PROBLEMS TO THE COURSE

GEOMETRY OF DIFFERENTIAL EQUATIONS

INDEPENDENT UNIVERSITY OF MOSCOW, february – april 1998

Joseph Krasil’shchik

Problem 1. Let N be a smooth manifold and 0 ≤ m ≤ dim N. The k-jet of a submanifold M⊂N, dim M = m, at a point x ∈ M is the set [M]_{x}^{k} of submanifolds M′ tangent to M at the point x with order k. The set of k-jets of m-dimensional submanifolds is denoted by J^{k}(N, m). Prove that

(a) J^{k}(N, m) is a smooth manifold;

(b) the natural mappings J^{k}(N, m) → J^{l}(N, m), k ≥ l, as well as J^{k}(N, m) → N, are smooth surjections;

(c) J^{k}(N, m) is locally diffeomorphic to a manifold J^{k}(π).

Problem 2. Define the Cartan distribution on J^{k}(N, m). Formulate and prove the Lie–Bäcklund theorem for these manifolds.

Problem 3. Two manifolds X and X′ with distributions D and D′ are called locally isomorphic, if there exists a local diffeomorphism

X ⊃ U ⟶^{f} U′⊂X′ satisfying f_{∗}(D_{x}) = D′_{f(x)}, x ∈ U. When the

manifolds J^{k}(π) and J^{k′}(π′) are locally isomorphic, if the corresponding distributions are the Cartan distributions?

Problem 4. Prove that the bundles π_{k,k−1} : J^{k}(π) → J^{k−1}(π) carry a natural affine structure, if (a) dim π > 1 and k ≥ 1 or (b) dim π = 1 and k > 1. Describe the fibers of these bundles.

Problem 5. What is a natural structure in the bundles J^{k}(N, m) → J^{k−1}(N, m)?

Problem 6. Let π: E → M, π′: E′ → M′ be smooth vector bundles and φ: M′ → M be a smooth mapping.

(a) Define the notion of a nonlinear differential operator Δ: Γ(π) → Γ(π′) over the mapping φ.

(b) Prove that if π′′: E′′ → M′′ is a smooth vector bundle, φ′: M′′ → M′ is a smooth mapping, and Δ: Γ(π) → Γ(π′), Δ′: Γ(π′) → Γ(π′′) are nonlinear operators of order k and k′ over the mappings φ and φ′ respectively, then Δ′ ∘ Δ is an operator of order k + k′ over φ′ ∘ φ.

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Problem 7. Let x ∈ S^{1} be a point on the circle and μ_{x}⊂C^{∞}(S^{1}) be the ideal of functions vanishing at the point x.

(a) Prove that μ_{x} is a projective C^{∞}(S^{1})-module.

(b) Describe the vector bundle over S^{1} corresponding to this module.

Problem 8. Construct an example of a unitary commutative algebra for which differential operators are not generated by operators of orders 0 and 1.

Problem 9. Let A = ℤ_{p}[x]/(x^{n}), n, p ∈ ℤ_{+}. Describe the algebra of differential operators A → A.

Problem 10. Prove that in the algebra of continuous functions over ℝ^{1} there exists no linear differential operators of order > 0.

Problem 11. Let A be a commutative unitary algebra and P be an A-module. The module of k-jets J^{k}(P) of the module P is defined as follows. A natural structure of an A-module is introduced into the tensor product A⊗_{K} P: a(b⊗p) = (ab)⊗p. For any a ∈ A, the operator δ^{a}: A ⊗_{K} P → A ⊗_{K} P is defined by δ^{a}(b ⊗ p) = ab ⊗ p − b ⊗ ap. Then J^{k}(P) = (A⊗_{K} P)/μ_{k}, where μ_{k} is the submodule in A⊗_{K} P generated by all elements of the form (δ^{a_{0}}∘⋯∘δ^{a_{k}})(b⊗p), a_{0}, …, a_{k}, b ∈ A, p ∈ P. There exists a natural mapping j_{k}: P → J^{k}(P), j_{k}(p) = 1 ⊗ p mod μ_{k}.

