Cusp (singularity) - More General Background

More General Background

Consider a smooth real-valued function of two variables, say f(x, y) where x and y are real numbers. So f is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target. This action splits the whole function space up into equivalence classes, i.e. orbits of the group action.

One such family of equivalence classes is denoted by A_k±, where k is a non-negative integer. This notation was introduced by V. I. Arnold. A function f is said to be of type A_k± if it lies in the orbit of x2 ± yk+1, i.e. there exists a diffeomorphic change of coordinate in source and target which takes f into one of these forms. These simple forms x2 ± yk+1 are said to give normal forms for the type A_k±-singularities. Notice that the A_2n+ are the same as the A_2n− since the diffeomorphic change of coordinate (x,y) → (x, −y) in the source takes x2 + y2n+1 to x2 − y2n+1. So we can drop the ± from A_2n± notation.

The cusps are then given by the zero-level-sets of the representatives of the A_2n equivalence classes, where n ≥ 1 is an integer.