Valeriy A. YUMAGUZHIN

Exercises of the course at the Join Russian-Italian Diffiety School,

Pereslavl-Zalessky (Russia), August 17 - 30, 1999

- Prove that for any horizontal subspace H Ì T
_{q}J^{0}M (H is horizontal if p_{0 *}|_{H}: H® T_{p(q)}M is an isomorphism) there exists q_{1}Î J^{1}M with k_{q1}= H. - Prove that
"
_{q Î J1M}dU_{1}|_{Cq}is a nondegenerate form. - Prove that a bilinear form (· , ·) is nondegenerate iff its matrix is nondegenerate.
- Prove that dimension of a vector space with a symplectic structure is even.
- Prove that dimension of a smooth manifold with a contact structure is odd.
- Let w be a contact form on a manifold M and
f Î C
^{¥}M be a nowhere vanishing function. Prove that:

1) f·w is a contact form,

2) dw differs from d(f·w) by a nonzero factor on any hyperplane,

kerw|_{x}Ì T_{x}M, x Î M. - Let (V
^{2n},w) be a symplectic vector space and W Ì V be an isotropic subspace (W is isotropic if "_{v,w Î W}w(v,w) = 0). Prove that dimW £ n. - Let N be an integral manifold of the Cartan distribution on J
^{1}M. Prove that dimN £ dimM. - Let w be a symplectic form on a vector space V and W Ì V be a hyperplane. Prove that the skew-orthogonal complementation of W is 1-dimensional and lies in W.
- Find the expression of a contact transformation f: J
^{1}M® J^{1}M in standard coordinates x^{1},¼,x^{n},u,p_{1},¼,p_{n}(n = 1,2). - Let A: J
^{0}M® J^{0}M be a point transformation defined in standard coordinates by Find explicit formulae defining the lift Aì

í

îX = X(x,y) Y = Y(x,y) ^{(1)}. - Check that the Legendre transformation
is contact and it cannot be obtained by lifting of a point transformation.ì

ï

ï

í

ï

ï

îX ^{i}= p _{i}U = n

å

i = 1p _{i}·x^{i}- uP _{i}= x ^{i} - Check that the mapping D(J
^{1}M)®L (J^{1}M), Y® dU_{1}(Y,·) defines an isomorphism between vector fields that lie in the Cartan distribution and 1-forms vanishing on X_{1}= ¶_{u}. - Find an explicit formula defining the Jacobi bracket of f,g Î C
^{¥}(J^{1}M) in standard coordinates. - Let
*E*= {F(x,u,p) = 0} Ì J^{1}M be a 1-st order PDE, let Y_{F}= X_{F}-F·X_{1}be the characteristic vector field of*E*, and let A_{t}be its flow. Prove that A_{t}takes the Cartan distribution C(*E*) on*E*to itself. - Let X = å
^{n}_{i = 1}a^{i}(x,u)¶x^{i}be a smooth vector field on J^{0}M. Find an explicit formula defining the lift X^{(1)}in standard coordinates. - Let
*E*= {F = å^{n}_{i = 1}a^{i}(x,u)p_{i}-b(x,u) = 0} Ì J^{1}M be a quasi-linear equation. Prove that Y_{F}|_{E}= X^{(1)}|_{E}, where X = å^{n}_{i = 1}a^{i}¶x^{i}+b¶u. - Solve the Cauchy problem for the equation
with the Cauchy datax ^{1}·u·p_{1}+x^{2}·u·p_{2}+x^{1}·x^{2}= 0ì

í

îg = {(x ^{1},x^{2}) | x^{2}= (x^{1})^{2}}j(x ^{1})= (x ^{1})^{3} - Prove the Jacobi identity for the Poisson bracket.
- Let M be a smooth n-dimensional manifold, f
_{1},¼,f_{k}Î C^{¥}(M) k < n, and let

M_{c}= { x Î M | f_{1}(x) = c_{1},¼,f_{k}(x) = c_{k}}, c = (c_{1},¼,c_{k}) Î**R**^{k}.

Assume that M_{c}is compact and "_{x Î Mc}the 1-forms df_{1}|_{x},¼,df_{k}|_{x}are linear independent. Prove that there exists a neighborhood of M_{c}diffeomorphic to M_{c}´B_{c}, where B_{c}Ì**R**^{k}is an open ball with center at c. - Let M
^{2n}be a symplectic manifold and f_{1},¼,f_{2n}Î C^{¥}(M^{2n}). Prove that if the function f_{1},¼,f_{2n}are functionally independent, then the 2n×2n-matrix of Poisson bracket ({f_{i},f_{j}}) is nondegenerate.

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