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

182(7.28) ⎧(R i 1u i 1u − i 1 κQk 1x k1 u = Π x,u,Q l′ ,Q l′i 1 x k,... ,Q l′′ x 1 k ′ ···x 1 k,Q l′′κ−1 u i′ 1 ,Rj′ ,Rx j′k,... ,R j′′1⎪⎨⎪⎩,R j′,R j′,... ,R j′,u i′ 1 x k ′ u i′ 1 x k ′ ···x k ′ u i′ 11 1 κ−2(R j u i 1u = Π x,u,Q l′ ,Q l′i 1 x k,... ,Q l′′ x k ′ ···x k,Q l′′1 1 κ−1 u i′ 1 ,Rj′ ,Rx j′k,... ,R j′′1)x k ′1 ···x k ′κ−1,x k ′1 ···x k ′κ−1,,R j′,R j′,... ,R j′, j ≠ iu i′ 1 x k ′ u i′ 1 x k ′ ···x k ′ u i′ 1 ,11 1 κ−2(2R i 2u i 1u − i 2 κQk 1x k1 u = Π x,u,Q l′ ,Q l′i 1 x k,... ,Q l′′ x 1 k ′ ···x 1 k,Q l′′κ−1 u i′ 1 ,Rj′ ,Rx j′k,... ,R′ x j′1k ′ ···x 1 k,′κ−1),R j′,R j′,... ,R j′, iu i′ 1 x k ′ u i′ 1 x k ′ ···x k ′ u i′ 1 ≠ i 2 ,11 1 κ−2(R j u i 1u = Π x,u,Q l′ ,Q l′i 2 x k,... ,Q l′′ x k ′ ···x k,Q l′′1 1 κ−1 u i′ 1 ,Rj′ ,Rx j′k,... ,R′ x j′k ′ ···x k,′1 1 κ−1),R j′,R j′,...,R j′u i′ 1 x k ′ u i′ 1 x k ′ ···x k ′ u i′ 11 1 κ−2Using the equations of (7.28) we already obtained (namely all except thesecond equation), using (7.28) and (7.28), we may simplify these four equations:⎧Π(x, u, J) = R i 1u⎪⎨i 1u , i 1Π(x, u, J) = R j u(7.28)i 1u , j ≠ i i 1 1,Π(x, u, J) = R i 1u⎪⎩i 1u , i i 2 1 ≠ i 2 ,Π(x, u, J) = R j u i 1u , i i 2 1 ≠ i 2 , j ≠ i 1 , j ≠ i 2 .This gives the second equation of (7.28), completing the proof of Lemma 8.1and consequently the proof of Theorem 6.4.REFERENCES[1] BAOUENDI, M.S.; EBENFELT, P.; ROTHSCHILD, L.P.: Real submanifolds in complexspace and their mappings. Princeton Mathematical Series, 47, Princeton UniversityPress Princeton, NJ, 1999, xii+404 pp.[2] BLUMAN, G.W.; KUMEI, S.: Symmetries and differential equations, Springer Verlag,Berlin, 1989.[3] CARTAN, É.: Sur la géométrie pseudo-conforme des hypersurfaces de l’espace dedeux variables complexes, I, Annali di Mat. 11 (1932), 17–90.[4] CHERN, S.S.; MOSER, J.K.: Real hypersurfaces in complex manifolds, Acta Math.133 (1974), no.2, 219–271.), i 1 ≠ i 2 , j ≠ i 2 , j ≠ i 2 .