Invariants différentiels d'un pseudogroupe de Lie. II

On the Equivalence Problem for Geometric Structures, Ii

“…Apparently, the most efficacious criteria are provided, on the one hand, by the proposition 21.5 of [6] that involves the bracket of formal and holonomic derivations and, on the other, by the proposition 25.4 that relies on the explicit formula (25.5) which, surprisingly, can already be found in Lie's work. The hypotheses underlying the last proposition are no other than conditions of transversality, in the strict sense, of the k − th order flows of local sections S of the fibration E −→ P with the trajectories of ∆ k such conditions being entirely antipodal to a formal transitivity hypothesis.…”

“…We infer the following corollaries by taking also into account the Theorem 13.1 as well as the Proposition 14.2 in [6].…”

“…, defines by means of the canonical identification ( [6], Proposition 14.2) a vector field ν(v) along the fiber A Z of J k+1 E above the point Z. This vector field admits a global 1-parameter group (ϕ t ) of affine transformations and the affine action of v on W ∈ A Z is defined by…”

“…We next recall ( [6], §19) that J k+1 E −→ J k E is an affine bundle, the underlying vector bundle being equal to (S k+1 T * P ⊗ V E) × E J k E, the affine action being defined as follows:…”

“…This matter will be discussed in more detail in later sections. However, it is also worthwhile to mention that we can employ, in this context, the method of characters by extending the reasoning of [6], §24, and relate the characters of the kernels of J ℓ+k L with those of ∆ k−1,k .…”