By Jarosz K. (ed.)

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7). 8) CONTACT EQUIVALENCE (K-EQUIVALENCE) Let G be the semidireat produat D mUltiplication rule n ~M n,p given by the where composition with h~I refers to all the entries of the matrix A2 • This group G will be also denoted by Kn,p or simply by K. 19 The K-equivalence is associated to the action k:GxE o n,p where the germs fand foh- 1 should be considered as column vectors wi th entries in the maximal ideal mn C En • The mul tiplicati on in the right-hand side is then matrix multiplication. A geometrie explanation for the use of the word contaat here can be found in [GG], p.

To the element x). One assurnes here Gx~Gy. Notation: x + y, Gx + Gy. 25 that x and only if there is areal analytic path c:[O,€) + + y if M for some e>O such that y=c(O) and c«O,e))CG·x. Hence y can be regarded as a limit of points xt=c(t)EG'x for t + O. 9). e. 12) as action a for d=l. ; i=l, ... ,k; j=l, ... ,n>+ J 1. 2) Note that dirn S=(n-k) (p-k) and hence G,u k is a smooth manifold of dimension np-dirn S. g. [GG], p. 3) PROPOSITION The specialization relations among the normal forms u k are the following other words: Guk =GU k U GU k _ 1 U •• ,U GU 1 U {O}.

The space Jk(n,p) is called the spaoe of k-jets of type (n,p). t i. e. J 0 J =J • If we analyze now aga in the results in Chapter 1, we see that essentially the Submersion Theorem (resp. Morse Lemma) says that the germ of the ma9ping f at the origin is equal (up to a coordinate change) to the germ associated to the first jet j1 f (resp. second jet j2 f ) of f. The atternpt to un- der stand and generalize these special cases will be our main concern in the sequel. Finally, we remark that for two manifolds (smooth or complex analytic) X and Y, there is a k-jet space Jk(X,y) which is a fiber bundle over XxY with typical fiber Jk(n,p), where n=dim X, p=dim Y, and which plays a basic role in studying the global mappings f:X [GGJ.