---

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)¹0 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 Î V\{0} there exist y Î V with w(x,y) ¹ 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,…,e_n} of V, the matrix (w(i,j))=(w(e_i, e_j)) is nondegenerate.
2. Let ( V²ⁿ, w ) be a symplectic space; then there exist a basis {e₁, … e_n, f₁, …, f_n} such that w = e₁^* Ù f₁^* + … + e_n^* Ù f_n^*.
3. Let (V²ⁿ, w) be a symplectic space and W Ì V be a subspace. Prove that
   W Ç W^{^} = ker (w½_W) and dim( W Ç W^{^} ) = rk ( w½_W ).
4. Prove that for any g Î Sp(2n, R), det g = 1.
5. Let g Î 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 Ì GL(n, R) be Lie group and g Ì gl(n, R) be its Lie algebra. Prove that for any x Î g^*, the set { bÎ g ½ 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 Ì J¹M be a submanifold. If 1) L is a Legendre's manifold and 2) p₁½_L : L ® M is a diffeomorphism,
    then $ fÎ 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]¹_x )=df½_x and let L Ì J¹M be a Legendre's manifold. Prove that " x Î L there exists a neighborhood V_x Ì L of x such that l ( V_x ) Ì T^*M is a Lagrangian submanifold and r ½_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 Ì T^*M there exist a Legendre's manifold L Ì J¹M so that l ½_L: L ® N is a covering.
18. Let M be a smooth manifold, N Ì M be a submanifold, and C₁, C₂ be contact structures on M with C₁½_T*xN = C₂½_T*xN. "xÎN. Prove that for some neighborhoods U_x, V_x of any point x Î N, there exist a diffeomorphism f: U_x ® V_x with f½_{Ux Ç N} = id such that it takes C₁½_Ux to C₂½_Vx.

---

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)