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

296In the case m = 1 (single dependent variable y ∈ K), the covering propertyalways hold with µ = µ ∗ = 2.§11. FORMAL AND SMOOTH EQUIVALENCES BETWEEN SUBMANIFOLDSOF SOLUTIONS11.1. Transformations of submanifolds of solutions. Lemma 7.3 showsthat every equivalence ϕ between two PDE systems (E ) and (E ′ ) lifts as atransformation which respects the separation between variables and parametersof the form(11.2)(x, y, a, b) ↦−→ ( φ(x, y), ψ(x, y), f(a, b), g(a, b) ) = ( ϕ(x, y), h(a, b) ) =: (x ′ , y ′ , a ′ , b ′ )from the source submanifolds of solutions M := V S (E ) to the targetM ′ := V S (E ′ ), whose equations are(11.3)y = Π(x, c) or dually b = Π ∗ (a, z) andy ′ = Π ′ (x ′ , c ′ ) or dually b ′ = Π ′∗ (a ′ , z ′ ).The study of transformations between submanifolds of solutions possessesstrong similarities with the study of CR mappings between CR manifolds([Pi1975, We1977, DW1980, BJT1985, DF1988, BER1999, Me2005a,Me2005b]). In fact, one may transfer the whole theory of the analytic reflectionprinciple to this more general context. In the present §10 and in the next§11, we select and establish some of the results that are useful to the Lie theory.Some accessible open questions will also be formulated.Maps of the form (11.2) send leaves of F v and of F p to leaves of F ′ v, andof F ′ p , respectively.MK n+2m+p 0F pK n+2m+pF ′ pcΓ ∗ ([ax] 3 )(ϕ, h)M ′(ϕ(z), h(c))F vΓ ∗ (a 1 )Γ ∗ ([ax] 2 )zF ′ vc ′ z ′0 ′