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

215Vanishing Hachtroudi curvatureand local equivalenceto the Heisenberg pseudosphereJoël MerkerAbstract. To any completely integrable second-order system of real or complexpartial differential equations:(y x k 1x k 2 = F k1 ,k 2 x 1 ,... ,x n ), y, y x 1,... ,y x nwith 1 k 1 , k 2 n and with F k1 ,k 2= F k2 ,k 1in n 2 independent variables(x 1 ,...,x n ) and in one dependent variable y, Mohsen Hachtroudi associated in1937 a normal projective (Cartan) connection, and he computed its curvature. Bymeans of a natural transfer of jet polynomials to the associated submanifold ofsolutions, what the vanishing of the Hachtroudi curvature gives can be preciselytranslated in order to characterize when both families of Segre varieties and of conjugateSegre varieties associated to a Levi nondegenerate real analytic hypersurfaceM in C n (n 3) can be straightened to be affine complex (conjugate) lines. Incontinuation to a previous paper devoted to the quite distinct C 2 -case, this thencharacterizes in an effective way those hypersurfaces of C n+1 in higher complexdimension n + 1 3 that are locally biholomorphic to a piece of the (2n + 1)-dimensional Heisenberg quadric, without any special assumption on their definingequations.arxiv.org/abs/0910.2861/Table of contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2. Segre varieties and differential equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .220.3. Geometry of associated submanifolds of solutions. . . . . . . . . . . . . . . . . . . . . . . . .224.4. Effective differential characterization of pseudosphericality in C n+1 . . . . . . 226.§1. INTRODUCTIONThe explicit characterization of pseudosphericality of an arbitrary real analyticlocal hypersurface sitting in the complex Euclidean space has been(re)studied recently by Isaev in [11], who employed the famous Chern(-Moser) tensorial approach [CM1974, Ch1975] to the concerned equivalenceproblem. But in the growing literature devoted to Lie-group symmetriesof Cauchy-Riemann manifolds, only a very few articles underline that, alreadyin his 1937 Ph.D. thesis [Ha1937] under the direction of his Élie Cartan— who was around the same period also the master of Chern —, the