R-manifolds and multi-valued solutions of PDE's
Nina Khor'kova
A program of the course at the 4-th Italian Diffiety School,
(Forino, July 17-29, 2000)
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Jet manifolds and PDE
Introduction: jet spaces J^{k}(n,m).
Jets of vector bundles J^{k}(p).
Jets of submanifolds J^{k}(E,n).
Jets of fibring.
Jets of mappings M® N.
Jets of functions.
Definitions of differential equation and its solution.
Geometric interpretation of solutions. R-manifolds.
The Cartan distribution on J^{k}(E,n).
Coordinate description of the Cartan distribution.
Geometric structure of the Cartan planes.
Description of integral submanifolds of the Cartan distribution.
Ray submanifolds, prolongations of integral submanifolds.
The structure of locally maximal integral submanifolds of the Cartan
distribution.
R-manifolds.
Definitions of differential equation and its solution.
Lie transformations (high order contact transformations).
Lie transformations. Point transformations.
Prolongations of point transformations.
Prolongations of Lie transformations.
Prolongations of morphisms of contact structures.
Lie-Bäcklund theorem.
Lie fields, liftings (prolongations) of Lie fields.
Infinitesimal Lie-Bäcklund theorem.
Extrinsic and intrinsic geometries of PDE.
External and internal points of view on PDE.
Problem of the reconstruction of the embedding
E®
J^{k} and the Cartan distribution on J^{k}.
Extrinsic and intrinsic symmetries of PDE.
Rigidity. Examples.
Examination problems
Let p:E^{n+m}® M^{n} be a smooth bundle. Prove
that J^{k}(p) is an open everywhere dense subset of J^{k}(E,n).
Let E Ì J^{2}(IR^{3},2) be the minimal surface equation.
Prove that p_{2,1}:E® J^{1}(IR^{3},2)
is a nontrivial 2-dimensional vector bundle.
Let k ³ l. Prove that J^{k}(E,n) is the manifold of (k-l)-jets
of submanifolds of the form L^{(l)} in J^{l}(E,n).
Prove that the Cartan distribution on J^{k}(E,n) is locally
determined by the set of the Cartan forms
w^{j}_{s} = dp^{j}_{s} -
n å
i = 1
dp^{j}_{s+1i}dx_{i}, | s | < k, j = 1,...,m.
Let
p:E® M be a fiber bundle and let
p_{1,0}:J^{1}(p)® E.
Show that sections of the bundle p_{1,0} are
connections in the bundle p, while the condition of zero curvature
determines a first order equation in the bundle p_{1,0} .
Consider the system of equations:
ì í
î
u_{x}
=
f(x,y,u)
u_{y}
=
g(x,y,u).
Prove that if this system is compatible, then the Cartan distribution
restricted to the corresponding surface is completely integrable.
Prove the following statements:
L^{(k)} is a locally maximal integral manifold of the Cartan
distribution.
Let Q Ì J^{k}(E,n) be an n-dimensional integral manifold
that is transversal to the fibers of the projection p_{k,k-1}.
Then, locally, Q is of the form L^{(k)}.
Let x = X(x,y,z)[(¶)/(¶x)] + Y(x,y,z)[(¶)/(¶y)] + Z(x,y,z)[(¶)/(¶z)] be a nonzero vector
at a point q Î J^{0}(2,1), and P the straight line
determined by this vector. Deduce equations describing the ray
manifold l(P).
Let W be a curve x = a(t),y = b(t),z = g(t).
Describe L(W).
Let W be a surface of the form z = f(x,y).
Describe L(W).
Prove that dimL(W) = r+m ((k+n-r-1) || (n-r-1)), where
r = dimW.
Let F:J^{0}(p) ® J^{0}(p) be a point transformation.
Prove that if the lifting F^{(1)} is defined in a neighborhood of
a point q Î J^{1}(p), then the lifting F^{(k)} is defined
in a neighborhood of any point q¢ such that
p_{k,1}(q¢) = q.
Prove that ordinary differential equations are non rigid.
Exercises
Prove that the family of neighborhoods p_{k}^{-1}(U) together
with the coordinate functions (x,p^{j}_{s}) determines a smooth
manifold structure in J^{k}(p). Prove that
dimJ^{k}(p) = n+m(n || (n+k))
Prove that p_{k}:J^{k}(p) ® M is a smooth locally
trivial vector bundle.
Prove that p_{k+1,k}:J^{k+1}(p) ® J^{k}(p)
is a smooth locally trivial bundle (not vector bundle).
Prove that
p_{l,s}°p_{k,l} = p_{k,s}, k ³ l ³ s;
p_{l}°p_{k,l} = p_{k}, k ³ l;
p_{k,l}°j_{k}(s) = j_{l}(s), k ³ l.
Prove that
J^{k}(E,n) is a smooth manifold of dimension n+m(n || (n+k));
p_{k,l}:J^{k}(E,n) ® J^{l}(E,n), k ³ l
is a smooth locally trivial bundle;
p_{l,s}°p_{k,l} = p_{k,s}, p_{k,l}°j_{k}(L) = j_{l}(L), k ³ l ³ s.
Let F:J^{k}(E,n)® J^{k}(E,n) be a Lie transformation.
Prove that
p_{k+l,k+s}°F^{(l)} = F^{(s)}°p_{k+s,k+l}, l ³ s;
id^{(s)} = id;
(F°G)^{(s)} = F^{(s)}°G^{(s)}.
Consider the Legendre transformation F in the space J^{1}(2,1):
[`x] = -p, [`y] = -q, [`u] = u-xp-yq, [`p] = x, [`q] = y.
Prove that F is a Lie transformation;
Describe the lifting F^{(1)};
Prove that F can not be represented in the form F = G^{(1)},
where G is a point transformation.
Let F:E® E¢ be a morphism over a diffeomorphism
[`F]:M® M of two bundles p and p¢ over M.
Define the prolongation F^{(k)}:J^{k}(p)® J^{k}(p¢)
in such a way that F^{(k)} will be a morphism of p_{k} in
p¢_{k} over [`F].
Let F:J^{k}(E,n)® J^{l}(E,n), k > l be a smooth surjection such
that F_{*}C^{(k)}_{q} = C^{(l)}_{F(q)}.
Define the prolongation F^{(k)}:J^{k}(E,n)® J^{k}(E,n).
For the vector field
X = x[(¶)/(¶u)] - u[(¶)/(¶x)]
on J^{0}(2,1) find X^{(2)}.
Passing the exam during the school required 6 solved problems. To
pass the exam by email one should solve 10 problems.
The exam has been passed by the following students:
Giovanni Manno (6 problems: 3,4,6,7,9,13)
Luca Vitagliano (9 problems: 1,3 - 8,10,13)
Questions and suggestions should go to
Jet NESTRUEV, jet @ diffiety.ac.ru.