Contact and symplectic multi-valued solutions of 1st order scalar differential equations
Valery Yumaguzhin
A program of the course at the 4-th Italian Diffiety School,
(Forino, July 17-29, 2000)

  1. Elements of linear symplectic geometry.
    Nondegeneracy of a sqew ortoghonal 2-form. Even dimension of a symplectic space. Symplectic basis. Sqew ortoghonal complement. Isotropic and Lagrangian subspaces. Symplectic group, its basis properties.
  2. Symplectic manifolds.
    Even dimension, orientability of symplectic manifolds. H2(M,R)0 for compact symplectic manifolds. Examples: symplest examples, orbits of the coadjoint representation, cotangent bundle. Darboux's theorem.
  3. Lagrangian manifolds.
    Lagrangian manifolds of a cotangent bundle.
  4. Multi-valued solutions of Hamilton-Jacobi equation.
  5. Elements of the contact geometry.
    Contact forms and a contact structure. Nonintegrability of a contact distribution. Odd dimension, orientability of contact manifolds. Legendre's submanifolds. Darboux's theorem. Bundle of 1-jets of functions. Legendre's submanifolds of this bundle. Contact transformations, Legendre's transformations.
  6. Geometrical interpretation of 1-order scalar differential operators.
  7. Multi-valued generalized solutions of 1-order equations.
  8. Relations between symplectic and contact geometry.
    Contactization of a symplectic manifold. Symplectification of a contact manifold.
Examination problems

  1. Let w be a sqew-symmetric 2-form on a vector space V. Prove that the following definition are equivalent:
    1) w is nondegenerate if for every x V\{0} there exist y V with w(x,y) 0;
    2) w is nondegenerate if the map V V* defined by x w (x, ) is an isomorphism of vector spaces;
    3) w is nondegenerate if for any basis {e1,,en} of V, the matrix (w(i,j))=(w(ei, ej)) is nondegenerate.
  2. Let ( V2n, w ) be a symplectic space; then there exist a basis {e1, en, f1, , fn} such that w = e1* f1* + + en* fn*.
  3. Let (V2n, w) be a symplectic space and W V be a subspace. Prove that
    W W^ = ker (wW) and dim( W W^ ) = rk ( wW ).
  4. Prove that for any g Sp(2n, R), det g = 1.
  5. Let g Sp(2n, R) and P(l) be its characteristic polinomial. Prove that P(l) = l2n P(1/l ). In particular, if l is an eigenvalue of multiplicity r then 1/l is an eigenvalue of multiplicity r too.
  6. Let ( M, w ) be a compact symplectic manifold; then the D'Ram's cohomology class of w is nontrivial.
  7. Let G GL(n, R) be Lie group and g gl(n, R) be its Lie algebra. Prove that for any x g*, the set { b g ad*b(x ) = 0 } is an isotropy algebra of x .
  8. Prove that the sqew symmetric 2-form w on an orbit of the coadjoint representation defined by
    w (h1, h2)= w ( ad*a1(x ), ad*a2(x ) )= x ( [a1, a2] ) is close.
  9. Prove that a symplectic map is an imbedding.
  10. Constract a multivalue solutions of some Hamilton-Jacobi equations.
  11. Let a be a contact form on a manifold M. Prove that M is orientable.
    Constract an example of nonorientable contact manifold.
  12. Let a be a contact form on M. Then there exist a unique vector field X on M with
    a (X) = 1 and da (X, ) = 0.
  13. Let p1: J1M M be 1-jet bundle of smooth functions of manifold M and let L J1M be a submanifold. If 1) L is a Legendre's manifold and 2) p1L : L M is a diffeomorphism,
    then $ f C (M) with L = Im j1f.
  14. Let f: J1M J1M be a diffeomorphism and U1 be a Cartan form on J1M. Prove that the folloving definitions are equivalent:
    1) f is a contact transformation if it takes the Cartan distribution of J1M to itself;
    2) f is a contact transformation if f*U1 = l U1, where l is a nowhere wanishing smooth function on M.
  15. Using the Legendre's transformation, constract a multivalue solutions of some 1-order scalar PDEs.
  16. Let l : J1M T*M be defined by l ( [f]1x )=dfx and let L J1M be a Legendre's manifold. Prove that " x L there exists a neighborhood Vx L of x such that l ( Vx ) T*M is a Lagrangian submanifold and r l ( Vx ) is an exact form ( r is the canonical differential 1-form on T*M ). Constract an example to show that this statement is not true globally.
  17. Prove, for any connected Lagrangian submanifold N T*M there exist a Legendre's manifold L J1M so that l L: L N is a covering.
  18. Let M be a smooth manifold, N M be a submanifold, and C1, C2 be contact structures on M with C1T*xN = C2T*xN. "xN. Prove that for some neighborhoods Ux, Vx of any point x N, there exist a diffeomorphism f: Ux Vx with fUx N = id such that it takes C1Ux to C2Vx.

Passing the exam during the school required 12 solved problems. To pass the exam by email one should solve 15 problems.
The exam has been passed by the following students:
  1. Giovanni Manno (14 problems)
  2. Luca Vitagliano (12 problems)