R-manifolds and multivalued solutions of PDE’s

Nina Khor’kova

A program of the course at the 4-th Italian Diffiety School, (Forino, July 17-29, 2000)

• Jet manifolds and PDE.

– Introduction: jet spaces J^{k}(n, m).

– Jets of vector bundles J^{k}(π).

– Jets of submanifolds J^{k}(E, n).

– Jets of fibring. Jets of mappings M → N. Jets of functions.

– Definitions of differential equation and its solution.

• Geometric interpretation of solutions. R-manifolds.

– The Cartan distribution on J^{k}(E, n).

– Coordinate description of the Cartan distribution.

– Geometric structure of the Cartan planes.

– Description of integral submanifolds of the Cartan distribution.

– Ray submanifolds, prolongations of integral submanifolds.

– The structure of locally maximal integral submanifolds of the Cartan distribution.

– R-manifolds.

– Definitions of differential equation and its solution.

• Lie transformations (high order contact transformations).

– Lie transformations. Point transformations.

– Prolongations of point transformations. Prolongations of Lie transformations.

– Prolongations of morphisms of contact structures.

– Lie-Bäcklund theorem.

– Lie fields, liftings (prolongations) of Lie fields.

– Infinitesimal Lie-Bäcklund theorem.

• Extrinsic and intrinsic geometries of PDE.

– External and internal points of view on PDE.

– Problem of the reconstruction of the embedding E → J^{k} and the Cartan distribution on J^{k}.

– Extrinsic and intrinsic symmetries of PDE.

– Rigidity. Examples.

Problems

1. Let π : E^{n+m} → M^{n} be a smooth bundle. Prove that J^{k}(π) is an open everywhere dense subset of J^{k}(E, n).

2. Let E ⊂ J^{2}(IR^{3}, 2) be the minimal surface equation. Prove that π_{2,1} : E → J^{1}(IR^{3}, 2) is a nontrivial 2-dimensional vector bundle.

3. Let k ≥ l. Prove that J^{k}(E, n) is the manifold of (k-l)-jets of submanifolds of the form L^{(l)} in J^{l}(E, n).

4. Prove that the Cartan distribution on J^{k}(E, n) is locally determined by the set of the Cartan forms

ω_{σ}^{j} = dp_{σ}^{j} − ∑_{i=1}^{n} dp_{σ+1_{i}}^{j}dx_{i}, | σ |< k, j = 1, …, m.

5. Let π : E → M be a fiber bundle and let π_{1,0} : J^{1}(π) → E. Show that sections of the bundle π_{1,0} are connections in the bundle π, while the condition of zero curvature determines a first order equation in the bundle π_{1,0} .

6. Consider the system of equations:

\{ \begin{matrix}u_{x} & = & f(x, y, u) \\ u_{y} & = & g(x, y, u). \end{matrix}

Prove that if this system is compatible, then the Cartan distribution restricted to the corresponding surface is completely integrable.

7. Prove the following statements:

(a) L^{(k)} is a locally maximal integral manifold of the Cartan distribution.

(b) Let Q ⊂ J^{k}(E, n) be an n-dimensional integral manifold that is transversal to the fibers of the projection π_{k,k−1}. Then, locally, Q is of the form L^{(k)}.

8. (a) Let ξ = X(x, y, z)\frac{∂}{∂x} + Y (x, y, z)\frac{∂}{∂y} + Z(x, y, z)\frac{∂}{∂z} be a nonzero vector at a point θ ∈ J^{0}(2, 1), and P the straight line determined by this vector. Deduce equations describing the ray manifold l(P).

(b) Let W be a curve x = α(t), y = β(t), z = γ(t). Describe L(W).

(c) Let W be a surface of the form z = f(x, y). Describe L(W).

9. Prove that dim L(W) = r + m\binom{k+n−r−1}{n−r−1}, where r = dim W.

10. Let F : J^{0}(π) → J^{0}(π) be a point transformation. Prove that if the lifting F^{(1)} is defined in a neighborhood of a point θ ∈ J^{1}(π), then the lifting F^{(k)} is defined in a neighborhood of any point θ′ such that π_{k,1}(θ′) = θ.

11. Prove that ordinary differential equations are non rigid.

Exercises

1. Prove that the family of neighborhoods π_{k}^{−1}(U) together with the coordinate functions (x, p_{σ}^{j}) determines a smooth manifold structure in J^{k}(π). Prove that dim J^{k}(π) = n + m\binom{n}{n+k}

2. Prove that π_{k} : J^{k}(π) → M is a smooth locally trivial vector bundle.

3. Prove that π_{k+1,k} : J^{k+1}(π) → J^{k}(π) is a smooth locally trivial bundle (not vector bundle).

4. Prove that

(a) π_{l,s} ∘ π_{k,l} = π_{k,s}, k ≥ l ≥ s;

(b) π_{l} ∘ π_{k,l} = π_{k}, k ≥ l;

(c) π_{k,l} ∘ j_{k}(s) = j_{l}(s), k ≥ l.

5. Prove that

(a) J^{k}(E, n) is a smooth manifold of dimension n + m\binom{n}{n+k};

(b) π_{k,l} : J^{k}(E, n) → J^{l}(E, n), k ≥ l is a smooth locally trivial bundle;

(c) π_{l,s} ∘ π_{k,l} = π_{k,s}, π_{k,l} ∘ j_{k}(L) = j_{l}(L), k ≥ l ≥ s.

6. Let F : J^{k}(E, n) → J^{k}(E, n) be a Lie transformation. Prove that

(a) π_{k+l,k+s} ∘ F^{(l)} = F^{(s)} ∘ π_{k+s,k+l}, l ≥ s;

(b) id^{(s)} = id;

(c) (F ∘ G)^{(s)} = F^{(s)} ∘ G^{(s)}.

7. Consider the Legendre transformation F in the space J^{1}(2, 1): x̄ = −p, ȳ = −q, ū = u − xp − yq, p̄ = x, q̄ = y.

(a) Prove that F is a Lie transformation;

(b) Describe the lifting F^{(1)};

(c) Prove that F can not be represented in the form F = G^{(1)}, where G is a point transformation.

8. Let F : E → E′ be a morphism over a diffeomorphism F̄ : M → M of two bundles π and π′ over M. Define the prolongation F^{(k)} : J^{k}(π) → J^{k}(π′) in such a way that F^{(k)} will be a morphism of π_{k} in π′_{k} over F̄.

9. Let F : J^{k}(E, n) → J^{l}(E, n), k > l be a smooth surjection such that F_{∗}C_{θ}^{(k)} = C_{F (θ)}^{(l)}. Define the prolongation F^{(k)} : J^{k}(E, n) → J^{k}(E, n).

10. For the vector field X = x\frac{∂}{∂u} − u\frac{∂}{∂x} on J^{0}(2, 1) find X^{(2)}.