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

to fix the ideas, it will be assumed throughout the paper — and recalledwhen necessary — that the CR dimension n is always 2.Locally in a neighborhood of one of its points p, the hypersurface M maybe represented, in any system of local holomorphic coordinates:t = (w, z) ∈ C n × Cvanishing at p for which the w-axis is not complex-tangent to M at p, by aso-called complex defining equation — Section 2 provides further informations— of the form:(7.28) w = Θ ( z, z, w) = Θ ( z, t ) ,or equivalently in a more expanded form which exhibits all the indices:w = Θ ( z 1 , . . .,z n , z 1 , . . ., z n , w ) = Θ ( z 1 , . . ., z n , t 1 , . . .,t n , t n+1).Then M localized at p is called pseudospherical (at p) if it is biholomorphicto a piece of one Heisenberg pseudosphere:(7.28) Im w ′ = |z ′ 1 |2 + · · · + |z ′ q |2 − |z ′ q+1 |2 − · · · − |z ′ n |2 ,for some q with 0 q n, the number of positive eigenvalues of thenondegenerate Levi form. Next, let us introduce the following Jacobian-likedeterminant:∆ :=∣Θ z1 · · · Θ zn Θ wΘ z1 z 1· · · Θ z1 z nΘ z1 w·· · · · ·· ··Θ znz 1· · · Θ znz nΘ znw=∣ ∣217∣Θ t1· · · Θ tnΘ tn+1∣∣∣∣∣∣∣Θ z1 t 1· · · Θ z1 t nΘ z1 t n+1.·· · · · ·· ··Θ znt 1· · · Θ znt nΘ zntn+1 For any index µ ∈ {1, . . ., n, n + 1} and for any index l ∈ {1, . . ., n}, letalso ∆ µ [0 1+l ]denote the same determinant, but with its µ-th column replacedby the transpose of the line (0 · · ·1 · · ·0) with 1 at the (1+l)-th place, and 0elsewhere, its other columns being untouched. One easily convinces oneself(but see also Section 2) that M is Levi-nondegenerate at p — which is theorigin of our system of coordinates — if and only if ∆ does not vanish at theorigin, whence ∆ is nowhere zero in some sufficiently small neighborhoodof the origin. Similarly, for any indices µ, ν, τ ∈ {1, . . .,n, n + 1}, denoteby ∆ τ the same determinant as ∆, but with only its τ-th column replaced[t µ t ν ]by the transpose of the line:( )Θt µ ν t Θ z1 t µ t ν · · · Θ znt µ t ν ,other columns being again untouched. All these determinants ∆, ∆ µ [0 1+l ] ,∆ τ [t µ t ν ]are visibly universal differential expressions depending upon thesecond-order jet J 2 z,z,w Θ and upon the third-order jet J3 z,z,w Θ.Main Theorem. An arbitrary, not necessarily rigid, real analytic hypersurfaceM ⊂ C n+1 with n 2 which is Levi nondegenerate at one of its points