On contact between submanifolds
Given two pairs of submanifold-germs at the origin in \(R^ n\), then the pairs have the same contact type if there is a diffeomorphism-germ of \((R^ n,0)\) taking one pair to the other. The main result of this paper is the following: For \(i=1,2\), let \(g_ i: (X_ i,x_ i)\to (R^ n,0)\) be immersion-germs and \(f_ i: (R^ n,0)\to (R^ p,0)\) be submersion-germs, with \(Y_ i=f_ i^{-1}(0)\). Then the pairs \((X_ 1,Y_ 1)\) and \((X_ 2,Y_ 2)\) have the same contact type if and only if \(f_ 1\circ g_ 1\) and \(f_ 2\circ g_ 2\) are K-equivalent. This relationship also serves to elucidate the work of \textit{I. R. Porteous} on the distance-squared function and the geometry of submanifolds of Euclidean space [J. Differ. Geom. 5, 543-564 (1971; Zbl 0226.53010)].
scientific article; zbMATH DE number 3969546
scientific article