Geometry of Differential Equations
Joseph KRASIL'SHCHIK
Program of the Second Italian
School
Forino, 17 July - 1 August 1998
1 Jet bundles
1.1 Vector bundles and sections.
Smooth manifolds. Smooth locally trivial vector bundles. Sections.
The C€(M)-module structure in G(p).
The jet [f]xk of a local section f at a point x ‘ M. The space
Jxk(p). Smooth structure in »x ‘ MJxk(p). Manifolds
Jk(p) and bundles pk : Jk(p)Æ M. The jet modules
Jk(p) = G(pk). Canonical coordinates usj in Jk(p)
associated to a local trivialization in p. Dimension of Jk(p). The
bundles pk,l : Jk(p)Æ Jl(p), k „ l. Graphs of jets.
R-planes. Presentation of points of Jk(p) as pairs (q
k-1,Lqk), where qk-1 ‘ Jk-1(p) and Lqk à Tqk-1Jk-1(p) is an R-plane.
1.3 Nonlinear differential operators.
Presentation of scalar operators as functions on Jk(p). Pull-backs
pk*(x) and nonlinear operators D : G(p)ÆG(x) as
sections of the bundles pk*(x). Presentation of operators as
morphisms Jk(p)Æ J0(x). The universal operator jk : G(p)ÆG(pk). Prolongations of nonlinear operators and their
correspondence to morphisms Jk+l(p)Æ Jl(x). Composition of
nonlinear operators.
1.4 Nonlinear equations.
Differential equations as submanifolds in E Ã Jk(p). Description
of equations by nonlinear operators. The first prolongation E1 Ã Jk+1(p). Three definitions of the l-the prolongation, there
equivalence. Solutions.
2&nb
sp; Geometry of the Cartan distribution in Jk(p)
2.1 The Cartan distribution.
The Cartan plane Cqk as the span of the set of R-planes at the
point q ‘ Jk(p). The distribution Ck : qÆCqk.
Description of Cqk in the form (pk,k-1)*-1(Lqk).
Local description of Ck by the Cartan forms wsj = dusj-Âius+1ijdxi. A local basis in Ck.
2.2 Maximal integral manifolds of the distribution Ck.
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.
2.3 The Lie-Bäcklund theorem.
Lie transformations as diffeomorphisms of Jk(p) preserving the
Cartan distribution. Lifting of Lie transformations from Jk(p) to
Jk+1(p). The case dim p > 1: correspondence between Lie
transformations and diffeomorphisms of J0(p). Th
e case dim p = 1:
the contact structure in J1(p), correspondence between Lie
transformations and contact transformations of J1(p) (inexceptional
case dim M ¼ 1 and exceptional case dim M = 1). Local formulas for
liftings of Lie transformations.
2.4 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 J1(p) and Lie fields for trivial one-dimensional
bundles. The jacobi bracket on C€(J1(p)). Local coordinate
formulas for Lie fields and Jacobi brackets in terms of generating functions.
Bundles of higher dimensions. The element rk(p) ‘ Jk(pk*(p)), its definition and properties. The Spencer complexes
ºÆJk(x)ŸLl(N) Æ SklJk-1(x)ŸLl+1(N)ƺ for a vector bundle x : PÆ N, their
exactness. The element Uk(p) = Sk0(rk(p)) ‘ Jk-1(p
k*(p))ŸL1(Jk(p)), its properties. Generating sections
f ‘ G(pk*(p)) as the result of construction of Lie fields with
U1(p). Jacobi brackets for generating sections. Local coordinates.
3 Classical symmetry theory for differential equations
3.1 Classical symmetries.
Finite and infinitesimal symmetries, definitions. ``Physical meaning''
of generating functions. Determining equations for coordinate computations.
An example: symmetries of the Burgers equation ut = uux+uxx.
3.2 Exterior and interior symmetries.
The restriction C(E)
of the Cartan distribution to E. Exterior Lie (E) and
interior Lieint(E) symmetries of an equation
E Ã Jk(p). The homomorphism r : Lie (E)ÆLieint(E).
Counterexamples.
4 Perspectives
Algebraic model. The basic constructions. Cohomological invariants.
Questions and suggestions should go to
diffiety @ tiros.dmi.unisa.it