Differential invariants and their applications to researching of differential equations
Valeriy A.Yumaguzhin
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 pseudogroups
Lie pseudogroups and their Lie algebras. Examples.
• Natural bundles and geometric structures
Natural bundles, first examples. Geometric structures, homogeneity.
Main examples:
1) Geometric structures connected with ordinary differential equations;
2) Web structures on the solutions of hydrodynamics type partial differential
equations;
3) Geometric structures connected with Monge-Ampere equations.
• Differential invariants
The 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 invariants
The general scheme: the action of a Lie pseudogroup of a manifold in jets of submanifolds. Differential invariants.
Examples:
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.
Open problems.
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.
Open problems.

References

1. D.V.Alekseevskiy, A.M.Vinogradov, V.V.Lychagin, Fundamental ideas and conceptions of differential geometry, Sovremennye problemy matematiki. Fundamental'nye napravleniy, Vol. 28} (Itogi nauki i techniki, VINITI, AN SSSR, Moscow, 1988 (Russian)) [English transl.: Encyclopedia of Math. Sciences, Vol.28 (Springer, Berlin, 1991)]
2. 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.
3. I.S.Krasil'shchik, V.V.Lychagin, A.M.Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986
4. V.N.Gusyatnikova, V.A.Yumaguzhin, Point transformations and linearisation of 2-order ordinary differential equations, Mat. Zametki Vol. 49 (1991), No. 1, pp.146-148 (in Russian, English translation in: Soviet Mat. Zametki).
5. 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.
6. V.V.Lychagin, Contact geometry and non-linear second-order differential equations, Russian Math. Surveys, Vol. 34, No. 1. pp. 149-180, 1979.
7. A.M.Vinogradov, Solution singularities of differential equations, to appear.
8. V.A.Yumaguzhin, Classification of 3-rd order linear ODEs up to equivalence, Journal of Differential Geometry and its Applications, 1996, Vol. 6, No. 4, pp. 343 - 350.
9. V.A.Yumaguzhin, Classification of linear ODEs up to equivalence, to appear.
10. A.M.Vinogradov, V.A.Yumaguzhin, Differential invariants of vebs on 2-dimensional manifolds, Mat. Zametki Vol. 48 (1990), No. 1, pp. 46-68 (in Russian, English translation in: Soviet Mat. Zametki).