A2 Course. Geometry of Differential Equations

Finite Jets

• The space J^k(E,n) of jets of submanifolds of the manifold E.
• The space J^k(π) of jets of sections of the bundle π.
• Canonical coordinates on J^k associated with adapted coordinate systems.
• Basic constructions with jets spaces.

Geometry of Jet Spaces

• R- planes.
• The Cartan distribution, Cartan fields, Cartan forms.
• Structure of the Cartan Distribution.
• Ray submanifolds.
• Structure of maximal integral manifolds of the Cartan distribution.

Geometry of PDE’s

• Differential equations.
• Prolongation of a differential equation.
• Multivalued solutions of PDE’s and their singularities.
• Classical symmetries of a differential equation.
• Rigidity theorem.

Higher Order Contact Trasformations

• Contact transformations.
• Lie transformations, point transformations.
• Lie-Bæcklund theorem.
• Lie Fields.
• Lifting of a Lie Fields.
• Genereting section of a Lie field.
• Jacobi brackets.

Geometric Theory of 1^st Order Equations in 1 Dependent Variable

• Contact structure on J¹(M).
• Contact transformations.
• Genereting function of a contact transformation.
• Jacobi brackets.
• Characteristics of a differential equation in J¹(M).
• Cauchy problem in J¹(M).

Infinite Jets

• The spaces J^∞ of infinite jets.
• Functions, differential forms and vector fields on J^∞.
• The Cartan distribution on J^∞.
• Maximal integral manifolds.
• Lifting of a differential operator.
• Infinite prolongation of a differential equation.
• Evolutionary derivations.
• Jacobi brackets.
• Universal Linearizations.
• Higher symmetries of a differential equation.

References

• I. S. Krasil’shchik, V. V. Lychagin, A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Differential Equations, Advanced Studies in Contemporary Mathematics, 1 (1986), Gordon and Breach, New York, London. xx+441 pp.
• V. N. Chetverikov, A. B. Bocharov, S. V. Duzhin, N. G. Khor’kova, I. S. Krasil’shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, A. M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, I. S. Krasil’shchik, and A. M. Vinogradov, Editors, AMS (1999), MMONO/182.