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Historically, jet bundles are attributed to Ehresmann, and were an advance on the method ( prolongation ) of Elie Cartan, of dealing geometrically with higher derivative s, by imposing differential form conditions on newly-introduced formal variables.
Let be the trivial bundle over B. Then sections of this bundle are smooth maps . Two such maps f and g are said to be equivalent at y in B if
(here d(x,y) denotes distance in any fixed Riemannian metric on X). The classes of equivalence of such maps at y form the fiber of the first jet bundle at y.
The n-th jet bundle is constructed by repeating this operation n times.
What follows is a generalization of this construction to an arbitrary fiber bundle E.
Given a differential manifold B and a fiber bundle E over B which is also a differential manifold. This means that the fiber Fx at a point x in B is also a differential manifold. Hence for any point y in Fx, the tangent space TyFx of Fx at y is a linear subspace of the full tangent space of E at y. TyFx is called the vertical subspace. The full tangent space can be decomposed into a direct sum of the vertical subspace and a complementary horizontal subspace. Now we can define a fiber bundle J over E whose fiber at a y is the set of all possible horizontal subspaces. Viewed as a fiber bundle over B, J is called the first order jet bundle over B.
The jet bundle of order n over B is now defined recursively as as the first order jet bundle over the jet bundle of order n-1 over B.
Given a smooth section of the n-1th jet bundle, it induces a unique section of the nth jet bundle by taking the horizontal subspace to be the tangent space to the section. Repeating this operation defines a unique section of the nth jet bundle out of section of original bundle, called the nth prolongation.
All sections which can be obtained this way are called holonomic.
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