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

224(z ′ , w ′ ) fixing the origin, both its Segre varieties and its conjugate Segrevarieties ([Me2005a, Me2005b, 18]):S z,w := { (z, w): w = Θ ( z, z, w )} and S z,w := { (z, w): w = Θ ( z, z, w )}are mapped to the Segre and conjugate Segre varieties of the Heisenbergpseudosphere:S ′ z ′ ,w = { w ′ = −w ′ + z ′ z ′} {and w ′ = −w ′ + z ′ z ′}′which, visibly, are plain complex affine lines.§3. GEOMETRY OF ASSOCIATED SUBMANIFOLDS OF SOLUTIONSCompletely integrable systems of partial differential equations. Thecharacterization of pseudosphericality we are dealing with holds in a contextmore general than just CR geometry 25 . Accordingly, let K denote eitherthe field C of complex numbers or the field R of real numbers, letx = (x 1 , . . ., x n ) ∈ K n with again n 2 — since the case n = 1 was alreadystudied in [21] —, let y ∈ K, and consider a system of the form (7.28).We will assume that it is completely integrable in the sense that the naturalcommutativity of partial derivatives enjoyed trivially by the left-hand sides:∂ 2 y x k 1x k 2/∂yx k 3 = ∂ 2 y x k 1x k 3/∂yx k 2(1 k 1 , k 2 , k 3 n)imposes immediately to the right-hand side functions F k1 ,k 2that they satisfythe so-called compatibility conditions:D k3(Fk1 ,k 2)= Dk2(Fk1 ,k 3),where we have introduced the following n total differentiation operators:D k := ∂∂x k + y x kn∑∂ + ∂y(1 k n)l=1F k,l∂∂y x lliving on the first-order jet space (x 1 , . . .,x n , y, y x 1, . . .,y x n). One verifiesthat these compatibility conditions amount to the fact that the n-dimensionaltangential distribution spanned by D 1 , . . .,D n in the (2n + 1)-dimensionalfirst-order [ ] jet space satisfies the classical Frobenius integrability conditionDk ′, D k ′′ = 0, and then the Clebsch-Frobenius theorem tells us thatthis distribution comes from a local foliation by n-dimensional manifoldsgraphed over the x-space that are naturally parametrized by n + 1 auxiliaryconstants (transversal directions) — call them a 1 , . . ., a n , b ∈ K —, namely25 We will be very brief here, the reader being referred to [19, 21] for the general theoreticalconsiderations.