A Lie-theoretic Construction of Cartan-Moser Chains

Abstract

Let $M^3\subset {\mathbb{C}}^2$ be a ${\mathcal{C}}^{\omega }$ 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: \[ Σ_0 \,:=\, \big\{ (x_1,y_1,x_2,y_2) \in \mathbb{R}^4 \colon\,\, x_2 = -2x_1^2y_1-2y_1^3,\,\,\, y_2 = 2x_1y_1^2 + 2x_1^3 \big\}. \] By plain translations, we may capture all points by working only at one point, the origin, and computations, although conceptually enlightening, become disappointingly simple.

This work was supported in part by the Polish National Science Centre (NCN) via the grant number 2018/29/B/ST1/02583

References in corpus (1)

• I: Lie symmetries of partial differential equations and CR geometry

Cited by in corpus (1)

• On Convergent Poincaré-Moser Reduction for Levi Degenerate Embedded $5$-Dimensional CR Manifolds