CONTENTS OF THE COURSE

GEOMETRY OF DIFFERENTIAL EQUATIONS

SECOND DIFFIETY SCHOOL, Forino, February – March 1999

Joseph Krasil’shchik

1. Jet bundles

1.1. Vector bundles and sections. Smooth manifolds. Smooth locally trivial vector bundles. Sections. The C^{∞}(M)-module structure in Γ(π).

1.2. Jets. The jet [f]_{x}^{k} of a local section f at a point x ∈ M. The space J_{x}^{k}(π). Smooth structure in ⋃_{x∈M} J_{x}^{k}(π). Manifolds J^{k}(π) and bundles π_{k}: J^{k}(π) → M. The jet modules J^{k}(π) = Γ(π_{k}). Canonical coordinates u_{σ}^{j} in J^{k}(π) associated to a local trivialization in π. Dimension of J^{k}(π). The bundles π_{k,l}: J^{k}(π) → J^{l}(π), k ≥ l. Graphs of jets. R-planes. Presentation of points of J^{k}(π) as pairs (θ_{k−1}, L_{θ_{k}}), where θ_{k−1} ∈ J^{k−1}(π) and L_{θ_{k}}⊂T_{θ_{k−1}}J^{k−1}(π) is an R-plane.

1.3. Nonlinear differential operators. Presentation of scalar operators as functions on J^{k}(π). Pull-backs π_{k}^{∗}(ξ) and nonlinear operators Δ: Γ(π) → Γ(ξ) as sections of the bundles π_{k}^{∗}(ξ). Presentation of operators as morphisms J^{k}(π) → J^{0}(ξ). The universal operator j_{k}: Γ(π) → Γ(π_{k}). Prolongations of nonlinear operators and their correspondence to morphisms J^{k+l}(π) → J^{l}(ξ). Composition of nonlinear operators.

1.4. Nonlinear equations. Differential equations as submanifolds in E⊂J^{k}(π). Description of equations by nonlinear operators. The first prolongation E^{1}⊂J^{k+1}(π). Three definitions of the l-the prolongation, there equivalence. Solutions.

2. Geometry of the Cartan distribution in J^{k}(π)

2.1. The Cartan distribution. The Cartan plane C_{θ}^{k} as the span of the set of R-planes at the point θ ∈ J^{k}(π). The distribution C^{k}: θ ↦ C_{θ}^{k}. Description of C_{θ}^{k} in the form (π_{k,k−1})_{∗}^{−1}(L_{θ_{k}}). Local description of C^{k} by the Cartan forms ω_{σ}^{j} = du_{σ}^{j} − ∑_{i} u_{σ+1_{i}}^{j}dx_{i}. A local basis in C^{k}.

2.2. Maximal integral manifolds of the distribution C^{k}. Involutive subspaces of the Cartan distribution. The theorem on maximal integral manifolds. The type of a maximal integral manifold. Computation of dimensions for maximal integral manifolds. Integral manifolds of maximal dimension in inexceptional cases.

2.3. The Lie–Bäcklund theorem. Lie transformations as diffeomorphisms of J^{k}(π) preserving the Cartan distribution. Lifting of Lie transformations from J^{k}(π) to J^{k+1}(π). The case dim π > 1: correspondence between Lie transformations and diffeomorphisms of J^{0}(π). The case dim π = 1: the contact structure in J^{1}(π), correspondence between Lie transformations and contact transformations of J^{1}(π) (inexceptional case dim M ≠ 1 and exceptional case dim M = 1). Local formulas for liftings of Lie transformations.

2.4. Infinitesimal theory. Lie fields. Local lifting formulas. Global nature of lifting for Lie fields. Infinitesimal analog for the Lie–Bäcklund theorem.

One-dimensional bundles. Generating functions of Lie fields. Correspondence between functions on J^{1}(π) and Lie fields for trivial onedimensional bundles. The jacobi bracket on C^{∞}(J^{1}(π)). Local coordinate formulas for Lie fields and Jacobi brackets in terms of generating functions.

Bundles of higher dimensions. The element ρ_{k}(π) ∈ J^{k}(π_{k}^{∗}(π)), its definition and properties. The Spencer complexes ⋯ → J^{k}(ξ) ⊗

Λ^{l}(N) ⟶^{S_{k}^{l}} J^{k−1}(ξ) ⊗ Λ^{l+1}(N) → ⋯ for a vector bundle ξ: P → N, their exactness. The element U_{k}(π) = S_{k}^{0}(ρ_{k}(π)) ∈ J^{k−1}(π_{k}^{∗}(π)) ⊗ Λ^{1}(J^{k}(π)), its properties. Generating sections f ∈ Γ(π_{k}^{∗}(π)) as the result of construction of Lie fields with U_{1}(π). Jacobi brackets for generating sections. Local coordinates.

3. Classical symmetry theory for differential equations

3.1. Classical symmetries. Finite and infinitesimal symmetries, definitions. “Physical meaning” of generating functions. Determining equations for coordinate computations. An example: symmetries of the Burgers equation u_{t} = uu_{x} + u_{xx}.

3.2. Exterior and interior symmetries. The restriction C(E) of the Cartan distribution to E. Exterior Lie (E) and interior Lie_{int}(E) symmetries of an equation E⊂J^{k}(π). The homomorphism r: Lie (E) → Lie_{int}(E). Counterexamples.

4. Perspectives

Algebraic model. The basic constructions. Cohomological invariants.