The significance of the cut locus is that the distance function from a point p is smooth, except on the cut locus of p and p itself. In particular, it makes sense to take the gradient and Hessian of the distance function away from the cut locus and p. This idea is used in the local Laplacian comparison theorem and the local Hessian comparison theorem. These are used in the proof of the local version of the Toponogov theorem, and many other important theorems in Riemannian geometry.
