R-manifolds and multi-valued solutions of PDE's
Nina Khor'kova
A program of the course at the 4-th Italian Diffiety School,
(Forino, July 17-29, 2000)
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Jet manifolds and PDE
  1. Introduction: jet spaces Jk(n,m).
  2. Jets of vector bundles Jk(p).
  3. Jets of submanifolds Jk(E,n).
  4. Jets of fibring. Jets of mappings M® N. Jets of functions.
  5. Definitions of differential equation and its solution.
Geometric interpretation of solutions. R-manifolds.
  1. The Cartan distribution on Jk(E,n).
  2. Coordinate description of the Cartan distribution.
  3. Geometric structure of the Cartan planes.
  4. Description of integral submanifolds of the Cartan distribution.
  5. Ray submanifolds, prolongations of integral submanifolds.
  6. The structure of locally maximal integral submanifolds of the Cartan distribution.
  7. R-manifolds.
  8. Definitions of differential equation and its solution.
Lie transformations (high order contact transformations).
  1. Lie transformations. Point transformations.
  2. Prolongations of point transformations. Prolongations of Lie
    transformations.
  3. Prolongations of morphisms of contact structures.
  4. Lie-Bäcklund theorem.
  5. Lie fields, liftings (prolongations) of Lie fields.
  6. Infinitesimal Lie-Bäcklund theorem.
Extrinsic and intrinsic geometries of PDE.
  1. External and internal points of view on PDE.
  2. Problem of the reconstruction of the embedding E® Jk and the Cartan distribution on Jk.
  3. Extrinsic and intrinsic symmetries of PDE.
  4. Rigidity. Examples.
Examination problems

  1. Let p:En+m® Mn be a smooth bundle. Prove that Jk(p) is an open everywhere dense subset of Jk(E,n).

  2. Let E Ì J2(IR3,2) be the minimal surface equation. Prove that p2,1:E® J1(IR3,2) is a nontrivial 2-dimensional vector bundle.

  3. Let k ³ l. Prove that Jk(E,n) is the manifold of (k-l)-jets of submanifolds of the form L(l) in Jl(E,n).

  4. Prove that the Cartan distribution on Jk(E,n) is locally determined by the set of the Cartan forms
    wjs = dpjs - n
    å
     i = 1 
    		 dpjs+1idxi,  | s | < k, j = 1,...,m.

  5. Let p:E® M be a fiber bundle and let p1,0:J1(p)® E. Show that sections of the bundle p1,0 are connections in the bundle p, while the condition of zero curvature determines a first order equation in the bundle p1,0 .

  6. Consider the system of equations:
    ì
    í
    î
    ux
    =
    f(x,y,u)
    uy
    =
    g(x,y,u).

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

  7. Prove the following statements:

    1. L(k) is a locally maximal integral manifold of the Cartan distribution.
    2. Let Q Ì Jk(E,n) be an n-dimensional integral manifold that is transversal to the fibers of the projection pk,k-1. Then, locally, Q is of the form L(k).

    1. Let x = X(x,y,z)[()/(x)] + Y(x,y,z)[()/(y)] + Z(x,y,z)[()/(z)] be a nonzero vector at a point q Î J0(2,1), and P the straight line determined by this vector. Deduce equations describing the ray manifold l(P).
    2. Let W be a curve x = a(t),y = b(t),z = g(t). Describe L(W).
    3. Let W be a surface of the form z = f(x,y). Describe L(W).

  9. Prove that dimL(W) = r+m ((k+n-r-1) || (n-r-1)), where r = dimW.

  10. Let F:J0(p) ® J0(p) be a point transformation. Prove that if the lifting F(1) is defined in a neighborhood of a point q Î J1(p), then the lifting F(k) is defined in a neighborhood of any point q¢ such that pk,1(q¢) = q.

  11. Prove that ordinary differential equations are non rigid.

Exercises

  1. Prove that the family of neighborhoods pk-1(U) together with the coordinate functions (x,pjs) determines a smooth manifold structure in Jk(p). Prove that dimJk(p) = n+m(n || (n+k))

  2. Prove that pk:Jk(p) ® M is a smooth locally trivial vector bundle.

  3. Prove that pk+1,k:Jk+1(p) ® Jk(p) is a smooth locally trivial bundle (not vector bundle).

  4. Prove that
    1. pl,s°pk,l = pk,s, k ³ l ³ s;
    2. pl°pk,l = pk, k ³ l;
    3. pk,l°jk(s) = jl(s), k ³ l.

  5. Prove that
    1. Jk(E,n) is a smooth manifold of dimension n+m(n || (n+k));
    2. pk,l:Jk(E,n) ® Jl(E,n), k ³ l is a smooth locally trivial bundle;
    3. pl,s°pk,l = pk,s,  pk,l°jk(L) = jl(L), k ³ l ³ s.

  6. Let F:Jk(E,n)® Jk(E,n) be a Lie transformation. Prove that
    1. pk+l,k+s°F(l) = F(s)°pk+s,k+l, l ³ s;
    2. id(s) = id;
    3. (F°G)(s) = F(s)°G(s).

  7. Consider the Legendre transformation F in the space J1(2,1): [`x] = -p, [`y] = -q, [`u] = u-xp-yq, [`p] = x, [`q] = y.
    1. Prove that F is a Lie transformation;
    2. Describe the lifting F(1);
    3. 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 [`F]:M® M of two bundles p and p¢ over M. Define the prolongation F(k):Jk(p)® Jk(p¢) in such a way that F(k) will be a morphism of pk in p¢k over [`F].

  9. Let F:Jk(E,n)® Jl(E,n), k > l be a smooth surjection such that F*C(k)q = C(l)F(q). Define the prolongation F(k):Jk(E,n)® Jk(E,n).

  10. For the vector field X = x[()/(u)] - u[()/(x)] on J0(2,1) find X(2).

Passing the exam during the school required 6 solved problems. To pass the exam by email one should solve 10 problems.
The exam has been passed by the following students:
  1. Giovanni Manno (6 problems: 3,4,6,7,9,13)
  2. Luca Vitagliano (9 problems: 1,3 - 8,10,13)
Questions and suggestions should go to Jet NESTRUEV, jet @ diffiety.ac.ru.