Morse-Smale Flows on a Tilted Torus

α

0.523

β

0.523

t

0.1

show axes

show level curves

This Demonstration shows the gradient flows of the height function on a tilted torus. It illustrates the basic concepts of Morse theory: the critical points of a Morse–Smale function and their stable and unstable manifolds. The position of a point on the tilted torus is determined by two parametrization angles

-π ≤ α ≤ π

and

-π ≤ β ≤ π

, which define a local coordinate system at each point of the torus. The parameter

t

(which can be negative) controls the duration of the flow.