https://hal-universite-paris-saclay.archives-ouvertes.fr/hal-03286288Merker, JoëlJoëlMerkerLMO - Laboratoire de Mathématiques d'Orsay - Université Paris-Saclay - CNRS - Centre National de la Recherche ScientifiqueA Lie-Theoretic Construction of Cartan-Moser ChainsHAL CCSD2021Lie prolongations of vector fieldsCauchy-Riemann manifoldsLocal biholomorphic equivalencesFormal and convergent normal forms[MATH] Mathematics [math]MERKER, Joël2021-07-14 10:02:192023-02-08 17:11:182021-07-16 18:52:14enJournal articlesapplication/pdf1Let M^3 in C^2 be a real-analytic Levi nondegenerate hypersurface. In the literature, Cartan-Moser chains are detected from rather advanced considerations: either from the construction of a Cartan connection associated with the CR equivalence problem; or from the construction of a formal or converging Poincaré-Moser normal form. This note provides an alternative direct elementary construction, based on the inspection of the Lie prolongations of 5 infinitesimal holomorphic automorphisms to the space of second order jets of CR-transversal curves. Within the 4-dimensional jet fiber, the orbits of these 5 prolonged fields happen to have a simple cubic 2-dimensional degenerate exceptional orbit, the chain locus:Sigma_0 := {(x1, y1, x2, y2) in R^4: x2 = -2*x1^2*y1 - 2*y1^3, y2 = 2*x1*y1^2 + 2*x1^3}.Using plain translations, we may capture all points by working only at one point, the origin, and computations become conceptually enlightening and simple.