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

Differentiating the first equation with respect to x and substituting, we getthe missing equation:(1.7)y 2 xx = F x + y 1 x F y 1 + y 2 x F y 2 + y 1 xx F y 1 x= F x + y 1 x F y 1 + y2 x F y 2 + G F y 1 x=: H ( x, y 1 , y 2 , y 1 x).Example 1.8. With n = 2, m = 1 and κ = 2, a system of the form{y x 2 = F ( )x 1 , x 2 , y, y x 1, y x(E 5 )1 x 1y x 1 x 1 x 1 = G( )x 1 , x 2 , y, y x 1, y x 1 x 1 .Here, five equations are missing. Differentiating the first equation with respectto x 1 and substituting:(1.9)y x 1 x 2 = F x 1 + y x 1 F y + y x 1 x 1 F y x 1 + y x 1 x 1 x 1 F y x 1 x 1= F x 1 + y x 1 F y + y x 1 x 1 F y x 1 + G F y x 1 x 1=: H ( x 1 , x 2 , y, y x 1, y x 1 x 1 ),and then similarly for y x 2 x 2, y x 1 x 1 x 2, y x 1 x 2 x 2, y x 2 x 2 x 2.1.10. Finitely nondegenerate generic submanifolds of C n+m . Examples1.3, 1.4, 1.6 and 1.8 (but not 1.5) are intrinsically linked to real submanifoldsof complex submanifolds.Let M be a real algebraic or analytic local generic CR 30 submanifoldof C n+m of codimension m 1 and of CR dimension n 1, and letp ∈ M. Classically, there exists local holomorphic coordinates t = (z, w) ∈C n × C m centered at p in which M is represented by(1.11) w j = Θ j (z, ¯z, ¯w), j = 1, . . .,m,for some local C-analytic map Θ = (Θ 1 , . . .,Θ m ) satisfying the identity(1.12) w ≡ Θ ( z, ¯z, Θ(¯z, z, w) ) ,reflecting the fact that M is real.Definition 1.13. ([BER1999, Me2005a, Me2005b, MP2005]) M is finitelynondegenerate if there exists an integer κ 1 such that the local holomorphicmap(1.14) (¯z, ¯w) ↦−→ ( Θ j z β(0, ¯z, ¯w)) 1jm|β|κis of rank n + m at (¯z, ¯w) = (0, 0).30 Fundamentals about Cauchy-Riemann geometry may be found in [Bo1991, BER1999,Me2005a, Me2005b, MP2005].235