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X. A sections of the jet bundle J1Y --+ X is said to be holonomic if it is the jet prolongation of some section of the fibre bundle Y -> X.

7). 35) Y _ ^a Adz A ... idxA' A ... A dx\'' 0 dy=. 2. 36) are not exterior forms. They exemplify vector-valued forms. Distributions An r-dimensional smooth distribution on a k-dimensional manifold Z is an rdimensional subbundle T of the tangent bundle TZ. We will say that a vector field v on Z is subordinate to a distribution T if it is a section of T --+ Z. A distribution T is said to be involutive if the Lie bracket [u, u'] is subordinate to T, whenever u and u' are subordinate to T. A connected submanifold N of a manifold Z is called an integral manifold of a distribution T on Z if the tangent spaces to N belong to the fibres of this distribution.

These charts cover the set JnZ, and transition functions between them are differentiable. 3. JET MANIFOLDS 31 It is convenient to use the following coordinate atlas of the jet manifold J, ',Z of n-dimensional submanifolds of Z. ,n+m. 19) Though J°Z = Z, let us provide J°Z with the atlas where every chart (U; zA) on a domain U C Z is replaced with the ^n + m'\ (n + m)! m! charts on the same domain U which correspond to different partitions of the collection (z1 . . 19) of Z, reduce to an exchange between coordinates xA and y.

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