Morse Theory: Gradient Flow & Topology

Critical points, gradient flow lines, and sublevel set topology on smooth surfaces

Minimum (index 0) Saddle (index 1) Maximum (index 2) Gradient flow
Minima: 0
Saddles: 0
Maxima: 0
Ļ‡ = Vāˆ’E+F: ā€”
Level: 0.00
Morse theory (Morse 1934): a smooth function f on a manifold encodes its topology via critical points. As the level h rises, the sublevel set {f ≤ h} changes topology only at critical values. Minimum: a disk appears. Saddle: a handle attaches (two disks merge or a hole opens). Maximum: a disk caps off. Morse inequality: #max - #saddle + #min = chi(M) (Euler characteristic). Gradient flow lines connect saddles to minima/maxima forming the Morse-Smale complex.