Geometry of Differential Equations
Joseph KRASIL'SHCHIK
A program of the course at the 2-nd Italian Diffiety School,
(Forino, February - March, 1999)
and the 2-nd russian Diffiety School
(Pereslavl-Zalessky, January 26 - February 5, 1999)

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## Jet bundles

### Vector bundles and sections.

Smooth manifolds. Smooth locally trivial vector bundles. Sections. The -module structure in .

### Jets.

The jet of a local section at a point . The space . Smooth structure in . Manifolds and bundles . The jet modules . Canonical coordinates in associated to a local trivialization in . Dimension of . The bundles , . Graphs of jets. -planes. Presentation of points of as pairs , where and is an -plane.

### Nonlinear differential operators.

Presentation of scalar operators as functions on . Pull-backs and nonlinear operators as sections of the bundles . Presentation of operators as morphisms . The universal operator . Prolongations of nonlinear operators and their correspondence to morphisms . Composition of nonlinear operators.

### Nonlinear equations.

Differential equations as submanifolds in . Description of equations by nonlinear operators. The first prolongation . Three definitions of the -the prolongation, there equivalence. Solutions.

## Geometry of the Cartan distribution in ### The Cartan distribution.

The Cartan plane as the span of the set of -planes at the point . The distribution . Description of in the form . Local description of by the Cartan forms . A local basis in .

### Maximal integral manifolds of the distribution .

Involutive subspaces of the Cartan distribution. The theorem on maximal integral manifolds. The type of a maximal integral manifold. Computation of dimensions for maximal integral manifolds. Integral manifolds of maximal dimension in inexceptional cases.

### The Lie-Bäcklund theorem.

Lie transformations as diffeomorphisms of preserving the Cartan distribution. Lifting of Lie transformations from to . The case : correspondence between Lie transformations and diffeomorphisms of . The case : the contact structure in , correspondence between Lie transformations and contact transformations of (inexceptional case and exceptional case ). Local formulas for liftings of Lie transformations.

### Infinitesimal theory.

Lie fields. Local lifting formulas. Global nature of lifting for Lie fields. Infinitesimal analog for the Lie-Bäcklund theorem. One-dimensional bundles. Generating functions of Lie fields. Correspondence between functions on and Lie fields for trivial one-dimensional bundles. The jacobi bracket on . Local coordinate formulas for Lie fields and Jacobi brackets in terms of generating functions. Bundles of higher dimensions. The element , its definition and properties. The Spencer complexes for a vector bundle , their exactness. The element , its properties. Generating sections as the result of construction of Lie fields with . Jacobi brackets for generating sections. Local coordinates.

## Classical symmetry theory for differential equations

### Classical symmetries.

Finite and infinitesimal symmetries, definitions. Physical meaning'' of generating functions. Determining equations for coordinate computations. An example: symmetries of the Burgers equation .

### Exterior and interior symmetries.

The restriction of the Cartan distribution to . Exterior and interior symmetries of an equation . The homomorphism . Counterexamples.

## Perspectives

Algebraic model. The basic constructions. Cohomological invariants.
Questions and suggestions should go to J. S. Krasil'shchik, josephk @ diffiety.ac.ru.