By Krasil'shchik J., Verbovetsky A.

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**Sample text**

Then the tangent plane to N at θk is a maximal involutive plane. Let its type be equal to r(θk ). 15. The number tp(N) = max r(θk ) is called the type of the θk ∈N maximal integral manifold N of the Cartan distribution. Obviously, the set g(N) = { θk ∈ N | r(θk ) = tp(N) } is everywhere dense in N. We call the points θk ∈ g(N) generic. Let θk be such a point and U be its neighborhood in N consisting of generic points. Then: (1) N ′ = πk,k−1(N) is an integral manifold of the Cartan distribution on J k−1 (π); (2) dim(N ′ ) = tp(N) and (3) πk−1 |N ′ : N ′ → M is an immersion.

Let Q be an A-module. Then a natural homomorˆˆ defined by ξQ (q)(ˆ phism ξQ : Q → Q q ) = q, qˆ exists. Consequently, to any ˆ operator ∆ : P → Q there corresponds the operator ∆◦ : Q → Pˆ , where ∆◦ = ∆∗ ◦ ξQ . This operator will also be called adjoint to ∆. 2. In the geometrical situation the two notions of adjointness coincide. 3. Let qˆ ∈ Q by a → aˆ q . The adjoint operator is qˆ itself understood as an element of HomA (Q, B). 5. The correspondence ∆ → ∆◦ possesses the following properties: ˆ and ∆(p) = [∇p ], where ∇p ∈ Diff(Q, Λi).

The correspondence jk : Γ(π) → Γ(πk ) is called the k-jet operator. 5) for any ϕ ∈ Γ(π). Let ϕ, ψ ∈ Γ(π) be two sections, x ∈ M and ϕ(x) = ψ(x) = θ ∈ E. It is a tautology to say that the manifolds Γϕ and Γψ are tangent to each other with order k + l at θ or that the manifolds Γjk (ϕ) , Γjk (ψ) ⊂ J k (π) are tangent with order l at the point θk = jk (ϕ)(x) = jk (ψ)(x). 4. Let θk ∈ J k (π). An R-plane at θk is an n-dimensional plane tangent to a manifold of the form Γjk (ϕ) such that [ϕ]kx = θk .

### Homological methods in equations of mathematical physics by Krasil'shchik J., Verbovetsky A.

by Thomas

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