(a) Prove that j_{k} is a linear differential operator of order ≤ k.

(b) Prove that any linear differential operator Δ ∈ Diff(P, Q), Q being an A-module, is uniquely represented in the form Δ = φ_{Δ} ∘ j_{k} and thus a natural isomorphism Diff(P, Q) = Hom_{A}(J^{k}(P), Q) is valid.

Problem 12. If in Problem 11 one sets P = A, k = 1, then the correspondence i: a ↦ a ⊗ 1 mod μ_{1} will define the A-module homomorphism i: A → J^{1}(A). The quotient module Λ^{1}(A) = J^{1}(A)/im i is called the module of 1-forms for the algebra A. Let us define the mapping d: A → Λ^{1}(A) as the composition of the operator j_{1} : A → J^{1}(A) with the natural projection J^{1}(A) → Λ^{1}(A).

(a) Prove that d is a derivation of the algebra A taking values in the A-module Λ^{1}(A).

(b) Prove that any derivation Δ ∈ D(Q), where Q is an A-module, is uniquely represented in the form Δ = ψ_{Δ} ∘ d and thus a natural isomorphism D(Q) = Hom_{A}(Λ^{1}(A), Q) takes place.

Problem 13. Prove that if A is an algebra over the field of rational numbers, while Λ^{1}(A) and P are projective modules of finite type, then J^{k}(P) is a projective finite type module for any k ≥ 0.

Problem 14. Describe the modules Λ^{1}(A) and J^{k}(A) for the algebra A from Problem 9.

Problem 15. Let A = C^{∞}(M), where M is a smooth manifold, and P be a A-module. Let us set G(P) = P/(∩_{x∈M}μ_{x}), where μ_{x} is the ideal of the point x. The module G(P) is called the geometrization of the module P.

(a) Prove that G(Λ^{1}(A)) (see Problem 12) coincides with the module of 1-forms of the manifold M.

(b) Let P = Γ(π) be the module of smooth sections of the vector bundle π: E → M. Prove that G(J^{k}(P)) (see Problem 11) coincides with the module of sections for the bundle of k-jets π_{k}: J^{k}(π) → M.

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Problem 16. Construct an example of an equation with the trivial Lie algebra of symmetries.

Problem 17. Consider a nonlinear operator Δ: Γ(π) → Γ(ξ) and let Φ_{Δ} : J^{∞}(π) → J^{∞}(ξ) be the morphism of the bundles π_{∞} and ξ_{∞} determined by this operator.

(a) Prove that Φ_{Δ} takes the Cartan distribution on J^{∞}(π) to that on J^{∞}(ξ).

(b) What are the conditions for Φ_{Δ}(J^{∞}(π)⊂J^{∞}(ξ) to be an infinitely prolonged equation?

(c) What Lie transformations (Lie fields) can be pulled down from J^{∞}(π) to J^{∞}(ξ)?

(d) What Lie transformations (Lie fields) can be lifted up from J^{∞}(ξ) to J^{∞}(π)?

Problem 18. Over every point x ∈ M in the fiber of the direct product M × ℝ^{m} a linear subspace L_{x}, dim L_{x} = n = const is given. Prove that the family {L_{x}}_{x∈M} is a vector bundle if and only if the mapping L: M → G_{m,n}, x ↦ L_{x}, is smooth.

Problem 19. Let ψ: J^{k}(π) → J^{k+1}(π) is a section of the bundle π_{k+1,k}. The graph G_{ψ} of this section is an equation of order k + 1.

(a) Describe the infinite prolongation of this equation.

(b) Describe its interior and exterior symmetries.

Problem 20. Any ψ from the Problem 19 determines a horizontal distribution D_{ψ} on J^{k}(π): D_{ψ}: θ ↦ L_{ψ(θ)}. Hence, the integrability conditions for the distribution D_{ψ} are a (nonlinear) equation I in the bundle π_{k+1,k}.

