Derivatives and Morse Functions

Example 5.1 manifolds Figure 5.4 displays the stable and unstable mani- folds of a sphere and a Morse function h. We show an uncompactified sphere: The boundary of the terrain is a minimum at negative infinity. Note that the stable manifold of a minimum and the unstable manifold of a maximum, are the critical points themselves, respectively. On the other hand, both the unsta- ble manifold of a minimum and the stable manifold of a maximum are 2-cells. A saddle has 1-cells as both stable and unstable manifolds. Also, observe that the stable manifolds of the saddles decompose M into the stable manifolds of the maxima. The unstable manifolds provide such a decomposition for the minima.

5.5 Morse-Smale Complex

We place one more restriction on Morse functions in order to be able to con- struct Morse-Smale complexes. Definition 5.16 Morse-Smale A Morse function is a Morse-Smale function if the stable and unstable manifolds intersect only transversally. In two dimensions, this means that stable and unstable 1-manifolds cross when they intersect. Their crossing point is necessarily a saddle, since crossing at a regular point would contradict property a in Theorem 5.2. Given a Morse- Smale function h, we intersect the stable and unstable manifolds to obtain the Morse-Smale complex. Definition 5.17 Morse-Smale complex Connected components of sets U p ∩ Sq for all critical points p, q ∈ M are Morse-Smale cells. We refer to the cells of dimension 0, 1, and 2 as vertices, arcs, and regions, respec- tively. The collection of Morse-Smale cells form a complex, the Morse-Smale complex . Note that U p ∩ Sp = {p}, and if p = q, then Up ∩ Sq is the set of regular points r ∈ M that lie on integral lines γ with org γ = p and dest γ = q. It is possible that the intersection of stable and unstable manifolds consists of more than one component, as seen in Figure 5.5. Example 5.2 Morse-Smale complex We continue with the manifold and Morse function in Example 5.1. Figure 5.5 shows the Morse-Smale com- plex we get by intersecting the stable and unstable manifolds displayed in Fig- ure 5.4. Each vertex of the Morse-Smale complex is a critical point, each arc is a Stable b Unstable minimum saddle maximum Fig. 5.4. The stable a and unstable b 1-manifolds, with dotted iso-lines h −1 c, for constants c. In the diagrams, all the saddle points have height between all minima and maxima. Regions of the 2-cells of maxima and minima are shown, including the critical point, and bounded by the dotted iso-lines. The underlying manifold is S 2 , and the outer 2-cell in b corresponds to the minimum at negative infinity.