Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

185Nonrigid sphericalreal analytic hypersurfaces in C 2Joël MerkerAbstract. A Levi nondegenerate real analytic hypersurface M of C 2 representedin local coordinates (z,w) ∈ C 2 by a complex defining equation of the formw = Θ(z,z,w) which satisfies an appropriate reality condition, is spherical if andonly if its complex graphing function Θ satisfies an explicitly written sixth-orderpolynomial complex partial differential equation. In the rigid case (known before),this system simplifies considerably, but in the general nonrigid case, its combinatorialcomplexity shows well why the two fundamental curvature tensors constructedby Élie Cartan in 1932 in his classification of hypersurfaces have, since then, neverbeen reached in parametric representation.arxiv.org/abs/0910.1694/Table of contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2. Segre varieties and partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 220.3. Geometry of associated submanifolds of solutions. . . . . . . . . . . . . . . . . . . . . . . . .224.4. Effective differential characterization of sphericality in C 2 . . . . . . . . . . . . . . . . 226.5. Some complete expansions: examples of expression swellings. . . . . . . . . . . . . .210.§1. INTRODUCTIONA real analytic hypersurface M in C 2 is called spherical at one of itspoints p if there exists a nonempty open neighborhood U p of p in C 2such that M ∩ U p is biholomorphic to a piece of the unit sphere S 3 ={(z, w): |z| 2 + |w| 2 = 1 } . When M is connected, sphericality at onepoint is known to propagate all over M, for it is equivalent to the vanishingof two certain real analytic curvature tensors that were constructed byÉlie Cartan in [3]. However, the intrinsic computational complexity, in theCauchy-Riemann (CR for short) context, of Élie Cartan’s algorithm to derivean absolute parallelism on some suitable eight-dimensional principalbundle P → M prevents from controlling explicitly all the appearing differentialforms. As a matter of fact, the effective computation, in terms of adefining equation for M, of the two fundamental differential invariants thevanishing of which characterizes sphericality, appears nowhere in the literature(see e.g. [23, 5, 11] and the references therein as well), except notablywhen one makes the assumption that, in some suitable local holomorphic