7 - Unfoldings: applications

Summary

‘Singularity is almost invariably a clue.’

(The Boscombe Valley Mystery)

In this chapter we apply the rather technical ideas of chapter 6 to some very concrete geometrical problems. Although the ideas are technical (and their proofs even more so), applying them is, fortunately, relatively easy. We shall concentrate on elucidating the local structure of bifurcation and discriminant sets (such as envelopes), though this is by no means the only application.

Here is the pattern. We start with a family F : ℝ × ℝ^r ↦ ℝ, where r = 2 or 3, whose bifurcation or discriminant set interests us. If x₀ is a point of one of these sets, with corresponding value t₀ (so ∂F/∂t = ∂²t/∂t² = 0 at (t₀, x₀) or F = ∂F/∂t = 0 at (t₀, x₀), respectively), then we work out the type A_k of the singularity which f = F_x0 has at t₀, by counting the number of derivatives of f which vanish at t₀. We then decide whether F versally unfolds f at t₀ by finding ∂F/∂x_i and using the matrix criterion, 6.1 Op or 6.10. If the criterion is satisfied then locally (near x₀) the bifurcation set or discriminant set is diflFeomorphic to the standard model applicable to the values of r and k in question. These are listed in chapter 6, (6.16p–6.18p and 6.16–6.18).

Lest the reader think that we now have the answer to all questions of this kind, we give several examples where these methods give little or no information.