Geometry of infinite jets and infinitely prolonged equations

Vladimir CHETVERIKOV

Program of the Joint Russian - Italian School
Pereslavl', 17 August - 30 August 1999

Introduction.

§ 1. Spaces of infinite jets.

1. The space J^€ p.
2. Jets of submanifolds.
3. Infinitely prolonged equations.

§ 2-3. Basic differential geometric structures on infinite-dimensional manifolds.

1. Smooth functions.
2. Smooth mappigs, Lie transformations.
3. Vector fields, Lie fields.
4. Differential forms.

§ 4. Differential operators.

1. Scalar and matrix differential operators.
2. Prolongation of differential operators. One more example of a smooth mapping.

§ 5-6. The Cartan distribution on J^€ p and E^€.

1. Distributions on infinite-dimensional manifolds and their automorphisms.
2. The Cartan connection.
3. Maximal integral manifolds of the Cartan distribution.
4. The structural element of an equation.

§ 7. Symmetries of the Cartan distribution on J^€ p.

1. Generating sections of symmetries of the Cartan distribution on J^€ p.
2. The structure of Sym (p).
3. Evolutionary derivations.
4. The higher Jacobi bracket.
5. Comparison with Lie fields.

§ 8. Higher symmetries of differential equations.

1. Exterior and interior higher sy mmetries.
2. C-differential operators.
3. Linearizations.
4. Defining equations for higher symmetries.

§ 9. Examples of computations.