Criteria for right-left equivalence of smooth map-germs (Q5939194)

Two \(C^{\infty}\)-map germs are said to be \(C^r\)-(right-left) equivalent if they are the same modulo \(C^r\)-reparametrizations of the source and target spaces, i.e. applying \(C^r\)-diffeomorphisms on them. The main problem addressed in the paper is the recognition of two germs being equivalent or not. The classical result in the area is the algebraic characterization of \(C^{\infty}\)-equivalence in case of stable germs, due to Mather. The non-stable range has also been studied by several authors, see references in the paper. The method used in this paper is an effort for constructing germs of \(C^r\)-diffeomorphisms of the source and target spaces -- directly. This approach turns out to be very fruitful, giving several sufficient conditions for \(C^r\)-equivalence for finite \(r\), as well as necessary and sufficient conditions when \(r=\infty\). The nature of the conditions obtained (i.e. the existence of \(C^r\)-contact equivalences with certain conditions) are probably best understood through the illuminating examples of the 13-page-long Section 2.