Geometry of infinite jets
and infinitely prolonged equations

Vladimir CHETVERIKOV
Exercises and problems of the course at the Join Russian-Italian Diffiety School,
Pereslavl-Zalessky (Russia), August 17  -  30, 1999

Exercise 1. What are J0(P,n) and J1(P,n)?

Exercise 2. Suppose that the l-th prolongation of an equation E is a smooth submanifold in Jk+l p. Show that in this case one has (E(l))(t) = E(l+t) for all t 0.

Exercise 3. Construct example(s) of an equation E such that E(1) is not a smooth submanifold in Jk+1 p.

Exercise 4. Construct example(s) of differential equation(s) for which the mapping pk+1,k :   E(1) E is not surjective.

Exercise 5. Let submanifolds N, N1 P be tangent to each other at a point a N N1 with order k, m = {f C(N) | f(a) = 0}, mk+1 be the (k+1)-st degree of the ideal m. Show that if g C(P) and g |N1 = 0 then g |N mk+1.

Problem 1. Consider a differential equation with delay of the form

ut(t) = f(t,u(t),u(t-a)),
where a = const, u is a scalar function. Try to invent a definition of space of generalized jets in such a way that this equation can be naturally interpreted as a submanifold of this space.

Hint: First describe all prolongations of this equation.

Exercise 6. Construct example(s) of map(s) G :   J pJx such that G*(F(x)) F(p) and for any l and k0 there exists an integer k k0 such that G*(Fk(x)) Fk+l(p).

Exercise 7. The Cartan distribution can be also defined on J p. What is the dimension of this distribution?

Exercise 8. Construct example(s) of derivation(s) X of the algebra F(p) such that for any l and k0 there exists an integer k k0 such that X(Fk(p)) Fk+l(p).

Exercise 9. Show that if X, Y D(p) and f F, then X+Y, fX, and [X,Y] = X Y - Y X are also vector fields on Jp. In addition, show that D (p) is a Lie algebra over IR and Dv(p) is its Lie subalgebra.

Problem 2. Let X D(p) and K(X) F(p) be the subalgebra generated by functions f, X(f), , Xk(f) = X(Xk-1(f)), , where f F0(p), k > 1. Prove that if the vector field X is integrable, then there exists an integer l 0 such that K(X) Fl(p).

Exercise 10. Using Problem 2, construct examples of

(i) integrable vector field(s) that raise(s) the filtration of the algebra F(p),

(ii) nonintegrable vector field(s).

Exercise 11. Show that the module D(p) is dual to L1(p), i.e.,

D(p) @ HomF(p)f(L1(p),F(p)),
where HomF(p)f denotes the module of filtered homomorphisms over F(p). (An F(p)-homomorphism h :  L1(p) F(p) is called filtered if there exists an integer l such that h(L1(pk)) Fk+l(p) for any k 0).

Exercise 12. Construct the map YD,x starting from the smooth map YD :  J(p)J(h).

Exercise 13. Show that Spec  injlimk Fk(p) = projlimk Spec  Fk(p).

Exercise 14. Let X be a derivation of the algebra F(p), l 0, and there exists an integer k0 such that X(Fk(p)) Fk+l(p) for any k k0. Show that X D(p).

Exercise 15. Prove that a distribution P is integrable iff d(PL*) PL*.

Exercise 16. Prove that X DP(p) iff [X,PD(p)] PD(p).

Exercise 17. Show that DP(p) is a Lie algebra and PD(p) is its ideal.

Exercise 18. Prove that the equality Cq(p) = Tq(j(s)(M)) holds for any point q = [s]x J(p).

Hint: Use the equalities Tq(j(s)(M)) = j(s)*(Tx(M)) and Cqkk = (pk,k-1)*-1(Lqk).

Exercise 19. Show that j(s)* [^X] = X j(s)* for any s G(p) and X D(M).

Exercise 20. Using Exercise 19, show that the Cartan connection is flat.

Exercise 21. Let X D(M), Dj :  G(p)C(M) be the scalar differential operator associated to a function j Fk(p). Show that the operator X Dj is associated to the function [^X](j): D[^X](j) = X Dj.

Exercise 22. Show that [^X] D(p), with deg ([^X]) = 1, for any X D(M).

Exercise 23. Prove that the Cartan distributions on J p and E are integrable.

Exercise 24. Show that X CD(E) iff X    UC(f) = 0 for any f F(E).

Exercise 25. Show that

UC(f) =

j,s 
f
usj
wsj,
where {xi, usj} are the special coordinates on J p, wsj = dusj-i = 1n usij dxi are the Cartan forms.

Exercise 26. Prove that UC(f g) = f UC(g) + g UC(f) for any f,g F(E).

Exercise 27. Write down the formula for UC in the case of the KdV equation ut = uxxx + u ux.

Exercise 28. Let p :  IR×IRIR be the trivial bundle. Show that any infinitesimal automorphism of the Cartan distribution on J p has the form

X = aD+

k = 0 
Dk(b)
uk
,
where (x,u0,...,uk,...) are the special coordinates on J p,
D =
x
+

k = 0 
uk+1
uk
is the total derivative along x.

Exercise 29. Prove that the module F(p,p) is a Lie IR-algebra with respect to the higher Jacobi bracket.

Exercise 30. Show that if D :  F(p,x)F(p,h) is a C-differential operator and D(p*(j)) = 0 for any j G(x), then D = 0.

Exercise 31. Show that the two definitions of linearization are equivalent.

Exercise 32. Let E be an evolution equation, j be a symmetry of E, ' jE and ljE be the restrictions of the corresponding operators to E. Prove that [ ' jE -ljElE] = 0 (the commutator relation).


Questions and suggestions should go to Jet Nestruev, jet @ diffiety.ac.ru.