Families of Morse Functions Parameterized by Maxima

摘要

A C~2 function f : M → R on a Riemannian manifold M is called a Morse func- tion if, at every critical point p, in local coordinates the Hessian matrix Hf_p is nonsingular. The classical Morse's lemma states that, in suitable local coordinates centered at a critical point, f is expressed as a sum and difference of squares of coordinates. Morse functions have been of great importance in differential topology and dy- Namical systems. Mobius's proof of the classification of surfaces decomposed a Surface into elementary pieces bounded by noncritical level sets of a Morse func- Tion (Mobius [9]; see Hirsch [e] for a modern presentation).