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

then by replacing the(so obtained)values for z and w in all second orderderivatives2 Θ ∂∂z k1 z k2z, z, w , see (7.28) below. Trivially, this systemis completely integrable, for we just derived it from its general solutionw(z) := Θ ( z, z, w ) , where (z, w) are understood as parameters.As we said, Hachtroudi showed that the curvature of the projective normal(Cartan) connection he associated with the system (7.28) vanishes if andonly if the right-hand side functions F k1 ,k 2satisfy the following explicit differentialsystem, which is linear in terms of their second-order derivatives(all of which, notably, appear only with respect to the y x l):(7.28)0 ≡ ∂2 F k1,k 2∂y x l 1 y x l 2−− 1n+2n∑l ′ =1(δ k1,l 1∂ 2 F l′ ,k 2∂y x l ′∂y x l 2[ ] ∑n 1+(n+1)(n+2) δk1,l 1δ k2,l 2+ δ k2,l 1δ k1,l 2l ′ =1∂ 2 F l′ ,k+ δ2∂ 2 F k1,lk1,l 2+ δ ′k2,l∂y x l 1 ∂y1x l ′ ∂y ′∂y x l x l 2n∑l ′′ =1219∂ 2 )F k1,l+ δ ′k2,l 2+∂y x l 1 ∂y x l ′∂ 2 F l ′ ,l ′′∂y x l ′∂y x l′′ (1 k 1, k 2 n)(1 l 1, l 2 n) .Hachtroudi also showed that this latter condition, better known nowadaysamongst the Several Complex Variables community as vanishing ofChern(-Moser) curvature to which it indeed amounts, characterizes thelocal equivalence, through a point transformation (x, y) ↦→ (x ′ , y ′ ) =(x ′ (x, y), y ′ (x, y) ) , to the simplest system: y ′ x ′k 1x ′k 2 (x′ ) = 0. We thenremind the semi-known fact that M is pseudospherical if and only if itsassociated second-order system (7.28) is equivalent, through a local biholomorphism(z, w) ↦→ (z ′ , w ′ ) = ( z ′ (z, w), w ′ (z, w) ) fixing the origin, to thesimplest system w ′ z ′ k 1z ′ k 2(z ′ ) = 0. So we may apply to the functions Φ k1 ,k 2Hachtroudi’s vanishing curvature equations (7.28), but still, the Φ k1 ,k 2arenot expressed in terms of Θ, for they were constructed by employing someunpleasant implicit functions when solving above for z and w. Fortunately,here similarly as in [21], we may apply the techniques of computational differentialalgebra sketched in [19] in order to explicitly express any algebraicexpressions in the second-order jet of the Φ k1 ,k 2in terms of the fourth-orderjet of Θ, and the appropriate general equation which we shall need:∂ 2 Φ k1,k 2= 1 n+1∑∂w zl1 ∂w zl2 ∆ 3n+1∑µ=1 ν=1∆ µ [0 1+l1 ]· ∆ν [0 1+l2 ]{∆ ·∂ 4 n+1Θ ∑∂z k1 ∂z k2 ∂t µ ∂t ν −τ=1∆ τ [t µ t ν ] ·will be obtained in Section 4 below, after rather lengthy but elementary calculations,parts of which are inspired from [17]. It is now essentially clearhow one obtains the (boxed) long fourth-order differential equations statedin the theorem, but in any case, some complete details will be provided atthe very end of the paper.}∂ 3 Θ∂z k1 ∂z k2 ∂t τ