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 $ C^\infty (M)$-module structure in $ {\Gamma}(\pi)$.

Jets.

The jet $ [f]_x^k$ of a local section $ f$ at a point $ x\in M$. The space $ J_x^k(\pi)$. Smooth structure in $ \bigcup_{x\in M}J_x^k(\pi)$. Manifolds $ J^k(\pi)$ and bundles $ \pi_k\colon J^k(\pi)\to M$. The jet modules $ {\mathcal J}^k(\pi)={\Gamma}(\pi_k)$. Canonical coordinates $ u_{\sigma}^j$ in $ J^k(\pi)$ associated to a local trivialization in $ \pi$. Dimension of $ J^k(\pi)$. The bundles $ \pi_{k,l}\colon J^k(\pi)\to J^l(\pi)$, $ k\ge l$. Graphs of jets. $ R$-planes. Presentation of points of $ J^k(\pi)$ as pairs $ ({\theta}_{k-1}, L_{{\theta}_k})$, where $ {\theta}_{k-1}\in J^{k-1}(\pi)$ and $ L_{{\theta}_k}{\subset} T_{{\theta}_{k-1}}J^{k-1}(\pi)$ is an $ R$-plane.

Nonlinear differential operators.

Presentation of scalar operators as functions on $ J^k(\pi)$. Pull-backs $ \pi_k^*(\xi)$ and nonlinear operators $ {\Delta}\colon{\Gamma}(\pi)\to{\Gamma}(\xi)$ as sections of the bundles $ \pi_k^*(\xi)$. Presentation of operators as morphisms $ J^k(\pi)\to J^0(\xi)$. The universal operator $ j_k\colon{\Gamma}(\pi) \to{\Gamma}(\pi_k)$. Prolongations of nonlinear operators and their correspondence to morphisms $ J^{k+l}(\pi)\to J^l(\xi)$. Composition of nonlinear operators.

Nonlinear equations.

Differential equations as submanifolds in $ {\mathcal E}{\subset}J^k(\pi)$. Description of equations by nonlinear operators. The first prolongation $ {\mathcal E}^1{\subset} J^{k+1}(\pi)$. Three definitions of the $ l$-the prolongation, there equivalence. Solutions.

Geometry of the Cartan distribution in $ J^k(\pi)$

The Cartan distribution.

The Cartan plane $ {\mathcal C}_{\theta}^k$ as the span of the set of $ R$-planes at the point $ {\theta}\in J^k(\pi)$. The distribution $ {\mathcal C}^k\colon{\theta}\mapsto{\mathcal C}_{\theta}^k$. Description of $ {\mathcal C}_{\theta}^k$ in the form $ (\pi_{k,k-1})_*^{-1}(L_{{\theta}_k})$. Local description of $ {\mathcal C}^k$ by the Cartan forms $ \omega _{\sigma}^j=du_{\sigma}^j-\sum_i u_{{\sigma}+1_i}^jdx_i$. A local basis in $ {\mathcal C}^k$.

Maximal integral manifolds of the distribution $ {\mathcal C}^k$.

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 $ J^k(\pi)$ preserving the Cartan distribution. Lifting of Lie transformations from $ J^k(\pi)$ to $ J^{k+1}(\pi)$. The case $ \dim\pi>1$: correspondence between Lie transformations and diffeomorphisms of $ J^0(\pi)$. The case $ \dim\pi=1$: the contact structure in $ J^1(\pi)$, correspondence between Lie transformations and contact transformations of $ J^1(\pi)$ (inexceptional case $ \dim M\neq1$ and exceptional case $ \dim M=1$). 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 $ J^1(\pi)$ and Lie fields for trivial one-dimensional bundles. The jacobi bracket on $ C^\infty (J^1(\pi))$. Local coordinate formulas for Lie fields and Jacobi brackets in terms of generating functions. Bundles of higher dimensions. The element $ \rho_k(\pi)\in {\mathcal J}^k(\pi_k^*(\pi))$, its definition and properties. The Spencer complexes $ \dotsb\to{\mathcal J}^k(\xi)\otimes \Lambda ^l(N)\xrightarrow{S_k^l}{\mathcal J}^{k-1}(\xi)\otimes \Lambda ^{l+1}(N)\to\dotsb$ for a vector bundle $ \xi\colon P\to N$, their exactness. The element $ U_k(\pi)=S_k^0(\rho_k(\pi))\in{\mathcal J}^{k-1} (\pi_k^*(\pi))\otimes \Lambda ^1(J^k(\pi))$, its properties. Generating sections $ f\in{\Gamma}(\pi_k^*(\pi))$ as the result of construction of Lie fields with $ U_1(\pi)$. 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 $ u_t=uu_x+u_{xx}$.

Exterior and interior symmetries.

The restriction $ {\mathcal C}({\mathcal E})$ of the Cartan distribution to $ {\mathcal E}$. Exterior $ \mathrm{Lie}\,({\mathcal E})$ and interior $ \mathrm{Lie}_{\mathrm{int}}({\mathcal E})$ symmetries of an equation $ {\mathcal E}{\subset}J^k(\pi)$. The homomorphism $ r\colon\mathrm{Lie}\,({\mathcal E})\to \mathrm{Lie}_{\mathrm{int}}({\mathcal E})$. Counterexamples.

Perspectives

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