Valery Yumaguzhin

A program of the course at the 4-th Italian Diffiety School,

(Forino, July 17-29, 2000)

**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.**Symplectic manifolds.**

Even dimension, orientability of symplectic manifolds. H^{2}(M,**R**)¹0 for compact symplectic manifolds. Examples: symplest examples, orbits of the coadjoint representation, cotangent bundle. Darboux's theorem.**Lagrangian manifolds.**

Lagrangian manifolds of a cotangent bundle.**Multi-valued solutions of Hamilton-Jacobi equation.****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.**Geometrical interpretation of 1-order scalar differential operators.****Multi-valued generalized solutions of 1-order equations.****Relations between symplectic and contact geometry.**

Contactization of a symplectic manifold. Symplectification of a contact manifold.

- 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 {e_{1},…,e_{n}} of V, the matrix (w(i,j))=(w(e_{i}, e_{j})) is nondegenerate. - Let ( V
^{2n}, w ) be a symplectic space; then there exist a basis {e_{1}, … e_{n}, f_{1}, …, f_{n}} such that w = e_{1}^{*}Ù f_{1}^{*}+ … + e_{n}^{*}Ù f_{n}^{*}. - Let (V
^{2n}, 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}). - Prove that for any g Î Sp(2n,
**R**), det g = 1. - Let g Î Sp(2n,
**R**) and P(l) be its characteristic polinomial. Prove that P(l) = l^{2n}P(1/l ). In particular, if l is an eigenvalue of multiplicity r then 1/l is an eigenvalue of multiplicity r too. - Let ( M, w ) be a compact symplectic manifold; then the D'Ram's cohomology class of w is nontrivial.
- 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 . - Prove that the sqew symmetric 2-form w on an orbit of the coadjoint representation defined by

w (h_{1}, h_{2})= w ( ad^{*}_{a1}(x ), ad^{*}_{a2}(x ) )= x ( [a_{1}, a_{2}] ) is close. - Prove that a symplectic map is an imbedding.
- Constract a multivalue solutions of some Hamilton-Jacobi equations.
- Let a be a contact form on a manifold M. Prove that M is orientable.

Constract an example of nonorientable contact manifold. - 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. - Let p
_{1}: J^{1}M ® M be 1-jet bundle of smooth functions of manifold M and let L Ì J^{1}M be a submanifold. If 1) L is a Legendre's manifold and 2) p_{1}½_{L}: L ® M is a diffeomorphism,

then $ fÎ C^{¥}(M) with L = Im j_{1}f. - Let f: J
^{1}M ® J^{1}M be a diffeomorphism and U_{1}be a Cartan form on J^{1}M. Prove that the folloving definitions are equivalent:

1) f is a contact transformation if it takes the Cartan distribution of J^{1}M to itself;

2) f is a contact transformation if f^{*}U_{1}= l U_{1}, where l is a nowhere wanishing smooth function on M. - Using the Legendre's transformation, constract a multivalue solutions of some 1-order scalar PDEs.
- Let l : J
^{1}M ® T^{*}M be defined by l ( [f]^{1}_{x})=df½_{x}and let L Ì J^{1}M 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. - Prove, for any connected Lagrangian submanifold N Ì T
^{*}M there exist a Legendre's manifold L Ì J^{1}M so that l ½_{L}: L ® N is a covering. - Let M be a smooth manifold, N Ì M be a submanifold, and C
_{1}, C_{2}be contact structures on M with C_{1}½_{T*xN}= C_{2}½_{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 C_{1}½_{Ux}to C_{2}½_{Vx}.

- Giovanni Manno (14 problems)
- Luca Vitagliano (12 problems)