Actions Of Jet Groups On Manifolds

. In the present paper, we deduce the sharp estimates on the order r of the jet groups G r m;n acting on a manifold of fixed dimension s which depend on m, n and s only. These estimates are essential for the theory of bundle functors on fibered manifolds and we find interesting that the dimensions m and n appear symmetrically in the outcome. In the case n = 0 we reprove the well known results by A. Zajtz. It is well known that the elements of classical geometric objects form associated bundles with the structure group G r m = invJ r 0 (R m ; R m ) 0 , the classical jet group (or differential group). This is described in the theory of bundle functors on the category Mfm of m-dimensional manifolds and local diffeomorphisms. But many geometric constructions are performed in other categories then Mfm . One of the most frequent examples of general bundle functors is the fuctor of general connections defined on the category FMm;n of fibered manifolds with m-dimensional bases and n..