The Kupka-Smale Theorem

Let M be a compact manifold of dimension m and Xr(M) the space of Cr vector fields on M, r ≥ 1, with a Cr norm. In Chapter 2 we showed that the set G₁ ⊂ Xr(M), consisting of fields whose singularities are hyperbolic, is open and dense in Xr(M). This is an example of a generic property, i.e. a property that is satisfied by almost all vector fields. In this chapter we shall analyse other generic properties in Xr(M). The original proof of the results dealt with here can be found in [44], [82] and [107].