Differential invariants and their applications to researching of
A program of the course at the 4-th Russian Diffiety School,
Pereslavl-Zalessky (Russia), January 31 - February 6, 2000
- Introduction: the equivalence problem and differential invariants
- Lie pseudogroupsLie pseudogroups and their Lie algebras. Examples.
- Natural bundles and geometric structuresNatural bundles, first examples. Geometric structures, homogeneity.
1) Geometric structures connected with ordinary differential equations;
2) Web structures on the solutions of hydrodynamics type partial differential
3) Geometric structures connected with Monge-Ampere equations.
- Differential invariantsThe equivalence problem.
The lifting of diffeomorphisms from a base up to Lie transformations in jet bundles.
Differential invariants, scalar differential invariants.
The lifting of vector fields from base up in jet bundles; the application to calculation of differential invariants.
Transitive pseudogroups. Differential groups and formal vector fields.
Actions of differential groups on fibers of jet bundles; the application to
calculation of invariants. Isotropy algebras; calculation of the number of
functionally independent scalar invariants on the k-jet bundle.
- Jets of submanifolds and differential invariantsThe general scheme: the action of a Lie pseudogroup of a manifold in jets of submanifolds. Differential invariants.
1) The action of the motion group of $\R^2 (\R^3)$ in jets of curves; the
calculation of the curvature (and the torsion);
2) The action of the motion group of $\R^3$ in jets of surfaces; the
calculation of the number of functionally independent scalar invariants on
the k-jet bundle.
- Application of differential invariants to the equivalence problem 1. Linear ordinary differential equation.
The Lagguere-Forsyth form of linear ODEs. The natural bundles of linear ODEs. Invariant differential forms. Scalar differential invariants.
The classification of linear ODEs up to equivalence.
2. Three webs.
Curvature form of a three web. Scalar differential invariants.
Differential correlations between scalar invariants. Algebra of scalar differential invariants; the description of the orbifold of three webs.
Classification of three webs up to equivalence.
Applications of differential invariants of three webs to find a solutions of hydrodynamics type PDEs.
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- A.M.Vinogradov, Scalar differential invariants, diffieties and characteristic classes, in: Mechanics, Analysis and Geometry: 200 Years after Lagrange, ed. M.Francaviglia (North-Holland), pp.379--414, 1991.
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- V.N.Gusyatnikova, V.A.Yumaguzhin, Contact transformations and local reducibility of ODEs to the form y'''=0, Acta Applicandae Mathematicae, 1999, Vol. 56, No. 2,3, pp. 155 - 179.
- V.V.Lychagin, Contact geometry and non-linear second-order differential equations, Russian Math. Surveys, Vol. 34, No. 1. pp. 149-180, 1979.
- A.M.Vinogradov, Solution singularities of differential equations, to appear.
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- V.A.Yumaguzhin, Classification of linear ODEs up to equivalence, to appear.
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