ to the cells and attaching maps in the CW-complex X determined by f (see Theorem 3.28).