Morse Theory - Formal Development

Formal Development

For a real-valued smooth function f : M → R on a differentiable manifold M, the points where the differential of f vanishes are called critical points of f and their images under f are called critical values. If at a critical point b, the matrix of second partial derivatives (the Hessian matrix) is non-singular, then b is called a non-degenerate critical point; if the Hessian is singular then b is a degenerate critical point.

For the functions

from R to R, f has a critical point at the origin if b=0, which is non-degenerate if c≠0 (i.e. f is of the form a+cx2+...) and degenerate if c=0 (i.e. f is of the form a+dx3+...). A less trivial example of a degenerate critical point is the origin of the monkey saddle.

The index of a non-degenerate critical point b of f is the dimension of the largest subspace of the tangent space to M at b on which the Hessian is negative definite. This corresponds to the intuitive notion that the index is the number of directions in which f decreases. The degeneracy and index of a critical point are independent of the choice the local coordinate system used, as shown by Sylvester's Law.