(a) Is the equation I formally integrable?

(b) Compute the Lie algebra of symmetries for the equation I.

Problem 21. Let F: J^{∞}(π) → J^{∞}(π) be a Lie transformation and �_{φ} be an evolutionary derivation on J^{∞}(π). Deduce the formulas transforming the generating section φ under the action of F.

Problem 22. Let π: ℝ × M → M be the trivial one-dimensional bundle and E⊂J^{1}(π) be a first-order equation. Consider the Cartan form ω ∈ Λ^{1}(J^{1}(π)) and let us set Ω = dω |_{E}. Then the 2-form Ω (almost everywhere) determines a symplectic structure on E. Study relations between symmetries of this structure and (exterior and interior) symmetries of the equation E.

Problem 23. Let the equation E be the same as in Problem 22. Assume that E is given by the operator f ∈ C^{∞}(J^{1}(π)).

(a) Prove that the contact field X_{f} is a symmetry of the equation E. An (n−1)-dimensional integral submanifold of the Cartan distribution on E is called the Cauchy data for the equation E. The Cauchy data is called noncharacteristic, if it is transversal to the field X_{f} (see (a)).

(b) Prove that the Cauchy problem is solvable for noncharacteristic Cauchy data, i.e., there exists a maximal n-dimensional manifold of the Cartan distribution containing the Cauchy data.

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Problem 24. Let U_{1} : F(π) → Λ^{1}(π) be the derivation determined by the Cartan connection in the bundle π_{∞}. It can be uniquely prolonged up to a complex d_{C}: Λ^{i}(π) → Λ^{i+1}(π), if one sets (1) d_{C} ∘ d = −d ∘ d_{C} and (2) d_{C}(ω ∧ θ) = d_{C}(ω) ∧ θ + (−1)^{|ω|}ω ∧ d_{C}(θ). Set d̄= d − d_{C}.

(a) Prove that the restriction of d̄ to Λ̄^{i}(π) coincides with the horizontal de Rham differential.

(b) Prove that d̄∘ d_{C} = d_{C} ∘ d̄.

Problem 25. By Problem 24, one has the direct sum decomposition of modules Λ^{i}(π) = ⨁_{p+q=i} Λ̄^{q}(π) ⊗_{F(π)} CΛ^{p}(π), where CΛ^{p}(π) is the submodule in Λ^{p}(π) generated by the forms d_{C}(f_{1})∧…∧d_{C}(f_{p}), f_{1}, …, f_{p} ∈ F(π).

(a) Prove that

d̄(Λ̄^{q}(π) ⊗_{F(π)} CΛ^{p}(π))⊂Λ̄^{q+1}(π) ⊗_{F(π)} CΛ^{p}(π),
d_{C}(Λ̄^{q}(π) ⊗_{F(π)} CΛ^{p}(π))⊂Λ̄^{q}(π) ⊗_{F(π)} CΛ^{p+1}(π).

(b) Prove that the composition of the operator d_{C}: Λ̄^{n}(π) → Λ̄^{n}(π) ⊗ CΛ^{1}(π) with the natural projection Λ̄^{n}(π) ⊗ CΛ^{1}(π) → Λ̄^{n}(π) ⊗ CΛ^{1}(π)/d̄(Λ̄^{n−1}(π) ⊗ CΛ^{1}(π)) is the Euler operator, n = dim M.

Problem 26. Prove that the first group H̄^{1}(E) of horizontal de Rham cohomology for the Korteweg–de Vries equation u_{t} = u_{xxx} + uu_{x} is an infinite-dimensional vector space over ℝ.

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Solutions of the last two problems are unknown to the author.

Problem 27. Explain or disprove the following experimental fact: there exists no differential equation possessing a finite number of higher (nonclassical) symmetries.

Problem 28. Explain or disprove the following experimental result: if an equation possesses infinite series of both higher and classical symmetries, then it is linear.