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We develop a theory of
representations of a locally trivial Cr groupoid, Z, on a Cr fiber bundle, E (with
fiber Y , Lie group G, and base space M = the set of units of Z).
A covariant functor, A, is defined, sendin gE into a locally trivial Cr groupoid
A(E) = the groupoid of admissible maps between fibers of E, with a natural Cr
structure. A Cγ bundle map h : E → E′ is sent into a Cr isorqiorphism
A(h) : A(E) → A(E′). Properties of tke functor A are studied.
A Cr representation of Z on E is defined as a Cr homomorphism ρ : Z → A(E).
Let Zee be the group of elements in Z with e as the left and right unit. We obtain
the important result that a Cr homomorphism ρe : Zee → A(Ee) has an
(essentially) unique extension to a Cr representation of Z on a Cr fiber bundle E′,
where E′ is determined by Z and ρe. This leads to interesting applications in
differential geometry. The representations of Lk, the (locally trivial C∞) groupoid
of invertible k-jets of C∞ maps of a C∞ manifold, M, into itself, provide
(but are not the same as) natural fiber bundles of order k in the sense of
Nijenhuis.
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