ABSTRACT
The investigation of degenerate critical points of smooth functions is initiated in this chapter by proving the Reduction Lemma. This lemma, which is also referred to in the literature as the splitting lemma, states that around a degenerate critical point p, a smooth function f of n variables can be expressed in suitable coordinates as a sum of two functions q and g of different variables. The function q is a nondegenerate normal quadratic form in r variables, whereas g is a smooth function of n − r variables that is totally degenerate at the corresponding critical point, i.e., g has rank zero there, so that g is a representative of the residual singularity of f at p. The number r is the rank of f at p, and the index of f at p is that of q. Due to the Reduction Lemma, the study of degenerate critical points is reduced to analyzing the behavior of the totally degenerate ones. Now it is possible to be more precise about the content of Thom’s first major theorem of Catastrophe Theory. This theorem classifies the totally degenerate critical points of functions of codimension at most 4. The notation codimension will be explained in Chapter 5.
