Critical point theory for distance functions

Morse theory is a basic tool in differential topology which also has many applications in Riemannian geometry. Roughly speaking, Morse theory relates the topology of M to the critical points of a Morse function on M . A Morse function is by definition a smooth function on M whose critical points are discrete and nondegenerate (the Hessian of f is nondegenerate). One of the most remarkable facts in Morse theory is the isotopy lemma, which says that the topology of M will not change without passing a critical point. Now let (M, g) be a Riemannian manifold, and p ∈M be a point. For geometric reasons, one would like to apply Morse theory to the distance function dp. However, one cannot apply Morse theory directly since