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

130It follows from the representation formula (6.20) that the mapping J κ 0is injective and from the Cauchy integral formula that J κ 0is continuouson its domain of definition H ρ 2,ρ 1M,κ 0 ,εendowed with the topology of uniformconvergence on compact sets.On the reverse side, let J κ 0∈ E. Then the mapping h(t) := H(t, J κ 0)defined for |t| < ρ 4 maps M ∩ ∆ n (ρ 4 ) into M. Applying Theorem 6.4to this mapping h(t), with ρ 1 replaced by ρ 4 , we deduce that there existsa radius ρ 6 < ρ 4 such that we can represent h(t) = H(t, J κ 0h(0)) for|t| < ρ 6 , with the same mapping H, as stated in the end of Theorem 6.4.By differentiating this representation with respect to t at t = 0, we deducethat J κ 0h(0) = ([∂t αH(t, Jκ 0h(0))] t=0 ) |α|≤κ0 . Consequently, sinceh(t) = H(t, J κ 0) by definition, we get J κ 0= ([∂t αH(t, Jκ 0)] t=0 ) |α|≤κ0 . Inconclusion, we proved that J κ 0(H(t, J κ 0)) = J κ 0for every J κ 0∈ E, soJ κ 0has a continuous local inverse on E, formally defined by H(t, J κ 0).It follows from the above two paragraphs that the mapping J κ 0is a localhomeomorphism from a neighborhood of the identity in H ρ 2,ρ 1M,κ 0 ,εonto itsimage E.Furthermore, we claim that the real algebraic subset E is in fact geometricallysmooth at every point, namely it is a real algebraic submanifold.Indeed, let J κ 01 be a regular point of E where E is of maximal geometricaldimension c 0 , with J κ 01 arbitrarily close to the identity jet J κ 0Id . Leth 1 ∈ H ρ 2,ρ 1M,κ 0 ,ε such that Jκ 01 = J κ 0(h 1 ). Let U 1 be a small neighborhoodof J κ 01 in C Nn,κ 0 in which E ∩ U1 is a regular c 0 -dimensional real algebraicsubmanifold and consider the complex algebraic mapping defined over U 1by(6.25) F 1 (J κ 0) := ([∂ α t (h −11 (H(t, J κ 0)))] t=0 ) |α|≤κ0 ∈ C Nn,κ 0 .We have F 1 (J κ 01 ) = Jκ 0Id and the restriction of F 1 to E∩U 1 induces a homeomorphismonto its image, which is a neighborhood of J κ 0Idin E. We remindthat the mapping J κ 0→ ([∂t α (H(t, J κ 0))] t=0 ) |α|≤κ0 restricted to E ∩ U 1 isthe identity and consequently of constant rank equal to c 0 . As h 1 is invertible,it follows from the chain rule by developing (6.25) that F 1 | E∩U1 is alsoof locally constant rank equal to c 0 . This proves that E is a c 0 -dimensionalreal algebraic submanifold in C Nn,κ 0 through Jκ 0Id. More generally, this reasoningshows that E is geometrically smooth at every point.Finally, applying Lemma 6.3 with the odd integer k = µ 0 = 2ν 0 + 1(instead of k = µ 0 + 1), we get a new, different representation formulah(t) = ˜H(t, J l 0µ 0¯h(0)) (notice ¯h(0)). Accordingly, we can define a realalgebraic submanifold Ẽ. It is clear that we can identify E and Ẽ, since theyboth parametrize the local biholomorphic self-mappings of M, so they arealgebraically equivalent by means of the natural projection from the l 0 (µ 0 +1)-th jet space onto the l 0 µ 0 -th jet space. Next, we see by differentiating