On Singularities of Smooth Maps to a Space with a Fixed Cone

We consider germs of smooth maps f : (R l ; 0) ! (R n ; c); c 2 C where C is the standard nondegenerate cone of some signature and classify their singularities under the actions of two natural groups of diieomorphisms preserving C. Occuring singularities are subdivided into 3 classes: regular, semiregular and irregular. In the regular case the classiication of singularities is reduced to the classiication of the usual singularities of germs of functions. We present the list of simple semiregular singularities and also analyze some irregular singularities. x0. Preliminaries and results The singularities of maps f : (R l ; 0) ! (R n ; s); s 2 S to a target space with some xed stratiied variety S were considered by several authors, see e.g. AVG],,A1], L],,Si]. In the case when S is a hyperplane they are called boundary singularities and were investigated in details in A1]. In particular, it was shown that the simple boundary singularities correspond to the A-, B-, C-, D-, E-series and F 4 in the classiication of simple Lie algebras. A much more general class of actions (including all actions on source and target with a stratiied S used in this note) was studied by J. Damon Da1-2]. He proved the existence of the versal unfoldings and nite determinacy of germs of maps in this situation. (Therefore, the normal forms we present can be achieved in formal, analytic and smooth categories.) In what follows we will always assume that the dimension of the source is less than the dimension of the target. Let s 2 S be a point on a stratiied variety S and let A S denote the product of the group of local diieomorphisms (R l ; 0) ! (R l ; 0) of the source by the group of local diieomorphisms (R n ; s) ! (R n ; s) of the target preserving S. By K we denote the group of contact transformations, i.e. elements of K are diieomor-phisms of (R l R n ; 0) preserving a) projection on R l (inducing diieomorphisms of R l) and b) the subspace (R l 0). Thus two germs f 1 and f 2 : (R l ; 0) ! (R n ; 0) are K-equivalent if there exists a diieomorphism of the source and a germ M : (R l ; 0) ! GL(R n) such …