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

292Every transformation (z, c) ↦→ ( ϕ(z), h(c) ) belonging to G v,p stabilizes both{z = cst.} and {c = cst.}. Accordingly, the two foliations of M(10.3) F v := ⋃ M ∩ { }c = c 0 and F p := ⋃ M ∩ { }z = z 0c 0 z 0are invariant under changes of coordinates. We call (F v , F p ) the fundamentalpair of foliations on M . The leaves of the foliation by variables F v are n-dimensional:(10.4) F v (c 0 ) = { (z, c 0 ) : y = Π(x, c 0 ) }The leaves of the foliation by parameters F p are p-dimensional:(10.5) F p (c 0 ) = { (z 0 , c) : b = Π ∗ (a, z 0 ) }We draw a diagram. In it, the positive codimension is invisible:(10.6) m = dimM − dimF v − dim F p 1cL ∗ 0F vF pzLMM10.7. Chains Γ k and dual chains Γ ∗ k . Similarly as in [GM2004, Me2005a,Me2005b, MP2005] (in a CR context), we introduce two collections(L k ) 1kn and (L ∗ q ) 1qp of vector fields whose integral manifolds coincidewith the leaves of F v and of F p :(10.8)⎧⎪⎨⎪⎩L k := ∂∂x k+m∑j=1L ∗ q := ∂∂a q + m∑j=1∂Π j∂x k(x, a, b) ∂∂y j ,k = 1, . . .,n,∂Π ∗j ∂(a, x, y) , q = 1, . . ., p.∂aq ∂bj